YES
proof of Rubio_04_bintrees.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   concat(leaf, Y) -> Y
   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
   lessleaves(X, leaf) -> false
   lessleaves(leaf, cons(W, Z)) -> true
   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:lessleaves_2 > true > false > concat_2 > cons_2 > leaf

and weight map:

   leaf=1
   false=2
   true=4
   concat_2=0
   cons_2=1
   lessleaves_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   concat(leaf, Y) -> Y
   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
   lessleaves(X, leaf) -> false
   lessleaves(leaf, cons(W, Z)) -> true
   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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