NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(f 1)
(g 1, 2, 3)
(p 1)
(q 1)
(a)
(b)
(c)
(d)
(fSNonEmpty)
(h 1, 2)
)
(RULES
f(g(x,a,b)) -> x
g(f(h(c,d)),x,y) -> h(p(x),q(x))
p(a) -> c
q(b) -> d
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(f 1)
(g 1, 2, 3)
(p 1)
(q 1)
(a)
(b)
(c)
(d)
(fSNonEmpty)
(h 1, 2)
)
(RULES
f(g(x,a,b)) -> x
g(f(h(c,d)),x,y) -> h(p(x),q(x))
p(a) -> c
q(b) -> d
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 f(g(x,a,b)) -> x
 g(f(h(c,d)),x,y) -> h(p(x),q(x))
 p(a) -> c
 q(b) -> d
-> Vars:
 x, x, y

-> Rlps:
 (rule: f(g(x,a,b)) -> x, id: 1, possubterms: f(g(x,a,b))->[], g(x,a,b)->[1], a->[1, 2], b->[1, 3])
 (rule: g(f(h(c,d)),x,y) -> h(p(x),q(x)), id: 2, possubterms: g(f(h(c,d)),x,y)->[], f(h(c,d))->[1], h(c,d)->[1, 1], c->[1, 1, 1], d->[1, 1, 2])
 (rule: p(a) -> c, id: 3, possubterms: p(a)->[], a->[1])
 (rule: q(b) -> d, id: 4, possubterms: q(b)->[], b->[1])

-> Unifications:
(R1 unifies with R2 at p: [1], l: f(g(x,a,b)), lp: g(x,a,b), sig: {x -> f(h(c,d)),x' -> a,y -> b}, l': g(f(h(c,d)),x',y), r: x, r': h(p(x'),q(x')))

-> Critical pairs info:
<f(h(p(a),q(a))),f(h(c,d))>   =>  Not trivial, Not overlay, Proper, NW0, N1

-> 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: f(h(p(a),q(a)))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(h(p(a),q(a)))', D0)
ID: 1 => ('f(h(c,q(a)))', D1, R3, P[1, 1], S{}), NR: 'c'
t: f(h(c,d))
Nodes: [0]
Edges: []
ID: 0 => ('f(h(c,d))', D0)
f(h(p(a),q(a))) ->* no union *<- f(h(c,d))
"Not joinable"

The problem is not confluent.