YES

Problem 1: 


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Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(c)
(f 1, 2)
(g 1)
(a)
(b)
(fSNonEmpty)
(h 1)
)
(RULES
c -> c
c -> f(a,h(b))
f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
f(x,y) -> f(y,x)
g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
)
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Problem 1: 

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

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(c)
(f 1, 2)
(g 1)
(a)
(b)
(fSNonEmpty)
(h 1)
)
(RULES
c -> f(a,h(b))
f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
f(x,y) -> f(y,x)
g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Commutativity Transform Procedure:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(c)
(f 1, 2)
(g 1)
(a)
(b)
(fSNonEmpty)
(h 1)
)
(RULES
c -> f(a,h(b))
f(g(f(g(b),x)),h(f(f(a,a),h(a)))) -> c
f(g(f(g(b),x)),h(f(h(a),f(a,a)))) -> c
f(g(f(x,g(b))),h(f(f(a,a),h(a)))) -> c
f(g(f(x,g(b))),h(f(h(a),f(a,a)))) -> c
f(h(f(f(a,a),h(a))),g(f(g(b),x))) -> c
f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
f(h(f(h(a),f(a,a))),g(f(g(b),x))) -> c
f(h(f(h(a),f(a,a))),g(f(x,g(b)))) -> c
g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(c)
(f 1, 2)
(g 1)
(a)
(b)
(fSNonEmpty)
(h 1)
)
(RULES
c -> f(a,h(b))
f(g(f(g(b),x)),h(f(f(a,a),h(a)))) -> c
f(g(f(g(b),x)),h(f(h(a),f(a,a)))) -> c
f(g(f(x,g(b))),h(f(f(a,a),h(a)))) -> c
f(g(f(x,g(b))),h(f(h(a),f(a,a)))) -> c
f(h(f(f(a,a),h(a))),g(f(g(b),x))) -> c
f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
f(h(f(h(a),f(a,a))),g(f(g(b),x))) -> c
f(h(f(h(a),f(a,a))),g(f(x,g(b)))) -> c
g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


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 x y)
(REPLACEMENT-MAP
(c)
(f 1, 2)
(g 1)
(a)
(b)
(fSNonEmpty)
(h 1)
)
(RULES
c -> f(a,h(b))
f(g(f(g(b),x)),h(f(f(a,a),h(a)))) -> c
f(g(f(g(b),x)),h(f(h(a),f(a,a)))) -> c
f(g(f(x,g(b))),h(f(f(a,a),h(a)))) -> c
f(g(f(x,g(b))),h(f(h(a),f(a,a)))) -> c
f(h(f(f(a,a),h(a))),g(f(g(b),x))) -> c
f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
f(h(f(h(a),f(a,a))),g(f(g(b),x))) -> c
f(h(f(h(a),f(a,a))),g(f(x,g(b)))) -> c
g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
)
Linear:YES
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Huet Levy Procedure:
-> Rules:
 c -> f(a,h(b))
 f(g(f(g(b),x)),h(f(f(a,a),h(a)))) -> c
 f(g(f(g(b),x)),h(f(h(a),f(a,a)))) -> c
 f(g(f(x,g(b))),h(f(f(a,a),h(a)))) -> c
 f(g(f(x,g(b))),h(f(h(a),f(a,a)))) -> c
 f(h(f(f(a,a),h(a))),g(f(g(b),x))) -> c
 f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c
 f(h(f(h(a),f(a,a))),g(f(g(b),x))) -> c
 f(h(f(h(a),f(a,a))),g(f(x,g(b)))) -> c
 g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a))))
-> Vars:
 x, x, x, x, x, x, x, x

