YES

Problem 1: 

(VAR x y z)
(THEORY 
(AC plus times))
(RULES
div(div(x,y),z) -> div(x,times(y,z))
div(0,y) -> 0
div(x,y) -> quot(x,y,y)
plus(0,y) -> y
plus(s(x),y) -> s(plus(x,y))
plus(x,0) -> x
quot(0,s(y),z) -> 0
quot(s(x),s(y),z) -> quot(x,y,z)
quot(x,0,s(z)) -> s(div(x,s(z)))
times(0,y) -> 0
times(s(0),y) -> y
times(s(x),y) -> plus(y,times(x,y))
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 DIV(div(x,y),z) -> DIV(x,times(y,z))
 DIV(div(x,y),z) -> TIMES(y,z)
 DIV(x,y) -> QUOT(x,y,y)
 PLUS(plus(0,y),x3) -> PLUS(y,x3)
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
 QUOT(s(x),s(y),z) -> QUOT(x,y,z)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(0),y),x3) -> TIMES(y,x3)
 TIMES(times(s(x),y),x3) -> PLUS(y,times(x,y))
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> PLUS(y,times(x,y))
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 DIV(div(x,y),z) -> DIV(x,times(y,z))
 DIV(div(x,y),z) -> TIMES(y,z)
 DIV(x,y) -> QUOT(x,y,y)
 PLUS(plus(0,y),x3) -> PLUS(y,x3)
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
 QUOT(s(x),s(y),z) -> QUOT(x,y,z)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(0),y),x3) -> TIMES(y,x3)
 TIMES(times(s(x),y),x3) -> PLUS(y,times(x,y))
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> PLUS(y,times(x,y))
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0,y),x3) -> PLUS(y,x3)
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->->Cycle:
->->-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(0),y),x3) -> TIMES(y,x3)
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->->Cycle:
->->-> Pairs:
 DIV(div(x,y),z) -> DIV(x,times(y,z))
 DIV(x,y) -> QUOT(x,y,y)
 QUOT(s(x),s(y),z) -> QUOT(x,y,z)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 Empty


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(0,y),x3) -> PLUS(y,x3)
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 2
[s](X) = X
[DIV](X1,X2) = 0
[PLUS](X1,X2) = X1 + X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(s(x),y),x3) -> PLUS(x,y)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 2
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 2
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 2
[s](X) = X + 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 1
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
 PLUS(s(x),y) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 2
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = X + 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 2
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 0
[s](X) = 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(0),y),x3) -> TIMES(y,x3)
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 1
[quot](X1,X2,X3) = 0
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2
[quot](X1,X2,X3) = 0
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(times(s(x),y),x3) -> TIMES(x,y)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 1
[quot](X1,X2,X3) = 0
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(s(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
 TIMES(s(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 1
[quot](X1,X2,X3) = 0
[times](X1,X2) = X1.X2 + X1 + X2
[0] = 1
[s](X) = X + 1
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(0,y),x3) -> TIMES(0,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = X1 + X2 + 3
[quot](X1,X2,X3) = 0
[times](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
[0] = 3/2
[s](X) = X + 2
[DIV](X1,X2) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 0
[TIMES](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 DIV(div(x,y),z) -> DIV(x,times(y,z))
 DIV(x,y) -> QUOT(x,y,y)
 QUOT(s(x),s(y),z) -> QUOT(x,y,z)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 Empty
->Projection:
 pi(DIV) = [1]
 pi(QUOT) = [1]

Problem 1.3: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 DIV(x,y) -> QUOT(x,y,y)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 DIV(x,y) -> QUOT(x,y,y)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 Empty

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> Pairs:
 DIV(x,y) -> QUOT(x,y,y)
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 Empty
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> Usable Rules:
 Empty
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div](X1,X2) = 0
[plus](X1,X2) = 0
[quot](X1,X2,X3) = 0
[times](X1,X2) = 0
[0] = 2
[s](X) = 0
[DIV](X1,X2) = 2.X2 + 1
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = X2 + X3
[TIMES](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 QUOT(x,0,s(z)) -> DIV(x,s(z))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 div(div(x,y),z) -> div(x,times(y,z))
 div(0,y) -> 0
 div(x,y) -> quot(x,y,y)
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(x,0) -> x
 quot(0,s(y),z) -> 0
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0,s(z)) -> s(div(x,s(z)))
 times(0,y) -> 0
 times(s(0),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.