details

property	value
status	complete
benchmark	2.41.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n042.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	Wanda
configuration	FirstOrder
runtime (wallclock)	0.056244134903 seconds
cpu usage	0.034891203
max memory	1720320.0

stage attributes

key	value
output-size	2934
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. 0 : [] --> o f : [o * o] --> o g : [o * o] --> o nil : [] --> o norm : [o] --> o rem : [o * o] --> o s : [o] --> o norm(nil) => 0 norm(g(X, Y)) => s(norm(X)) f(X, nil) => g(nil, X) f(X, g(Y, Z)) => g(f(X, Y), Z) rem(nil, X) => nil rem(g(X, Y), 0) => g(X, Y) rem(g(X, Y), s(Z)) => rem(X, Z) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): norm(nil) >? 0 norm(g(X, Y)) >? s(norm(X)) f(X, nil) >? g(nil, X) f(X, g(Y, Z)) >? g(f(X, Y), Z) rem(nil, X) >? nil rem(g(X, Y), 0) >? g(X, Y) rem(g(X, Y), s(Z)) >? rem(X, Z) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 f = \y0y1.2 + 2y0 + 3y1 g = \y0y1.2 + y0 + 2y1 nil = 0 norm = \y0.y0 rem = \y0y1.3 + y1 + 2y0 s = \y0.1 + y0 Using this interpretation, the requirements translate to: [[norm(nil)]] = 0 >= 0 = [[0]] [[norm(g(_x0, _x1))]] = 2 + x0 + 2x1 > 1 + x0 = [[s(norm(_x0))]] [[f(_x0, nil)]] = 2 + 2x0 >= 2 + 2x0 = [[g(nil, _x0)]] [[f(_x0, g(_x1, _x2))]] = 8 + 2x0 + 3x1 + 6x2 > 4 + 2x0 + 2x2 + 3x1 = [[g(f(_x0, _x1), _x2)]] [[rem(nil, _x0)]] = 3 + x0 > 0 = [[nil]] [[rem(g(_x0, _x1), 0)]] = 7 + 2x0 + 4x1 > 2 + x0 + 2x1 = [[g(_x0, _x1)]] [[rem(g(_x0, _x1), s(_x2))]] = 8 + x2 + 2x0 + 4x1 > 3 + x2 + 2x0 = [[rem(_x0, _x2)]] We can thus remove the following rules: norm(g(X, Y)) => s(norm(X)) f(X, g(Y, Z)) => g(f(X, Y), Z) rem(nil, X) => nil rem(g(X, Y), 0) => g(X, Y) rem(g(X, Y), s(Z)) => rem(X, Z) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): norm(nil) >? 0 f(X, nil) >? g(nil, X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 f = \y0y1.3 + y0 + 3y1 g = \y0y1.y0 + y1 nil = 0 norm = \y0.3 + y0 Using this interpretation, the requirements translate to: [[norm(nil)]] = 3 > 0 = [[0]] [[f(_x0, nil)]] = 3 + x0 > x0 = [[g(nil, _x0)]] We can thus remove the following rules: norm(nil) => 0 f(X, nil) => g(nil, 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 +++ popout

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