/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

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


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


The TRS R consists of the following rules:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

S is empty.
Rewrite Strategy: INNERMOST
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(3) 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]
transitions: 
Nil0() -> 0
walk_xs0() -> 0
Cons0(0, 0) -> 0
comp_f_g0(0, 0) -> 0
walk_xs_30(0) -> 0
walk#10(0) -> 1
comp_f_g#10(0, 0, 0) -> 2
main0(0) -> 3
walk_xs1() -> 1
walk#11(0) -> 4
walk_xs_31(0) -> 5
comp_f_g1(4, 5) -> 1
Cons1(0, 0) -> 6
comp_f_g#11(0, 0, 6) -> 2
Cons1(0, 0) -> 2
Nil1() -> 3
walk#11(0) -> 7
walk_xs_31(0) -> 8
Nil1() -> 9
comp_f_g#11(7, 8, 9) -> 3
walk_xs1() -> 4
walk_xs1() -> 7
comp_f_g1(4, 5) -> 4
comp_f_g1(4, 5) -> 7
Cons1(0, 6) -> 6
Cons1(0, 6) -> 2
Cons2(0, 9) -> 10
comp_f_g#12(4, 5, 10) -> 3
Cons2(0, 9) -> 3
Cons2(0, 10) -> 10
Cons2(0, 10) -> 3

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(4)
BOUNDS(1, n^1)

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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

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

The rewrite sequence

walk#1(Cons(x4, x3)) ->^+ comp_f_g(walk#1(x3), walk_xs_3(x4))

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

The pumping substitution is [x3 / Cons(x4, x3)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

S is empty.
Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

S is empty.
Rewrite Strategy: INNERMOST