/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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NO
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Outermost Termination of the given OTRS could be disproven:

(0) OTRS
(1) OutermostNonTerminationProof [COMPLETE, 57 ms]
(2) NO


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

   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   sel(0, cons(X, Z)) -> X
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   sel1(0, cons(X, Z)) -> quote(X)
   first1(0, Z) -> nil1
   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
   quote(0) -> 01
   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
   quote1(nil) -> nil1
   quote(s(X)) -> s1(quote(X))
   quote(sel(X, Z)) -> sel1(X, Z)
   quote1(first(X, Z)) -> first1(X, Z)
   unquote(01) -> 0
   unquote(s1(X)) -> s(unquote(X))
   unquote1(nil1) -> nil
   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
   fcons(X, Z) -> cons(X, Z)



Outermost Strategy.

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(1) OutermostNonTerminationProof (COMPLETE)
Term rewrite system R:
The TRS R consists of the following rules:

   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   sel(0, cons(X, Z)) -> X
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))
   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
   sel1(0, cons(X, Z)) -> quote(X)
   first1(0, Z) -> nil1
   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
   quote(0) -> 01
   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
   quote1(nil) -> nil1
   quote(s(X)) -> s1(quote(X))
   quote(sel(X, Z)) -> sel1(X, Z)
   quote1(first(X, Z)) -> first1(X, Z)
   unquote(01) -> 0
   unquote(s1(X)) -> s(unquote(X))
   unquote1(nil1) -> nil
   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
   fcons(X, Z) -> cons(X, Z)



Outermost Strategy.

---------- Loop: ----------

from(X) -> cons(X, from(s(X))) with rule from(X') -> cons(X', from(s(X'))) at position [] and matcher [X' / X]

Now an instance of the first term with Matcher [X / s(X)] occurs in the last term at position [1].

Context: cons(X, [])

We used [THIEMANN_LOOPS_UNDER_STRATEGIES] to show that this Loop is an Outermost-Loop.
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(2)
NO