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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270227
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
n002.star.cs.uiowa.edu
space
Kop_11
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.719557 seconds
cpu usage
1.59787
user time
1.48098
system time
0.116882
max virtual memory
7017092.0
max residence set size
109888.0
stage attributes
key
value
starexec-result
YES
output
1.56/0.71 YES 1.56/0.71 We consider the system theBenchmark. 1.56/0.71 1.56/0.71 Alphabet: 1.56/0.71 1.56/0.71 0 : [] --> nat 1.56/0.71 apply : [nat * nat] --> nat 1.56/0.71 avg : [nat * nat] --> nat 1.56/0.71 check : [nat] --> nat 1.56/0.71 fun : [nat -> nat] --> nat 1.56/0.71 s : [nat] --> nat 1.56/0.71 1.56/0.71 Rules: 1.56/0.71 1.56/0.71 avg(s(x), y) => avg(x, s(y)) 1.56/0.71 avg(x, s(s(s(y)))) => s(avg(s(x), y)) 1.56/0.71 avg(0, 0) => 0 1.56/0.71 avg(0, s(0)) => 0 1.56/0.71 avg(0, s(s(0))) => s(0) 1.56/0.71 apply(fun(f), x) => f check(x) 1.56/0.71 check(s(x)) => s(check(x)) 1.56/0.71 check(0) => 0 1.56/0.71 1.56/0.71 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 1.56/0.71 1.56/0.71 We observe that the rules contain a first-order subset: 1.56/0.71 1.56/0.71 avg(s(X), Y) => avg(X, s(Y)) 1.56/0.71 avg(X, s(s(s(Y)))) => s(avg(s(X), Y)) 1.56/0.71 avg(0, 0) => 0 1.56/0.71 avg(0, s(0)) => 0 1.56/0.71 avg(0, s(s(0))) => s(0) 1.56/0.71 check(s(X)) => s(check(X)) 1.56/0.71 check(0) => 0 1.56/0.71 1.56/0.71 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. 1.56/0.71 1.56/0.71 According to the external first-order termination prover, this system is indeed Ce-terminating: 1.56/0.71 1.56/0.71 || proof of resources/system.trs 1.56/0.71 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 1.56/0.71 || 1.56/0.71 || 1.56/0.71 || Termination w.r.t. Q of the given QTRS could be proven: 1.56/0.71 || 1.56/0.71 || (0) QTRS 1.56/0.71 || (1) QTRSRRRProof [EQUIVALENT] 1.56/0.71 || (2) QTRS 1.56/0.71 || (3) RisEmptyProof [EQUIVALENT] 1.56/0.71 || (4) YES 1.56/0.71 || 1.56/0.71 || 1.56/0.71 || ---------------------------------------- 1.56/0.71 || 1.56/0.71 || (0) 1.56/0.71 || Obligation: 1.56/0.71 || Q restricted rewrite system: 1.56/0.71 || The TRS R consists of the following rules: 1.56/0.71 || 1.56/0.71 || avg(s(%X), %Y) -> avg(%X, s(%Y)) 1.56/0.71 || avg(%X, s(s(s(%Y)))) -> s(avg(s(%X), %Y)) 1.56/0.71 || avg(0, 0) -> 0 1.56/0.71 || avg(0, s(0)) -> 0 1.56/0.71 || avg(0, s(s(0))) -> s(0) 1.56/0.71 || check(s(%X)) -> s(check(%X)) 1.56/0.71 || check(0) -> 0 1.56/0.71 || ~PAIR(%X, %Y) -> %X 1.56/0.71 || ~PAIR(%X, %Y) -> %Y 1.56/0.71 || 1.56/0.71 || Q is empty. 1.56/0.71 || 1.56/0.71 || ---------------------------------------- 1.56/0.71 || 1.56/0.71 || (1) QTRSRRRProof (EQUIVALENT) 1.56/0.71 || Used ordering: 1.56/0.71 || Knuth-Bendix order [KBO] with precedence:check_1 > ~PAIR_2 > 0 > s_1 > avg_2 1.56/0.71 || 1.56/0.71 || and weight map: 1.56/0.71 || 1.56/0.71 || 0=1 1.56/0.71 || s_1=1 1.56/0.71 || check_1=0 1.56/0.71 || avg_2=0 1.56/0.71 || ~PAIR_2=0 1.56/0.71 || 1.56/0.71 || The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 1.56/0.71 || 1.56/0.71 || avg(s(%X), %Y) -> avg(%X, s(%Y)) 1.56/0.71 || avg(%X, s(s(s(%Y)))) -> s(avg(s(%X), %Y)) 1.56/0.71 || avg(0, 0) -> 0 1.56/0.71 || avg(0, s(0)) -> 0 1.56/0.71 || avg(0, s(s(0))) -> s(0) 1.56/0.71 || check(s(%X)) -> s(check(%X)) 1.56/0.71 || check(0) -> 0 1.56/0.71 || ~PAIR(%X, %Y) -> %X 1.56/0.71 || ~PAIR(%X, %Y) -> %Y 1.56/0.71 || 1.56/0.71 || 1.56/0.71 || 1.56/0.71 ||
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