YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(g 1)
(c)
(fSNonEmpty)
(h 1)
)
(RULES
a -> b
b -> c
f(g(g(x))) -> a
f(g(h(x))) -> b
f(h(g(x))) -> b
f(h(h(x))) -> c
g(x) -> h(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
Linearity Procedure:
Linear?
YES

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(g 1)
(c)
(fSNonEmpty)
(h 1)
)
(RULES
a -> b
b -> c
f(g(g(x))) -> a
f(g(h(x))) -> b
f(h(g(x))) -> b
f(h(h(x))) -> c
g(x) -> h(x)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

-> Rlps:
 (rule: a -> b, id: 1, possubterms: a->[])
 (rule: b -> c, id: 2, possubterms: b->[])
 (rule: f(g(g(x))) -> a, id: 3, possubterms: f(g(g(x)))->[], g(g(x))->[1], g(x)->[1, 1])
 (rule: f(g(h(x))) -> b, id: 4, possubterms: f(g(h(x)))->[], g(h(x))->[1], h(x)->[1, 1])
 (rule: f(h(g(x))) -> b, id: 5, possubterms: f(h(g(x)))->[], h(g(x))->[1], g(x)->[1, 1])
 (rule: f(h(h(x))) -> c, id: 6, possubterms: f(h(h(x)))->[], h(h(x))->[1], h(x)->[1, 1])
 (rule: g(x) -> h(x), id: 7, possubterms: g(x)->[])

-> Unifications:
(R3 unifies with R7 at p: [1], l: f(g(g(x))), lp: g(g(x)), sig: {x' -> g(x)}, l': g(x'), r: a, r': h(x'))
(R3 unifies with R7 at p: [1,1], l: f(g(g(x))), lp: g(x), sig: {x -> x'}, l': g(x'), r: a, r': h(x'))
(R4 unifies with R7 at p: [1], l: f(g(h(x))), lp: g(h(x)), sig: {x' -> h(x)}, l': g(x'), r: b, r': h(x'))
(R5 unifies with R7 at p: [1,1], l: f(h(g(x))), lp: g(x), sig: {x -> x'}, l': g(x'), r: b, r': h(x'))

-> Critical pairs info:
<f(h(h(x'))),b>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(g(h(x'))),a>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(h(h(x))),b>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(h(g(x))),a>   =>  Not trivial, Not overlay, Proper, NW0, N4

-> 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 



The problem is decomposed in 4 subproblems.

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

Strong Confluence Procedure:
-> Rewritings:
s: f(h(h(x')))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(h(h(x')))', D0)
ID: 1 => ('c', D1, R6, P[], S{x7 -> x'}), NR: 'c'
t: b
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b', D0)
ID: 1 => ('c', D1, R2, P[], S{}), NR: 'c'
SNodesPath1: f(h(h(x'))) -> c
TNodesPath1: b -> c
SNodesPath2: f(h(h(x'))) -> c
TNodesPath2: b -> c
f(h(h(x'))) ->= c *<- b
b ->= c *<- f(h(h(x')))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(g 1)
(c)
(fSNonEmpty)
(h 1)
)
(RULES
a -> b
b -> c
f(g(g(x))) -> a
f(g(h(x))) -> b
f(h(g(x))) -> b
f(h(h(x))) -> c
g(x) -> h(x)
)
Critical Pairs:
<f(g(h(x'))),a>   =>  Not trivial, Not overlay, Proper, NW0, N2
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(g(h(x')))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('f(g(h(x')))', D0)
ID: 1 => ('b', D1, R4, P[], S{x5 -> x'}), NR: 'b'
ID: 2 => ('f(h(h(x')))', D1, R7, P[1], S{x8 -> h(x')}), NR: 'h(h(x'))'
ID: 3 => ('c', D2, R2, P[], S{}), NR: 'c'
t: a
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a', D0)
ID: 1 => ('b', D1, R1, P[], S{}), NR: 'b'
ID: 2 => ('c', D2, R2, P[], S{}), NR: 'c'
SNodesPath1: f(g(h(x'))) -> b
TNodesPath1: a -> b
SNodesPath2: f(g(h(x'))) -> b
TNodesPath2: a -> b
f(g(h(x'))) ->= b *<- a
a ->= b *<- f(g(h(x')))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(g 1)
(c)
(fSNonEmpty)
(h 1)
)
(RULES
a -> b
b -> c
f(g(g(x))) -> a
f(g(h(x))) -> b
f(h(g(x))) -> b
f(h(h(x))) -> c
g(x) -> h(x)
)
Critical Pairs:
<f(h(h(x))),b>   =>  Not trivial, Not overlay, Proper, NW0, N3
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(h(h(x)))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(h(h(x)))', D0)
ID: 1 => ('c', D1, R6, P[], S{x7 -> x}), NR: 'c'
t: b
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('b', D0)
ID: 1 => ('c', D1, R2, P[], S{}), NR: 'c'
SNodesPath1: f(h(h(x))) -> c
TNodesPath1: b -> c
SNodesPath2: f(h(h(x))) -> c
TNodesPath2: b -> c
f(h(h(x))) ->= c *<- b
b ->= c *<- f(h(h(x)))
"Strongly closed critical pair"

The problem is confluent.

Problem 1.4: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(g 1)
(c)
(fSNonEmpty)
(h 1)
)
(RULES
a -> b
b -> c
f(g(g(x))) -> a
f(g(h(x))) -> b
f(h(g(x))) -> b
f(h(h(x))) -> c
g(x) -> h(x)
)
Critical Pairs:
<f(h(g(x))),a>   =>  Not trivial, Not overlay, Proper, NW0, N4
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(h(g(x)))
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(2,3)]
ID: 0 => ('f(h(g(x)))', D0)
ID: 1 => ('b', D1, R5, P[], S{x6 -> x}), NR: 'b'
ID: 2 => ('f(h(h(x)))', D1, R7, P[1, 1], S{x8 -> x}), NR: 'h(x)'
ID: 3 => ('c', D2, R2, P[], S{}), NR: 'c'
t: a
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('a', D0)
ID: 1 => ('b', D1, R1, P[], S{}), NR: 'b'
ID: 2 => ('c', D2, R2, P[], S{}), NR: 'c'
SNodesPath1: f(h(g(x))) -> b
TNodesPath1: a -> b
SNodesPath2: f(h(g(x))) -> b
TNodesPath2: a -> b
f(h(g(x))) ->= b *<- a
a ->= b *<- f(h(g(x)))
"Strongly closed critical pair"

The problem is confluent.