Runtime Complexity (innermost) proof of /tmp/tmpKKSgHz/PALINDROME_nokinds-noand

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtUnreachableProof (⇔, 0 ms)

↳4 CdtProblem

↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 2 ms)

↳6 CdtProblem

↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1193 ms)

↳10 CdtProblem

↳11 CdtKnowledgeProof (⇔, 0 ms)

↳12 CdtProblem

↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1201 ms)

↳14 CdtProblem

↳15 SIsEmptyProof (⇔, 0 ms)

↳16 BOUNDS(O(1), O(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 CpxTRS 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))
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))
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(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(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))
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)

S tuples:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
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))
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(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(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))
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)

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', U71', ISLIST, ISNELIST, ISNEPAL, ISPAL, ACTIVATE

Compound Symbols:

c, c5, c8, c10, c13, c16, c18, c19, c20, c21, c22, c23, c24, c37, c38, c39, c40, c41, c42, c43

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

__'(__(z0, z1), z2) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ISNEPAL(n____(z0, __(z1, z0))) → c23(U71'(isQid(activate(z0)), activate(z1)), ISQID(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

(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))
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(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(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))
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
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)

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))
U71'(tt, z0) → c13(U72'(isPal(activate(z0))), ISPAL(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(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))
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
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)

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', U71', ISLIST, ISNELIST, ISNEPAL, ISPAL, ACTIVATE

Compound Symbols:

c5, c8, c10, c13, c16, c18, c19, c20, c21, c22, c24, c37, c38, c39, c40, c41, c42, c43

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

Removed 11 trailing nodes:

ACTIVATE(n__a) → c39(A)
ACTIVATE(n__i) → c41(I)
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))
ACTIVATE(n__nil) → c37(NIL)
ACTIVATE(n____(z0, z1)) → c38(__'(z0, z1))
ACTIVATE(n__e) → c40(E)
ACTIVATE(n__u) → c43(U)
ISPAL(z0) → c24(U81'(isNePal(activate(z0))), ISNEPAL(activate(z0)), ACTIVATE(z0))
ISNEPAL(z0) → c22(U61'(isQid(activate(z0))), ISQID(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__o) → c42(O)

(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(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

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

Removed 14 trailing tuple parts

(8) 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

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

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

U51'(tt, z0) → c10(ISLIST(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)))

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

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)) = [4]x₁
POL(ISNELIST(x₁)) = [4]x₁
POL(U11(x₁)) = [3]
POL(U21(x₁, x₂)) = [2] + [4]x₁ + [2]x₂
POL(U21'(x₁, x₂)) = [4]x₂
POL(U22(x₁)) = [3] + [3]x₁
POL(U31(x₁)) = [2]
POL(U41(x₁, x₂)) = [1] + [4]x₁ + [4]x₂
POL(U41'(x₁, x₂)) = [4]x₂
POL(U42(x₁)) = [5] + [2]x₁
POL(U51(x₁, x₂)) = [3] + [3]x₁ + x₂
POL(U51'(x₁, x₂)) = [4] + [4]x₂
POL(U52(x₁)) = [3]
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)) = 0
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) = [2]
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = [2]

(10) 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)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

K tuples:

U51'(tt, z0) → c10(ISLIST(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)))

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

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(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)) → c20(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

(12) 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:

ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

K tuples:

U51'(tt, z0) → c10(ISLIST(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)))
U21'(tt, z0) → c5(ISLIST(activate(z0)))
U41'(tt, z0) → c8(ISNELIST(activate(z0)))
ISLIST(z0) → c16(ISNELIST(activate(z0)))

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

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

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

ISNELIST(n____(z0, z1)) → c21(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

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

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)) = [1] + [4]x₁
POL(ISNELIST(x₁)) = [1] + [4]x₁
POL(U11(x₁)) = 0
POL(U21(x₁, x₂)) = [3] + x₂
POL(U21'(x₁, x₂)) = [1] + [4]x₂
POL(U22(x₁)) = 0
POL(U31(x₁)) = [2] + [2]x₁
POL(U41(x₁, x₂)) = [3] + x₂
POL(U41'(x₁, x₂)) = [4] + [4]x₂
POL(U42(x₁)) = [2] + [3]x₁
POL(U51(x₁, x₂)) = [3] + [3]x₁ + [4]x₂
POL(U51'(x₁, x₂)) = [2] + [4]x₂
POL(U52(x₁)) = [2]
POL(__(x₁, x₂)) = [4] + x₁ + x₂
POL(a) = [4]
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) = [4]
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = [4]x₁
POL(isQid(x₁)) = [3] + [3]x₁
POL(n____(x₁, x₂)) = [4] + x₁ + x₂
POL(n__a) = [4]
POL(n__e) = 0
POL(n__i) = [4]
POL(n__nil) = 0
POL(n__o) = [1]
POL(n__u) = 0
POL(nil) = 0
POL(o) = [1]
POL(tt) = 0
POL(u) = 0

(14) 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:none
K tuples:

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

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

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtUnreachableProof (EQUIVALENT transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtKnowledgeProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

(15) SIsEmptyProof (EQUIVALENT transformation)

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