/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 31 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   plus(0, X) -> X
   plus(s(X), Y) -> plus(X, s(Y))
   fib(0) -> pair(s(0), 0)

The conditional TRS C consists of the following conditional rules:

   fib(s(X)) -> pair(W, Y) <= fib(X) -> pair(Y, Z), plus(Y, Z) -> W


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(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   fib(s(X)) -> U1(fib(X))
   U1(pair(Y, Z)) -> U2(plus(Y, Z), Y)
   U2(W, Y) -> pair(W, Y)
   plus(0, X) -> X
   plus(s(X), Y) -> plus(X, s(Y))
   fib(0) -> pair(s(0), 0)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
fib_1 > [U1_1, U2_2, plus_2] > s_1
fib_1 > [U1_1, U2_2, plus_2] > pair_2
fib_1 > 0


Status:
fib_1: multiset status
s_1: multiset status
U1_1: [1]
pair_2: multiset status
U2_2: [2,1]
plus_2: [1,2]
0: multiset status

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

   fib(s(X)) -> U1(fib(X))
   U1(pair(Y, Z)) -> U2(plus(Y, Z), Y)
   U2(W, Y) -> pair(W, Y)
   plus(0, X) -> X
   plus(s(X), Y) -> plus(X, s(Y))
   fib(0) -> pair(s(0), 0)




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES