NO

Problem 1: 


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Confluence Problem:
(VAR X)
(REPLACEMENT-MAP
(a)
(f 1)
(g 1)
(h 1)
(b)
)
(RULES
a -> b
f(X,X) -> h(a)
g(a,X) -> f(b,X)
h(X) -> g(X,X)
)

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Problem 1: 

Problem 1: 

Problem 1: 
CS-TRS Procedure:
R is a CS-TRS

Problem 1: 
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Confluence Problem:
(VAR X)
(REPLACEMENT-MAP
(a)
(f 1)
(g 1)
(h 1)
(b)
)
(RULES
a -> b
f(X,X) -> h(a)
g(a,X) -> f(b,X)
h(X) -> g(X,X)
)

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LH u-Critical Pair Instances Procedure [JLAMP21]:
->LH u-Critical Pair Instance:
 Rule 2 (l :-> r) => f(X,X) -> h(a)
 Rule 1 (l' :-> r') => a -> b
 Var => X
 Pos X in l => [1]
 Sigma => {X -> a}
 s => f(b,a)
 t => h(a)
 NW => 0


->LH u-Critical Pair Instance:
 Rule 2 (l :-> r) => f(X,X) -> h(a)
 Rule 2 (l' :-> r') => f(X',X') -> h(a)
 Var => X
 Pos X in l => [1]
 Sigma => {X -> f(X',X')}
 s => f(h(a),f(X',X'))
 t => h(a)
 NW => 0


->LH u-Critical Pair Instance:
 Rule 2 (l :-> r) => f(X,X) -> h(a)
 Rule 3 (l' :-> r') => g(a,X') -> f(b,X')
 Var => X
 Pos X in l => [1]
 Sigma => {X -> g(a,X')}
 s => f(f(b,X'),g(a,X'))
 t => h(a)
 NW => 0


->LH u-Critical Pair Instance:
 Rule 2 (l :-> r) => f(X,X) -> h(a)
 Rule 4 (l' :-> r') => h(X') -> g(X',X')
 Var => X
 Pos X in l => [1]
 Sigma => {X -> h(X')}
 s => f(g(X',X'),h(X'))
 t => h(a)
 NW => 0


->LH u-Critical Pair Instance:
 Rule 4 (l :-> r) => h(X) -> g(X,X)
 Rule 1 (l' :-> r') => a -> b
 Var => X
 Pos X in l => [1]
 Sigma => {X -> a}
 s => h(b)
 t => g(a,a)
 NW => 0


->LH u-Critical Pair Instance:
 Rule 4 (l :-> r) => h(X) -> g(X,X)
 Rule 2 (l' :-> r') => f(X',X') -> h(a)
 Var => X
 Pos X in l => [1]
 Sigma => {X -> f(X',X')}
 s => h(h(a))
 t => g(f(X',X'),f(X',X'))
 NW => 0


->LH u-Critical Pair Instance:
 Rule 4 (l :-> r) => h(X) -> g(X,X)
 Rule 3 (l' :-> r') => g(a,X') -> f(b,X')
 Var => X
 Pos X in l => [1]
 Sigma => {X -> g(a,X')}
 s => h(f(b,X'))
 t => g(g(a,X'),g(a,X'))
 NW => 0


->LH u-Critical Pair Instance:
 Rule 4 (l :-> r) => h(X) -> g(X,X)
 Rule 4 (l' :-> r') => h(X') -> g(X',X')
 Var => X
 Pos X in l => [1]
 Sigma => {X -> h(X')}
 s => h(g(X',X'))
 t => g(h(X'),h(X'))
 NW => 0


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR X X')
(REPLACEMENT-MAP
(a)
(f 1)
(g 1)
(h 1)
(b)
)
(RULES
a -> b
f(X,X) -> h(a)
g(a,X) -> f(b,X)
h(X) -> g(X,X)
)

Critical Pairs:
<f(b,a),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N1
<f(h(a),f(X',X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(f(b,X'),g(a,X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(g(X',X'),h(X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N4
<h(b),g(a,a)>   =>  Not trivial, Not overlay, Proper, NW0, N5
<h(h(a)),g(f(X',X'),f(X',X'))>   =>  Not trivial, Not overlay, Proper, NW0, N6
<h(f(b,X')),g(g(a,X'),g(a,X'))>   =>  Not trivial, Not overlay, Proper, NW0, N7
<h(g(X',X')),g(h(X'),h(X'))>   =>  Not trivial, Not overlay, Proper, NW0, N8
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Huet Levy Procedure:
-> Rules:
 a -> b
 f(X,X) -> h(a)
 g(a,X) -> f(b,X)
 h(X) -> g(X,X)
-> Vars:
 X, X, X
-> UVars:
 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [X], UV-RActive: [], UV-LFrozen: [X], UV-RFrozen: [])
 (UV-RuleId: 3, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X], UV-RFrozen: [X])
 (UV-RuleId: 4, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: [X])

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

-> Unifications:
(R3 unifies with R1 at p: [1], l: g(a,X), lp: a, sig: {}, l': a, r: f(b,X), r': b)

-> Critical pairs info:
<h(g(X',X')),g(h(X'),h(X'))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<h(b),g(a,a)>   =>  Not trivial, Not overlay, Proper, NW0, N2
<h(h(a)),g(f(X',X'),f(X',X'))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(f(b,X'),g(a,X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N4
<f(g(X',X'),h(X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N5
<f(b,a),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N6
<g(b,X),f(b,X)>   =>  Not trivial, Not overlay, Proper, NW0, N7
<h(f(b,X')),g(g(a,X'),g(a,X'))>   =>  Not trivial, Not overlay, Proper, NW0, N8
<f(h(a),f(X',X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N9

-> Problem conclusions:
 Not left linear, Not right linear, Not linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal, Not strongly orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a CS-TRS, Not left-homogeneous u-replacing variables
 Maybe confluent


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR X X')
(REPLACEMENT-MAP
(a)
(f 1)
(g 1)
(h 1)
(b)
)
(RULES
a -> b
f(X,X) -> h(a)
g(a,X) -> f(b,X)
h(X) -> g(X,X)
)

Critical Pairs:
<h(g(X',X')),g(h(X'),h(X'))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<h(b),g(a,a)>   =>  Not trivial, Not overlay, Proper, NW0, N2
<h(h(a)),g(f(X',X'),f(X',X'))>   =>  Not trivial, Not overlay, Proper, NW0, N3
<f(f(b,X'),g(a,X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N4
<f(g(X',X'),h(X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N5
<f(b,a),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N6
<g(b,X),f(b,X)>   =>  Not trivial, Not overlay, Proper, NW0, N7
<h(f(b,X')),g(g(a,X'),g(a,X'))>   =>  Not trivial, Not overlay, Proper, NW0, N8
<f(h(a),f(X',X')),h(a)>   =>  Not trivial, Not overlay, Proper, NW0, N9
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Critical Pairs Distributor Processor:


Problem 1: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: h(g(X',X'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('h(g(X',X'))', D0)
ID: 1 => ('g(g(X',X'),g(X',X'))', D1, R4, P[], S{x4 -> g(X',X')}), NR: 'g(g(X',X'),g(X',X'))'
t: g(h(X'),h(X'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('g(h(X'),h(X'))', D0)
ID: 1 => ('g(g(X',X'),h(X'))', D1, R4, P[1], S{x4 -> X'}), NR: 'g(X',X')'
h(g(X',X')) ->* no union *<- g(h(X'),h(X'))
"Not joinable"

The problem is not confluent.