TRS Standard pair #615298360

details

property	value
status	complete
benchmark	t001.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n010.star.cs.uiowa.edu
space	HirokawaMiddeldorp_04

run statistics

property	value
solver	AProVE23
configuration	standard
runtime (wallclock)	2.07239794731 seconds
cpu usage	5.779421579
max memory	3.02772224E8

stage attributes

key	value
output-size	13962
starexec-result	"YES"

output

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


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 363435ff5b19173ec8a83b45cfdec073678dabcc jan-christoph 20230815 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) UsableRulesProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 25 ms]
        (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 11 ms]
        (24) QDP
        (25) PisEmptyProof [EQUIVALENT, 0 ms]
        (26) YES
    (27) QDP
        (28) UsableRulesProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (31) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   if(true, x, y) -> true
   if(false, x, y) -> false
   pow(x, y) -> f(x, y, s(0))
   f(x, 0, z) -> z
   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   -^1(s(x), s(y)) -> -^1(x, y)
   *^1(x, s(y)) -> *^1(x, y)
   ODD(s(s(x))) -> ODD(x)
   HALF(s(s(x))) -> HALF(x)
   POW(x, y) -> F(x, y, s(0))
   F(x, s(y), z) -> IF(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
   F(x, s(y), z) -> ODD(s(y))
   F(x, s(y), z) -> F(x, y, *(x, z))
   F(x, s(y), z) -> *^1(x, z)
   F(x, s(y), z) -> F(*(x, x), half(s(y)), z)
   F(x, s(y), z) -> *^1(x, x)
   F(x, s(y), z) -> HALF(s(y))

The TRS R consists of the following rules:

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