Runtime Complexity (innermost) proof of /tmp/tmpEe9aBD/PALINDROME_nokinds-noand

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

0 CpxTRS

↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 43 ms)

↳2 CpxTRS

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

↳4 CdtProblem

↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

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

↳8 CdtProblem

↳9 CdtUsableRulesProof (⇔, 0 ms)

↳10 CdtProblem

↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 276 ms)

↳12 CdtProblem

↳13 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 142 ms)

↳16 CdtProblem

↳17 CdtKnowledgeProof (⇔, 0 ms)

↳18 BOUNDS(1, 1)

(0) 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:

__(__(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) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of U31: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of isQid: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of activate: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U22: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 0th argument of isList: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U41: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 1th argument of U41: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U42: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 0th argument of isNeList: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U51: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 1th argument of U51: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U72: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of isPal: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U52: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 0th argument of U11: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 0th argument of U21: a, U31, U52, U11, U41, isQid, isPal, __, U61, e, isList, U42, U72, isNePal, U81, U21, activate, u, nil, U71, U22, o, isNeList, i, U51
The following defined symbols can occur below the 1th argument of U21: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U61: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U81: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of isNePal: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 0th argument of U71: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e
The following defined symbols can occur below the 1th argument of U71: a, U72, isNePal, U81, activate, u, nil, isQid, U71, o, i, isPal, __, U61, e

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X

(2) 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:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

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

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

Removed 35 trailing nodes:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISLIST(z0) → c25(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c27(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c40(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))

S tuples:

U21'(tt, z0) → c17(U22'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISLIST(z0) → c25(U11'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c27(U42'(isNeList(activate(z0))), ISNELIST(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c40(U52'(isList(activate(z0))), ISLIST(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

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

Removed 14 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

S tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

S tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))

We considered the (Usable) Rules:

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

And the Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

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

POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = x₁
POL(U11(x₁)) = [1]
POL(U21(x₁, x₂)) = 0
POL(U21'(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U41'(x₁, x₂)) = [1] + x₂
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U51'(x₁, x₂)) = x₂
POL(U52(x₁)) = 0
POL(__(x₁, x₂)) = [1] + x₁ + x₂
POL(a) = 0
POL(activate(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c25(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = [1]
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) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

S tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

K tuples:

ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

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

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

U21'(tt, z0) → c17(ISLIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

S tuples:

ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))

K tuples:

ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U21'(tt, z0) → c17(ISLIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

We considered the (Usable) Rules:

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

And the Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

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

POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21'(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = [1]
POL(U41'(x₁, x₂)) = x₂
POL(U42(x₁)) = [1]
POL(U51(x₁, x₂)) = 0
POL(U51'(x₁, x₂)) = [1] + x₂
POL(U52(x₁)) = 0
POL(__(x₁, x₂)) = [1] + x₁ + x₂
POL(a) = 0
POL(activate(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c25(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = [1] + x₁
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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U21'(tt, z0) → c17(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISLIST(z0) → c25(ISNELIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))

S tuples:

ISLIST(z0) → c25(ISNELIST(activate(z0)))

K tuples:

ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))
ISLIST(n____(z0, z1)) → c23(U21'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
U41'(tt, z0) → c27(ISNELIST(activate(z0)))
U21'(tt, z0) → c17(ISLIST(activate(z0)))
U51'(tt, z0) → c40(ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c17, c18, c20, c23, c25, c27, c40

(17) CdtKnowledgeProof (EQUIVALENT transformation)

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

ISLIST(z0) → c25(ISNELIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c18(U41'(isList(activate(z0)), activate(z1)), ISLIST(activate(z0)))
ISNELIST(n____(z0, z1)) → c20(U51'(isNeList(activate(z0)), activate(z1)), ISNELIST(activate(z0)))

Now S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtUsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(16) Obligation:

(17) CdtKnowledgeProof (EQUIVALENT transformation)

(18) BOUNDS(1, 1)