/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 31 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QReductionProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QReductionProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) QDP
        (22) UsableRulesProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QReductionProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (27) YES
    (28) QDP
        (29) UsableRulesProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) QReductionProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (34) YES
    (35) QDP
        (36) UsableRulesProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QReductionProof [EQUIVALENT, 0 ms]
        (39) QDP
        (40) QDPOrderProof [EQUIVALENT, 108 ms]
        (41) QDP
        (42) DependencyGraphProof [EQUIVALENT, 0 ms]
        (43) TRUE


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

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

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

Q is empty.

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

(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

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

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)
   APP(add(n, x), y) -> APP(x, y)
   LOW(n, add(m, x)) -> IF_LOW(le(m, n), n, add(m, x))
   LOW(n, add(m, x)) -> LE(m, n)
   IF_LOW(true, n, add(m, x)) -> LOW(n, x)
   IF_LOW(false, n, add(m, x)) -> LOW(n, x)
   HIGH(n, add(m, x)) -> IF_HIGH(le(m, n), n, add(m, x))
   HIGH(n, add(m, x)) -> LE(m, n)
   IF_HIGH(true, n, add(m, x)) -> HIGH(n, x)
   IF_HIGH(false, n, add(m, x)) -> HIGH(n, x)
   QUICKSORT(x) -> IF_QS(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   QUICKSORT(x) -> ISEMPTY(x)
   QUICKSORT(x) -> LOW(head(x), tail(x))
   QUICKSORT(x) -> HEAD(x)
   QUICKSORT(x) -> TAIL(x)
   QUICKSORT(x) -> HIGH(head(x), tail(x))
   IF_QS(false, x, n, y) -> APP(quicksort(x), add(n, quicksort(y)))
   IF_QS(false, x, n, y) -> QUICKSORT(x)
   IF_QS(false, x, n, y) -> QUICKSORT(y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 8 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(add(n, x), y) -> APP(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(add(n, x), y) -> APP(x, y)

R is empty.
The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(add(n, x), y) -> APP(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(12) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP(add(n, x), y) -> APP(x, y)
The graph contains the following edges 1 > 1, 2 >= 2


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

(13)
YES

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)

R is empty.
The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LE(s(x), s(y)) -> LE(x, y)
The graph contains the following edges 1 > 1, 2 > 2


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

(20)
YES

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   HIGH(n, add(m, x)) -> IF_HIGH(le(m, n), n, add(m, x))
   IF_HIGH(true, n, add(m, x)) -> HIGH(n, x)
   IF_HIGH(false, n, add(m, x)) -> HIGH(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(22) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   HIGH(n, add(m, x)) -> IF_HIGH(le(m, n), n, add(m, x))
   IF_HIGH(true, n, add(m, x)) -> HIGH(n, x)
   IF_HIGH(false, n, add(m, x)) -> HIGH(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(24) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   HIGH(n, add(m, x)) -> IF_HIGH(le(m, n), n, add(m, x))
   IF_HIGH(true, n, add(m, x)) -> HIGH(n, x)
   IF_HIGH(false, n, add(m, x)) -> HIGH(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(26) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*HIGH(n, add(m, x)) -> IF_HIGH(le(m, n), n, add(m, x))
The graph contains the following edges 1 >= 2, 2 >= 3


*IF_HIGH(true, n, add(m, x)) -> HIGH(n, x)
The graph contains the following edges 2 >= 1, 3 > 2


*IF_HIGH(false, n, add(m, x)) -> HIGH(n, x)
The graph contains the following edges 2 >= 1, 3 > 2


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

(27)
YES

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

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LOW(n, add(m, x)) -> IF_LOW(le(m, n), n, add(m, x))
   IF_LOW(true, n, add(m, x)) -> LOW(n, x)
   IF_LOW(false, n, add(m, x)) -> LOW(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LOW(n, add(m, x)) -> IF_LOW(le(m, n), n, add(m, x))
   IF_LOW(true, n, add(m, x)) -> LOW(n, x)
   IF_LOW(false, n, add(m, x)) -> LOW(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(31) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LOW(n, add(m, x)) -> IF_LOW(le(m, n), n, add(m, x))
   IF_LOW(true, n, add(m, x)) -> LOW(n, x)
   IF_LOW(false, n, add(m, x)) -> LOW(n, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LOW(n, add(m, x)) -> IF_LOW(le(m, n), n, add(m, x))
The graph contains the following edges 1 >= 2, 2 >= 3


*IF_LOW(true, n, add(m, x)) -> LOW(n, x)
The graph contains the following edges 2 >= 1, 3 > 2


*IF_LOW(false, n, add(m, x)) -> LOW(n, x)
The graph contains the following edges 2 >= 1, 3 > 2


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

(34)
YES

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

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   QUICKSORT(x) -> IF_QS(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   IF_QS(false, x, n, y) -> QUICKSORT(x)
   IF_QS(false, x, n, y) -> QUICKSORT(y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(36) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   QUICKSORT(x) -> IF_QS(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   IF_QS(false, x, n, y) -> QUICKSORT(x)
   IF_QS(false, x, n, y) -> QUICKSORT(y)

The TRS R consists of the following rules:

   isempty(nil) -> true
   isempty(add(n, x)) -> false
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   app(nil, x0)
   app(add(x0, x1), x2)
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(38) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   app(nil, x0)
   app(add(x0, x1), x2)
   quicksort(x0)
   if_qs(true, x0, x1, x2)
   if_qs(false, x0, x1, x2)


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

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   QUICKSORT(x) -> IF_QS(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   IF_QS(false, x, n, y) -> QUICKSORT(x)
   IF_QS(false, x, n, y) -> QUICKSORT(y)

The TRS R consists of the following rules:

   isempty(nil) -> true
   isempty(add(n, x)) -> false
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(40) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   IF_QS(false, x, n, y) -> QUICKSORT(x)
   IF_QS(false, x, n, y) -> QUICKSORT(y)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO,RATPOLO]:

   POL(0) = 0
   POL(IF_QS(x_1, x_2, x_3, x_4)) = [1/2]x_1 + [4]x_2 + [1/4]x_3 + [4]x_4
   POL(QUICKSORT(x_1)) = [4]x_1
   POL(add(x_1, x_2)) = [4] + [2]x_1 + [4]x_2
   POL(false) = [1/4]
   POL(head(x_1)) = [1/2]x_1
   POL(high(x_1, x_2)) = x_2
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = [2]x_3
   POL(isempty(x_1)) = [1/4]x_1
   POL(le(x_1, x_2)) = 0
   POL(low(x_1, x_2)) = [2]x_2
   POL(nil) = [1/4]
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1/4]x_1
   POL(true) = 0
The value of delta used in the strict ordering is 1/8.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isempty(nil) -> true
   isempty(add(n, x)) -> false
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_high(false, n, add(m, x)) -> add(m, high(n, x))


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

(41)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   QUICKSORT(x) -> IF_QS(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))

The TRS R consists of the following rules:

   isempty(nil) -> true
   isempty(add(n, x)) -> false
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   low(x0, nil)
   low(x0, add(x1, x2))
   if_low(true, x0, add(x1, x2))
   if_low(false, x0, add(x1, x2))
   high(x0, nil)
   high(x0, add(x1, x2))
   if_high(true, x0, add(x1, x2))
   if_high(false, x0, add(x1, x2))
   head(add(x0, x1))
   tail(add(x0, x1))
   isempty(nil)
   isempty(add(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(42) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(43)
TRUE