Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

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

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

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

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

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

a____#( X , nil ) → mark#( X )

a____#( nil , X ) → mark#( X )

a__and#( tt , X ) → mark#( X )

a__isList#( V ) → a__isNeList#( V )

a__isList#( __( V1 , V2 ) ) → a__and#( a__isList( V1 ) , isList( V2 ) )

a__isList#( __( V1 , V2 ) ) → a__isList#( V1 )

a__isNeList#( V ) → a__isQid#( V )

a__isNeList#( __( V1 , V2 ) ) → a__and#( a__isList( V1 ) , isNeList( V2 ) )

a__isNeList#( __( V1 , V2 ) ) → a__isList#( V1 )

a__isNeList#( __( V1 , V2 ) ) → a__and#( a__isNeList( V1 ) , isList( V2 ) )

a__isNeList#( __( V1 , V2 ) ) → a__isNeList#( V1 )

a__isNePal#( V ) → a__isQid#( V )

a__isNePal#( __( I , __( P , I ) ) ) → a__and#( a__isQid( I ) , isPal( P ) )

a__isNePal#( __( I , __( P , I ) ) ) → a__isQid#( I )

a__isPal#( V ) → a__isNePal#( V )

mark#( __( X1 , X2 ) ) → a____#( mark( X1 ) , mark( X2 ) )

mark#( __( X1 , X2 ) ) → mark#( X1 )

mark#( __( X1 , X2 ) ) → mark#( X2 )

mark#( and( X1 , X2 ) ) → a__and#( mark( X1 ) , X2 )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isList( X ) ) → a__isList#( X )

mark#( isNeList( X ) ) → a__isNeList#( X )

mark#( isQid( X ) ) → a__isQid#( X )

mark#( isNePal( X ) ) → a__isNePal#( X )

mark#( isPal( X ) ) → a__isPal#( X )

1.1: dependency graph processor

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

The 1st component contains the pair(s)

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

mark#( __( X1 , X2 ) ) → a____#( mark( X1 ) , mark( X2 ) )

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

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

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

mark#( __( X1 , X2 ) ) → mark#( X1 )

mark#( __( X1 , X2 ) ) → mark#( X2 )

mark#( and( X1 , X2 ) ) → a__and#( mark( X1 ) , X2 )

a__and#( tt , X ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isList( X ) ) → a__isList#( X )

a__isList#( V ) → a__isNeList#( V )

a__isNeList#( __( V1 , V2 ) ) → a__and#( a__isList( V1 ) , isNeList( V2 ) )

a__isNeList#( __( V1 , V2 ) ) → a__isList#( V1 )

a__isList#( __( V1 , V2 ) ) → a__and#( a__isList( V1 ) , isList( V2 ) )

a__isList#( __( V1 , V2 ) ) → a__isList#( V1 )

a__isNeList#( __( V1 , V2 ) ) → a__and#( a__isNeList( V1 ) , isList( V2 ) )

a__isNeList#( __( V1 , V2 ) ) → a__isNeList#( V1 )

mark#( isNeList( X ) ) → a__isNeList#( X )

mark#( isNePal( X ) ) → a__isNePal#( X )

a__isNePal#( __( I , __( P , I ) ) ) → a__and#( a__isQid( I ) , isPal( P ) )

mark#( isPal( X ) ) → a__isPal#( X )

a__isPal#( V ) → a__isNePal#( V )

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

a____#( X , nil ) → mark#( X )

a____#( nil , X ) → mark#( X )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a] = 0

[__ (x₁, x₂) ] = x₁ + x₂ + 2

[a__isQid (x₁) ] = 0

[isNePal (x₁) ] = 0

[a____ (x₁, x₂) ] = x₁ + x₂ + 2

[i] = 0

[a__isPal# (x₁) ] = 0

[and (x₁, x₂) ] = x₁ + x₂

[u] = 1

[a__isNeList# (x₁) ] = 2 x₁

[isNeList (x₁) ] = 2 x₁

[a__isPal (x₁) ] = 0

[isPal (x₁) ] = 0

[isList (x₁) ] = 2 x₁

[a____# (x₁, x₂) ] = x₁ + x₂

[mark (x₁) ] = x₁

[a__isList# (x₁) ] = 2 x₁

[a__and# (x₁, x₂) ] = x₁

[mark# (x₁) ] = x₁

[nil] = 0

[a__isNeList (x₁) ] = 2 x₁

[tt] = 0

[o] = 0

[a__and (x₁, x₂) ] = x₁ + x₂

[e] = 0

[a__isNePal# (x₁) ] = 0

[a__isNePal (x₁) ] = 0

[isQid (x₁) ] = 0

[a__isList (x₁) ] = 2 x₁

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

one remains with the following pair(s).

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

mark#( and( X1 , X2 ) ) → a__and#( mark( X1 ) , X2 )

a__and#( tt , X ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isList( X ) ) → a__isList#( X )

a__isList#( V ) → a__isNeList#( V )

mark#( isNeList( X ) ) → a__isNeList#( X )

mark#( isNePal( X ) ) → a__isNePal#( X )

a__isNePal#( __( I , __( P , I ) ) ) → a__and#( a__isQid( I ) , isPal( P ) )

mark#( isPal( X ) ) → a__isPal#( X )

a__isPal#( V ) → a__isNePal#( V )

a____#( X , nil ) → mark#( X )

a____#( nil , X ) → mark#( X )

1.1.1.1: dependency graph processor

The dependency pairs are split into 2 component(s).

The 1st component contains the pair(s)

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

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a] = 2

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[mark (x₁) ] = x₁

[isNePal (x₁) ] = 0

[a__isQid (x₁) ] = 0

[a____ (x₁, x₂) ] = 2 x₁ + x₂ + 1

[i] = 2

[nil] = 0

[a__isNeList (x₁) ] = 0

[tt] = 0

[o] = 1

[a__and (x₁, x₂) ] = x₁

[e] = 0

[u] = 0

[and (x₁, x₂) ] = x₁

[a__isNePal (x₁) ] = 0

[isNeList (x₁) ] = 0

[a__isPal (x₁) ] = 0

[isQid (x₁) ] = 0

[a__isList (x₁) ] = 0

[isPal (x₁) ] = 0

[isList (x₁) ] = 0

[a____# (x₁, x₂) ] = x₁

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

one remains with the following pair(s).

none

1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

a__and#( tt , X ) → mark#( X )

mark#( and( X1 , X2 ) ) → a__and#( mark( X1 ) , X2 )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isNePal( X ) ) → a__isNePal#( X )

a__isNePal#( __( I , __( P , I ) ) ) → a__and#( a__isQid( I ) , isPal( P ) )

mark#( isPal( X ) ) → a__isPal#( X )

a__isPal#( V ) → a__isNePal#( V )

1.1.1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a] = 1

[__ (x₁, x₂) ] = 2 x₁ + x₂

[isNePal (x₁) ] = 2 x₁

[a__isQid (x₁) ] = x₁

[a____ (x₁, x₂) ] = 2 x₁ + x₂

[i] = 1

[a__isPal# (x₁) ] = 2 x₁ + 1

[u] = 1

[and (x₁, x₂) ] = 2 x₁ + x₂

[isNeList (x₁) ] = x₁

[a__isPal (x₁) ] = 2 x₁

[isPal (x₁) ] = 2 x₁

[isList (x₁) ] = x₁

[mark (x₁) ] = x₁

[a__and# (x₁, x₂) ] = 2 x₁ + x₂

[mark# (x₁) ] = x₁ + 1

[nil] = 1

[a__isNeList (x₁) ] = x₁

[tt] = 1

[a__and (x₁, x₂) ] = 2 x₁ + x₂

[o] = 1

[e] = 1

[a__isNePal# (x₁) ] = x₁ + 1

[a__isNePal (x₁) ] = 2 x₁

[isQid (x₁) ] = x₁

[a__isList (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).

