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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsProof [FINISHED, 5 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 2 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 457 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs


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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(a))) -> mark(f(g(f(a))))
   active(f(X)) -> f(active(X))
   f(mark(X)) -> mark(f(X))
   proper(f(X)) -> f(proper(X))
   proper(a) -> ok(a)
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(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(a))) -> mark(f(g(f(a))))
active(f(X)) -> f(active(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))

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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:

   f(mark(X)) -> mark(f(X))
   proper(a) -> ok(a)
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(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))
   proper(a) -> ok(a)
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(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 2. 
The certificate found is represented by the following graph.

"[10, 11, 17, 18, 19, 20, 21, 22, 23]
{(10,11,[f_1|0, proper_1|0, g_1|0, top_1|0]), (10,17,[mark_1|1]), (10,18,[ok_1|1]), (10,19,[ok_1|1]), (10,20,[ok_1|1]), (10,21,[top_1|1]), (10,22,[top_1|1]), (10,23,[top_1|2]), (11,11,[mark_1|0, a|0, ok_1|0, active_1|0]), (17,11,[f_1|1]), (17,17,[mark_1|1]), (17,18,[ok_1|1]), (18,11,[f_1|1]), (18,17,[mark_1|1]), (18,18,[ok_1|1]), (19,11,[a|1]), (20,11,[g_1|1]), (20,20,[ok_1|1]), (21,11,[proper_1|1]), (21,19,[ok_1|1]), (22,11,[active_1|1]), (23,19,[active_1|2])}"
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(6)
BOUNDS(1, n^1)

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(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(f(f(a))) -> mark(f(g(f(a))))
   active(f(X)) -> f(active(X))
   f(mark(X)) -> mark(f(X))
   proper(f(X)) -> f(proper(X))
   proper(a) -> ok(a)
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
TRS:
Rules:
active(f(f(a))) -> mark(f(g(f(a))))
active(f(X)) -> f(active(X))
f(mark(X)) -> mark(f(X))
proper(f(X)) -> f(proper(X))
proper(a) -> ok(a)
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: a:mark:ok -> a:mark:ok
f :: a:mark:ok -> a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok -> a:mark:ok
g :: a:mark:ok -> a:mark:ok
proper :: a:mark:ok -> a:mark:ok
ok :: a:mark:ok -> a:mark:ok
top :: a:mark:ok -> top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat -> a:mark:ok

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(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, f, g, proper, top

They will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top

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(12)
Obligation:
TRS:
Rules:
active(f(f(a))) -> mark(f(g(f(a))))
active(f(X)) -> f(active(X))
f(mark(X)) -> mark(f(X))
proper(f(X)) -> f(proper(X))
proper(a) -> ok(a)
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: a:mark:ok -> a:mark:ok
f :: a:mark:ok -> a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok -> a:mark:ok
g :: a:mark:ok -> a:mark:ok
proper :: a:mark:ok -> a:mark:ok
ok :: a:mark:ok -> a:mark:ok
top :: a:mark:ok -> top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat -> a:mark:ok


Generator Equations:
gen_a:mark:ok3_0(0) <=> a
gen_a:mark:ok3_0(+(x, 1)) <=> mark(gen_a:mark:ok3_0(x))


The following defined symbols remain to be analysed:
f, active, g, proper, top

They will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_a:mark:ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
f(gen_a:mark:ok3_0(+(1, 0)))

Induction Step:
f(gen_a:mark:ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
mark(f(gen_a:mark:ok3_0(+(1, n5_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(f(f(a))) -> mark(f(g(f(a))))
active(f(X)) -> f(active(X))
f(mark(X)) -> mark(f(X))
proper(f(X)) -> f(proper(X))
proper(a) -> ok(a)
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: a:mark:ok -> a:mark:ok
f :: a:mark:ok -> a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok -> a:mark:ok
g :: a:mark:ok -> a:mark:ok
proper :: a:mark:ok -> a:mark:ok
ok :: a:mark:ok -> a:mark:ok
top :: a:mark:ok -> top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat -> a:mark:ok


Generator Equations:
gen_a:mark:ok3_0(0) <=> a
gen_a:mark:ok3_0(+(x, 1)) <=> mark(gen_a:mark:ok3_0(x))


The following defined symbols remain to be analysed:
f, active, g, proper, top

They will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
TRS:
Rules:
active(f(f(a))) -> mark(f(g(f(a))))
active(f(X)) -> f(active(X))
f(mark(X)) -> mark(f(X))
proper(f(X)) -> f(proper(X))
proper(a) -> ok(a)
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: a:mark:ok -> a:mark:ok
f :: a:mark:ok -> a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok -> a:mark:ok
g :: a:mark:ok -> a:mark:ok
proper :: a:mark:ok -> a:mark:ok
ok :: a:mark:ok -> a:mark:ok
top :: a:mark:ok -> top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat -> a:mark:ok


Lemmas:
f(gen_a:mark:ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_a:mark:ok3_0(0) <=> a
gen_a:mark:ok3_0(+(x, 1)) <=> mark(gen_a:mark:ok3_0(x))


The following defined symbols remain to be analysed:
g, active, proper, top

They will be analysed ascendingly in the following order:
g < active
active < top
g < proper
proper < top