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

(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, 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))

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, 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:

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

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

(6) BOUNDS(1, n^1)