/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 696 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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(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 cons: active, proper, cons
The following defined symbols can occur below the 1th argument of cons: active, proper, cons
The following defined symbols can occur below the 0th argument of top: active, proper, cons
The following defined symbols can occur below the 0th argument of proper: active, proper, cons
The following defined symbols can occur below the 0th argument of active: active, proper, cons

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U11(tt, L)) -> mark(s(length(L)))
active(and(tt, X)) -> mark(X)
active(isNat(0)) -> mark(tt)
active(isNat(length(V1))) -> mark(isNatList(V1))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNatIList(V)) -> mark(isNatList(V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) -> mark(0)
active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(and(X1, X2)) -> and(active(X1), X2)
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(isNatIList(X)) -> isNatIList(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(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(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(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(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 5. 

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, 10, 11]
transitions: 
zeros0() -> 0
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
tt0() -> 0
nil0() -> 0
active0(0) -> 1
cons0(0, 0) -> 2
U110(0, 0) -> 3
s0(0) -> 4
length0(0) -> 5
and0(0, 0) -> 6
proper0(0) -> 7
isNat0(0) -> 8
isNatList0(0) -> 9
isNatIList0(0) -> 10
top0(0) -> 11
01() -> 13
zeros1() -> 14
cons1(13, 14) -> 12
mark1(12) -> 1
cons1(0, 0) -> 15
mark1(15) -> 2
U111(0, 0) -> 16
mark1(16) -> 3
s1(0) -> 17
mark1(17) -> 4
length1(0) -> 18
mark1(18) -> 5
and1(0, 0) -> 19
mark1(19) -> 6
zeros1() -> 20
ok1(20) -> 7
01() -> 21
ok1(21) -> 7
tt1() -> 22
ok1(22) -> 7
nil1() -> 23
ok1(23) -> 7
cons1(0, 0) -> 24
ok1(24) -> 2
U111(0, 0) -> 25
ok1(25) -> 3
s1(0) -> 26
ok1(26) -> 4
length1(0) -> 27
ok1(27) -> 5
and1(0, 0) -> 28
ok1(28) -> 6
isNat1(0) -> 29
ok1(29) -> 8
isNatList1(0) -> 30
ok1(30) -> 9
isNatIList1(0) -> 31
ok1(31) -> 10
proper1(0) -> 32
top1(32) -> 11
active1(0) -> 33
top1(33) -> 11
mark1(12) -> 33
mark1(15) -> 15
mark1(15) -> 24
mark1(16) -> 16
mark1(16) -> 25
mark1(17) -> 17
mark1(17) -> 26
mark1(18) -> 18
mark1(18) -> 27
mark1(19) -> 19
mark1(19) -> 28
ok1(20) -> 32
ok1(21) -> 32
ok1(22) -> 32
ok1(23) -> 32
ok1(24) -> 15
ok1(24) -> 24
ok1(25) -> 16
ok1(25) -> 25
ok1(26) -> 17
ok1(26) -> 26
ok1(27) -> 18
ok1(27) -> 27
ok1(28) -> 19
ok1(28) -> 28
ok1(29) -> 29
ok1(30) -> 30
ok1(31) -> 31
proper2(12) -> 34
top2(34) -> 11
active2(20) -> 35
top2(35) -> 11
active2(21) -> 35
active2(22) -> 35
active2(23) -> 35
02() -> 37
zeros2() -> 38
cons2(37, 38) -> 36
mark2(36) -> 35
proper2(13) -> 39
proper2(14) -> 40
cons2(39, 40) -> 34
zeros2() -> 41
ok2(41) -> 40
02() -> 42
ok2(42) -> 39
proper3(36) -> 43
top3(43) -> 11
proper3(37) -> 44
proper3(38) -> 45
cons3(44, 45) -> 43
cons3(42, 41) -> 46
ok3(46) -> 34
zeros3() -> 47
ok3(47) -> 45
03() -> 48
ok3(48) -> 44
active3(46) -> 49
top3(49) -> 11
cons4(48, 47) -> 50
ok4(50) -> 43
active4(42) -> 51
cons4(51, 41) -> 49
active4(50) -> 52
top4(52) -> 11
active5(48) -> 53
cons5(53, 47) -> 52

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

(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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(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

isNatIList(ok(X)) ->^+ ok(isNatIList(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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL