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

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
__(z0, z1) → n____(z0, z1)
U11(tt) → tt
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U31(tt) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt
U61(tt) → tt
U71(tt, z0) → U72(isPal(activate(z0)))
U72(tt) → tt
U81(tt) → tt
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
isNePal(z0) → U61(isQid(activate(z0)))
isNePal(n____(z0, __(z1, z0))) → U71(isQid(activate(z0)), activate(z1))
isPal(z0) → U81(isNePal(activate(z0)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0

Tuples:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
__'(z0, nil) → c1
__'(nil, z0) → c2
__'(z0, z1) → c3
U11'(tt) → c4
U21'(tt, z0) → c5(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U22'(tt) → c6
U31'(tt) → c7
U41'(tt, z0) → c8(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U42'(tt) → c9
U51'(tt, z0) → c10(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U52'(tt) → c11
U61'(tt) → c12
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(activate(z0)), ACTIVATE(z0))
U72'(tt) → c14
U81'(tt) → c15
ISLIST(z0) → c16(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
ISLIST(n__nil) → c17
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(z0) → c19(U31'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNEPAL(z0) → c22(U61'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
ISNEPAL(n____(z0, __(z1, z0))) → c23(U71'(isQid(activate(z0)), activate(z1)), ISQID(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
ISPAL(n__nil) → c25
ISQID(n__a) → c26
ISQID(n__e) → c27
ISQID(n__i) → c28
ISQID(n__o) → c29
ISQID(n__u) → c30
NIL → c31
A → c32
E → c33
I → c34
O → c35
U → c36
ACTIVATE(n__nil) → c37(NIL)
ACTIVATE(n____(z0, z1)) → c38(__'(z0, z1))
ACTIVATE(n__a) → c39(A)
ACTIVATE(n__e) → c40(E)
ACTIVATE(n__i) → c41(I)
ACTIVATE(n__o) → c42(O)
ACTIVATE(n__u) → c43(U)
ACTIVATE(z0) → c44

S tuples:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
__'(z0, nil) → c1
__'(nil, z0) → c2
__'(z0, z1) → c3
U11'(tt) → c4
U21'(tt, z0) → c5(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U22'(tt) → c6
U31'(tt) → c7
U41'(tt, z0) → c8(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U42'(tt) → c9
U51'(tt, z0) → c10(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U52'(tt) → c11
U61'(tt) → c12
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(activate(z0)), ACTIVATE(z0))
U72'(tt) → c14
U81'(tt) → c15
ISLIST(z0) → c16(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
ISLIST(n__nil) → c17
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(z0) → c19(U31'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNEPAL(z0) → c22(U61'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
ISNEPAL(n____(z0, __(z1, z0))) → c23(U71'(isQid(activate(z0)), activate(z1)), ISQID(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
ISPAL(n__nil) → c25
ISQID(n__a) → c26
ISQID(n__e) → c27
ISQID(n__i) → c28
ISQID(n__o) → c29
ISQID(n__u) → c30
NIL → c31
A → c32
E → c33
I → c34
O → c35
U → c36
ACTIVATE(n__nil) → c37(NIL)
ACTIVATE(n____(z0, z1)) → c38(__'(z0, z1))
ACTIVATE(n__a) → c39(A)
ACTIVATE(n__e) → c40(E)
ACTIVATE(n__i) → c41(I)
ACTIVATE(n__o) → c42(O)
ACTIVATE(n__u) → c43(U)
ACTIVATE(z0) → c44

K tuples:none
Defined Rule Symbols:

__, U11, U21, U22, U31, U41, U42, U51, U52, U61, U71, U72, U81, isList, isNeList, isNePal, isPal, isQid, nil, a, e, i, o, u, activate

Defined Pair Symbols:

__', U11', U21', U22', U31', U41', U42', U51', U52', U61', U71', U72', U81', ISLIST, ISNELIST, ISNEPAL, ISPAL, ISQID, NIL, A, E, I, O, U, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44

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

Removed 38 trailing nodes:

