TRS Conditional pair #516978715

details

property	value
status	complete
benchmark	quotrem.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n113.star.cs.uiowa.edu
space	Mixed_CTRS

run statistics

property	value
solver	AProVE21
configuration	standard
runtime (wallclock)	2.51429796219 seconds
cpu usage	6.712745223
max memory	4.82902016E8

stage attributes

key	value
output-size	14920
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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) Overlay + Local Confluence [EQUIVALENT, 0 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) QReductionProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) UsableRulesProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QReductionProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) UsableRulesProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) QReductionProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPOrderProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) DependencyGraphProof [EQUIVALENT, 0 ms]
        (31) TRUE


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

   less(x, 0) -> false
   less(0, s(x)) -> true
   less(s(x), s(y)) -> less(x, y)
   minus(0, s(y)) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quotrem(0, s(y)) -> pair(0, 0)

The conditional TRS C consists of the following conditional rules:

   quotrem(s(x), s(y)) -> pair(0, s(x)) <= less(x, y) -> true
   quotrem(s(x), s(y)) -> pair(s(q), r) <= less(x, y) -> false, quotrem(minus(x, y), s(y)) -> pair(q, r)


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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:

   quotrem(s(x), s(y)) -> U1(less(x, y), x, y)
   U1(true, x, y) -> pair(0, s(x))
   U1(false, x, y) -> U2(quotrem(minus(x, y), s(y)))
   U2(pair(q, r)) -> pair(s(q), r)
   less(x, 0) -> false
   less(0, s(x)) -> true
   less(s(x), s(y)) -> less(x, y)
   minus(0, s(y)) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quotrem(0, s(y)) -> pair(0, 0)

Q is empty.

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(3) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(4)

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