/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 3 ms]
(4) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   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, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

Q is empty.

----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:norm_1 > 0 > f_2 > rem_2 > s_1 > g_2 > nil

and weight map:

   nil=2
   0=4
   norm_1=2
   s_1=1
   g_2=0
   f_2=0
   rem_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   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, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

----------------------------------------

(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(4)
YES