/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 42 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:

   rev_l#2(x8, x10) -> Cons(x10, x8)
   step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
   step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5)
   foldr#3(Nil) -> fleft_op_e_xs_1
   foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6))
   main(Nil) -> Nil
   main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), 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:

   rev_l#2(x8, x10) -> Cons(x10, x8)
   step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
   step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5)
   foldr#3(Nil) -> fleft_op_e_xs_1
   foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6))
   main(Nil) -> Nil
   main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), 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 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
Cons0(0, 0) -> 0
rev_l0() -> 0
step_x_f0(0, 0, 0) -> 0
fleft_op_e_xs_10() -> 0
Nil0() -> 0
rev_l#20(0, 0) -> 1
step_x_f#10(0, 0, 0, 0) -> 2
foldr#30(0) -> 3
main0(0) -> 4
Cons1(0, 0) -> 1
rev_l#21(0, 0) -> 5
step_x_f#11(0, 0, 0, 5) -> 2
rev_l#21(0, 0) -> 2
fleft_op_e_xs_11() -> 3
rev_l1() -> 6
foldr#31(0) -> 7
step_x_f1(6, 0, 7) -> 3
Nil1() -> 4
rev_l1() -> 8
foldr#31(0) -> 9
Nil1() -> 10
step_x_f#11(8, 0, 9, 10) -> 4
Cons2(0, 0) -> 2
Cons2(0, 0) -> 5
rev_l#21(5, 0) -> 5
rev_l#21(5, 0) -> 2
fleft_op_e_xs_11() -> 7
fleft_op_e_xs_11() -> 9
step_x_f1(6, 0, 7) -> 7
step_x_f1(6, 0, 7) -> 9
Cons2(0, 5) -> 2
Cons2(0, 5) -> 5
rev_l#22(10, 0) -> 11
step_x_f#12(6, 0, 7, 11) -> 4
rev_l#22(10, 0) -> 4
Cons3(0, 10) -> 4
Cons3(0, 10) -> 11
rev_l#22(11, 0) -> 11
rev_l#22(11, 0) -> 4
Cons3(0, 11) -> 4
Cons3(0, 11) -> 11

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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:

   rev_l#2(x8, x10) -> Cons(x10, x8)
   step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
   step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5)
   foldr#3(Nil) -> fleft_op_e_xs_1
   foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6))
   main(Nil) -> Nil
   main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), 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

foldr#3(Cons(x16, x6)) ->^+ step_x_f(rev_l, x16, foldr#3(x6))

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

The pumping substitution is [x6 / Cons(x16, x6)].

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:

   rev_l#2(x8, x10) -> Cons(x10, x8)
   step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
   step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5)
   foldr#3(Nil) -> fleft_op_e_xs_1
   foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6))
   main(Nil) -> Nil
   main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), 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:

   rev_l#2(x8, x10) -> Cons(x10, x8)
   step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
   step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5)
   foldr#3(Nil) -> fleft_op_e_xs_1
   foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6))
   main(Nil) -> Nil
   main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil)

S is empty.
Rewrite Strategy: INNERMOST