YES
proof of SK90_2.27.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 63 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:

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[fib_1, 0, g_1] > [sp_1, np_1, +_2] > s_1 > pair_2


Status:
fib_1: [1]
0: multiset status
s_1: [1]
sp_1: [1]
g_1: [1]
pair_2: [2,1]
np_1: [1]
+_2: [1,2]

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))




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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