/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 15 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 544 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, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U51(X1, X2, X3)) -> U51(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(U61(X)) -> U61(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U61(X)) -> U61(proper(X))
   proper(0) -> ok(0)
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
active(U12(tt, V2)) -> mark(U13(isNat(V2)))
active(U13(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
active(U32(tt, V2)) -> mark(U33(isNat(V2)))
active(U33(tt)) -> mark(tt)
active(U41(tt, N)) -> mark(N)
active(U51(tt, M, N)) -> mark(s(plus(N, M)))
active(U61(tt)) -> mark(0)
active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0)) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) -> mark(tt)
active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U13(X)) -> U13(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U51(X1, X2, X3)) -> U51(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(U61(X)) -> U61(active(X))
active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U61(X)) -> U61(proper(X))
proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatKind(X)) -> isNatKind(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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(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(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(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(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(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 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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0, 0) -> 1
U120(0, 0) -> 2
U130(0) -> 3
U210(0, 0) -> 4
U220(0) -> 5
U310(0, 0, 0) -> 6
U320(0, 0) -> 7
U330(0) -> 8
U410(0, 0) -> 9
U510(0, 0, 0) -> 10
s0(0) -> 11
plus0(0, 0) -> 12
U610(0) -> 13
U710(0, 0, 0) -> 14
x0(0, 0) -> 15
and0(0, 0) -> 16
proper0(0) -> 17
isNat0(0) -> 18
isNatKind0(0) -> 19
top0(0) -> 20
U111(0, 0, 0) -> 21
mark1(21) -> 1
U121(0, 0) -> 22
mark1(22) -> 2
U131(0) -> 23
mark1(23) -> 3
U211(0, 0) -> 24
mark1(24) -> 4
U221(0) -> 25
mark1(25) -> 5
U311(0, 0, 0) -> 26
mark1(26) -> 6
U321(0, 0) -> 27
mark1(27) -> 7
U331(0) -> 28
mark1(28) -> 8
U411(0, 0) -> 29
mark1(29) -> 9
U511(0, 0, 0) -> 30
mark1(30) -> 10
s1(0) -> 31
mark1(31) -> 11
plus1(0, 0) -> 32
mark1(32) -> 12
U611(0) -> 33
mark1(33) -> 13
U711(0, 0, 0) -> 34
mark1(34) -> 14
x1(0, 0) -> 35
mark1(35) -> 15
and1(0, 0) -> 36
mark1(36) -> 16
tt1() -> 37
ok1(37) -> 17
01() -> 38
ok1(38) -> 17
U111(0, 0, 0) -> 39
ok1(39) -> 1
U121(0, 0) -> 40
ok1(40) -> 2
isNat1(0) -> 41
ok1(41) -> 18
U131(0) -> 42
ok1(42) -> 3
U211(0, 0) -> 43
ok1(43) -> 4
U221(0) -> 44
ok1(44) -> 5
U311(0, 0, 0) -> 45
ok1(45) -> 6
U321(0, 0) -> 46
ok1(46) -> 7
U331(0) -> 47
ok1(47) -> 8
U411(0, 0) -> 48
ok1(48) -> 9
U511(0, 0, 0) -> 49
ok1(49) -> 10
s1(0) -> 50
ok1(50) -> 11
plus1(0, 0) -> 51
ok1(51) -> 12
U611(0) -> 52
ok1(52) -> 13
U711(0, 0, 0) -> 53
ok1(53) -> 14
x1(0, 0) -> 54
ok1(54) -> 15
and1(0, 0) -> 55
ok1(55) -> 16
isNatKind1(0) -> 56
ok1(56) -> 19
proper1(0) -> 57
top1(57) -> 20
active1(0) -> 58
top1(58) -> 20
mark1(21) -> 21
mark1(21) -> 39
mark1(22) -> 22
mark1(22) -> 40
mark1(23) -> 23
mark1(23) -> 42
mark1(24) -> 24
mark1(24) -> 43
mark1(25) -> 25
mark1(25) -> 44
mark1(26) -> 26
mark1(26) -> 45
mark1(27) -> 27
mark1(27) -> 46
mark1(28) -> 28
mark1(28) -> 47
mark1(29) -> 29
mark1(29) -> 48
mark1(30) -> 30
mark1(30) -> 49
mark1(31) -> 31
mark1(31) -> 50
mark1(32) -> 32
mark1(32) -> 51
mark1(33) -> 33
mark1(33) -> 52
mark1(34) -> 34
mark1(34) -> 53
mark1(35) -> 35
mark1(35) -> 54
mark1(36) -> 36
mark1(36) -> 55
ok1(37) -> 57
ok1(38) -> 57
ok1(39) -> 21
ok1(39) -> 39
ok1(40) -> 22
ok1(40) -> 40
ok1(41) -> 41
ok1(42) -> 23
ok1(42) -> 42
ok1(43) -> 24
ok1(43) -> 43
ok1(44) -> 25
ok1(44) -> 44
ok1(45) -> 26
ok1(45) -> 45
ok1(46) -> 27
ok1(46) -> 46
ok1(47) -> 28
ok1(47) -> 47
ok1(48) -> 29
ok1(48) -> 48
ok1(49) -> 30
ok1(49) -> 49
ok1(50) -> 31
ok1(50) -> 50
ok1(51) -> 32
ok1(51) -> 51
ok1(52) -> 33
ok1(52) -> 52
ok1(53) -> 34
ok1(53) -> 53
ok1(54) -> 35
ok1(54) -> 54
ok1(55) -> 36
ok1(55) -> 55
ok1(56) -> 56
active2(37) -> 59
top2(59) -> 20
active2(38) -> 59

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

(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, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U51(X1, X2, X3)) -> U51(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(U61(X)) -> U61(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U61(X)) -> U61(proper(X))
   proper(0) -> ok(0)
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(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

U32(mark(X1), X2) ->^+ mark(U32(X1, X2))

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

The pumping substitution is [X1 / mark(X1)].

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, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U51(X1, X2, X3)) -> U51(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(U61(X)) -> U61(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U61(X)) -> U61(proper(X))
   proper(0) -> ok(0)
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U51(X1, X2, X3)) -> U51(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(U61(X)) -> U61(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U51(mark(X1), X2, X3) -> mark(U51(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))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U61(X)) -> U61(proper(X))
   proper(0) -> ok(0)
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL