/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 84 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), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 425 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(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: FULL
----------------------------------------

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of active: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of f: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of start: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of match: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of check: active, start, proper, top, f, check, match

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
match(f(x), f(y)) -> f(match(x, y))
proper(f(x)) -> f(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:

   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(X, x) -> proper(x)
   proper(c) -> ok(c)
   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: 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:

   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(X, x) -> proper(x)
   proper(c) -> ok(c)
   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: FULL
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7]
transitions: 
mark0(0) -> 0
c0() -> 0
X0() -> 0
ok0(0) -> 0
found0(0) -> 0
active0(0) -> 1
top0(0) -> 2
check0(0) -> 3
match0(0, 0) -> 4
proper0(0) -> 5
f0(0) -> 6
start0(0) -> 7
check1(0) -> 8
top1(8) -> 2
X1() -> 11
f1(11) -> 10
match1(10, 0) -> 9
start1(9) -> 3
proper1(0) -> 4
c1() -> 12
ok1(12) -> 5
f1(0) -> 13
ok1(13) -> 6
found1(0) -> 7
f1(0) -> 14
found1(14) -> 6
active1(0) -> 15
top1(15) -> 2
f1(0) -> 16
mark1(16) -> 6
c1() -> 18
mark1(18) -> 17
top1(17) -> 2
X2() -> 21
f2(21) -> 20
match2(20, 0) -> 19
start2(19) -> 8
ok1(12) -> 4
ok1(13) -> 13
ok1(13) -> 14
ok1(13) -> 16
found1(14) -> 13
found1(14) -> 14
found1(14) -> 16
mark1(16) -> 13
mark1(16) -> 14
mark1(16) -> 16
check2(18) -> 22
top2(22) -> 2
X3() -> 25
f3(25) -> 24
match3(24, 18) -> 23
start3(23) -> 22

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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(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: FULL
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
TRS:
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))

Types:
active :: mark:c:X:ok:found -> mark:c:X:ok:found
f :: mark:c:X:ok:found -> mark:c:X:ok:found
mark :: mark:c:X:ok:found -> mark:c:X:ok:found
top :: mark:c:X:ok:found -> top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found -> mark:c:X:ok:found
start :: mark:c:X:ok:found -> mark:c:X:ok:found
match :: mark:c:X:ok:found -> mark:c:X:ok:found -> mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found -> mark:c:X:ok:found
ok :: mark:c:X:ok:found -> mark:c:X:ok:found
found :: mark:c:X:ok:found -> mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat -> mark:c:X:ok:found

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

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

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(12)
Obligation:
TRS:
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))

Types:
active :: mark:c:X:ok:found -> mark:c:X:ok:found
f :: mark:c:X:ok:found -> mark:c:X:ok:found
mark :: mark:c:X:ok:found -> mark:c:X:ok:found
top :: mark:c:X:ok:found -> top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found -> mark:c:X:ok:found
start :: mark:c:X:ok:found -> mark:c:X:ok:found
match :: mark:c:X:ok:found -> mark:c:X:ok:found -> mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found -> mark:c:X:ok:found
ok :: mark:c:X:ok:found -> mark:c:X:ok:found
found :: mark:c:X:ok:found -> mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat -> mark:c:X:ok:found


Generator Equations:
gen_mark:c:X:ok:found3_0(0) <=> c
gen_mark:c:X:ok:found3_0(+(x, 1)) <=> mark(gen_mark:c:X:ok:found3_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
f(gen_mark:c:X:ok:found3_0(+(1, 0)))

Induction Step:
f(gen_mark:c:X:ok:found3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
mark(f(gen_mark:c:X:ok:found3_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(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))

Types:
active :: mark:c:X:ok:found -> mark:c:X:ok:found
f :: mark:c:X:ok:found -> mark:c:X:ok:found
mark :: mark:c:X:ok:found -> mark:c:X:ok:found
top :: mark:c:X:ok:found -> top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found -> mark:c:X:ok:found
start :: mark:c:X:ok:found -> mark:c:X:ok:found
match :: mark:c:X:ok:found -> mark:c:X:ok:found -> mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found -> mark:c:X:ok:found
ok :: mark:c:X:ok:found -> mark:c:X:ok:found
found :: mark:c:X:ok:found -> mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat -> mark:c:X:ok:found


Generator Equations:
gen_mark:c:X:ok:found3_0(0) <=> c
gen_mark:c:X:ok:found3_0(+(x, 1)) <=> mark(gen_mark:c:X:ok:found3_0(x))


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

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

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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(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))

Types:
active :: mark:c:X:ok:found -> mark:c:X:ok:found
f :: mark:c:X:ok:found -> mark:c:X:ok:found
mark :: mark:c:X:ok:found -> mark:c:X:ok:found
top :: mark:c:X:ok:found -> top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found -> mark:c:X:ok:found
start :: mark:c:X:ok:found -> mark:c:X:ok:found
match :: mark:c:X:ok:found -> mark:c:X:ok:found -> mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found -> mark:c:X:ok:found
ok :: mark:c:X:ok:found -> mark:c:X:ok:found
found :: mark:c:X:ok:found -> mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat -> mark:c:X:ok:found


Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_mark:c:X:ok:found3_0(0) <=> c
gen_mark:c:X:ok:found3_0(+(x, 1)) <=> mark(gen_mark:c:X:ok:found3_0(x))


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

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