/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 283 ms]
(4) CpxRelTRS
(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


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

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


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(encArg(x_1))


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

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


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

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

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(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, INF).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

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

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(encArg(x_1))

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

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

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

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

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(encArg(x_1))

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

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

The rewrite sequence

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

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

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

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
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, INF).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

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

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(encArg(x_1))

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

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

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

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


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

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

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(c) -> c
   encArg(X) -> X
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(found(x_1)) -> found(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encArg(cons_match(x_1, x_2)) -> match(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_start(x_1)) -> start(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_c -> c
   encode_check(x_1) -> check(encArg(x_1))
   encode_start(x_1) -> start(encArg(x_1))
   encode_match(x_1, x_2) -> match(encArg(x_1), encArg(x_2))
   encode_X -> X
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_found(x_1) -> found(encArg(x_1))

Rewrite Strategy: INNERMOST