YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y z)
(REPLACEMENT-MAP
(from 1)
(sel 1, 2)
(0)
(: 1, 2)
(fSNonEmpty)
(s 1)
)
(RULES
from(x) -> :(x,from(s(x)))
sel(0,:(y,z)) -> y
sel(s(x),:(y,z)) -> sel(x,z)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 

Modular Confluence Combinations Decomposition Procedure:
TRS combination:
{from(x) -> :(x,from(s(x)))}
{sel(0,:(y,z)) -> y
 sel(s(x),:(y,z)) -> sel(x,z)}
Not disjoint
Constructor-sharing
Not composable
Left linear
Not layer-preserving
TRS1
Just (VAR x)
(STRATEGY CONTEXTSENSITIVE
(from 1)
(: 1 2)
(s 1)
)
(RULES
from(x) -> :(x,from(s(x)))
)

TRS2
Just (VAR x y z)
(STRATEGY CONTEXTSENSITIVE
(sel 1 2)
(0)
(: 1 2)
(s 1)
)
(RULES
sel(0,:(y,z)) -> y
sel(s(x),:(y,z)) -> sel(x,z)
)


The problem is decomposed in 2 subproblems.

Problem 1.1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(from 1)
(: 1, 2)
(s 1)
)
(RULES
from(x) -> :(x,from(s(x)))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.1: 

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

Problem 1.1: 
Linearity Procedure:
Linear?
NO

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(from 1)
(: 1, 2)
(s 1)
)
(RULES
from(x) -> :(x,from(s(x)))
)
Linear:NO
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 from(x) -> :(x,from(s(x)))
-> Vars:
 x

-> Rlps:
 (rule: from(x) -> :(x,from(s(x))), id: 1, possubterms: from(x)->[])

-> Unifications:


-> Critical pairs info:


-> Problem conclusions:
 Left linear, Not right linear, Not 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:
(VAR x y z)
(REPLACEMENT-MAP
(sel 1, 2)
(0)
(: 1, 2)
(s 1)
)
(RULES
sel(0,:(y,z)) -> y
sel(s(x),:(y,z)) -> sel(x,z)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


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:
(VAR x y z)
(REPLACEMENT-MAP
(sel 1, 2)
(0)
(: 1, 2)
(s 1)
)
(RULES
sel(0,:(y,z)) -> y
sel(s(x),:(y,z)) -> sel(x,z)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 sel(0,:(y,z)) -> y
 sel(s(x),:(y,z)) -> sel(x,z)
-> Vars:
 y, z, x, y, z

-> Rlps:
 (rule: sel(0,:(y,z)) -> y, id: 1, possubterms: sel(0,:(y,z))->[], 0->[1], :(y,z)->[2])
 (rule: sel(s(x),:(y,z)) -> sel(x,z), id: 2, possubterms: sel(s(x),:(y,z))->[], s(x)->[1], :(y,z)->[2])

-> 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.