YES
proof of SK90_2.43.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 72 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:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[merge_2, <_2] > [._2, if_3]
nil > [._2, if_3]
++_2 > [._2, if_3]
true > [._2, if_3]
false > [._2, if_3]


Status:
merge_2: [1,2]
nil: multiset status
._2: [1,2]
if_3: [2,1,3]
<_2: [1,2]
++_2: multiset status
true: multiset status
false: multiset status

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

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x




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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