/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(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [FINISHED, 0 ms]
(4) BOUNDS(EXP, INF)


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


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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).


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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [z / cons(y, z)].

The result substitution is [ ].



The rewrite sequence

min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))

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

The pumping substitution is [z / cons(y, z)].

The result substitution is [x / y].




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(4)
BOUNDS(EXP, INF)