YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 76 ms]
(2) RelTRS
(3) RIsEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   f(el(x), y) -> f(x, el(y))

The relative TRS consists of the following S rules:

   f(x, y) -> f(l(x), y)
   f(x, y) -> f(x, r(y))
   l(el(x)) -> el(l(x))
   el(r(x)) -> r(el(x))


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(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
f/2(YES,YES)
el/1(YES)
l/1)YES(
r/1)YES(

Quasi precedence:
f_2 > el_1


Status:
f_2: [1,2]
el_1: [1]

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   f(el(x), y) -> f(x, el(y))
Rules from S:
none




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(2)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   f(x, y) -> f(l(x), y)
   f(x, y) -> f(x, r(y))
   l(el(x)) -> el(l(x))
   el(r(x)) -> r(el(x))


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(3) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES