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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(U11(X)) → U11(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(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:

U52(mark(X)) → mark(U52(X))
top(ok(X)) → top(active(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
__(mark(X1), X2) → mark(__(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
proper(i) → ok(i)
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(o) → ok(o)
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U42(mark(X)) → mark(U42(X))
proper(u) → ok(u)
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
isPal(ok(X)) → ok(isPal(X))
U72(mark(X)) → mark(U72(X))
U72(ok(X)) → ok(U72(X))
U22(mark(X)) → mark(U22(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U22(ok(X)) → ok(U22(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
proper(a) → ok(a)
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X)) → ok(U31(X))
U31(mark(X)) → mark(U31(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U81(ok(X)) → ok(U81(X))
U81(mark(X)) → mark(U81(X))
top(mark(X)) → top(proper(X))
U52(ok(X)) → ok(U52(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:

U52(mark(X)) → mark(U52(X))
top(ok(X)) → top(active(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
__(mark(X1), X2) → mark(__(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
proper(i) → ok(i)
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(o) → ok(o)
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U42(mark(X)) → mark(U42(X))
proper(u) → ok(u)
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
isPal(ok(X)) → ok(isPal(X))
U72(mark(X)) → mark(U72(X))
U72(ok(X)) → ok(U72(X))
U22(mark(X)) → mark(U22(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U22(ok(X)) → ok(U22(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
proper(a) → ok(a)
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X)) → ok(U31(X))
U31(mark(X)) → mark(U31(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U81(ok(X)) → ok(U81(X))
U81(mark(X)) → mark(U81(X))
top(mark(X)) → top(proper(X))
U52(ok(X)) → ok(U52(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:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
i0() → 0
o0() → 0
u0() → 0
e0() → 0
nil0() → 0
tt0() → 0
a0() → 0
U520(0) → 1
top0(0) → 2
isNeList0(0) → 3
isQid0(0) → 4
U510(0, 0) → 5
isNePal0(0) → 6
__0(0, 0) → 7
U610(0) → 8
proper0(0) → 9
U410(0, 0) → 10
U420(0) → 11
U210(0, 0) → 12
isList0(0) → 13
isPal0(0) → 14
U720(0) → 15
U220(0) → 16
U710(0, 0) → 17
U110(0) → 18
U310(0) → 19
U810(0) → 20
U521(0) → 21
mark1(21) → 1
active1(0) → 22
top1(22) → 2
isNeList1(0) → 23
ok1(23) → 3
isQid1(0) → 24
ok1(24) → 4
U511(0, 0) → 25
mark1(25) → 5
isNePal1(0) → 26
ok1(26) → 6
__1(0, 0) → 27
mark1(27) → 7
U611(0) → 28
mark1(28) → 8
U611(0) → 29
ok1(29) → 8
i1() → 30
ok1(30) → 9
U511(0, 0) → 31
ok1(31) → 5
o1() → 32
ok1(32) → 9
U411(0, 0) → 33
mark1(33) → 10
U421(0) → 34
ok1(34) → 11
U211(0, 0) → 35
mark1(35) → 12
U421(0) → 36
mark1(36) → 11
u1() → 37
ok1(37) → 9
e1() → 38
ok1(38) → 9
isList1(0) → 39
ok1(39) → 13
nil1() → 40
ok1(40) → 9
tt1() → 41
ok1(41) → 9
isPal1(0) → 42
ok1(42) → 14
U721(0) → 43
mark1(43) → 15
U721(0) → 44
ok1(44) → 15
U221(0) → 45
mark1(45) → 16
__1(0, 0) → 46
ok1(46) → 7
U221(0) → 47
ok1(47) → 16
U411(0, 0) → 48
ok1(48) → 10
U711(0, 0) → 49
mark1(49) → 17
a1() → 50
ok1(50) → 9
U111(0) → 51
mark1(51) → 18
U111(0) → 52
ok1(52) → 18
U211(0, 0) → 53
ok1(53) → 12
U311(0) → 54
ok1(54) → 19
U311(0) → 55
mark1(55) → 19
U711(0, 0) → 56
ok1(56) → 17
U811(0) → 57
ok1(57) → 20
U811(0) → 58
mark1(58) → 20
proper1(0) → 59
top1(59) → 2
U521(0) → 60
ok1(60) → 1
mark1(21) → 21
mark1(21) → 60
ok1(23) → 23
ok1(24) → 24
mark1(25) → 25
mark1(25) → 31
ok1(26) → 26
mark1(27) → 27
mark1(27) → 46
mark1(28) → 28
mark1(28) → 29
ok1(29) → 28
ok1(29) → 29
ok1(30) → 59
ok1(31) → 25
ok1(31) → 31
ok1(32) → 59
mark1(33) → 33
mark1(33) → 48
ok1(34) → 34
ok1(34) → 36
mark1(35) → 35
mark1(35) → 53
mark1(36) → 34
mark1(36) → 36
ok1(37) → 59
ok1(38) → 59
ok1(39) → 39
ok1(40) → 59
ok1(41) → 59
ok1(42) → 42
mark1(43) → 43
mark1(43) → 44
ok1(44) → 43
ok1(44) → 44
mark1(45) → 45
mark1(45) → 47
ok1(46) → 27
ok1(46) → 46
ok1(47) → 45
ok1(47) → 47
ok1(48) → 33
ok1(48) → 48
mark1(49) → 49
mark1(49) → 56
ok1(50) → 59
mark1(51) → 51
mark1(51) → 52
ok1(52) → 51
ok1(52) → 52
ok1(53) → 35
ok1(53) → 53
ok1(54) → 54
ok1(54) → 55
mark1(55) → 54
mark1(55) → 55
ok1(56) → 49
ok1(56) → 56
ok1(57) → 57
ok1(57) → 58
mark1(58) → 57
mark1(58) → 58
ok1(60) → 21
ok1(60) → 60
active2(30) → 61
top2(61) → 2
active2(32) → 61
active2(37) → 61
active2(38) → 61
active2(40) → 61
active2(41) → 61
active2(50) → 61

(6) BOUNDS(1, n^1)