NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(* 1, 2)
(+ 1, 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
*(x,p(y)) -> -(*(x,y),x)
*(x,s(y)) -> +(*(x,y),x)
*(x,0) -> 0
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
-(x,0) -> x
p(s(x)) -> x
s(p(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(* 1, 2)
(+ 1, 2)
(- 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
*(x,p(y)) -> -(*(x,y),x)
*(x,s(y)) -> +(*(x,y),x)
*(x,0) -> 0
+(x,p(y)) -> p(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,0) -> x
-(x,p(y)) -> s(-(x,y))
-(x,s(y)) -> p(-(x,y))
-(x,0) -> x
p(s(x)) -> x
s(p(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 *(x,p(y)) -> -(*(x,y),x)
 *(x,s(y)) -> +(*(x,y),x)
 *(x,0) -> 0
 +(x,p(y)) -> p(+(x,y))
 +(x,s(y)) -> s(+(x,y))
 +(x,0) -> x
 -(x,p(y)) -> s(-(x,y))
 -(x,s(y)) -> p(-(x,y))
 -(x,0) -> x
 p(s(x)) -> x
 s(p(x)) -> x
-> Vars:
 x, y, x, y, x, x, y, x, y, x, x, y, x, y, x, x, x

-> Rlps:
 (rule: *(x,p(y)) -> -(*(x,y),x), id: 1, possubterms: *(x,p(y))->[], p(y)->[2])
 (rule: *(x,s(y)) -> +(*(x,y),x), id: 2, possubterms: *(x,s(y))->[], s(y)->[2])
 (rule: *(x,0) -> 0, id: 3, possubterms: *(x,0)->[], 0->[2])
 (rule: +(x,p(y)) -> p(+(x,y)), id: 4, possubterms: +(x,p(y))->[], p(y)->[2])
 (rule: +(x,s(y)) -> s(+(x,y)), id: 5, possubterms: +(x,s(y))->[], s(y)->[2])
 (rule: +(x,0) -> x, id: 6, possubterms: +(x,0)->[], 0->[2])
 (rule: -(x,p(y)) -> s(-(x,y)), id: 7, possubterms: -(x,p(y))->[], p(y)->[2])
 (rule: -(x,s(y)) -> p(-(x,y)), id: 8, possubterms: -(x,s(y))->[], s(y)->[2])
 (rule: -(x,0) -> x, id: 9, possubterms: -(x,0)->[], 0->[2])
 (rule: p(s(x)) -> x, id: 10, possubterms: p(s(x))->[], s(x)->[1])
 (rule: s(p(x)) -> x, id: 11, possubterms: s(p(x))->[], p(x)->[1])

-> Unifications:
(R1 unifies with R10 at p: [2], l: *(x,p(y)), lp: p(y), sig: {y -> s(x')}, l': p(s(x')), r: -(*(x,y),x), r': x')
(R2 unifies with R11 at p: [2], l: *(x,s(y)), lp: s(y), sig: {y -> p(x')}, l': s(p(x')), r: +(*(x,y),x), r': x')
(R4 unifies with R10 at p: [2], l: +(x,p(y)), lp: p(y), sig: {y -> s(x')}, l': p(s(x')), r: p(+(x,y)), r': x')
(R5 unifies with R11 at p: [2], l: +(x,s(y)), lp: s(y), sig: {y -> p(x')}, l': s(p(x')), r: s(+(x,y)), r': x')
(R7 unifies with R10 at p: [2], l: -(x,p(y)), lp: p(y), sig: {y -> s(x')}, l': p(s(x')), r: s(-(x,y)), r': x')
(R8 unifies with R11 at p: [2], l: -(x,s(y)), lp: s(y), sig: {y -> p(x')}, l': s(p(x')), r: p(-(x,y)), r': x')
(R10 unifies with R11 at p: [1], l: p(s(x)), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: x, r': x')
(R11 unifies with R10 at p: [1], l: s(p(x)), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: x, r': x')

-> Critical pairs info:
<-(x,x'),s(-(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<*(x,x'),+(*(x,p(x')),x)>   =>  Not trivial, Not overlay, Proper, NW0, N2
<+(x,x'),s(+(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<*(x,x'),-(*(x,s(x')),x)>   =>  Not trivial, Not overlay, Proper, NW0, N4
<-(x,x'),p(-(x,p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N5
<s(x'),s(x')>   =>  Trivial, Not overlay, Proper, NW0, N6
<p(x'),p(x')>   =>  Trivial, Not overlay, Proper, NW0, N7
<+(x,x'),p(+(x,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N8

-> Problem conclusions:
 Left linear, Not right linear, Not linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a TRS 


Problem 1: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: *(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('*(x,x')', D0)
t: +(*(x,p(x')),x)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('+(*(x,p(x')),x)', D0)
ID: 1 => ('+(-(*(x,x'),x),x)', D1, R1, P[1], S{x5 -> x, x6 -> x'}), NR: '-(*(x,x'),x)'
*(x,x') ->* no union *<- +(*(x,p(x')),x)
"Not joinable"

The problem is not confluent.