/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 6 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) 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:

   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(X)))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(f(X))) -> mark(c(f(g(f(X)))))
active(c(X)) -> mark(d(X))
active(h(X)) -> mark(c(d(X)))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
proper(f(X)) -> f(proper(X))
proper(c(X)) -> c(proper(X))
proper(g(X)) -> g(proper(X))
proper(d(X)) -> d(proper(X))
proper(h(X)) -> h(proper(X))

----------------------------------------

(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:

   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
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:

   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]
{(33,34,[f_1|0, h_1|0, c_1|0, g_1|0, d_1|0, top_1|0]), (33,35,[mark_1|1]), (33,36,[ok_1|1]), (33,37,[mark_1|1]), (33,38,[ok_1|1]), (33,39,[ok_1|1]), (33,40,[ok_1|1]), (33,41,[ok_1|1]), (33,42,[top_1|1]), (33,43,[top_1|1]), (34,34,[mark_1|0, ok_1|0, proper_1|0, active_1|0]), (35,34,[f_1|1]), (35,35,[mark_1|1]), (35,36,[ok_1|1]), (36,34,[f_1|1]), (36,35,[mark_1|1]), (36,36,[ok_1|1]), (37,34,[h_1|1]), (37,37,[mark_1|1]), (37,38,[ok_1|1]), (38,34,[h_1|1]), (38,37,[mark_1|1]), (38,38,[ok_1|1]), (39,34,[c_1|1]), (39,39,[ok_1|1]), (40,34,[g_1|1]), (40,40,[ok_1|1]), (41,34,[d_1|1]), (41,41,[ok_1|1]), (42,34,[proper_1|1]), (43,34,[active_1|1])}"
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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(8)
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:

   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(X)))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

The rewrite sequence

c(ok(X)) ->^+ ok(c(X))

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

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

The result substitution is [ ].




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(10)
Complex Obligation (BEST)

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(11)
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:

   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(X)))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
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:

   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(X)))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL