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


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YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 38 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:

   even(0) -> true
   odd(0) -> false

The conditional TRS C consists of the following conditional rules:

   even(s(x)) -> false <= odd(x) -> false
   even(s(x)) -> true <= odd(x) -> true
   odd(s(x)) -> false <= even(x) -> false
   odd(s(x)) -> true <= even(x) -> true


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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:

   even(s(x)) -> U1(odd(x))
   U1(true) -> true
   U1(false) -> false
   odd(s(x)) -> U2(even(x))
   U2(true) -> true
   U2(false) -> false
   even(0) -> true
   odd(0) -> false

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:U2_1 > 0 > odd_1 > false > even_1 > true > s_1 > U1_1

and weight map:

   true=3
   false=3
   0=1
   even_1=2
   s_1=1
   U1_1=1
   odd_1=2
   U2_1=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:

   even(s(x)) -> U1(odd(x))
   U1(true) -> true
   U1(false) -> false
   odd(s(x)) -> U2(even(x))
   U2(true) -> true
   U2(false) -> false
   even(0) -> true
   odd(0) -> false




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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