/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 45 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(and(tt, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(isNePal(X)) -> isNePal(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isNePal(X)) -> isNePal(proper(X))
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isNePal(__(I, __(P, I)))) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
active(isNePal(X)) -> isNePal(active(X))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNePal(X)) -> isNePal(proper(X))

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(nil) -> ok(nil)
   proper(tt) -> ok(tt)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(nil) -> ok(nil)
   proper(tt) -> ok(tt)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5]
transitions: 
mark0(0) -> 0
nil0() -> 0
ok0(0) -> 0
tt0() -> 0
active0(0) -> 0
__0(0, 0) -> 1
and0(0, 0) -> 2
isNePal0(0) -> 3
proper0(0) -> 4
top0(0) -> 5
__1(0, 0) -> 6
mark1(6) -> 1
and1(0, 0) -> 7
mark1(7) -> 2
isNePal1(0) -> 8
mark1(8) -> 3
nil1() -> 9
ok1(9) -> 4
tt1() -> 10
ok1(10) -> 4
__1(0, 0) -> 11
ok1(11) -> 1
and1(0, 0) -> 12
ok1(12) -> 2
isNePal1(0) -> 13
ok1(13) -> 3
proper1(0) -> 14
top1(14) -> 5
active1(0) -> 15
top1(15) -> 5
mark1(6) -> 6
mark1(6) -> 11
mark1(7) -> 7
mark1(7) -> 12
mark1(8) -> 8
mark1(8) -> 13
ok1(9) -> 14
ok1(10) -> 14
ok1(11) -> 6
ok1(11) -> 11
ok1(12) -> 7
ok1(12) -> 12
ok1(13) -> 8
ok1(13) -> 13
active2(9) -> 16
top2(16) -> 5
active2(10) -> 16

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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(and(tt, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(isNePal(X)) -> isNePal(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isNePal(X)) -> isNePal(proper(X))
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

__(mark(X1), X2) ->^+ mark(__(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / mark(X1)].

The result substitution is [ ].




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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(and(tt, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(isNePal(X)) -> isNePal(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isNePal(X)) -> isNePal(proper(X))
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(and(tt, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(isNePal(X)) -> isNePal(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   isNePal(mark(X)) -> mark(isNePal(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isNePal(X)) -> isNePal(proper(X))
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL