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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

S is empty.
Rewrite Strategy: FULL
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(3) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
The certificate found is represented by the following graph.

"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
{(1,2,[a__f_1|0, a__c_1|0, a__h_1|0, mark_1|0, f_1|1, d_1|1, c_1|1, h_1|1, a__c_1|1, g_1|1, d_1|2, c_1|2]), (1,3,[a__c_1|1, d_1|2, c_1|2]), (1,6,[a__c_1|1, d_1|2, c_1|2]), (1,7,[a__f_1|1, f_1|2]), (1,8,[a__h_1|1, h_1|2]), (1,9,[a__c_1|2, d_1|3, c_1|3]), (1,10,[a__c_1|2, d_1|3, c_1|3]), (2,2,[f_1|0, g_1|0, d_1|0, c_1|0, h_1|0]), (3,4,[f_1|1]), (4,5,[g_1|1]), (5,2,[f_1|1]), (6,2,[d_1|1]), (7,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (7,7,[a__f_1|1, f_1|2]), (7,8,[a__h_1|1, h_1|2]), (7,9,[a__c_1|2, d_1|3, c_1|3]), (7,10,[a__c_1|2, d_1|3, c_1|3]), (8,2,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (8,7,[a__f_1|1, f_1|2]), (8,8,[a__h_1|1, h_1|2]), (8,9,[a__c_1|2, d_1|3, c_1|3]), (8,10,[a__c_1|2, d_1|3, c_1|3]), (9,8,[d_1|2]), (10,11,[f_1|2]), (11,12,[g_1|2]), (12,7,[f_1|2])}"
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

S is empty.
Rewrite Strategy: FULL
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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

mark(f(X)) ->^+ a__f(mark(X))

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

The pumping substitution is [X / f(X)].

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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

S is empty.
Rewrite Strategy: FULL