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 )

and#( tt , X ) → activate#( X )

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

isList#( V ) → activate#( V )

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

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

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

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

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

isNeList#( V ) → activate#( V )

isNeList#( n____( V1 , V2 ) ) → and#( isList( activate( V1 ) ) , n__isNeList( 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 ) ) → and#( isNeList( activate( V1 ) ) , n__isList( activate( V2 ) ) )

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

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

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

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

isNePal#( V ) → activate#( V )

isNePal#( n____( I , n____( P , I ) ) ) → and#( isQid( activate( I ) ) , n__isPal( 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 ) → 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__isList( X ) ) → isList#( X )

activate#( n__isNeList( X ) ) → isNeList#( X )

activate#( n__isPal( X ) ) → isPal#( X )

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 2 component(s).

The 1st component contains the pair(s)

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

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

activate#( n__isList( X ) ) → isList#( X )

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

isNeList#( V ) → activate#( V )

activate#( n__isNeList( X ) ) → isNeList#( X )

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

and#( tt , X ) → activate#( X )

activate#( n__isPal( X ) ) → isPal#( X )

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

isNePal#( V ) → activate#( V )

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

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

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

isPal#( V ) → activate#( V )

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

isList#( V ) → activate#( V )

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

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

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

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

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

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

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

[isList# (x₁) ] = x₁

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[a] = 0

[isNePal (x₁) ] = 2 x₁

[isPal# (x₁) ] = 2 x₁

[i] = 0

[activate (x₁) ] = x₁

[n__isList (x₁) ] = x₁

[and (x₁, x₂) ] = x₁ + x₂ + 1

[u] = 0

[n__o] = 0

[isNeList (x₁) ] = x₁

[isPal (x₁) ] = 2 x₁

[n____ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[isList (x₁) ] = x₁

[and# (x₁, x₂) ] = x₁

[n__e] = 1

[n__nil] = 1

[n__isPal (x₁) ] = 2 x₁

[nil] = 1

[tt] = 0

[o] = 0

[e] = 1

[n__a] = 0

[n__i] = 0

[isNeList# (x₁) ] = x₁

[activate# (x₁) ] = x₁

[isNePal# (x₁) ] = 2 x₁

[isQid (x₁) ] = 0

[n__isNeList (x₁) ] = x₁

[n__u] = 0

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

one remains with the following pair(s).

activate#( n__isList( X ) ) → isList#( X )

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

isNeList#( V ) → activate#( V )

activate#( n__isNeList( X ) ) → isNeList#( X )

and#( tt , X ) → activate#( X )

activate#( n__isPal( X ) ) → isPal#( X )

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

isNePal#( V ) → activate#( V )

isPal#( V ) → activate#( V )

isList#( V ) → activate#( V )

1.1.1.1: dependency graph processor

The dependency pairs are split into 1 component(s).

The 1st component contains the pair(s)

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

isNeList#( V ) → activate#( V )

activate#( n__isList( X ) ) → isList#( X )

isList#( V ) → activate#( V )

activate#( n__isNeList( X ) ) → isNeList#( X )

activate#( n__isPal( X ) ) → isPal#( X )

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

isNePal#( V ) → activate#( V )

isPal#( V ) → activate#( V )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[isList# (x₁) ] = x₁ + 2

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 3

[a] = 0

[isNePal (x₁) ] = 2 x₁

[isPal# (x₁) ] = 2 x₁ + 2

[i] = 0

[activate (x₁) ] = x₁

[n__isList (x₁) ] = x₁ + 1

[and (x₁, x₂) ] = x₁

[u] = 0

[n__o] = 0

[isNeList (x₁) ] = x₁ + 1

[isPal (x₁) ] = 2 x₁ + 3

[n____ (x₁, x₂) ] = 2 x₁ + x₂ + 3

[isList (x₁) ] = x₁ + 1

[n__e] = 0

[n__nil] = 0

[n__isPal (x₁) ] = 2 x₁ + 3

[nil] = 0

[tt] = 0

[o] = 0

[e] = 0

[n__a] = 0

[isNeList# (x₁) ] = x₁ + 1

[n__i] = 0

[activate# (x₁) ] = x₁ + 1

[isNePal# (x₁) ] = x₁ + 1

[n__u] = 0

[isQid (x₁) ] = x₁

[n__isNeList (x₁) ] = x₁ + 1

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

one remains with the following pair(s).

isNeList#( V ) → activate#( V )

activate#( n__isList( X ) ) → isList#( X )

isNePal#( V ) → activate#( V )

1.1.1.1.1.1: dependency graph processor

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

The 2nd component contains the pair(s)

__#( __( X , Y ) , Z ) → __#( Y , Z )

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

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a] = 0

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[isNePal (x₁) ] = 0

[__# (x₁, x₂) ] = 2 x₁ + x₂

[i] = 2

[activate (x₁) ] = 2 x₁

[n__isList (x₁) ] = 0

[and (x₁, x₂) ] = 3 x₁

[u] = 0

[n__o] = 0

[isNeList (x₁) ] = 0

[isPal (x₁) ] = 0

[n____ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[isList (x₁) ] = 0

[n__e] = 0

[n__nil] = 2

[n__isPal (x₁) ] = 0

[nil] = 2

[tt] = 0

[o] = 0

[e] = 0

[n__a] = 0

[n__i] = 1

[n__u] = 0

[n__isNeList (x₁) ] = 0

[isQid (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

1.1.2.1: P is empty

All dependency pairs have been removed.

__#( __( X , Y ) , Z )	→	__#( X , __( Y , Z ) )
__#( __( X , Y ) , Z )	→	__#( Y , Z )
and#( tt , X )	→	activate#( X )
isList#( V )	→	isNeList#( activate( V ) )
isList#( V )	→	activate#( V )
isList#( n____( V1 , V2 ) )	→	and#( isList( activate( V1 ) ) , n__isList( activate( V2 ) ) )
isList#( n____( V1 , V2 ) )	→	isList#( activate( V1 ) )
isList#( n____( V1 , V2 ) )	→	activate#( V1 )
isList#( n____( V1 , V2 ) )	→	activate#( V2 )
isNeList#( V )	→	isQid#( activate( V ) )
isNeList#( V )	→	activate#( V )
isNeList#( n____( V1 , V2 ) )	→	and#( isList( activate( V1 ) ) , n__isNeList( 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 ) )	→	and#( isNeList( activate( V1 ) ) , n__isList( activate( V2 ) ) )
isNeList#( n____( V1 , V2 ) )	→	isNeList#( activate( V1 ) )
isNeList#( n____( V1 , V2 ) )	→	activate#( V1 )
isNeList#( n____( V1 , V2 ) )	→	activate#( V2 )
isNePal#( V )	→	isQid#( activate( V ) )
isNePal#( V )	→	activate#( V )
isNePal#( n____( I , n____( P , I ) ) )	→	and#( isQid( activate( I ) ) , n__isPal( 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 )	→	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__isList( X ) )	→	isList#( X )
activate#( n__isNeList( X ) )	→	isNeList#( X )
activate#( n__isPal( X ) )	→	isPal#( X )
activate#( n__a )	→	a#
activate#( n__e )	→	e#
activate#( n__i )	→	i#
activate#( n__o )	→	o#
activate#( n__u )	→	u#

[isList# (x₁) ]	=	x₁
[__ (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[a]	=	0
[isNePal (x₁) ]	=	2 x₁
[isPal# (x₁) ]	=	2 x₁
[i]	=	0
[activate (x₁) ]	=	x₁
[n__isList (x₁) ]	=	x₁
[and (x₁, x₂) ]	=	x₁ + x₂ + 1
[u]	=	0
[n__o]	=	0
[isNeList (x₁) ]	=	x₁
[isPal (x₁) ]	=	2 x₁
[n____ (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[isList (x₁) ]	=	x₁
[and# (x₁, x₂) ]	=	x₁
[n__e]	=	1
[n__nil]	=	1
[n__isPal (x₁) ]	=	2 x₁
[nil]	=	1
[tt]	=	0
[o]	=	0
[e]	=	1
[n__a]	=	0
[n__i]	=	0
[isNeList# (x₁) ]	=	x₁
[activate# (x₁) ]	=	x₁
[isNePal# (x₁) ]	=	2 x₁
[isQid (x₁) ]	=	0
[n__isNeList (x₁) ]	=	x₁
[n__u]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[isList# (x₁) ]	=	x₁ + 2
[__ (x₁, x₂) ]	=	2 x₁ + x₂ + 3
[a]	=	0
[isNePal (x₁) ]	=	2 x₁
[isPal# (x₁) ]	=	2 x₁ + 2
[i]	=	0
[activate (x₁) ]	=	x₁
[n__isList (x₁) ]	=	x₁ + 1
[and (x₁, x₂) ]	=	x₁
[u]	=	0
[n__o]	=	0
[isNeList (x₁) ]	=	x₁ + 1
[isPal (x₁) ]	=	2 x₁ + 3
[n____ (x₁, x₂) ]	=	2 x₁ + x₂ + 3
[isList (x₁) ]	=	x₁ + 1
[n__e]	=	0
[n__nil]	=	0
[n__isPal (x₁) ]	=	2 x₁ + 3
[nil]	=	0
[tt]	=	0
[o]	=	0
[e]	=	0
[n__a]	=	0
[isNeList# (x₁) ]	=	x₁ + 1
[n__i]	=	0
[activate# (x₁) ]	=	x₁ + 1
[isNePal# (x₁) ]	=	x₁ + 1
[n__u]	=	0
[isQid (x₁) ]	=	x₁
[n__isNeList (x₁) ]	=	x₁ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n