WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 90 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


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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(U11(tt, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   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(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(and(tt, X)) -> mark(X)
active(isNat(0)) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(plus(N, 0)) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(and(X1, X2)) -> and(active(X1), X2)
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(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:

   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(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:

   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(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 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]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0) -> 1
U210(0, 0, 0) -> 2
s0(0) -> 3
plus0(0, 0) -> 4
and0(0, 0) -> 5
proper0(0) -> 6
isNat0(0) -> 7
top0(0) -> 8
U111(0, 0) -> 9
mark1(9) -> 1
U211(0, 0, 0) -> 10
mark1(10) -> 2
s1(0) -> 11
mark1(11) -> 3
plus1(0, 0) -> 12
mark1(12) -> 4
and1(0, 0) -> 13
mark1(13) -> 5
tt1() -> 14
ok1(14) -> 6
01() -> 15
ok1(15) -> 6
U111(0, 0) -> 16
ok1(16) -> 1
U211(0, 0, 0) -> 17
ok1(17) -> 2
s1(0) -> 18
ok1(18) -> 3
plus1(0, 0) -> 19
ok1(19) -> 4
and1(0, 0) -> 20
ok1(20) -> 5
isNat1(0) -> 21
ok1(21) -> 7
proper1(0) -> 22
top1(22) -> 8
active1(0) -> 23
top1(23) -> 8
mark1(9) -> 9
mark1(9) -> 16
mark1(10) -> 10
mark1(10) -> 17
mark1(11) -> 11
mark1(11) -> 18
mark1(12) -> 12
mark1(12) -> 19
mark1(13) -> 13
mark1(13) -> 20
ok1(14) -> 22
ok1(15) -> 22
ok1(16) -> 9
ok1(16) -> 16
ok1(17) -> 10
ok1(17) -> 17
ok1(18) -> 11
ok1(18) -> 18
ok1(19) -> 12
ok1(19) -> 19
ok1(20) -> 13
ok1(20) -> 20
ok1(21) -> 21
active2(14) -> 24
top2(24) -> 8
active2(15) -> 24

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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(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, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(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 [ ].




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(10)
Complex Obligation (BEST)

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(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, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(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, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL