/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) Overlay + Local Confluence [EQUIVALENT, 13 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 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) QDPOrderProof [EQUIVALENT, 13 ms]
        (27) QDP
        (28) DependencyGraphProof [EQUIVALENT, 0 ms]
        (29) 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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))


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

(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)
   MINUS(x, s(y)) -> PRED(minus(x, y))
   MINUS(x, s(y)) -> MINUS(x, y)
   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
   MOD(s(x), s(y)) -> LE(y, x)
   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))
   IF_MOD(true, s(x), s(y)) -> MINUS(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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   MINUS(x, s(y)) -> MINUS(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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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:

   MINUS(x, s(y)) -> MINUS(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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))


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

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

   MINUS(x, s(y)) -> MINUS(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:
*MINUS(x, s(y)) -> MINUS(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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))


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

(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:

   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

The set Q consists of the following terms:

   le(0, x0)
   le(s(x0), 0)
   le(s(x0), s(x1))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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:

   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   pred(s(x)) -> x
   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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))
   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))

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].

   mod(0, x0)
   mod(s(x0), 0)
   mod(s(x0), s(x1))
   if_mod(true, s(x0), s(x1))
   if_mod(false, s(x0), s(x1))


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

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

   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   pred(s(x)) -> x
   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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( IF_MOD_3(x_1, ..., x_3) ) = max{0, x_2 - 1}
POL( le_2(x_1, x_2) ) = 2x_2
POL( 0 ) = 2
POL( true ) = 2
POL( s_1(x_1) ) = x_1 + 2
POL( false ) = 1
POL( MOD_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( minus_2(x_1, x_2) ) = x_1
POL( pred_1(x_1) ) = max{0, x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   pred(s(x)) -> x


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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   pred(s(x)) -> x
   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))
   pred(s(x0))
   minus(x0, 0)
   minus(x0, s(x1))

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

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

(29)
TRUE