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isNePal( X ) ) → a__isNePal#( X )

mark#( isPal( X ) ) → a__isPal#( X )

a__isPal#( V ) → a__isNePal#( V )

1.1.1.1.2.1: dependency graph processor

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

The 1st component contains the pair(s)

mark#( and( X1 , X2 ) ) → mark#( X1 )

1.1.1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a] = 3

[__ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[mark (x₁) ] = x₁

[isNePal (x₁) ] = 2 x₁

[a__isQid (x₁) ] = x₁

[a____ (x₁, x₂) ] = 2 x₁ + x₂ + 2

[mark# (x₁) ] = 2 x₁

[i] = 3

[nil] = 0

[a__isNeList (x₁) ] = 2 x₁

[tt] = 0

[o] = 1

[a__and (x₁, x₂) ] = 2 x₁ + x₂ + 1

[e] = 3

[and (x₁, x₂) ] = 2 x₁ + x₂ + 1

[u] = 3

[a__isNePal (x₁) ] = 2 x₁

[isNeList (x₁) ] = 2 x₁

[a__isPal (x₁) ] = 2 x₁

[isQid (x₁) ] = x₁

[a__isList (x₁) ] = 2 x₁

[isPal (x₁) ] = 2 x₁

[isList (x₁) ] = 2 x₁

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

one remains with the following pair(s).

none

1.1.1.1.2.1.1.1: P is empty

All dependency pairs have been removed.

a____#( __( X , Y ) , Z )	→	a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )
a____#( __( X , Y ) , Z )	→	mark#( X )
a____#( __( X , Y ) , Z )	→	a____#( mark( Y ) , mark( Z ) )
a____#( __( X , Y ) , Z )	→	mark#( Y )
a____#( __( X , Y ) , Z )	→	mark#( Z )
a____#( X , nil )	→	mark#( X )
a____#( nil , X )	→	mark#( X )
a__and#( tt , X )	→	mark#( X )
a__isList#( V )	→	a__isNeList#( V )
a__isList#( __( V1 , V2 ) )	→	a__and#( a__isList( V1 ) , isList( V2 ) )
a__isList#( __( V1 , V2 ) )	→	a__isList#( V1 )
a__isNeList#( V )	→	a__isQid#( V )
a__isNeList#( __( V1 , V2 ) )	→	a__and#( a__isList( V1 ) , isNeList( V2 ) )
a__isNeList#( __( V1 , V2 ) )	→	a__isList#( V1 )
a__isNeList#( __( V1 , V2 ) )	→	a__and#( a__isNeList( V1 ) , isList( V2 ) )
a__isNeList#( __( V1 , V2 ) )	→	a__isNeList#( V1 )
a__isNePal#( V )	→	a__isQid#( V )
a__isNePal#( __( I , __( P , I ) ) )	→	a__and#( a__isQid( I ) , isPal( P ) )
a__isNePal#( __( I , __( P , I ) ) )	→	a__isQid#( I )
a__isPal#( V )	→	a__isNePal#( V )
mark#( __( X1 , X2 ) )	→	a____#( mark( X1 ) , mark( X2 ) )
mark#( __( X1 , X2 ) )	→	mark#( X1 )
mark#( __( X1 , X2 ) )	→	mark#( X2 )
mark#( and( X1 , X2 ) )	→	a__and#( mark( X1 ) , X2 )
mark#( and( X1 , X2 ) )	→	mark#( X1 )
mark#( isList( X ) )	→	a__isList#( X )
mark#( isNeList( X ) )	→	a__isNeList#( X )
mark#( isQid( X ) )	→	a__isQid#( X )
mark#( isNePal( X ) )	→	a__isNePal#( X )
mark#( isPal( X ) )	→	a__isPal#( X )

