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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270269
details
property
value
status
complete
benchmark
eval.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n091.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.77251 seconds
cpu usage
1.67877
user time
1.55448
system time
0.124289
max virtual memory
7017980.0
max residence set size
109432.0
stage attributes
key
value
starexec-result
YES
output
1.65/0.76 YES 1.65/0.77 We consider the system theBenchmark. 1.65/0.77 1.65/0.77 Alphabet: 1.65/0.77 1.65/0.77 dom : [N * N * N] --> N 1.65/0.77 eval : [N * N] --> N 1.65/0.77 fun : [N -> N * N * N] --> N 1.65/0.77 o : [] --> N 1.65/0.77 s : [N] --> N 1.65/0.77 1.65/0.77 Rules: 1.65/0.77 1.65/0.77 eval(fun(f, x, y), z) => f dom(x, y, z) 1.65/0.77 dom(s(x), s(y), s(z)) => s(dom(x, y, z)) 1.65/0.77 dom(o, s(x), s(y)) => s(dom(o, x, y)) 1.65/0.77 dom(x, y, o) => x 1.65/0.77 dom(o, o, x) => o 1.65/0.77 1.65/0.77 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 1.65/0.77 1.65/0.77 We observe that the rules contain a first-order subset: 1.65/0.77 1.65/0.77 dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 1.65/0.77 dom(o, s(X), s(Y)) => s(dom(o, X, Y)) 1.65/0.77 dom(X, Y, o) => X 1.65/0.77 dom(o, o, X) => o 1.65/0.77 1.65/0.77 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. 1.65/0.77 1.65/0.77 According to the external first-order termination prover, this system is indeed Ce-terminating: 1.65/0.77 1.65/0.77 || proof of resources/system.trs 1.65/0.77 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 1.65/0.77 || 1.65/0.77 || 1.65/0.77 || Termination w.r.t. Q of the given QTRS could be proven: 1.65/0.77 || 1.65/0.77 || (0) QTRS 1.65/0.77 || (1) QTRSRRRProof [EQUIVALENT] 1.65/0.77 || (2) QTRS 1.65/0.77 || (3) RisEmptyProof [EQUIVALENT] 1.65/0.77 || (4) YES 1.65/0.77 || 1.65/0.77 || 1.65/0.77 || ---------------------------------------- 1.65/0.77 || 1.65/0.77 || (0) 1.65/0.77 || Obligation: 1.65/0.77 || Q restricted rewrite system: 1.65/0.77 || The TRS R consists of the following rules: 1.65/0.77 || 1.65/0.77 || dom(s(%X), s(%Y), s(%Z)) -> s(dom(%X, %Y, %Z)) 1.65/0.77 || dom(o, s(%X), s(%Y)) -> s(dom(o, %X, %Y)) 1.65/0.77 || dom(%X, %Y, o) -> %X 1.65/0.77 || dom(o, o, %X) -> o 1.65/0.77 || ~PAIR(%X, %Y) -> %X 1.65/0.77 || ~PAIR(%X, %Y) -> %Y 1.65/0.77 || 1.65/0.77 || Q is empty. 1.65/0.77 || 1.65/0.77 || ---------------------------------------- 1.65/0.77 || 1.65/0.77 || (1) QTRSRRRProof (EQUIVALENT) 1.65/0.77 || Used ordering: 1.65/0.77 || Polynomial interpretation [POLO]: 1.65/0.77 || 1.65/0.77 || POL(dom(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 1.65/0.77 || POL(o) = 2 1.65/0.77 || POL(s(x_1)) = 1 + x_1 1.65/0.77 || POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2 1.65/0.77 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 1.65/0.77 || 1.65/0.77 || dom(s(%X), s(%Y), s(%Z)) -> s(dom(%X, %Y, %Z)) 1.65/0.77 || dom(o, s(%X), s(%Y)) -> s(dom(o, %X, %Y)) 1.65/0.77 || dom(%X, %Y, o) -> %X 1.65/0.77 || dom(o, o, %X) -> o 1.65/0.77 || ~PAIR(%X, %Y) -> %X 1.65/0.77 || ~PAIR(%X, %Y) -> %Y 1.65/0.77 || 1.65/0.77 || 1.65/0.77 || 1.65/0.77 || 1.65/0.77 || ---------------------------------------- 1.65/0.77 || 1.65/0.77 || (2) 1.65/0.77 || Obligation: 1.65/0.77 || Q restricted rewrite system: 1.65/0.77 || R is empty. 1.65/0.77 || Q is empty. 1.65/0.77 || 1.65/0.77 || ---------------------------------------- 1.65/0.77 || 1.65/0.77 || (3) RisEmptyProof (EQUIVALENT) 1.65/0.77 || The TRS R is empty. Hence, termination is trivially proven. 1.65/0.77 || ---------------------------------------- 1.65/0.77 || 1.65/0.77 || (4) 1.65/0.77 || YES 1.65/0.77 ||
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