/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 10 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) TransformationProof [EQUIVALENT, 3 ms]
        (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) TransformationProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) TransformationProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) TransformationProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QDPOrderProof [EQUIVALENT, 41 ms]
        (30) QDP
        (31) DependencyGraphProof [EQUIVALENT, 0 ms]
        (32) TRUE


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

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

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, 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:

   CONCAT(cons(u, v), y) -> CONCAT(v, y)
   LESS_LEAVES(u, v) -> IF1(isLeaf(u), isLeaf(v), u, v)
   LESS_LEAVES(u, v) -> ISLEAF(u)
   LESS_LEAVES(u, v) -> ISLEAF(v)
   IF1(b, false, u, v) -> IF2(b, u, v)
   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
   IF2(false, u, v) -> CONCAT(left(u), right(u))
   IF2(false, u, v) -> LEFT(u)
   IF2(false, u, v) -> RIGHT(u)
   IF2(false, u, v) -> CONCAT(left(v), right(v))
   IF2(false, u, v) -> LEFT(v)
   IF2(false, u, v) -> RIGHT(v)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, x1)

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   CONCAT(cons(u, v), y) -> CONCAT(v, y)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, 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:

   CONCAT(cons(u, v), y) -> CONCAT(v, y)

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

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, 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].

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, x1)


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

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

   CONCAT(cons(u, v), y) -> CONCAT(v, 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:
*CONCAT(cons(u, v), y) -> CONCAT(v, 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:

   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
   LESS_LEAVES(u, v) -> IF1(isLeaf(u), isLeaf(v), u, v)
   IF1(b, false, u, v) -> IF2(b, u, v)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, 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:

   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
   LESS_LEAVES(u, v) -> IF1(isLeaf(u), isLeaf(v), u, v)
   IF1(b, false, u, v) -> IF2(b, u, v)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)
   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, 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].

   less_leaves(x0, x1)
   if1(x0, true, x1, x2)
   if1(x0, false, x1, x2)
   if2(true, x0, x1)
   if2(false, x0, x1)


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

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

   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
   LESS_LEAVES(u, v) -> IF1(isLeaf(u), isLeaf(v), u, v)
   IF1(b, false, u, v) -> IF2(b, u, v)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

(19) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule LESS_LEAVES(u, v) -> IF1(isLeaf(u), isLeaf(v), u, v) at position [1] we obtained the following new rules [LPAR04]:

   (LESS_LEAVES(y0, leaf) -> IF1(isLeaf(y0), true, y0, leaf),LESS_LEAVES(y0, leaf) -> IF1(isLeaf(y0), true, y0, leaf))
   (LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1)),LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1)))


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
   IF1(b, false, u, v) -> IF2(b, u, v)
   LESS_LEAVES(y0, leaf) -> IF1(isLeaf(y0), true, y0, leaf)
   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))
   IF1(b, false, u, v) -> IF2(b, u, v)
   IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

(23) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule IF2(false, u, v) -> LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v))) at position [1] we obtained the following new rules [LPAR04]:

   (IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1)))),IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1)))))
   (IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1)),IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1)))


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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))
   IF1(b, false, u, v) -> IF2(b, u, v)
   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1))))
   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1))

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

(25) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1)))) at position [1,1] we obtained the following new rules [LPAR04]:

   (IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1)),IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1)))


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

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))
   IF1(b, false, u, v) -> IF2(b, u, v)
   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1))
   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

(27) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1)) at position [1,0] we obtained the following new rules [LPAR04]:

   (IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1)),IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1)))


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

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

   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))
   IF1(b, false, u, v) -> IF2(b, u, v)
   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

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


The following pairs can be oriented strictly and are deleted.

   IF2(false, y0, cons(x0, x1)) -> LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
LESS_LEAVES(x1, x2)  =  x2

cons(x1, x2)  =  cons(x1, x2)

IF1(x1, x2, x3, x4)  =  x4

IF2(x1, x2, x3)  =  x3

concat(x1, x2)  =  concat(x1, x2)

leaf  =  leaf


Knuth-Bendix order [KBO] with precedence:concat_2 > cons_2

and weight map:

   cons_2=2
   concat_2=1
   leaf=1

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

   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))


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

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

   LESS_LEAVES(y0, cons(x0, x1)) -> IF1(isLeaf(y0), false, y0, cons(x0, x1))
   IF1(b, false, u, v) -> IF2(b, u, v)

The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))

The set Q consists of the following terms:

   isLeaf(leaf)
   isLeaf(cons(x0, x1))
   left(cons(x0, x1))
   right(cons(x0, x1))
   concat(leaf, x0)
   concat(cons(x0, x1), x2)

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

(31) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(32)
TRUE