(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, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(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(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(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))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(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(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
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))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(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, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(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(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(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(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(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(ok(X1), ok(X2)) → ok(U11(X1, X2))
top(ok(X)) → top(active(X))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U61(ok(X)) → ok(U61(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
proper(tt) → ok(tt)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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))
x(X1, mark(X2)) → mark(x(X1, X2))
U32(ok(X)) → ok(U32(X))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
top(mark(X)) → top(proper(X))

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))
top(ok(X)) → top(active(X))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U61(ok(X)) → ok(U61(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
proper(tt) → ok(tt)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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))
x(X1, mark(X2)) → mark(x(X1, X2))
U32(ok(X)) → ok(U32(X))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
top(mark(X)) → top(proper(X))

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, 12, 13, 14, 15, 16, 17]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
00() → 0
U110(0, 0) → 1
top0(0) → 2
U120(0) → 3
isNat0(0) → 4
U710(0, 0, 0) → 5
U610(0) → 6
U520(0, 0, 0) → 7
U720(0, 0, 0) → 8
U510(0, 0, 0) → 9
U410(0, 0) → 10
plus0(0, 0) → 11
U210(0) → 12
proper0(0) → 13
U320(0) → 14
U310(0, 0) → 15
x0(0, 0) → 16
s0(0) → 17
U111(0, 0) → 18
ok1(18) → 1
active1(0) → 19
top1(19) → 2
U121(0) → 20
mark1(20) → 3
isNat1(0) → 21
ok1(21) → 4
U711(0, 0, 0) → 22
ok1(22) → 5
U121(0) → 23
ok1(23) → 3
U611(0) → 24
mark1(24) → 6
U711(0, 0, 0) → 25
mark1(25) → 5
U611(0) → 26
ok1(26) → 6
U521(0, 0, 0) → 27
mark1(27) → 7
U721(0, 0, 0) → 28
mark1(28) → 8
U511(0, 0, 0) → 29
mark1(29) → 9
U521(0, 0, 0) → 30
ok1(30) → 7
U411(0, 0) → 31
mark1(31) → 10
plus1(0, 0) → 32
ok1(32) → 11
plus1(0, 0) → 33
mark1(33) → 11
U211(0) → 34
mark1(34) → 12
U211(0) → 35
ok1(35) → 12
U721(0, 0, 0) → 36
ok1(36) → 8
tt1() → 37
ok1(37) → 13
U511(0, 0, 0) → 38
ok1(38) → 9
U321(0) → 39
mark1(39) → 14
U311(0, 0) → 40
ok1(40) → 15
x1(0, 0) → 41
mark1(41) → 16
U411(0, 0) → 42
ok1(42) → 10
x1(0, 0) → 43
ok1(43) → 16
s1(0) → 44
ok1(44) → 17
U111(0, 0) → 45
mark1(45) → 1
s1(0) → 46
mark1(46) → 17
U321(0) → 47
ok1(47) → 14
01() → 48
ok1(48) → 13
U311(0, 0) → 49
mark1(49) → 15
proper1(0) → 50
top1(50) → 2
ok1(18) → 18
ok1(18) → 45
mark1(20) → 20
mark1(20) → 23
ok1(21) → 21
ok1(22) → 22
ok1(22) → 25
ok1(23) → 20
ok1(23) → 23
mark1(24) → 24
mark1(24) → 26
mark1(25) → 22
mark1(25) → 25
ok1(26) → 24
ok1(26) → 26
mark1(27) → 27
mark1(27) → 30
mark1(28) → 28
mark1(28) → 36
mark1(29) → 29
mark1(29) → 38
ok1(30) → 27
ok1(30) → 30
mark1(31) → 31
mark1(31) → 42
ok1(32) → 32
ok1(32) → 33
mark1(33) → 32
mark1(33) → 33
mark1(34) → 34
mark1(34) → 35
ok1(35) → 34
ok1(35) → 35
ok1(36) → 28
ok1(36) → 36
ok1(37) → 50
ok1(38) → 29
ok1(38) → 38
mark1(39) → 39
mark1(39) → 47
ok1(40) → 40
ok1(40) → 49
mark1(41) → 41
mark1(41) → 43
ok1(42) → 31
ok1(42) → 42
ok1(43) → 41
ok1(43) → 43
ok1(44) → 44
ok1(44) → 46
mark1(45) → 18
mark1(45) → 45
mark1(46) → 44
mark1(46) → 46
ok1(47) → 39
ok1(47) → 47
ok1(48) → 50
mark1(49) → 40
mark1(49) → 49
active2(37) → 51
top2(51) → 2
active2(48) → 51

(6) BOUNDS(1, n^1)