/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 146 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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0)) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0)) -> mark(0)
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))

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

(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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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 2. 

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, 8, 9]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0, 0) -> 1
U120(0, 0, 0) -> 2
s0(0) -> 3
plus0(0, 0) -> 4
U210(0, 0, 0) -> 5
U220(0, 0, 0) -> 6
x0(0, 0) -> 7
proper0(0) -> 8
top0(0) -> 9
U111(0, 0, 0) -> 10
mark1(10) -> 1
U121(0, 0, 0) -> 11
mark1(11) -> 2
s1(0) -> 12
mark1(12) -> 3
plus1(0, 0) -> 13
mark1(13) -> 4
U211(0, 0, 0) -> 14
mark1(14) -> 5
U221(0, 0, 0) -> 15
mark1(15) -> 6
x1(0, 0) -> 16
mark1(16) -> 7
tt1() -> 17
ok1(17) -> 8
01() -> 18
ok1(18) -> 8
U111(0, 0, 0) -> 19
ok1(19) -> 1
U121(0, 0, 0) -> 20
ok1(20) -> 2
s1(0) -> 21
ok1(21) -> 3
plus1(0, 0) -> 22
ok1(22) -> 4
U211(0, 0, 0) -> 23
ok1(23) -> 5
U221(0, 0, 0) -> 24
ok1(24) -> 6
x1(0, 0) -> 25
ok1(25) -> 7
proper1(0) -> 26
top1(26) -> 9
active1(0) -> 27
top1(27) -> 9
mark1(10) -> 10
mark1(10) -> 19
mark1(11) -> 11
mark1(11) -> 20
mark1(12) -> 12
mark1(12) -> 21
mark1(13) -> 13
mark1(13) -> 22
mark1(14) -> 14
mark1(14) -> 23
mark1(15) -> 15
mark1(15) -> 24
mark1(16) -> 16
mark1(16) -> 25
ok1(17) -> 26
ok1(18) -> 26
ok1(19) -> 10
ok1(19) -> 19
ok1(20) -> 11
ok1(20) -> 20
ok1(21) -> 12
ok1(21) -> 21
ok1(22) -> 13
ok1(22) -> 22
ok1(23) -> 14
ok1(23) -> 23
ok1(24) -> 15
ok1(24) -> 24
ok1(25) -> 16
ok1(25) -> 25
active2(17) -> 28
top2(28) -> 9
active2(18) -> 28

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

(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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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

plus(X1, mark(X2)) ->^+ mark(plus(X1, X2))

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

The pumping substitution is [X2 / mark(X2)].

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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL