Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

__#( __( X , Y ) , Z ) → __#( X , __( Y , Z ) )

__#( __( X , Y ) , Z ) → __#( Y , Z )

U21#( tt , V2 ) → U22#( isList( activate( V2 ) ) )

U21#( tt , V2 ) → isList#( activate( V2 ) )

U21#( tt , V2 ) → activate#( V2 )

U41#( tt , V2 ) → U42#( isNeList( activate( V2 ) ) )

U41#( tt , V2 ) → isNeList#( activate( V2 ) )

U41#( tt , V2 ) → activate#( V2 )

U51#( tt , V2 ) → U52#( isList( activate( V2 ) ) )

U51#( tt , V2 ) → isList#( activate( V2 ) )

U51#( tt , V2 ) → activate#( V2 )

U71#( tt , P ) → U72#( isPal( activate( P ) ) )

U71#( tt , P ) → isPal#( activate( P ) )

U71#( tt , P ) → activate#( P )

isList#( V ) → U11#( isNeList( activate( V ) ) )

isList#( V ) → isNeList#( activate( V ) )

isList#( V ) → activate#( V )

isList#( n____( V1 , V2 ) ) → U21#( isList( activate( V1 ) ) , activate( V2 ) )

isList#( n____( V1 , V2 ) ) → isList#( activate( V1 ) )

isList#( n____( V1 , V2 ) ) → activate#( V1 )

isList#( n____( V1 , V2 ) ) → activate#( V2 )

isNeList#( V ) → U31#( isQid( activate( V ) ) )

isNeList#( V ) → isQid#( activate( V ) )

isNeList#( V ) → activate#( V )

isNeList#( n____( V1 , V2 ) ) → U41#( isList( activate( V1 ) ) , activate( V2 ) )

isNeList#( n____( V1 , V2 ) ) → isList#( activate( V1 ) )

isNeList#( n____( V1 , V2 ) ) → activate#( V1 )

isNeList#( n____( V1 , V2 ) ) → activate#( V2 )

isNeList#( n____( V1 , V2 ) ) → U51#( isNeList( activate( V1 ) ) , activate( V2 ) )

isNeList#( n____( V1 , V2 ) ) → isNeList#( activate( V1 ) )

isNeList#( n____( V1 , V2 ) ) → activate#( V1 )

isNeList#( n____( V1 , V2 ) ) → activate#( V2 )

isNePal#( V ) → U61#( isQid( activate( V ) ) )

isNePal#( V ) → isQid#( activate( V ) )

isNePal#( V ) → activate#( V )

isNePal#( n____( I , n____( P , I ) ) ) → U71#( isQid( activate( I ) ) , activate( P ) )

isNePal#( n____( I , n____( P , I ) ) ) → isQid#( activate( I ) )

isNePal#( n____( I , n____( P , I ) ) ) → activate#( I )

isNePal#( n____( I , n____( P , I ) ) ) → activate#( P )

isPal#( V ) → U81#( isNePal( activate( V ) ) )

isPal#( V ) → isNePal#( activate( V ) )

isPal#( V ) → activate#( V )

activate#( n__nil ) → nil#

activate#( n____( X1 , X2 ) ) → __#( activate( X1 ) , activate( X2 ) )

activate#( n____( X1 , X2 ) ) → activate#( X1 )

activate#( n____( X1 , X2 ) ) → activate#( X2 )

activate#( n__a ) → a#

activate#( n__e ) → e#

activate#( n__i ) → i#

activate#( n__o ) → o#

activate#( n__u ) → u#

1.1: dependency graph processor

The dependency pairs are split into 4 component(s).

The 1st component contains the pair(s)

U21#( tt , V2 ) → isList#( activate( V2 ) )

isList#( V ) → isNeList#( activate( V ) )

isNeList#( n____( V1 , V2 ) ) → U41#( isList( activate( V1 ) ) , activate( V2 ) )

U41#( tt , V2 ) → isNeList#( activate( V2 ) )

isNeList#( n____( V1 , V2 ) ) → isList#( activate( V1 ) )

isList#( n____( V1 , V2 ) ) → U21#( isList( activate( V1 ) ) , activate( V2 ) )

isList#( n____( V1 , V2 ) ) → isList#( activate( V1 ) )

isNeList#( n____( V1 , V2 ) ) → U51#( isNeList( activate( V1 ) ) , activate( V2 ) )

U51#( tt , V2 ) → isList#( activate( V2 ) )

isNeList#( n____( V1 , V2 ) ) → isNeList#( activate( V1 ) )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U81 (x₁) ] = 1

