WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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), 875 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 1 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:

   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))

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

   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

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:

   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))

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

   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

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:

   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))

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

   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

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

SPLITMERGE(Cons(z0, z1), z2, z3) ->^+ c11(SPLITMERGE(z1, Cons(z0, z3), z2))

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

The pumping substitution is [z1 / Cons(z0, z1)].

The result substitution is [z2 / Cons(z0, z3), z3 / z2].




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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:

   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))

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

   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

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:

   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))

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

   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

Rewrite Strategy: INNERMOST