NO

Problem 1: 


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


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

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

Huet Levy Ordered by Num of Vars and Symbs Procedure:
-> Rules:
 a -> h(a,f(b))
 f(f(h(a,a))) -> f(f(f(a)))
 f(h(a,h(a,a))) -> f(a)
 f(h(h(b,a),h(f(c),c))) -> c
-> Vars:


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

-> Unifications:
(R2 unifies with R1 at p: [1,1,1], l: f(f(h(a,a))), lp: a, sig: {}, l': a, r: f(f(f(a))), r': h(a,f(b)))
(R2 unifies with R1 at p: [1,1,2], l: f(f(h(a,a))), lp: a, sig: {}, l': a, r: f(f(f(a))), r': h(a,f(b)))
(R3 unifies with R1 at p: [1,1], l: f(h(a,h(a,a))), lp: a, sig: {}, l': a, r: f(a), r': h(a,f(b)))
(R3 unifies with R1 at p: [1,2,1], l: f(h(a,h(a,a))), lp: a, sig: {}, l': a, r: f(a), r': h(a,f(b)))
(R3 unifies with R1 at p: [1,2,2], l: f(h(a,h(a,a))), lp: a, sig: {}, l': a, r: f(a), r': h(a,f(b)))
(R4 unifies with R1 at p: [1,1,2], l: f(h(h(b,a),h(f(c),c))), lp: a, sig: {}, l': a, r: c, r': h(a,f(b)))

-> Critical pairs info:
<f(f(h(a,h(a,f(b))))),f(f(f(a)))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(f(h(h(a,f(b)),a))),f(f(f(a)))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(h(h(b,h(a,f(b))),h(f(c),c))),c>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(h(a,h(h(a,f(b)),a))),f(a)>   =>  Not trivial, Not overlay, Proper, NW0, N4
<f(h(a,h(a,h(a,f(b))))),f(a)>   =>  Not trivial, Not overlay, Proper, NW0, N5
<f(h(h(a,f(b)),h(a,a))),f(a)>   =>  Not trivial, Not overlay, Proper, NW0, N6

-> 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 


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(f 1)
(b)
(c)
(fSNonEmpty)
(h 1, 2)
)
(RULES
a -> h(a,f(b))
f(f(h(a,a))) -> f(f(f(a)))
f(h(a,h(a,a))) -> f(a)
f(h(h(b,a),h(f(c),c))) -> c
)
Critical Pairs:
<f(f(h(a,h(a,f(b))))),f(f(f(a)))>   =>  Not trivial, Not overlay, Proper, NW0, N1
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1)
(b)
(c)
(fSNonEmpty)
(h 1 2)
)
(RULES
a -> h(a,f(b))
f(f(h(a,a))) -> f(f(f(a)))
f(h(a,h(a,a))) -> f(a)
f(h(h(b,a),h(f(c),c))) -> c
)]
Infeasibility Conditions:
f(f(h(a,h(a,f(b))))) ->* z0, f(f(f(a))) ->* z0

Problem 1: 

Obtaining a model using AGES:

 -> Theory (Usable Rules):
a -> h(a,f(b))
f(f(h(a,a))) -> f(f(f(a)))
f(h(a,h(a,a))) -> f(a)
f(h(h(b,a),h(f(c),c))) -> c

 -> AGES Output:




The problem is infeasible.


The problem is not confluent.