YES

Problem 1: 

(VAR x y)
(THEORY 
(AC plus times))
(RULES
div(0,s(y)) -> 0
div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
minus(x,0) -> x
p(s(x)) -> x
plus(x,0) -> x
plus(x,s(y)) -> s(plus(x,y))
times(x,0) -> 0
times(x,s(y)) -> plus(x,times(x,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:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
 DIV(s(x),s(y)) -> MINUS(x,y)
 MINUS(s(x),s(y)) -> MINUS(p(s(x)),p(s(y)))
 MINUS(s(x),s(y)) -> P(s(x))
 MINUS(s(x),s(y)) -> P(s(y))
 PLUS(plus(x,0),x2) -> PLUS(x,x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(y)) -> PLUS(x,y)
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> PLUS(x,times(x,y))
 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(y)) -> PLUS(x,times(x,y))
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
 DIV(s(x),s(y)) -> MINUS(x,y)
 MINUS(s(x),s(y)) -> MINUS(p(s(x)),p(s(y)))
 MINUS(s(x),s(y)) -> P(s(x))
 MINUS(s(x),s(y)) -> P(s(y))
 PLUS(plus(x,0),x2) -> PLUS(x,x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(y)) -> PLUS(x,y)
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> PLUS(x,times(x,y))
 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(y)) -> PLUS(x,times(x,y))
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(plus(x,0),x2) -> PLUS(x,x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->->Cycle:
->->-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(p(s(x)),p(s(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)
-> 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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(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)
-> 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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 Empty


The problem is decomposed in 4 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(plus(x,0),x2) -> PLUS(x,x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> 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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[0] = 2
[s](X) = X + 2
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(plus(x,s(y)),x2) -> PLUS(x,y)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> 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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> 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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 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(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2)
 PLUS(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
-> SRules:
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](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(x,s(y)),x2) -> PLUS(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(x,s(y)),x2) -> PLUS(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 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(x,s(y)),x2) -> PLUS(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
-> SRules:
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[0] = 0
[s](X) = 2
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> TIMES(plus(x,times(x,y)),x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(times(x,s(y)),x2) -> TIMES(x,y)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,0),x2) -> TIMES(0,x2)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,x2)
 TIMES(x,s(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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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(x,0),x2) -> TIMES(0,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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(x,0),x2) -> TIMES(0,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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 0
[plus](X1,X2) = X1 + X2 + 1/3
[times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
[0] = 1/2
[s](X) = X + 1/3
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 0
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2

Problem 1.2: 

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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,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.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(p(s(x)),p(s(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:
 Empty
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 p(s(x)) -> x
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[div](X1,X2) = 0
[minus](X1,X2) = 0
[p](X) = 2/3.X + 1/3
[plus](X1,X2) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = 2.X + 3
[DIV](X1,X2) = 0
[MINUS](X1,X2) = 1/2.X1 + 3/2.X2
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 0

Problem 1.3: 

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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(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:
 Empty
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> Usable Rules:
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[minus](X1,X2) = 2.X1
[p](X) = X
[plus](X1,X2) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = 2.X + 2
[DIV](X1,X2) = X1
[MINUS](X1,X2) = 0
[P](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 0

Problem 1.4: 

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:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
 minus(x,0) -> x
 p(s(x)) -> x
 plus(x,0) -> x
 plus(x,s(y)) -> s(plus(x,y))
 times(x,0) -> 0
 times(x,s(y)) -> plus(x,times(x,y))
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.