ACTIVATE(z0) → c44
ISQID(n__o) → c29
ISLIST(n__nil) → c17
ISQID(n__i) → c28
U81'(tt) → c15
O → c35
ISQID(n__u) → c30
U42'(tt) → c9
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(activate(z0)), ACTIVATE(z0))
ISNELIST(z0) → c19(U31'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
U22'(tt) → c6
__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ACTIVATE(n__i) → c41(I)
ACTIVATE(n__u) → c43(U)
ACTIVATE(n__e) → c40(E)
ISNEPAL(z0) → c22(U61'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
__'(z0, z1) → c3
U11'(tt) → c4
U52'(tt) → c11
U61'(tt) → c12
U31'(tt) → c7
ACTIVATE(n__nil) → c37(NIL)
ISPAL(n__nil) → c25
ACTIVATE(n____(z0, z1)) → c38(__'(z0, z1))
U72'(tt) → c14
ACTIVATE(n__a) → c39(A)
__'(z0, nil) → c1
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
U → c36
A → c32
ISNEPAL(n____(z0, __(z1, z0))) → c23(U71'(isQid(activate(z0)), activate(z1)), ISQID(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
NIL → c31
ISQID(n__a) → c26
I → c34
__'(nil, z0) → c2
ACTIVATE(n__o) → c42(O)
E → c33
ISQID(n__e) → c27

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
__(z0, z1) → n____(z0, z1)
U11(tt) → tt
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U31(tt) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt
U61(tt) → tt
U71(tt, z0) → U72(isPal(activate(z0)))
U72(tt) → tt
U81(tt) → tt
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
isNePal(z0) → U61(isQid(activate(z0)))
isNePal(n____(z0, __(z1, z0))) → U71(isQid(activate(z0)), activate(z1))
isPal(z0) → U81(isNePal(activate(z0)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0

Tuples:

U21'(tt, z0) → c5(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c8(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c10(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
ISLIST(z0) → c16(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

U21'(tt, z0) → c5(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c8(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c10(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
ISLIST(z0) → c16(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

__, U11, U21, U22, U31, U41, U42, U51, U52, U61, U71, U72, U81, isList, isNeList, isNePal, isPal, isQid, nil, a, e, i, o, u, activate

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 14 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
__(z0, z1) → n____(z0, z1)
U11(tt) → tt
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U31(tt) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt
U61(tt) → tt
U71(tt, z0) → U72(isPal(activate(z0)))
U72(tt) → tt
U81(tt) → tt
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
isNePal(z0) → U61(isQid(activate(z0)))
isNePal(n____(z0, __(z1, z0))) → U71(isQid(activate(z0)), activate(z1))
isPal(z0) → U81(isNePal(activate(z0)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0

Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

S tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

K tuples:none
Defined Rule Symbols:

__, U11, U21, U22, U31, U41, U42, U51, U52, U61, U71, U72, U81, isList, isNeList, isNePal, isPal, isQid, nil, a, e, i, o, u, activate

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

__(__(z0, z1), z2) → __(z0, __(z1, z2))
__(z0, nil) → z0
__(nil, z0) → z0
U61(tt) → tt
U71(tt, z0) → U72(isPal(activate(z0)))
U72(tt) → tt
U81(tt) → tt
isNePal(z0) → U61(isQid(activate(z0)))
isNePal(n____(z0, __(z1, z0))) → U71(isQid(activate(z0)), activate(z1))
isPal(z0) → U81(isNePal(activate(z0)))
isPal(n__nil) → tt

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0
nil → n__nil
__(z0, z1) → n____(z0, z1)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
U11(tt) → tt
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
U31(tt) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt

Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

S tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

K tuples:none
Defined Rule Symbols:

activate, nil, __, a, e, i, o, u, isList, U11, isNeList, U31, isQid, U41, U21, U22, U42, U51, U52

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(tt, z0) → c5(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

We considered the (Usable) Rules:

nil → n__nil
activate(z0) → z0
u → n__u
o → n__o
activate(n__o) → o
i → n__i
__(z0, z1) → n____(z0, z1)
e → n__e
activate(n__i) → i
a → n__a
activate(n__nil) → nil
activate(n__u) → u
activate(n__a) → a
activate(n__e) → e
activate(n____(z0, z1)) → __(z0, z1)

And the Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ISLIST(x₁)) = [3] + [4]x₁   
POL(ISNELIST(x₁)) = [4]x₁   
POL(U11(x₁)) = [3] + [3]x₁   
POL(U21(x₁, x₂)) = [5] + [5]x₁   
POL(U21'(x₁, x₂)) = [4] + [4]x₂   
POL(U22(x₁)) = [3] + [3]x₁   
POL(U31(x₁)) = [4]   
POL(U41(x₁, x₂)) = [3] + [2]x₁   
POL(U41'(x₁, x₂)) = [4]x₂   
POL(U42(x₁)) = [4] + [2]x₁   
POL(U51(x₁, x₂)) = [3] + [4]x₁   
POL(U51'(x₁, x₂)) = [3] + [4]x₂   
POL(U52(x₁)) = [1] + [3]x₁   
POL(__(x₁, x₂)) = [1] + x₁ + x₂   
POL(a) = 0   
POL(activate(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(c21(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(e) = 0   
POL(i) = 0   
POL(isList(x₁)) = 0   
POL(isNeList(x₁)) = [4]   
POL(isQid(x₁)) = 0   
POL(n____(x₁, x₂)) = [1] + x₁ + x₂   
POL(n__a) = 0   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__nil) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0
nil → n__nil
__(z0, z1) → n____(z0, z1)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
U11(tt) → tt
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
U31(tt) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt

Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

S tuples:

U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

K tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

Defined Rule Symbols:

activate, nil, __, a, e, i, o, u, isList, U11, isNeList, U31, isQid, U41, U21, U22, U42, U51, U52

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0
nil → n__nil
__(z0, z1) → n____(z0, z1)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
U11(tt) → tt
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
U31(tt) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt

Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

S tuples:

ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

K tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))

Defined Rule Symbols:

activate, nil, __, a, e, i, o, u, isList, U11, isNeList, U31, isQid, U41, U21, U22, U42, U51, U52

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(13) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

We considered the (Usable) Rules:

nil → n__nil
activate(z0) → z0
u → n__u
o → n__o
activate(n__o) → o
i → n__i
__(z0, z1) → n____(z0, z1)
e → n__e
activate(n__i) → i
a → n__a
activate(n__nil) → nil
activate(n__u) → u
activate(n__a) → a
activate(n__e) → e
activate(n____(z0, z1)) → __(z0, z1)

And the Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ISLIST(x₁)) = [2] + [2]x₁   
POL(ISNELIST(x₁)) = [2]x₁   
POL(U11(x₁)) = [3] + [3]x₁   
POL(U21(x₁, x₂)) = [5]   
POL(U21'(x₁, x₂)) = [4] + [2]x₂   
POL(U22(x₁)) = [5]   
POL(U31(x₁)) = [3]   
POL(U41(x₁, x₂)) = [3] + [5]x₁   
POL(U41'(x₁, x₂)) = [2] + [2]x₂   
POL(U42(x₁)) = [1] + [3]x₁   
POL(U51(x₁, x₂)) = [5] + [4]x₁   
POL(U51'(x₁, x₂)) = [4] + [2]x₂   
POL(U52(x₁)) = [2]   
POL(__(x₁, x₂)) = [5] + x₁ + x₂   
POL(a) = [3]   
POL(activate(x₁)) = [1] + x₁   
POL(c10(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(c21(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(e) = 0   
POL(i) = [1]   
POL(isList(x₁)) = 0   
POL(isNeList(x₁)) = 0   
POL(isQid(x₁)) = 0   
POL(n____(x₁, x₂)) = [5] + x₁ + x₂   
POL(n__a) = [3]   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__nil) = 0   
POL(n__o) = [3]   
POL(n__u) = [3]   
POL(nil) = [1]   
POL(o) = [3]   
POL(tt) = 0   
POL(u) = [4]   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__nil) → nil
activate(n____(z0, z1)) → __(z0, z1)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(z0) → z0
nil → n__nil
__(z0, z1) → n____(z0, z1)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
isList(z0) → U11(isNeList(activate(z0)))
isList(n__nil) → tt
isList(n____(z0, z1)) → U21(isList(activate(z0)), activate(z1))
U11(tt) → tt
isNeList(z0) → U31(isQid(activate(z0)))
isNeList(n____(z0, z1)) → U41(isList(activate(z0)), activate(z1))
isNeList(n____(z0, z1)) → U51(isNeList(activate(z0)), activate(z1))
U31(tt) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
U41(tt, z0) → U42(isNeList(activate(z0)))
U21(tt, z0) → U22(isList(activate(z0)))
U22(tt) → tt
U42(tt) → tt
U51(tt, z0) → U52(isList(activate(z0)))
U52(tt) → tt

Tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

S tuples:none
K tuples:

U21'(tt, z0) → c5(ISLIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
U51'(tt, z0) → c10(ISLIST(activate(z0)))
ISLIST(n____(z0, z1)) → c18(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

Defined Rule Symbols:

activate, nil, __, a, e, i, o, u, isList, U11, isNeList, U31, isQid, U41, U21, U22, U42, U51, U52

Defined Pair Symbols:

U21', U41', U51', ISLIST, ISNELIST

Compound Symbols:

c5, c8, c10, c16, c18, c20, c21

(15) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))