TRS Standard pair #487069213

details

property	value
status	complete
benchmark	quotminus.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n191.star.cs.uiowa.edu
space	Rubio_04

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	1.74908 seconds
cpu usage	4.11191
user time	3.94338
system time	0.168528
max virtual memory	1.8409712E7
max residence set size	250864.0

stage attributes

key	value
starexec-result	YES

output

YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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) MNOCProof [EQUIVALENT, 1 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) QDPOrderProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) PisEmptyProof [EQUIVALENT, 0 ms]
        (21) YES


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

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

   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   min(X, 0) -> X
   min(s(X), s(Y)) -> min(X, Y)
   min(min(X, Y), Z) -> min(X, plus(Y, Z))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

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:

   PLUS(s(X), Y) -> PLUS(X, Y)
   MIN(s(X), s(Y)) -> MIN(X, Y)
   MIN(min(X, Y), Z) -> MIN(X, plus(Y, Z))
   MIN(min(X, Y), Z) -> PLUS(Y, Z)
   QUOT(s(X), s(Y)) -> QUOT(min(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MIN(X, Y)

The TRS R consists of the following rules:

   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   min(X, 0) -> X
   min(s(X), s(Y)) -> min(X, Y)
   min(min(X, Y), Z) -> min(X, plus(Y, Z))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(s(X), Y) -> PLUS(X, Y)

The TRS R consists of the following rules:

   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))

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