/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

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

   s(p(x)) -> x
   p(s(x)) -> x
   pos(0) -> false
   pos(s(0)) -> true

The conditional TRS C consists of the following conditional rules:

   pos(s(x)) -> true <= pos(x) -> true
   pos(p(x)) -> false <= pos(x) -> false


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

(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   pos(s(x)) -> U1(pos(x))
   U1(true) -> true
   pos(p(x)) -> U2(pos(x))
   U2(false) -> false
   s(p(x)) -> x
   p(s(x)) -> x
   pos(0) -> false
   pos(s(0)) -> true

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:0 > pos_1 > false > p_1 > s_1 > U2_1 > true > U1_1

and weight map:

   true=1
   false=4
   0=1
   pos_1=3
   s_1=1
   U1_1=1
   p_1=2
   U2_1=1

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

   pos(s(x)) -> U1(pos(x))
   U1(true) -> true
   pos(p(x)) -> U2(pos(x))
   U2(false) -> false
   s(p(x)) -> x
   p(s(x)) -> x
   pos(0) -> false
   pos(s(0)) -> true




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

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(6)
YES