/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), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 251 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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


The TRS R consists of the following rules:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

The (relative) TRS S consists of the following rules:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

The (relative) TRS S consists of the following rules:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

The (relative) TRS S consists of the following rules:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST
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(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

rw(Op(Op(x1_0, y2_0), y), c) ->^+ rw[Let](Op(Op(x1_0, y2_0), y), c, rw(x1_0, y2_0))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2].

The pumping substitution is [x1_0 / Op(Op(x1_0, y2_0), y)].

The result substitution is [c / y2_0].




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(6)
Complex Obligation (BEST)

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(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

The (relative) TRS S consists of the following rules:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rw(Val(n), c) -> Op(Val(n), rewrite(c))
   rewrite(Op(x, y)) -> rw(x, y)
   rw(Op(x, y), c) -> rw[Let](Op(x, y), c, rewrite(x))
   rewrite(Val(n)) -> Val(n)
   second(Op(x, y)) -> y
   isOp(Val(n)) -> False
   isOp(Op(x, y)) -> True
   first(Val(n)) -> Val(n)
   first(Op(x, y)) -> x
   assrewrite(exp) -> rewrite(exp)

The (relative) TRS S consists of the following rules:

   rw[Let](Op(x, y), c, a1) -> rw[Let][Let](Op(x, y), c, a1, rewrite(y))
   rw[Let][Let](ab, c, a1, b1) -> rw[Let][Let][Let](c, a1, b1, rewrite(c))
   rw[Let][Let][Let](c, a1, b1, c1) -> rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST