YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(H 1)
(K 1)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
CSR Converter From Canonical u-Replacement Map Procedure [CSUR20]:

Original Replacement Map:
(H 1)
(K 1)
(fSNonEmpty)

New Replacement Map:
(H 1)
(K)
(fSNonEmpty)

New problem:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(H 1)
(K)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(H 1)
(K)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 H(H(x)) -> K(x)
 H(K(x)) -> K(H(x))
-> Unhiding Rules:
 Empty

Problem 1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 H(H(x)) -> K(x)
 H(K(x)) -> K(H(x))
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(H 1)
(K)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy NW Procedure:
-> Rules:
 H(H(x)) -> K(x)
 H(K(x)) -> K(H(x))
-> Vars:
 x, x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: H(H(x)) -> K(x), id: 1, possubterms: H(H(x))->[], H(x)->[1])
 (rule: H(K(x)) -> K(H(x)), id: 2, possubterms: H(K(x))->[], K(x)->[1])

-> Unifications:
(R1 unifies with R1 at p: [1], l: H(H(x)), lp: H(x), sig: {x -> H(x')}, l': H(H(x')), r: K(x), r': K(x'))
(R1 unifies with R2 at p: [1], l: H(H(x)), lp: H(x), sig: {x -> K(x')}, l': H(K(x')), r: K(x), r': K(H(x')))

-> Critical pairs info:
<H(K(H(x'))),K(K(x'))>   =>  Not trivial, Not overlay, Proper, NW1, N1
<H(K(x')),K(H(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N2

-> 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 CS-TRS, Left-homogeneous u-replacing variables



The problem is decomposed in 2 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(H 1)
(K)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)
Critical Pairs:
<H(K(H(x'))),K(K(x'))>   =>  Not trivial, Not overlay, Proper, NW1, N1
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: H(K(H(x')))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('H(K(H(x')))', D0)
ID: 1 => ('K(H(H(x')))', D1, R2, P[], S{x5 -> H(x')}), NR: 'K(H(H(x')))'
ID: 2 => ('K(K(x'))', D2, R1, P[1], S{x4 -> x'}), NR: 'K(x')'
t: K(K(x'))
Nodes: [0]
Edges: []
ID: 0 => ('K(K(x'))', D0)
SNodesPath: H(K(H(x'))) -> K(H(H(x'))) -> K(K(x'))
TNodesPath: K(K(x'))
H(K(H(x'))) ->* K(K(x')) *<- K(K(x'))
"Joinable"

The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x x')
(REPLACEMENT-MAP
(H 1)
(K)
(fSNonEmpty)
)
(RULES
H(H(x)) -> K(x)
H(K(x)) -> K(H(x))
)
Critical Pairs:
<H(K(x')),K(H(x'))>   =>  Not trivial, Not overlay, Proper, NW0, N2
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Brute Force Joinability Procedure:
-> Rewritings:
s: H(K(x'))
Nodes: [0,1]
Edges: [(0,1)]
ID: 0 => ('H(K(x'))', D0)
ID: 1 => ('K(H(x'))', D1, R2, P[], S{x5 -> x'}), NR: 'K(H(x'))'
t: K(H(x'))
Nodes: [0]
Edges: []
ID: 0 => ('K(H(x'))', D0)
SNodesPath: H(K(x')) -> K(H(x'))
TNodesPath: K(H(x'))
H(K(x')) ->* K(H(x')) *<- K(H(x'))
"Joinable"

The problem is confluent.