NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y z)
(REPLACEMENT-MAP
(f 1, 2)
(1)
(fSNonEmpty)
)
(RULES
f(f(x,y),z) -> f(x,f(y,z))
f(x,1) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y z)
(REPLACEMENT-MAP
(f 1, 2)
(1)
(fSNonEmpty)
)
(RULES
f(f(x,y),z) -> f(x,f(y,z))
f(x,1) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 f(f(x,y),z) -> f(x,f(y,z))
 f(x,1) -> x
-> Vars:
 x, y, z, x

-> Rlps:
 (rule: f(f(x,y),z) -> f(x,f(y,z)), id: 1, possubterms: f(f(x,y),z)->[], f(x,y)->[1])
 (rule: f(x,1) -> x, id: 2, possubterms: f(x,1)->[], 1->[2])

-> Unifications:
(R1 unifies with R1 at p: [1], l: f(f(x,y),z), lp: f(x,y), sig: {x -> f(x',y'),y -> z'}, l': f(f(x',y'),z'), r: f(x,f(y,z)), r': f(x',f(y',z')))
(R1 unifies with R2 at p: [1], l: f(f(x,y),z), lp: f(x,y), sig: {x -> x',y -> 1}, l': f(x',1), r: f(x,f(y,z)), r': x')
(R2 unifies with R1 at p: [], l: f(x,1), lp: f(x,1), sig: {x -> f(x',y),z -> 1}, l': f(f(x',y),z), r: x, r': f(x',f(y,z)))

-> Critical pairs info:
<f(x',z),f(x',f(1,z))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(f(x',f(y',z')),z),f(f(x',y'),f(z',z))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(x',f(y,1)),f(x',y)>   =>  Not trivial, Overlay, Proper, NW0, N3

-> Problem conclusions:
 Left linear, Right linear, 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: f(x',z)
Nodes: [0]
Edges: []
ID: 0 => ('f(x',z)', D0)
t: f(x',f(1,z))
Nodes: [0]
Edges: []
ID: 0 => ('f(x',f(1,z))', D0)
f(x',z) ->* no union *<- f(x',f(1,z))
"Not joinable"

The problem is not confluent.