YES

Problem 1: 


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


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

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


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

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

-> Critical pairs info:
<f(d),f(c)>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(d),b>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(c),b>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(c),b>   =>  Not trivial, 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:
(REPLACEMENT-MAP
(a)
(d)
(f 1)
(b)
(c)
(fSNonEmpty)
)
(RULES
a -> d
d -> c
f(a) -> f(c)
f(a) -> b
f(d) -> b
f(c) -> b
)
Critical Pairs:
<f(d),f(c)>   =>  Not trivial, Not overlay, Proper, NW0, N1
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Strong Confluence Procedure:
-> Rewritings:
s: f(d)
Nodes: [0,1,2]
Edges: [(0,1),(0,2),(1,2)]
ID: 0 => ('f(d)', D0)
ID: 1 => ('f(c)', D1, R2, P[1], S{}), NR: 'c'
ID: 2 => ('b', D1, R5, P[], S{}), NR: 'b'
t: f(c)
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('f(c)', D0)
ID: 1 => ('b', D1, R6, P[], S{}), NR: 'b'
SNodesPath1: f(d) -> f(c)
TNodesPath1: f(c)
SNodesPath2: f(d) -> f(c)
TNodesPath2: f(c)
f(d) ->= f(c) *<- f(c)
f(c) ->= f(c) *<- f(d)
"Strongly closed critical pair"

The problem is confluent.

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

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

The problem is confluent.

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

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

The problem is confluent.