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HRS union beta 16688 pair #381734168
details
property
value
status
complete
benchmark
average.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n095.star.cs.uiowa.edu
space
Kop_11
run statistics
property
value
solver
Wanda
configuration
HigherOrder
runtime (wallclock)
0.694977045059 seconds
cpu usage
1.599470541
max memory
7.3117696E7
stage attributes
key
value
output-size
6828
starexec-result
YES
output
/export/starexec/sandbox/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat apply : [nat * nat] --> nat avg : [nat * nat] --> nat check : [nat] --> nat fun : [nat -> nat] --> nat s : [nat] --> nat Rules: avg(s(x), y) => avg(x, s(y)) avg(x, s(s(s(y)))) => s(avg(s(x), y)) avg(0, 0) => 0 avg(0, s(0)) => 0 avg(0, s(s(0))) => s(0) apply(fun(f), x) => f check(x) check(s(x)) => s(check(x)) check(0) => 0 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We observe that the rules contain a first-order subset: avg(s(X), Y) => avg(X, s(Y)) avg(X, s(s(s(Y)))) => s(avg(s(X), Y)) avg(0, 0) => 0 avg(0, s(0)) => 0 avg(0, s(s(0))) => s(0) check(s(X)) => s(check(X)) check(0) => 0 Moreover, the system is finitely branching. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed Ce-terminating: || proof of resources/system.trs || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 || || || Termination w.r.t. Q of the given QTRS could be proven: || || (0) QTRS || (1) QTRSRRRProof [EQUIVALENT] || (2) QTRS || (3) QTRSRRRProof [EQUIVALENT] || (4) QTRS || (5) RisEmptyProof [EQUIVALENT] || (6) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || avg(s(%X), %Y) -> avg(%X, s(%Y)) || avg(%X, s(s(s(%Y)))) -> s(avg(s(%X), %Y)) || avg(0, 0) -> 0 || avg(0, s(0)) -> 0 || avg(0, s(s(0))) -> s(0) || check(s(%X)) -> s(check(%X)) || check(0) -> 0 || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || || ---------------------------------------- || || (1) QTRSRRRProof (EQUIVALENT) || Used ordering: || Polynomial interpretation [POLO]: || || POL(0) = 1 || POL(avg(x_1, x_2)) = 2*x_1 + x_2 || POL(check(x_1)) = 2 + 2*x_1 || POL(s(x_1)) = 1 + x_1 || POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: || || avg(s(%X), %Y) -> avg(%X, s(%Y)) || avg(0, 0) -> 0 || avg(0, s(0)) -> 0 || avg(0, s(s(0))) -> s(0) || check(s(%X)) -> s(check(%X)) || check(0) -> 0 || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y
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