[isList# (x₁) ] = 3 x₁ + 1

[__ (x₁, x₂) ] = x₁ + x₂ + 2

[a] = 1

[isNePal (x₁) ] = 2

[U22 (x₁) ] = 3

[i] = 2

[activate (x₁) ] = x₁

[U72 (x₁) ] = 1

[U21 (x₁, x₂) ] = x₁ + 2

[u] = 1

[U71 (x₁, x₂) ] = 1

[n__o] = 2

[isNeList (x₁) ] = x₁

[U52 (x₁) ] = 1

[isPal (x₁) ] = x₁ + 3

[U11 (x₁) ] = x₁

[n____ (x₁, x₂) ] = x₁ + x₂ + 2

[isList (x₁) ] = x₁

[U21# (x₁, x₂) ] = 3 x₁ + 1

[n__e] = 1

[n__nil] = 1

[U41 (x₁, x₂) ] = x₁ + x₂ + 2

[U61 (x₁) ] = 1

[U41# (x₁, x₂) ] = x₁ + 3 x₂

[U31 (x₁) ] = x₁

[U51 (x₁, x₂) ] = 2

[nil] = 1

[tt] = 1

[o] = 2

[e] = 1

[n__a] = 1

[isNeList# (x₁) ] = 3 x₁ + 1

[n__i] = 2

[U51# (x₁, x₂) ] = x₁ + 3 x₂

[isQid (x₁) ] = x₁

[n__u] = 1

[U42 (x₁) ] = x₁ + 3

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

U21#( tt , V2 ) → isList#( activate( V2 ) )

isList#( V ) → isNeList#( activate( V ) )

U41#( tt , V2 ) → isNeList#( activate( V2 ) )

U51#( tt , V2 ) → isList#( activate( V2 ) )

1.1.1.1: dependency graph processor

The dependency pairs are split into 0 component(s).

The 2nd component contains the pair(s)

U71#( tt , P ) → isPal#( activate( P ) )

isPal#( V ) → isNePal#( activate( V ) )

isNePal#( n____( I , n____( P , I ) ) ) → U71#( isQid( activate( I ) ) , activate( P ) )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U81 (x₁) ] = x₁

[__ (x₁, x₂) ] = 2 x₁ + x₂

[a] = 2

[isPal# (x₁) ] = 2 x₁

[isNePal (x₁) ] = x₁

[U22 (x₁) ] = 1

[i] = 2

[activate (x₁) ] = x₁

[U72 (x₁) ] = 2

[U21 (x₁, x₂) ] = 1

[u] = 2

[U71 (x₁, x₂) ] = 3 x₁ + 2 x₂

[n__o] = 1

[isNeList (x₁) ] = x₁

[U52 (x₁) ] = 1

[isPal (x₁) ] = 2 x₁

[U11 (x₁) ] = 1

[n____ (x₁, x₂) ] = 2 x₁ + x₂

[isList (x₁) ] = 1

[n__e] = 1

[n__nil] = 2

[U41 (x₁, x₂) ] = x₁

[U61 (x₁) ] = x₁

[U31 (x₁) ] = x₁

[U51 (x₁, x₂) ] = x₁ + x₂

[nil] = 2

[tt] = 1

[o] = 1

[e] = 1

[n__a] = 2

[n__i] = 2

[isNePal# (x₁) ] = 2 x₁

[isQid (x₁) ] = x₁

[n__u] = 2

[U71# (x₁, x₂) ] = 2 x₁ + 2 x₂

[U42 (x₁) ] = x₁

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

isPal#( V ) → isNePal#( activate( V ) )

isNePal#( n____( I , n____( P , I ) ) ) → U71#( isQid( activate( I ) ) , activate( P ) )

1.1.2.1: dependency graph processor

The dependency pairs are split into 0 component(s).

The 3rd component contains the pair(s)

activate#( n____( X1 , X2 ) ) → activate#( X2 )

activate#( n____( X1 , X2 ) ) → activate#( X1 )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U81 (x₁) ] = 0

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[a] = 2

[isNePal (x₁) ] = 2 x₁

[U22 (x₁) ] = 0

[i] = 3

[activate (x₁) ] = 2 x₁

[U72 (x₁) ] = 3

[U21 (x₁, x₂) ] = 0

[u] = 0

[U71 (x₁, x₂) ] = x₁ + 3

[n__o] = 1

[isNeList (x₁) ] = 2 x₁

[U52 (x₁) ] = 2

[isPal (x₁) ] = 1

[U11 (x₁) ] = 0

[n____ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[isList (x₁) ] = 0

[n__e] = 0

[n__nil] = 1

[U41 (x₁, x₂) ] = 0

[U61 (x₁) ] = 0

[U31 (x₁) ] = 0

[U51 (x₁, x₂) ] = x₁ + 3

[nil] = 1

[tt] = 0

[o] = 1

[e] = 0

[n__a] = 1

[n__i] = 2

[activate# (x₁) ] = 2 x₁

[isQid (x₁) ] = 0

[n__u] = 0

[U42 (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.3.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

