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HRS union beta 16688 pair #381734240
details
property
value
status
complete
benchmark
noneating.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n010.star.cs.uiowa.edu
space
Kop_11
run statistics
property
value
solver
Wanda
configuration
HigherOrder
runtime (wallclock)
0.693852901459 seconds
cpu usage
1.294713293
max memory
9.0161152E7
stage attributes
key
value
output-size
4675
starexec-result
YES
output
/export/starexec/sandbox2/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: 0 : [] --> o a : [] --> o f : [o -> o] --> o g : [o] --> o h : [o * o] --> o s : [o] --> o Rules: a => f(/\x.g(x)) f(/\x.y) => a g(x) => h(x, x) h(0, x) => x h(s(x), 0) => g(x) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a >? f(/\x.g(x)) f(/\x.X) >? a g(X) >? h(X, X) h(0, X) >? X h(s(X), 0) >? g(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 a = 0 f = \G0.G0(0) g = \y0.3y0 h = \y0y1.y0 + 2y1 s = \y0.3 + 3y0 Using this interpretation, the requirements translate to: [[a]] = 0 >= 0 = [[f(/\x.g(x))]] [[f(/\x._x0)]] = x0 >= 0 = [[a]] [[g(_x0)]] = 3x0 >= 3x0 = [[h(_x0, _x0)]] [[h(0, _x0)]] = 3 + 2x0 > x0 = [[_x0]] [[h(s(_x0), 0)]] = 9 + 3x0 > 3x0 = [[g(_x0)]] We can thus remove the following rules: h(0, X) => X h(s(X), 0) => g(X) We observe that the rules contain a first-order subset: g(X) => h(X, X) 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) RisEmptyProof [EQUIVALENT] || (4) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || g(%X) -> h(%X, %X) || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || || ---------------------------------------- || || (1) QTRSRRRProof (EQUIVALENT) || Used ordering:
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