details

property	value
status	complete
benchmark	findzero.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n140.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.992763 seconds
cpu usage	0.99033
user time	0.900462
system time	0.089868
max virtual memory	138588.0
max residence set size	24304.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat find0 : [nat -> nat * nat * nat] --> nat if : [nat * nat * nat] --> nat min : [nat * nat] --> nat nul : [nat -> nat * nat] --> nat s : [nat] --> nat Rules: min(s(x), s(y)) => min(x, y) min(x, 0) => 0 min(0, x) => 0 min(nul(f, x), y) => nul(f, min(x, y)) nul(f, x) => find0(f, 0, x) find0(f, x, 0) => x find0(f, x, s(y)) => if(f x, find0(f, s(x), y), x) if(s(x), y, z) => y if(0, x, y) => y 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: if(s(X), Y, Z) => Y if(0, X, Y) => Y 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: if(s(PeRCenTX),PeRCenTY,PeRCenTZ) -> PeRCenTY || 2: if(0(),PeRCenTX,PeRCenTY) -> PeRCenTY || 3: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 4: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 4 || Direct POLO(bPol) ... removes: 4 1 3 2 || TIlDePAIR w: x1 + 2 * x2 + 1 || s w: x1 + 1 || if w: 2 * x1 + 2 * x2 + 2 * x3 || 0 w: 1 || 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] min#(s(X), s(Y)) =#> min#(X, Y) 1] min#(nul(F, X), Y) =#> nul#(F, min(X, Y)) 2] min#(nul(F, X), Y) =#> min#(X, Y) 3] nul#(F, X) =#> find0#(F, 0, X) 4] find0#(F, X, s(Y)) =#> if#(F X, find0(F, s(X), Y), X) 5] find0#(F, X, s(Y)) =#> F(X) 6] find0#(F, X, s(Y)) =#> find0#(F, s(X), Y) Rules R_0: min(s(X), s(Y)) => min(X, Y) min(X, 0) => 0 min(0, X) => 0 min(nul(F, X), Y) => nul(F, min(X, Y)) nul(F, X) => find0(F, 0, X) find0(F, X, 0) => X find0(F, X, s(Y)) => if(F X, find0(F, s(X), Y), X) if(s(X), Y, Z) => Y if(0, X, Y) => Y 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, 2 * 1 : 3 * 2 : 0, 1, 2 * 3 : 4, 5, 6 * 4 : * 5 : 0, 1, 2, 3, 4, 5, 6 * 6 : 4, 5, 6 This graph has the following strongly connected components: P_1: min#(s(X), s(Y)) =#> min#(X, Y) min#(nul(F, X), Y) =#> nul#(F, min(X, Y)) min#(nul(F, X), Y) =#> min#(X, Y) nul#(F, X) =#> find0#(F, 0, X) find0#(F, X, s(Y)) =#> F(X) find0#(F, X, s(Y)) =#> find0#(F, s(X), Y) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f). popout

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all output return to Higher Order Rewriting Union Beta