__#( __( X , Y ) , Z ) → __#( Y , Z )

__#( __( X , Y ) , Z ) → __#( X , __( Y , Z ) )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U81 (x₁) ] = 0

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[a] = 3

[isNePal (x₁) ] = 0

[U22 (x₁) ] = 1

[__# (x₁, x₂) ] = 2 x₁ + x₂

[i] = 3

[activate (x₁) ] = 2 x₁

[U72 (x₁) ] = 0

[U21 (x₁, x₂) ] = x₁ + 1

[u] = 0

[U71 (x₁, x₂) ] = 0

[n__o] = 2

[isNeList (x₁) ] = 0

[U52 (x₁) ] = 0

[isPal (x₁) ] = 3 x₁

[U11 (x₁) ] = 0

[n____ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[isList (x₁) ] = x₁

[n__e] = 2

[n__nil] = 3

[U41 (x₁, x₂) ] = 0

[U61 (x₁) ] = 0

[U31 (x₁) ] = 0

[U51 (x₁, x₂) ] = 0

[nil] = 3

[tt] = 0

[o] = 3

[e] = 3

[n__a] = 2

[n__i] = 2

[isQid (x₁) ] = 0

[n__u] = 0

[U42 (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.4.1: P is empty

All dependency pairs have been removed.

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

[U81 (x₁) ]	=	1
[isList# (x₁) ]	=	3 x₁ + 1
[__ (x₁, x₂) ]	=	x₁ + x₂ + 2
[a]	=	1
[isNePal (x₁) ]	=	2
[U22 (x₁) ]	=	3
[i]	=	2
[activate (x₁) ]	=	x₁
[U72 (x₁) ]	=	1
[U21 (x₁, x₂) ]	=	x₁ + 2
[u]	=	1
[U71 (x₁, x₂) ]	=	1
[n__o]	=	2
[isNeList (x₁) ]	=	x₁
[U52 (x₁) ]	=	1
[isPal (x₁) ]	=	x₁ + 3
[U11 (x₁) ]	=	x₁
[n____ (x₁, x₂) ]	=	x₁ + x₂ + 2
[isList (x₁) ]	=	x₁
[U21# (x₁, x₂) ]	=	3 x₁ + 1
[n__e]	=	1
[n__nil]	=	1
[U41 (x₁, x₂) ]	=	x₁ + x₂ + 2
[U61 (x₁) ]	=	1
[U41# (x₁, x₂) ]	=	x₁ + 3 x₂
[U31 (x₁) ]	=	x₁
[U51 (x₁, x₂) ]	=	2
[nil]	=	1
[tt]	=	1
[o]	=	2
[e]	=	1
[n__a]	=	1
[isNeList# (x₁) ]	=	3 x₁ + 1
[n__i]	=	2
[U51# (x₁, x₂) ]	=	x₁ + 3 x₂
[isQid (x₁) ]	=	x₁
[n__u]	=	1
[U42 (x₁) ]	=	x₁ + 3
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U81 (x₁) ]	=	0
[__ (x₁, x₂) ]	=	2 x₁ + x₂ + 2
[a]	=	2
[isNePal (x₁) ]	=	2 x₁
[U22 (x₁) ]	=	0
[i]	=	3
[activate (x₁) ]	=	2 x₁
[U72 (x₁) ]	=	3
[U21 (x₁, x₂) ]	=	0
[u]	=	0
[U71 (x₁, x₂) ]	=	x₁ + 3
[n__o]	=	1
[isNeList (x₁) ]	=	2 x₁
[U52 (x₁) ]	=	2
[isPal (x₁) ]	=	1
[U11 (x₁) ]	=	0
[n____ (x₁, x₂) ]	=	2 x₁ + x₂ + 2
[isList (x₁) ]	=	0
[n__e]	=	0
[n__nil]	=	1
[U41 (x₁, x₂) ]	=	0
[U61 (x₁) ]	=	0
[U31 (x₁) ]	=	0
[U51 (x₁, x₂) ]	=	x₁ + 3
[nil]	=	1
[tt]	=	0
[o]	=	1
[e]	=	0
[n__a]	=	1
[n__i]	=	2
[activate# (x₁) ]	=	2 x₁
[isQid (x₁) ]	=	0
[n__u]	=	0
[U42 (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n