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TRS Stand 20472 pair #381713314
details
property
value
status
complete
benchmark
wst99.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n081.star.cs.uiowa.edu
space
Rubio_04
run statistics
property
value
solver
Wanda
configuration
FirstOrder
runtime (wallclock)
0.40074801445 seconds
cpu usage
0.393979818
max memory
1.3021184E7
stage attributes
key
value
output-size
17428
starexec-result
YES
output
/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. der : [o] --> o din : [o] --> o dout : [o] --> o plus : [o * o] --> o times : [o * o] --> o u21 : [o * o * o] --> o u22 : [o * o * o * o] --> o u31 : [o * o * o] --> o u32 : [o * o * o * o] --> o u41 : [o * o] --> o u42 : [o * o * o] --> o din(der(plus(X, Y))) => u21(din(der(X)), X, Y) u21(dout(X), Y, Z) => u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) => dout(plus(U, X)) din(der(times(X, Y))) => u31(din(der(X)), X, Y) u31(dout(X), Y, Z) => u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) => dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) => u41(din(der(X)), X) u41(dout(X), Y) => u42(din(der(X)), Y, X) u42(dout(X), Y, Z) => dout(X) As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95]. Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows: der : [sd] --> sd din : [sd] --> yd dout : [sd] --> yd plus : [sd * sd] --> sd times : [sd * sd] --> sd u21 : [yd * sd * sd] --> yd u22 : [yd * sd * sd * sd] --> yd u31 : [yd * sd * sd] --> yd u32 : [yd * sd * sd * sd] --> yd u41 : [yd * sd] --> yd u42 : [yd * sd * sd] --> yd We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]). We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) 1] din#(der(plus(X, Y))) =#> din#(der(X)) 2] u21#(dout(X), Y, Z) =#> u22#(din(der(Z)), Y, Z, X) 3] u21#(dout(X), Y, Z) =#> din#(der(Z)) 4] din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) 5] din#(der(times(X, Y))) =#> din#(der(X)) 6] u31#(dout(X), Y, Z) =#> u32#(din(der(Z)), Y, Z, X) 7] u31#(dout(X), Y, Z) =#> din#(der(Z)) 8] din#(der(der(X))) =#> u41#(din(der(X)), X) 9] din#(der(der(X))) =#> din#(der(X)) 10] u41#(dout(X), Y) =#> u42#(din(der(X)), Y, X) 11] u41#(dout(X), Y) =#> din#(der(X)) Rules R_0: din(der(plus(X, Y))) => u21(din(der(X)), X, Y) u21(dout(X), Y, Z) => u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) => dout(plus(U, X)) din(der(times(X, Y))) => u31(din(der(X)), X, Y) u31(dout(X), Y, Z) => u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) => dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) => u41(din(der(X)), X) u41(dout(X), Y) => u42(din(der(X)), Y, X) u42(dout(X), Y, Z) => dout(X) 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 : 2, 3 * 1 : 0, 1, 4, 5, 8, 9 * 2 : * 3 : 0, 1, 4, 5, 8, 9 * 4 : 6, 7 * 5 : 0, 1, 4, 5, 8, 9 * 6 : * 7 : 0, 1, 4, 5, 8, 9 * 8 : 10, 11 * 9 : 0, 1, 4, 5, 8, 9 * 10 : * 11 : 0, 1, 4, 5, 8, 9 This graph has the following strongly connected components: P_1:
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