TRS Standard pair #615305976

details

property	value
status	complete
benchmark	cade13t.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n150.star.cs.uiowa.edu
space	GTSSK07

run statistics

property	value
solver	AProVE23
configuration	standard
runtime (wallclock)	2.59772586823 seconds
cpu usage	7.161778139
max memory	3.33545472E8

stage attributes

key	value
output-size	16972
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# AProVE Commit ID: 363435ff5b19173ec8a83b45cfdec073678dabcc jan-christoph 20230815 unpublished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) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 4 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QReductionProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QReductionProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) QDP
        (22) UsableRulesProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QReductionProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) NonInfProof [EQUIVALENT, 18 ms]
        (27) AND
            (28) QDP
                (29) DependencyGraphProof [EQUIVALENT, 0 ms]
                (30) TRUE
            (31) QDP
                (32) DependencyGraphProof [EQUIVALENT, 0 ms]
                (33) TRUE


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

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

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

Q is empty.

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

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

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

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

The set Q consists of the following terms:

   div(x0, s(x1))
   d(x0, s(x1), x2)
   cond(true, x0, x1, x2)
   cond(false, x0, x1, x2)
   ge(x0, 0)
   ge(0, s(x0))
   ge(s(x0), s(x1))
   plus(x0, 0)
   plus(x0, s(x1))

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