YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(f 1)
(g 1)
(fSNonEmpty)
(h 1, 2)
(s 1)
)
(RULES
f(g(x)) -> h(g(x),g(x))
f(s(x)) -> h(s(x),s(x))
g(x) -> s(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
CSR Converter From Canonical u-Replacement Map Procedure [CSUR20]:

Original Replacement Map:
(f 1)
(g 1)
(fSNonEmpty)
(h 1 2)
(s 1)

New Replacement Map:
(f 1)
(g)
(fSNonEmpty)
(h)
(s)

New problem:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(f 1)
(g)
(fSNonEmpty)
(h)
(s)
)
(RULES
f(g(x)) -> h(g(x),g(x))
f(s(x)) -> h(s(x),s(x))
g(x) -> s(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(f 1)
(g)
(fSNonEmpty)
(h)
(s)
)
(RULES
f(g(x)) -> h(g(x),g(x))
f(s(x)) -> h(s(x),s(x))
g(x) -> s(x)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 f(g(x)) -> h(g(x),g(x))
 f(s(x)) -> h(s(x),s(x))
 g(x) -> s(x)
-> Unhiding Rules:
 Empty

Problem 1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 f(g(x)) -> h(g(x),g(x))
 f(s(x)) -> h(s(x),s(x))
 g(x) -> s(x)
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(f 1)
(g)
(fSNonEmpty)
(h)
(s)
)
(RULES
f(g(x)) -> h(g(x),g(x))
f(s(x)) -> h(s(x),s(x))
g(x) -> s(x)
)
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy NW Procedure:
-> Rules:
 f(g(x)) -> h(g(x),g(x))
 f(s(x)) -> h(s(x),s(x))
 g(x) -> s(x)
-> Vars:
 x, x, x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 3, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: f(g(x)) -> h(g(x),g(x)), id: 1, possubterms: f(g(x))->[], g(x)->[1])
 (rule: f(s(x)) -> h(s(x),s(x)), id: 2, possubterms: f(s(x))->[], s(x)->[1])
 (rule: g(x) -> s(x), id: 3, possubterms: g(x)->[])

-> Unifications:
(R1 unifies with R3 at p: [1], l: f(g(x)), lp: g(x), sig: {x -> x'}, l': g(x'), r: h(g(x),g(x)), r': s(x'))

-> Critical pairs info:
<f(s(x')),h(g(x'),g(x'))>   =>  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 CS-TRS, Left-homogeneous u-replacing variables

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(f 1)
(g)
(fSNonEmpty)
(h)
(s)
)
(RULES
f(g(x)) -> h(g(x),g(x))
f(s(x)) -> h(s(x),s(x))
g(x) -> s(x)
)
Critical Pairs:
<f(s(x')),h(g(x'),g(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N1
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: f(s(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(s(x'))', D0)
ID: 1 => ('h(s(x'),s(x'))', D1, R2, P[], S{x5 -> x'}), NR: 'h(s(x'),s(x'))'
t: h(g(x'),g(x'))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('h(g(x'),g(x'))', D0)
ID: 1 => ('h(s(x'),g(x'))', D1, R3, P[1], S{x6 -> x'}), NR: 's(x')'
ID: 2 => ('h(g(x'),s(x'))', D1, R3, P[2], S{x6 -> x'}), NR: 's(x')'
ID: 3 => ('h(s(x'),s(x'))', D2, R3, P[2], S{x6 -> x'}), NR: 's(x')'
SNodesPath: f(s(x')) -> h(s(x'),s(x'))
TNodesPath: h(g(x'),g(x')) -> h(s(x'),g(x')) -> h(s(x'),s(x'))
f(s(x')) ->* h(s(x'),s(x')) *<- h(g(x'),g(x'))
"Joinable"

The problem is confluent.