YES

Problem 1: 


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


Problem 1: 

CleanTRS Procedure:
R was updated by simple cleaning of the TRS

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


Problem 1: 

Modular Confluence Combinations Decomposition Procedure:
TRS combination:
{a -> b}
{f(c) -> b
 f(b) -> c}
Not disjoint
Constructor-sharing
Not composable
Left linear
Not layer-preserving
TRS1
Just (STRATEGY CONTEXTSENSITIVE
(a)
(b)
)
(RULES
a -> b
)

TRS2
Just (STRATEGY CONTEXTSENSITIVE
(c)
(f 1)
(b)
)
(RULES
f(c) -> b
f(b) -> c
)


The problem is decomposed in 2 subproblems.

Problem 1.1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
)
(RULES
a -> b
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.1: 

Problem 1.1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1.1: 
Linearity Procedure:
Linear?
YES

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
)
(RULES
a -> b
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 a -> b
-> Vars:


-> Rlps:
 (rule: a -> b, id: 1, possubterms: a->[])

-> Unifications:


-> Critical pairs info:


-> Problem conclusions:
 Left linear, Right linear, Linear
 Weakly orthogonal, Almost orthogonal, Orthogonal
 Huet-Levy confluent, Not Newman confluent
 R is a TRS 


The problem is confluent.

Problem 1.2: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(c)
(f 1)
(b)
)
(RULES
f(c) -> b
f(b) -> c
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.2: 

Problem 1.2: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1.2: 
Linearity Procedure:
Linear?
YES

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(c)
(f 1)
(b)
)
(RULES
f(c) -> b
f(b) -> c
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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


-> Rlps:
 (rule: f(c) -> b, id: 1, possubterms: f(c)->[], c->[1])
 (rule: f(b) -> c, id: 2, possubterms: f(b)->[], b->[1])

-> Unifications:


-> Critical pairs info:


-> Problem conclusions:
 Left linear, Right linear, Linear
 Weakly orthogonal, Almost orthogonal, Orthogonal
 Huet-Levy confluent, Not Newman confluent
 R is a TRS 


The problem is confluent.