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


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 56 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   add(x, 0) -> x
   add(x, s(y)) -> s(add(x, y))

The conditional TRS C consists of the following conditional rules:

   quad(x) -> z <= add(x, x) -> y, add(y, y) -> z


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

(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:

   quad(x) -> U1(add(x, x))
   U1(y) -> U2(add(y, y))
   U2(z) -> z
   add(x, 0) -> x
   add(x, s(y)) -> s(add(x, y))

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
quad/1(YES)
U1/1(YES)
add/2(YES,YES)
U2/1)YES(
0/0)
s/1)YES(

Quasi precedence:
quad_1 > U1_1 > add_2
0 > add_2


Status:
quad_1: multiset status
U1_1: multiset status
add_2: multiset status
0: multiset status

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

   quad(x) -> U1(add(x, x))
   U1(y) -> U2(add(y, y))
   add(x, 0) -> x




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   U2(z) -> z
   add(x, s(y)) -> s(add(x, y))

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:add_2 > s_1 > U2_1

and weight map:

   U2_1=1
   s_1=1
   add_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:

   U2(z) -> z
   add(x, s(y)) -> s(add(x, y))




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

(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(8)
YES