YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(b)
(d)
(f 1, 2)
(c)
(fSNonEmpty)
)
(RULES
a -> c
b -> c
d -> f(a,c)
d -> f(c,b)
f(a,b) -> d
f(c,x) -> f(c,c)
f(x,c) -> f(c,c)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


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)
(d)
(f 1, 2)
(c)
(fSNonEmpty)
)
(RULES
a -> c
b -> c
d -> f(a,c)
d -> f(c,b)
f(a,b) -> d
f(c,x) -> f(c,c)
f(x,c) -> f(c,c)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

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

-> Unifications:
(R4 unifies with R3 at p: [], l: d, lp: d, sig: {}, l': d, r: f(c,b), r': f(a,c))
(R5 unifies with R1 at p: [1], l: f(a,b), lp: a, sig: {}, l': a, r: d, r': c)
(R5 unifies with R2 at p: [2], l: f(a,b), lp: b, sig: {}, l': b, r: d, r': c)
(R7 unifies with R6 at p: [], l: f(x,c), lp: f(x,c), sig: {x -> c,x' -> c}, l': f(c,x'), r: f(c,c), r': f(c,c))

-> Critical pairs info:
<f(c,c),f(c,c)>   =>  Trivial, Overlay, Proper, NW0, N1
<f(a,c),f(c,b)>   =>  Not trivial, Overlay, Proper, NW0, N2
<f(a,c),d>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(c,b),d>   =>  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 3 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(b)
(d)
(f 1, 2)
(c)
(fSNonEmpty)
)
(RULES
a -> c
b -> c
d -> f(a,c)
d -> f(c,b)
f(a,b) -> d
f(c,x) -> f(c,c)
f(x,c) -> f(c,c)
)
Critical Pairs:
<f(a,c),f(c,b)>   =>  Not trivial, Overlay, Proper, NW0, N2
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(b)
(d)
(f 1, 2)
(c)
(fSNonEmpty)
)
(RULES
a -> c
b -> c
d -> f(a,c)
d -> f(c,b)
f(a,b) -> d
f(c,x) -> f(c,c)
f(x,c) -> f(c,c)
)
Critical Pairs:
<f(a,c),d>   =>  Not trivial, Not overlay, Proper, NW0, N3
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(a,c)
Nodes: [0,1]
Edges: [(0,1),(0,1),(1,1),(1,1)]
ID: 0 => ('f(a,c)', D0)
ID: 1 => ('f(c,c)', D1, R1, P[1], S{}), NR: 'c'
t: d
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(1,3),(2,3),(2,3),(3,3),(3,3)]
ID: 0 => ('d', D0)
ID: 1 => ('f(a,c)', D1, R3, P[], S{}), NR: 'f(a,c)'
ID: 2 => ('f(c,b)', D1, R4, P[], S{}), NR: 'f(c,b)'
ID: 3 => ('f(c,c)', D2, R1, P[1], S{}), NR: 'c'
SNodesPath1: f(a,c)
TNodesPath1: d -> f(a,c)
SNodesPath2: f(a,c)
TNodesPath2: d -> f(a,c)
f(a,c) ->= f(a,c) *<- d
d ->= f(a,c) *<- f(a,c)
"Strongly closed critical pair"

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(b)
(d)
(f 1, 2)
(c)
(fSNonEmpty)
)
(RULES
a -> c
b -> c
d -> f(a,c)
d -> f(c,b)
f(a,b) -> d
f(c,x) -> f(c,c)
f(x,c) -> f(c,c)
)
Critical Pairs:
<f(c,b),d>   =>  Not trivial, Not overlay, Proper, NW0, N4
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(c,b)
Nodes: [0,1]
Edges: [(0,1),(0,1),(1,1),(1,1)]
ID: 0 => ('f(c,b)', D0)
ID: 1 => ('f(c,c)', D1, R2, P[2], S{}), NR: 'c'
t: d
Nodes: [0,1,2,3]
Edges: [(0,1),(0,2),(1,3),(1,3),(2,3),(2,3),(3,3),(3,3)]
ID: 0 => ('d', D0)
ID: 1 => ('f(a,c)', D1, R3, P[], S{}), NR: 'f(a,c)'
ID: 2 => ('f(c,b)', D1, R4, P[], S{}), NR: 'f(c,b)'
ID: 3 => ('f(c,c)', D2, R1, P[1], S{}), NR: 'c'
SNodesPath1: f(c,b)
TNodesPath1: 
SNodesPath2: f(c,b)
TNodesPath2: 
f(c,b) ->= f(c,b) *<- d
d ->= f(c,b) *<- f(c,b)
"Strongly closed critical pair"

The problem is confluent.