YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(F 1, 2)
(G 1)
(h 1)
(A)
(c 1)
(fSNonEmpty)
)
(RULES
F(x,y) -> c(A)
G(x) -> x
h(x) -> c(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 

Modular Confluence Combinations Decomposition Procedure:
TRS combination:
{F(x,y) -> c(A)}
{G(x) -> x
 h(x) -> c(x)}
Not disjoint
Constructor-sharing
Not composable
Left linear
Not layer-preserving
TRS1
Just (VAR x y)
(STRATEGY CONTEXTSENSITIVE
(F 1 2)
(A)
(c 1)
)
(RULES
F(x,y) -> c(A)
)

TRS2
Just (VAR x)
(STRATEGY CONTEXTSENSITIVE
(G 1)
(h 1)
(c 1)
)
(RULES
G(x) -> x
h(x) -> c(x)
)


The problem is decomposed in 2 subproblems.

Problem 1.1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x y)
(REPLACEMENT-MAP
(F 1, 2)
(A)
(c 1)
)
(RULES
F(x,y) -> c(A)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


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:
(VAR x y)
(REPLACEMENT-MAP
(F 1, 2)
(A)
(c 1)
)
(RULES
F(x,y) -> c(A)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 F(x,y) -> c(A)
-> Vars:
 x, y

-> Rlps:
 (rule: F(x,y) -> c(A), id: 1, possubterms: F(x,y)->[])

-> 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:
(VAR x)
(REPLACEMENT-MAP
(G 1)
(h 1)
(c 1)
)
(RULES
G(x) -> x
h(x) -> c(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.2: 

Problem 1.2: 

Modular Confluence Combinations Decomposition Procedure:
TRS combination:
{G(x) -> x}
{h(x) -> c(x)}
Disjoint
Constructor-sharing
Not composable
Left linear
Not layer-preserving
TRS1
Just (VAR x)
(STRATEGY CONTEXTSENSITIVE
(G 1)
)
(RULES
G(x) -> x
)

TRS2
Just (VAR x)
(STRATEGY CONTEXTSENSITIVE
(h 1)
(c 1)
)
(RULES
h(x) -> c(x)
)


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(G 1)
)
(RULES
G(x) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.2.1: 

Problem 1.2.1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1.2.1: 
Linearity Procedure:
Linear?
YES

Problem 1.2.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(G 1)
)
(RULES
G(x) -> x
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 G(x) -> x
-> Vars:
 x

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

-> 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 1.2.2: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(h 1)
(c 1)
)
(RULES
h(x) -> c(x)
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1.2.2: 

Problem 1.2.2: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1.2.2: 
Linearity Procedure:
Linear?
YES

Problem 1.2.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(h 1)
(c 1)
)
(RULES
h(x) -> c(x)
)
Linear:YES
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 h(x) -> c(x)
-> Vars:
 x

-> Rlps:
 (rule: h(x) -> c(x), id: 1, possubterms: h(x)->[])

-> 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.