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	n148.star.cs.uiowa.edu
space	Kop_11

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.139407 seconds
cpu usage	0.128991
user time	0.090489
system time	0.038502
max virtual memory	113188.0
max residence set size	6128.0

stage attributes

key	value
starexec-result	YES

output

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: || Input TRS: || 1: avg(s(PeRCenTX),PeRCenTY) -> avg(PeRCenTX,s(PeRCenTY)) || 2: avg(PeRCenTX,s(s(s(PeRCenTY)))) -> s(avg(s(PeRCenTX),PeRCenTY)) || 3: avg(0(),0()) -> 0() || 4: avg(0(),s(0())) -> 0() || 5: avg(0(),s(s(0()))) -> s(0()) || 6: check(s(PeRCenTX)) -> s(check(PeRCenTX)) || 7: check(0()) -> 0() || 8: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 9: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 9 || Direct POLO(bPol) ... removes: 4 8 1 3 5 7 9 6 || TIlDePAIR w: 2 * x1 + 2 * x2 + 1 || s w: x1 + 1 || check w: 2 * x1 + 1 || 0 w: 1 || avg w: 2 * x1 + x2 || Number of strict rules: 1 || Direct POLO(bPol) ... removes: 2 || TIlDePAIR w: 2 * x1 + 2 * x2 + 1 || s w: x1 + 1 || check w: 2 * x1 + 1 || 0 w: 1 || avg w: 2 * x1 + 2 * x2 || Number of strict rules: 0 || We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] apply#(fun(F), X) =#> F(check(X)) 1] apply#(fun(F), X) =#> check#(X) Rules R_0: 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 Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_0, minimal, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : 0, 1 * 1 : This graph has the following strongly connected components: P_1: apply#(fun(F), X) =#> F(check(X)) popout

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actions

all output return to Higher Order Rewriting Union Beta