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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) CpxTrsMatchBoundsTAProof [FINISHED, 145 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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

(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, g(X), Y)) -> mark(f(Y, Y, Y))
   active(g(b)) -> mark(c)
   active(b) -> mark(c)
   active(g(X)) -> g(active(X))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(g(X)) -> g(proper(X))
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   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(X, g(X), Y)) -> mark(f(Y, Y, Y))
active(g(b)) -> mark(c)
active(g(X)) -> g(active(X))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(g(X)) -> g(proper(X))

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

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

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

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

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

(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, 5]
transitions: 
b0() -> 0
mark0(0) -> 0
c0() -> 0
ok0(0) -> 0
active0(0) -> 1
g0(0) -> 2
proper0(0) -> 3
f0(0, 0, 0) -> 4
top0(0) -> 5
c1() -> 6
mark1(6) -> 1
g1(0) -> 7
mark1(7) -> 2
b1() -> 8
ok1(8) -> 3
c1() -> 9
ok1(9) -> 3
f1(0, 0, 0) -> 10
ok1(10) -> 4
g1(0) -> 11
ok1(11) -> 2
proper1(0) -> 12
top1(12) -> 5
active1(0) -> 13
top1(13) -> 5
mark1(6) -> 13
mark1(7) -> 7
mark1(7) -> 11
ok1(8) -> 12
ok1(9) -> 12
ok1(10) -> 10
ok1(11) -> 7
ok1(11) -> 11
proper2(6) -> 14
top2(14) -> 5
active2(8) -> 15
top2(15) -> 5
active2(9) -> 15
c2() -> 16
mark2(16) -> 15
c2() -> 17
ok2(17) -> 14
proper3(16) -> 18
top3(18) -> 5
active3(17) -> 19
top3(19) -> 5
c3() -> 20
ok3(20) -> 18
active4(20) -> 21
top4(21) -> 5

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

(6)
BOUNDS(1, n^1)

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

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

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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

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

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

The rewrite sequence

g(ok(X)) ->^+ ok(g(X))

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

The pumping substitution is [X / ok(X)].

The result substitution is [ ].




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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following 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, g(X), Y)) -> mark(f(Y, Y, Y))
   active(g(b)) -> mark(c)
   active(b) -> mark(c)
   active(g(X)) -> g(active(X))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
   proper(g(X)) -> g(proper(X))
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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

S is empty.
Rewrite Strategy: FULL