TRS Conditional pair #516978567

details

property	value
status	complete
benchmark	378.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n007.star.cs.uiowa.edu
space	COPS

run statistics

property	value
solver	AProVE21
configuration	standard
runtime (wallclock)	2.64224600792 seconds
cpu usage	7.195049556
max memory	5.05860096E8

stage attributes

key	value
output-size	9116
starexec-result	YES

output

/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 81 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) UsableRulesProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) MRRProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 34 ms]
        (18) QDP
        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
        (20) TRUE


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

   cons(x, cons(x, rest)) -> cons(x, rest)
   orient(s(x), 0) -> pair(0, s(x))

The conditional TRS C consists of the following conditional rules:

   cons(x, cons(y, rest)) -> cons(z1, cons(z2, rest)) <= orient(x, y) -> pair(z1, z2)
   orient(s(x), s(y)) -> pair(s(z1), s(z2)) <= orient(x, y) -> pair(z1, z2)


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(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   cons(x, cons(y, rest)) -> U1(orient(x, y), rest)
   U1(pair(z1, z2), rest) -> cons(z1, cons(z2, rest))
   orient(s(x), s(y)) -> U2(orient(x, y))
   U2(pair(z1, z2)) -> pair(s(z1), s(z2))
   cons(x, cons(x, rest)) -> cons(x, rest)
   orient(s(x), 0) -> pair(0, s(x))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U2(x_1)) = x_1
   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(orient(x_1, x_2)) = x_1 + x_2
   POL(pair(x_1, x_2)) = 2*x_1 + x_2
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   cons(x, cons(x, rest)) -> cons(x, rest)




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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   cons(x, cons(y, rest)) -> U1(orient(x, y), rest)

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