/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 50 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) TRS for Loop Detection


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


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

S is empty.
Rewrite Strategy: INNERMOST
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(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
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:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

S is empty.
Rewrite Strategy: INNERMOST
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(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. 
The certificate found is represented by the following graph.

"[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
{(5,6,[f_1|0, encArg_1|0, encode_f_1|0, encode_0|0, encode_s_1|0, 0|1]), (5,7,[s_1|1, f_1|1]), (5,8,[f_1|1]), (5,10,[s_1|2]), (5,12,[s_1|2]), (5,13,[f_1|2]), (5,17,[s_1|3]), (6,6,[0|0, s_1|0, cons_f_1|0]), (7,6,[0|1, encArg_1|1]), (7,7,[s_1|1, f_1|1]), (7,10,[s_1|2]), (7,13,[f_1|2]), (7,12,[s_1|2]), (7,17,[s_1|3]), (8,9,[f_1|1]), (8,11,[s_1|1]), (8,8,[f_1|1]), (8,12,[s_1|2]), (9,6,[s_1|1]), (10,6,[0|2]), (11,6,[0|1]), (12,6,[0|2]), (13,14,[f_1|2]), (13,12,[s_1|2]), (13,13,[f_1|2]), (13,15,[f_1|3]), (13,17,[s_1|3]), (13,18,[s_1|4]), (14,7,[s_1|2]), (14,10,[s_1|2]), (14,12,[s_1|2]), (14,17,[s_1|2]), (15,16,[f_1|3]), (15,17,[s_1|3]), (15,18,[s_1|4]), (16,10,[s_1|3]), (16,12,[s_1|3]), (16,17,[s_1|3]), (17,6,[0|3]), (18,6,[0|4])}"
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(8)
BOUNDS(1, n^1)

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(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(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, n^1).


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

The rewrite sequence

f(s(s(x))) ->^+ f(f(s(x)))

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

The pumping substitution is [x / s(x)].

The result substitution is [ ].




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(12)
Complex Obligation (BEST)

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(13)
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, n^1).


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST