/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 109 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(a, X, X)) -> mark(f(X, b, b))
   active(b) -> mark(a)
   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   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(a, X, X)) -> mark(f(X, b, b))
active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))

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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(b) -> mark(a)
   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   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:

   active(b) -> mark(a)
   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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 4. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
b0() -> 0
mark0(0) -> 0
a0() -> 0
ok0(0) -> 0
active0(0) -> 1
f0(0, 0, 0) -> 2
proper0(0) -> 3
top0(0) -> 4
a1() -> 5
mark1(5) -> 1
f1(0, 0, 0) -> 6
mark1(6) -> 2
a1() -> 7
ok1(7) -> 3
b1() -> 8
ok1(8) -> 3
f1(0, 0, 0) -> 9
ok1(9) -> 2
proper1(0) -> 10
top1(10) -> 4
active1(0) -> 11
top1(11) -> 4
mark1(5) -> 11
mark1(6) -> 6
mark1(6) -> 9
ok1(7) -> 10
ok1(8) -> 10
ok1(9) -> 6
ok1(9) -> 9
proper2(5) -> 12
top2(12) -> 4
active2(7) -> 13
top2(13) -> 4
active2(8) -> 13
a2() -> 14
mark2(14) -> 13
a2() -> 15
ok2(15) -> 12
proper3(14) -> 16
top3(16) -> 4
active3(15) -> 17
top3(17) -> 4
a3() -> 18
ok3(18) -> 16
active4(18) -> 19
top4(19) -> 4

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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(a, X, X)) -> mark(f(X, b, b))
   active(b) -> mark(a)
   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(a) -> ok(a)
   proper(b) -> ok(b)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   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(a, X, X)) -> mark(f(X, b, b))
active(b) -> mark(a)
active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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

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

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

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(12)
Obligation:
TRS:
Rules:
active(f(a, X, X)) -> mark(f(X, b, b))
active(b) -> mark(a)
active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


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


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

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

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

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

Induction Step:
f(gen_a:b:mark:ok3_0(a), gen_a:b:mark:ok3_0(+(1, +(n5_0, 1))), gen_a:b:mark:ok3_0(c)) ->_R^Omega(1)
mark(f(gen_a:b:mark:ok3_0(a), gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(c))) ->_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(a, X, X)) -> mark(f(X, b, b))
active(b) -> mark(a)
active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


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


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

They will be analysed ascendingly in the following order:
f < active
active < top
f < 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(a, X, X)) -> mark(f(X, b, b))
active(b) -> mark(a)
active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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


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


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


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

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