(0) 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(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(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(ok(X1), ok(X2)) → ok(U11(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
top(mark(X)) → top(proper(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

Rewrite Strategy: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
top(mark(X)) → top(proper(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

Rewrite Strategy: INNERMOST

(5) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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]
transitions:
ok0(0) → 0
mark0(0) → 0
active0(0) → 0
tt0() → 0
00() → 0
U110(0, 0) → 1
U420(0, 0, 0) → 2
top0(0) → 3
proper0(0) → 4
U310(0, 0) → 5
U120(0) → 6
isNat0(0) → 7
s0(0) → 8
plus0(0, 0) → 9
U410(0, 0, 0) → 10
U210(0) → 11
U111(0, 0) → 12
ok1(12) → 1
U421(0, 0, 0) → 13
mark1(13) → 2
active1(0) → 14
top1(14) → 3
tt1() → 15
ok1(15) → 4
U311(0, 0) → 16
ok1(16) → 5
U121(0) → 17
mark1(17) → 6
isNat1(0) → 18
ok1(18) → 7
U421(0, 0, 0) → 19
ok1(19) → 2
U121(0) → 20
ok1(20) → 6
s1(0) → 21
ok1(21) → 8
U111(0, 0) → 22
mark1(22) → 1
s1(0) → 23
mark1(23) → 8
plus1(0, 0) → 24
mark1(24) → 9
U411(0, 0, 0) → 25
mark1(25) → 10
01() → 26
ok1(26) → 4
U311(0, 0) → 27
mark1(27) → 5
plus1(0, 0) → 28
ok1(28) → 9
proper1(0) → 29
top1(29) → 3
U211(0) → 30
mark1(30) → 11
U211(0) → 31
ok1(31) → 11
U411(0, 0, 0) → 32
ok1(32) → 10
ok1(12) → 12
ok1(12) → 22
mark1(13) → 13
mark1(13) → 19
ok1(15) → 29
ok1(16) → 16
ok1(16) → 27
mark1(17) → 17
mark1(17) → 20
ok1(18) → 18
ok1(19) → 13
ok1(19) → 19
ok1(20) → 17
ok1(20) → 20
ok1(21) → 21
ok1(21) → 23
mark1(22) → 12
mark1(22) → 22
mark1(23) → 21
mark1(23) → 23
mark1(24) → 24
mark1(24) → 28
mark1(25) → 25
mark1(25) → 32
ok1(26) → 29
mark1(27) → 16
mark1(27) → 27
ok1(28) → 24
ok1(28) → 28
mark1(30) → 30
mark1(30) → 31
ok1(31) → 30
ok1(31) → 31
ok1(32) → 25
ok1(32) → 32
active2(15) → 33
top2(33) → 3
active2(26) → 33

(6) BOUNDS(1, n^1)