[a]	=	0
[__ (x₁, x₂) ]	=	x₁ + x₂ + 2
[a__isQid (x₁) ]	=	0
[isNePal (x₁) ]	=	0
[a____ (x₁, x₂) ]	=	x₁ + x₂ + 2
[i]	=	0
[a__isPal# (x₁) ]	=	0
[and (x₁, x₂) ]	=	x₁ + x₂
[u]	=	1
[a__isNeList# (x₁) ]	=	2 x₁
[isNeList (x₁) ]	=	2 x₁
[a__isPal (x₁) ]	=	0
[isPal (x₁) ]	=	0
[isList (x₁) ]	=	2 x₁
[a____# (x₁, x₂) ]	=	x₁ + x₂
[mark (x₁) ]	=	x₁
[a__isList# (x₁) ]	=	2 x₁
[a__and# (x₁, x₂) ]	=	x₁
[mark# (x₁) ]	=	x₁
[nil]	=	0
[a__isNeList (x₁) ]	=	2 x₁
[tt]	=	0
[o]	=	0
[a__and (x₁, x₂) ]	=	x₁ + x₂
[e]	=	0
[a__isNePal# (x₁) ]	=	0
[a__isNePal (x₁) ]	=	0
[isQid (x₁) ]	=	0
[a__isList (x₁) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a]	=	2
[__ (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[mark (x₁) ]	=	x₁
[isNePal (x₁) ]	=	0
[a__isQid (x₁) ]	=	0
[a____ (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[i]	=	2
[nil]	=	0
[a__isNeList (x₁) ]	=	0
[tt]	=	0
[o]	=	1
[a__and (x₁, x₂) ]	=	x₁
[e]	=	0
[u]	=	0
[and (x₁, x₂) ]	=	x₁
[a__isNePal (x₁) ]	=	0
[isNeList (x₁) ]	=	0
[a__isPal (x₁) ]	=	0
[isQid (x₁) ]	=	0
[a__isList (x₁) ]	=	0
[isPal (x₁) ]	=	0
[isList (x₁) ]	=	0
[a____# (x₁, x₂) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a]	=	1
[__ (x₁, x₂) ]	=	2 x₁ + x₂
[isNePal (x₁) ]	=	2 x₁
[a__isQid (x₁) ]	=	x₁
[a____ (x₁, x₂) ]	=	2 x₁ + x₂
[i]	=	1
[a__isPal# (x₁) ]	=	2 x₁ + 1
[u]	=	1
[and (x₁, x₂) ]	=	2 x₁ + x₂
[isNeList (x₁) ]	=	x₁
[a__isPal (x₁) ]	=	2 x₁
[isPal (x₁) ]	=	2 x₁
[isList (x₁) ]	=	x₁
[mark (x₁) ]	=	x₁
[a__and# (x₁, x₂) ]	=	2 x₁ + x₂
[mark# (x₁) ]	=	x₁ + 1
[nil]	=	1
[a__isNeList (x₁) ]	=	x₁
[tt]	=	1
[a__and (x₁, x₂) ]	=	2 x₁ + x₂
[o]	=	1
[e]	=	1
[a__isNePal# (x₁) ]	=	x₁ + 1
[a__isNePal (x₁) ]	=	2 x₁
[isQid (x₁) ]	=	x₁
[a__isList (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a]	=	3
[__ (x₁, x₂) ]	=	2 x₁ + x₂ + 2
[mark (x₁) ]	=	x₁
[isNePal (x₁) ]	=	2 x₁
[a__isQid (x₁) ]	=	x₁
[a____ (x₁, x₂) ]	=	2 x₁ + x₂ + 2
[mark# (x₁) ]	=	2 x₁
[i]	=	3
[nil]	=	0
[a__isNeList (x₁) ]	=	2 x₁
[tt]	=	0
[o]	=	1
[a__and (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[e]	=	3
[and (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[u]	=	3
[a__isNePal (x₁) ]	=	2 x₁
[isNeList (x₁) ]	=	2 x₁
[a__isPal (x₁) ]	=	2 x₁
[isQid (x₁) ]	=	x₁
[a__isList (x₁) ]	=	2 x₁
[isPal (x₁) ]	=	2 x₁
[isList (x₁) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n