/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 7 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), 474 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 146 ms]
            (20) 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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
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(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   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(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   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) 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
mark0(0) -> 0
ok0(0) -> 0
proper0(0) -> 0
active0(0) -> 0
f0(0, 0) -> 1
g0(0) -> 2
top0(0) -> 3
f1(0, 0) -> 4
mark1(4) -> 1
g1(0) -> 5
mark1(5) -> 2
f1(0, 0) -> 6
ok1(6) -> 1
g1(0) -> 7
ok1(7) -> 2
proper1(0) -> 8
top1(8) -> 3
active1(0) -> 9
top1(9) -> 3
mark1(4) -> 4
mark1(4) -> 6
mark1(5) -> 5
mark1(5) -> 7
ok1(6) -> 4
ok1(6) -> 6
ok1(7) -> 5
ok1(7) -> 7

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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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
g(mark(X)) -> mark(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok -> mark:ok
g :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> 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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
g(mark(X)) -> mark(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok -> mark:ok
g :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_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_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
f(gen_mark:ok3_0(+(1, 0)), gen_mark:ok3_0(b))

Induction Step:
f(gen_mark:ok3_0(+(1, +(n5_0, 1))), gen_mark:ok3_0(b)) ->_R^Omega(1)
mark(f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b))) ->_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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
g(mark(X)) -> mark(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok -> mark:ok
g :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_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(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
g(mark(X)) -> mark(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok -> mark:ok
g :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


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


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_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

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_mark:ok3_0(+(1, n594_0))) -> *4_0, rt in Omega(n594_0)

Induction Base:
g(gen_mark:ok3_0(+(1, 0)))

Induction Step:
g(gen_mark:ok3_0(+(1, +(n594_0, 1)))) ->_R^Omega(1)
mark(g(gen_mark:ok3_0(+(1, n594_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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
g(mark(X)) -> mark(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:ok -> mark:ok
f :: mark:ok -> mark:ok -> mark:ok
g :: mark:ok -> mark:ok
mark :: mark:ok -> mark:ok
proper :: mark:ok -> mark:ok
ok :: mark:ok -> mark:ok
top :: mark:ok -> top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat -> mark:ok


Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
g(gen_mark:ok3_0(+(1, n594_0))) -> *4_0, rt in Omega(n594_0)


Generator Equations:
gen_mark:ok3_0(0) <=> hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) <=> mark(gen_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