-> Rlps:
 (rule: c -> f(a,h(b)), id: 1, possubterms: c->[])
 (rule: f(g(f(g(b),x)),h(f(f(a,a),h(a)))) -> c, id: 2, possubterms: f(g(f(g(b),x)),h(f(f(a,a),h(a))))->[], g(f(g(b),x))->[1], f(g(b),x)->[1, 1], g(b)->[1, 1, 1], b->[1, 1, 1, 1], h(f(f(a,a),h(a)))->[2], f(f(a,a),h(a))->[2, 1], f(a,a)->[2, 1, 1], a->[2, 1, 1, 1], a->[2, 1, 1, 2], h(a)->[2, 1, 2], a->[2, 1, 2, 1])
 (rule: f(g(f(g(b),x)),h(f(h(a),f(a,a)))) -> c, id: 3, possubterms: f(g(f(g(b),x)),h(f(h(a),f(a,a))))->[], g(f(g(b),x))->[1], f(g(b),x)->[1, 1], g(b)->[1, 1, 1], b->[1, 1, 1, 1], h(f(h(a),f(a,a)))->[2], f(h(a),f(a,a))->[2, 1], h(a)->[2, 1, 1], a->[2, 1, 1, 1], f(a,a)->[2, 1, 2], a->[2, 1, 2, 1], a->[2, 1, 2, 2])
 (rule: f(g(f(x,g(b))),h(f(f(a,a),h(a)))) -> c, id: 4, possubterms: f(g(f(x,g(b))),h(f(f(a,a),h(a))))->[], g(f(x,g(b)))->[1], f(x,g(b))->[1, 1], g(b)->[1, 1, 2], b->[1, 1, 2, 1], h(f(f(a,a),h(a)))->[2], f(f(a,a),h(a))->[2, 1], f(a,a)->[2, 1, 1], a->[2, 1, 1, 1], a->[2, 1, 1, 2], h(a)->[2, 1, 2], a->[2, 1, 2, 1])
 (rule: f(g(f(x,g(b))),h(f(h(a),f(a,a)))) -> c, id: 5, possubterms: f(g(f(x,g(b))),h(f(h(a),f(a,a))))->[], g(f(x,g(b)))->[1], f(x,g(b))->[1, 1], g(b)->[1, 1, 2], b->[1, 1, 2, 1], h(f(h(a),f(a,a)))->[2], f(h(a),f(a,a))->[2, 1], h(a)->[2, 1, 1], a->[2, 1, 1, 1], f(a,a)->[2, 1, 2], a->[2, 1, 2, 1], a->[2, 1, 2, 2])
 (rule: f(h(f(f(a,a),h(a))),g(f(g(b),x))) -> c, id: 6, possubterms: f(h(f(f(a,a),h(a))),g(f(g(b),x)))->[], h(f(f(a,a),h(a)))->[1], f(f(a,a),h(a))->[1, 1], f(a,a)->[1, 1, 1], a->[1, 1, 1, 1], a->[1, 1, 1, 2], h(a)->[1, 1, 2], a->[1, 1, 2, 1], g(f(g(b),x))->[2], f(g(b),x)->[2, 1], g(b)->[2, 1, 1], b->[2, 1, 1, 1])
 (rule: f(h(f(f(a,a),h(a))),g(f(x,g(b)))) -> c, id: 7, possubterms: f(h(f(f(a,a),h(a))),g(f(x,g(b))))->[], h(f(f(a,a),h(a)))->[1], f(f(a,a),h(a))->[1, 1], f(a,a)->[1, 1, 1], a->[1, 1, 1, 1], a->[1, 1, 1, 2], h(a)->[1, 1, 2], a->[1, 1, 2, 1], g(f(x,g(b)))->[2], f(x,g(b))->[2, 1], g(b)->[2, 1, 2], b->[2, 1, 2, 1])
 (rule: f(h(f(h(a),f(a,a))),g(f(g(b),x))) -> c, id: 8, possubterms: f(h(f(h(a),f(a,a))),g(f(g(b),x)))->[], h(f(h(a),f(a,a)))->[1], f(h(a),f(a,a))->[1, 1], h(a)->[1, 1, 1], a->[1, 1, 1, 1], f(a,a)->[1, 1, 2], a->[1, 1, 2, 1], a->[1, 1, 2, 2], g(f(g(b),x))->[2], f(g(b),x)->[2, 1], g(b)->[2, 1, 1], b->[2, 1, 1, 1])
 (rule: f(h(f(h(a),f(a,a))),g(f(x,g(b)))) -> c, id: 9, possubterms: f(h(f(h(a),f(a,a))),g(f(x,g(b))))->[], h(f(h(a),f(a,a)))->[1], f(h(a),f(a,a))->[1, 1], h(a)->[1, 1, 1], a->[1, 1, 1, 1], f(a,a)->[1, 1, 2], a->[1, 1, 2, 1], a->[1, 1, 2, 2], g(f(x,g(b)))->[2], f(x,g(b))->[2, 1], g(b)->[2, 1, 2], b->[2, 1, 2, 1])
 (rule: g(g(a)) -> f(h(g(f(c,c))),f(f(g(c),a),g(f(a,a)))), id: 10, possubterms: g(g(a))->[], g(a)->[1], a->[1, 1])

-> Unifications:
(R4 unifies with R2 at p: [], l: f(g(f(x,g(b))),h(f(f(a,a),h(a)))), lp: f(g(f(x,g(b))),h(f(f(a,a),h(a)))), sig: {x -> g(b),x' -> g(b)}, l': f(g(f(g(b),x')),h(f(f(a,a),h(a)))), r: c, r': c)
(R5 unifies with R3 at p: [], l: f(g(f(x,g(b))),h(f(h(a),f(a,a)))), lp: f(g(f(x,g(b))),h(f(h(a),f(a,a)))), sig: {x -> g(b),x' -> g(b)}, l': f(g(f(g(b),x')),h(f(h(a),f(a,a)))), r: c, r': c)
(R7 unifies with R6 at p: [], l: f(h(f(f(a,a),h(a))),g(f(x,g(b)))), lp: f(h(f(f(a,a),h(a))),g(f(x,g(b)))), sig: {x -> g(b),x' -> g(b)}, l': f(h(f(f(a,a),h(a))),g(f(g(b),x'))), r: c, r': c)
(R9 unifies with R8 at p: [], l: f(h(f(h(a),f(a,a))),g(f(x,g(b)))), lp: f(h(f(h(a),f(a,a))),g(f(x,g(b)))), sig: {x -> g(b),x' -> g(b)}, l': f(h(f(h(a),f(a,a))),g(f(g(b),x'))), r: c, r': c)

-> Critical pairs info:
<c,c>   =>  Trivial, Overlay, Proper, NW0, N1
<c,c>   =>  Trivial, Overlay, Proper, NW0, N2
<c,c>   =>  Trivial, Overlay, Proper, NW0, N3
<c,c>   =>  Trivial, Overlay, Proper, NW0, N4

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


The problem is confluent.