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TRS Stand 20472 pair #381715305
details
property
value
status
complete
benchmark
Ex6_Luc98_L.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n050.star.cs.uiowa.edu
space
Transformed_CSR_04
run statistics
property
value
solver
Wanda
configuration
FirstOrder
runtime (wallclock)
0.0226988792419 seconds
cpu usage
0.019116962
max memory
1294336.0
stage attributes
key
value
output-size
2407
starexec-result
YES
output
/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o cons : [o] --> o first : [o * o] --> o from : [o] --> o nil : [] --> o s : [o] --> o first(0, X) => nil first(s(X), cons(Y)) => cons(Y) from(X) => cons(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: 0 : [] --> o cons : [ba] --> ba first : [o * ba] --> ba from : [ba] --> ba nil : [] --> ba s : [ba] --> o We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): first(0, X) >? nil first(s(X), cons(Y)) >? cons(Y) from(X) >? cons(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 cons = \y0.y0 first = \y0y1.3 + 3y0 + 3y1 from = \y0.3 + 3y0 nil = 0 s = \y0.3 + y0 Using this interpretation, the requirements translate to: [[first(0, _x0)]] = 12 + 3x0 > 0 = [[nil]] [[first(s(_x0), cons(_x1))]] = 12 + 3x0 + 3x1 > x1 = [[cons(_x1)]] [[from(_x0)]] = 3 + 3x0 > x0 = [[cons(_x0)]] We can thus remove the following rules: first(0, X) => nil first(s(X), cons(Y)) => cons(Y) from(X) => cons(X) All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [FuhGieParSchSwi11] C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski. Proving Termination by Dependency Pairs and Inductive Theorem Proving. In volume 47(2) of Journal of Automated Reasoning. 133--160, 2011. [Gra95] B. Gramlich. Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems. In volume 24(1-2) of Fundamentae Informaticae. 3--23, 1995. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.
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