TRS Stand 20472 pair #381710817

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	n096.star.cs.uiowa.edu
space	Rubio_04

run statistics

property	value
solver	AProVE
configuration	standard
runtime (wallclock)	1.76490187645 seconds
cpu usage	3.90471347
max memory	2.02702848E8

stage attributes

key	value
output-size	8051
starexec-result	YES

output

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


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 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.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(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.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
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(4)
Complex Obligation (AND)

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

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