details

property	value
status	complete
benchmark	h09.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n150.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.661876 seconds
cpu usage	0.651494
user time	0.572256
system time	0.079238
max virtual memory	141448.0
max residence set size	26232.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> N false : [] --> B g : [] --> N -> B h : [] --> N -> (N -> B) -> N -> B iszero : [] --> N -> N -> B rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B true : [] --> B Rules: rec f (i 0) => i g x => true h x f y => false iszero x y => rec h (g x) y Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: Alphabet: 0 : [] --> N false : [] --> B g : [] --> N -> B h : [] --> N -> (N -> B) -> N -> B iszero : [N] --> N -> B rec : [N -> (N -> B) -> N -> B * B] --> N -> B true : [] --> B ~AP1 : [N -> B * N] --> B Rules: rec(F, ~AP1(G, 0)) => G g X => true h X F Y => false iszero(X) Y => ~AP1(rec(h, g X), Y) rec(F, g 0) => g rec(F, h X G 0) => h X G rec(F, iszero(X) 0) => iszero(X) ~AP1(F, X) => F X We observe that the rules contain a first-order subset: g X => true 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: || Input TRS: || 1: g(PeRCenTX) -> true() || 2: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 3: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 3 || Direct POLO(bPol) ... removes: 1 3 2 || TIlDePAIR w: 2 * x1 + 2 * x2 + 1 || true w: 0 || g w: x1 + 1 || Number of strict rules: 0 || We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]). In order to do so, we start by eta-expanding the system, which gives: rec(F, ~AP1(G, 0), X) => G X g(X) => true h(X, F, Y) => false iszero(X, Y) => ~AP1(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y) rec(F, g(0), X) => g(X) rec(F, h(X, G, 0), Y) => h(X, G, Y) rec(F, iszero(X, 0), Y) => iszero(X, Y) ~AP1(F, X) => F X We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: 0] iszero#(X, Y) =#> ~AP1#(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y) 1] iszero#(X, Y) =#> rec#(/\x./\f./\y.h(x, /\z.f z, y), g(X), Z) 2] iszero#(X, Y) =#> h#(Z, /\x.F x, U) 3] iszero#(X, Y) =#> g#(X) 4] rec#(F, g(0), X) =#> g#(X) 5] rec#(F, h(X, G, 0), Y) =#> h#(X, G, Y) 6] rec#(F, iszero(X, 0), Y) =#> iszero#(X, Y) Rules R_0: rec(F, ~AP1(G, 0), X) => G X g(X) => true h(X, F, Y) => false iszero(X, Y) => ~AP1(/\x.rec(/\y./\f./\z.h(y, /\u.f u, z), g(X), x), Y) rec(F, g(0), X) => g(X) rec(F, h(X, G, 0), Y) => h(X, G, Y) rec(F, iszero(X, 0), Y) => iszero(X, Y) ~AP1(F, X) => F X popout

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all output return to Higher Order Rewriting Union Beta