YES

Problem 1: 


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

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


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

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

-> Critical pairs info:
<f(a',b),c>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(a,b'),f(a',b)>   =>  Not trivial, Overlay, Proper, NW0, N2
<f(a,b),f(a,b')>   =>  Not trivial, Overlay, Proper, NW0, N3
<f(a,b'),c>   =>  Not trivial, Not overlay, Proper, NW0, N4
<f(a,b),f(a',b)>   =>  Not trivial, Overlay, Proper, NW0, N5

-> 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 5 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1, 2)
(a')
(b')
(fSNonEmpty)
)
(RULES
a -> a'
b -> b'
c -> f(a,b)
c -> f(a,b')
c -> f(a',b)
f(a,b) -> c
)
Critical Pairs:
<f(a',b),c>   =>  Not trivial, Not overlay, Proper, NW0, N1
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(a',b)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(a',b)', D0)
ID: 1 => ('f(a',b')', D1, R2, P[2], S{}), NR: 'b''
t: c
Nodes: [0,1,2,3,4]
Edges: [(0,1),(0,2),(0,3),(1,0),(1,2),(1,3),(2,4),(3,4)]
ID: 0 => ('c', D0)
ID: 1 => ('f(a,b)', D1, R3, P[], S{}), NR: 'f(a,b)'
ID: 2 => ('f(a,b')', D1, R4, P[], S{}), NR: 'f(a,b')'
ID: 3 => ('f(a',b)', D1, R5, P[], S{}), NR: 'f(a',b)'
ID: 4 => ('f(a',b')', D2, R1, P[1], S{}), NR: 'a''
SNodesPath1: f(a',b)
TNodesPath1: 
SNodesPath2: f(a',b)
TNodesPath2: 
f(a',b) ->= f(a',b) *<- c
c ->= f(a',b) *<- f(a',b)
"Strongly closed critical pair"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1, 2)
(a')
(b')
(fSNonEmpty)
)
(RULES
a -> a'
b -> b'
c -> f(a,b)
c -> f(a,b')
c -> f(a',b)
f(a,b) -> c
)
Critical Pairs:
<f(a,b'),f(a',b)>   =>  Not trivial, Overlay, Proper, NW0, N2
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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

The problem is confluent.

Problem 1.3: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1, 2)
(a')
(b')
(fSNonEmpty)
)
(RULES
a -> a'
b -> b'
c -> f(a,b)
c -> f(a,b')
c -> f(a',b)
f(a,b) -> c
)
Critical Pairs:
<f(a,b),f(a,b')>   =>  Not trivial, Overlay, Proper, NW0, N3
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(a,b)
Nodes: [0,1,2,3,4]
Edges: [(0,1),(0,2),(0,3),(1,4),(2,4),(3,0),(3,1),(3,2)]
ID: 0 => ('f(a,b)', D0)
ID: 1 => ('f(a',b)', D1, R1, P[1], S{}), NR: 'a''
ID: 2 => ('f(a,b')', D1, R2, P[2], S{}), NR: 'b''
ID: 3 => ('c', D1, R6, P[], S{}), NR: 'c'
ID: 4 => ('f(a',b')', D2, R2, P[2], S{}), NR: 'b''
t: f(a,b')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(a,b')', D0)
ID: 1 => ('f(a',b')', D1, R1, P[1], S{}), NR: 'a''
SNodesPath1: 
TNodesPath1: f(a,b')
SNodesPath2: 
TNodesPath2: f(a,b')
f(a,b) ->= f(a,b') *<- f(a,b')
f(a,b') ->= f(a,b') *<- f(a,b)
"Strongly closed critical pair"

The problem is confluent.

Problem 1.4: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1, 2)
(a')
(b')
(fSNonEmpty)
)
(RULES
a -> a'
b -> b'
c -> f(a,b)
c -> f(a,b')
c -> f(a',b)
f(a,b) -> c
)
Critical Pairs:
<f(a,b'),c>   =>  Not trivial, Not overlay, Proper, NW0, N4
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(a,b')
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(a,b')', D0)
ID: 1 => ('f(a',b')', D1, R1, P[1], S{}), NR: 'a''
t: c
Nodes: [0,1,2,3,4]
Edges: [(0,1),(0,2),(0,3),(1,0),(1,2),(1,3),(2,4),(3,4)]
ID: 0 => ('c', D0)
ID: 1 => ('f(a,b)', D1, R3, P[], S{}), NR: 'f(a,b)'
ID: 2 => ('f(a,b')', D1, R4, P[], S{}), NR: 'f(a,b')'
ID: 3 => ('f(a',b)', D1, R5, P[], S{}), NR: 'f(a',b)'
ID: 4 => ('f(a',b')', D2, R1, P[1], S{}), NR: 'a''
SNodesPath1: f(a,b')
TNodesPath1: c -> f(a,b) -> f(a,b')
SNodesPath2: f(a,b')
TNodesPath2: c -> f(a,b) -> f(a,b')
f(a,b') ->= f(a,b') *<- c
c ->= f(a,b') *<- f(a,b')
"Strongly closed critical pair"

The problem is confluent.

Problem 1.5: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1, 2)
(a')
(b')
(fSNonEmpty)
)
(RULES
a -> a'
b -> b'
c -> f(a,b)
c -> f(a,b')
c -> f(a',b)
f(a,b) -> c
)
Critical Pairs:
<f(a,b),f(a',b)>   =>  Not trivial, Overlay, Proper, NW0, N5
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(a,b)
Nodes: [0,1,2,3,4]
Edges: [(0,1),(0,2),(0,3),(1,4),(2,4),(3,0),(3,1),(3,2)]
ID: 0 => ('f(a,b)', D0)
ID: 1 => ('f(a',b)', D1, R1, P[1], S{}), NR: 'a''
ID: 2 => ('f(a,b')', D1, R2, P[2], S{}), NR: 'b''
ID: 3 => ('c', D1, R6, P[], S{}), NR: 'c'
ID: 4 => ('f(a',b')', D2, R2, P[2], S{}), NR: 'b''
t: f(a',b)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(a',b)', D0)
ID: 1 => ('f(a',b')', D1, R2, P[2], S{}), NR: 'b''
SNodesPath1: f(a,b) -> f(a',b)
TNodesPath1: f(a',b)
SNodesPath2: f(a,b) -> f(a',b)
TNodesPath2: f(a',b)
f(a,b) ->= f(a',b) *<- f(a',b)
f(a',b) ->= f(a',b) *<- f(a,b)
"Strongly closed critical pair"

The problem is confluent.