Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

active#( terms( N ) ) → cons#( recip( sqr( N ) ) , terms( s( N ) ) )

active#( terms( N ) ) → recip#( sqr( N ) )

active#( terms( N ) ) → sqr#( N )

active#( terms( N ) ) → terms#( s( N ) )

active#( terms( N ) ) → s#( N )

active#( sqr( 0 ) ) → mark#( 0 )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

active#( sqr( s( X ) ) ) → s#( add( sqr( X ) , dbl( X ) ) )

active#( sqr( s( X ) ) ) → add#( sqr( X ) , dbl( X ) )

active#( sqr( s( X ) ) ) → sqr#( X )

active#( sqr( s( X ) ) ) → dbl#( X )

active#( dbl( 0 ) ) → mark#( 0 )

active#( dbl( s( X ) ) ) → mark#( s( s( dbl( X ) ) ) )

active#( dbl( s( X ) ) ) → s#( s( dbl( X ) ) )

active#( dbl( s( X ) ) ) → s#( dbl( X ) )

active#( dbl( s( X ) ) ) → dbl#( X )

active#( add( 0 , X ) ) → mark#( X )

active#( add( s( X ) , Y ) ) → mark#( s( add( X , Y ) ) )

active#( add( s( X ) , Y ) ) → s#( add( X , Y ) )

active#( add( s( X ) , Y ) ) → add#( X , Y )

active#( first( 0 , X ) ) → mark#( nil )

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → cons#( Y , first( X , Z ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → first#( X , Z )

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

mark#( terms( X ) ) → terms#( mark( X ) )

mark#( terms( X ) ) → mark#( X )

mark#( cons( X1 , X2 ) ) → active#( cons( mark( X1 ) , X2 ) )

mark#( cons( X1 , X2 ) ) → cons#( mark( X1 ) , X2 )

mark#( cons( X1 , X2 ) ) → mark#( X1 )

mark#( recip( X ) ) → active#( recip( mark( X ) ) )

mark#( recip( X ) ) → recip#( mark( X ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

mark#( sqr( X ) ) → sqr#( mark( X ) )

mark#( sqr( X ) ) → mark#( X )

mark#( s( X ) ) → active#( s( X ) )

mark#( s( X ) ) → s#( X )

mark#( 0 ) → active#( 0 )

mark#( add( X1 , X2 ) ) → active#( add( mark( X1 ) , mark( X2 ) ) )

mark#( add( X1 , X2 ) ) → add#( mark( X1 ) , mark( X2 ) )

mark#( add( X1 , X2 ) ) → mark#( X1 )

mark#( add( X1 , X2 ) ) → mark#( X2 )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( dbl( X ) ) → dbl#( mark( X ) )

mark#( dbl( X ) ) → mark#( X )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

mark#( first( X1 , X2 ) ) → first#( mark( X1 ) , mark( X2 ) )

mark#( first( X1 , X2 ) ) → mark#( X1 )

mark#( first( X1 , X2 ) ) → mark#( X2 )

mark#( nil ) → active#( nil )

terms#( mark( X ) ) → terms#( X )

terms#( active( X ) ) → terms#( X )

cons#( mark( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( X1 , mark( X2 ) ) → cons#( X1 , X2 )

cons#( active( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( X1 , active( X2 ) ) → cons#( X1 , X2 )

recip#( mark( X ) ) → recip#( X )

recip#( active( X ) ) → recip#( X )

sqr#( mark( X ) ) → sqr#( X )

sqr#( active( X ) ) → sqr#( X )

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

add#( mark( X1 ) , X2 ) → add#( X1 , X2 )

add#( X1 , mark( X2 ) ) → add#( X1 , X2 )

add#( active( X1 ) , X2 ) → add#( X1 , X2 )

add#( X1 , active( X2 ) ) → add#( X1 , X2 )

dbl#( mark( X ) ) → dbl#( X )

dbl#( active( X ) ) → dbl#( X )

first#( mark( X1 ) , X2 ) → first#( X1 , X2 )

first#( X1 , mark( X2 ) ) → first#( X1 , X2 )

first#( active( X1 ) , X2 ) → first#( X1 , X2 )

first#( X1 , active( X2 ) ) → first#( X1 , X2 )

1.1: dependency graph processor

The dependency pairs are split into 9 component(s).

The 1st component contains the pair(s)

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

mark#( terms( X ) ) → mark#( X )

mark#( cons( X1 , X2 ) ) → active#( cons( mark( X1 ) , X2 ) )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

mark#( cons( X1 , X2 ) ) → mark#( X1 )

mark#( recip( X ) ) → active#( recip( mark( X ) ) )

active#( dbl( s( X ) ) ) → mark#( s( s( dbl( X ) ) ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

active#( add( 0 , X ) ) → mark#( X )

mark#( sqr( X ) ) → mark#( X )

mark#( s( X ) ) → active#( s( X ) )

active#( add( s( X ) , Y ) ) → mark#( s( add( X , Y ) ) )

mark#( add( X1 , X2 ) ) → active#( add( mark( X1 ) , mark( X2 ) ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

mark#( add( X1 , X2 ) ) → mark#( X1 )

mark#( add( X1 , X2 ) ) → mark#( X2 )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( dbl( X ) ) → mark#( X )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

mark#( first( X1 , X2 ) ) → mark#( X1 )

mark#( first( X1 , X2 ) ) → mark#( X2 )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[dbl (x₁) ] = 1

[first (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 1

[terms (x₁) ] = 1

[recip (x₁) ] = 1

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 1

[s (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).

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

mark#( terms( X ) ) → mark#( X )

mark#( cons( X1 , X2 ) ) → active#( cons( mark( X1 ) , X2 ) )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

mark#( cons( X1 , X2 ) ) → mark#( X1 )

mark#( recip( X ) ) → active#( recip( mark( X ) ) )

active#( dbl( s( X ) ) ) → mark#( s( s( dbl( X ) ) ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

active#( add( 0 , X ) ) → mark#( X )

mark#( sqr( X ) ) → mark#( X )

active#( add( s( X ) , Y ) ) → mark#( s( add( X , Y ) ) )

mark#( add( X1 , X2 ) ) → active#( add( mark( X1 ) , mark( X2 ) ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

mark#( add( X1 , X2 ) ) → mark#( X1 )

mark#( add( X1 , X2 ) ) → mark#( X2 )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( dbl( X ) ) → mark#( X )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

mark#( first( X1 , X2 ) ) → mark#( X1 )

mark#( first( X1 , X2 ) ) → mark#( X2 )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[dbl (x₁) ] = 1

[first (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 1

[recip (x₁) ] = 0

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 1

[s (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).

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

mark#( terms( X ) ) → mark#( X )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

mark#( cons( X1 , X2 ) ) → mark#( X1 )

active#( dbl( s( X ) ) ) → mark#( s( s( dbl( X ) ) ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

active#( add( 0 , X ) ) → mark#( X )

mark#( sqr( X ) ) → mark#( X )

active#( add( s( X ) , Y ) ) → mark#( s( add( X , Y ) ) )

mark#( add( X1 , X2 ) ) → active#( add( mark( X1 ) , mark( X2 ) ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

mark#( add( X1 , X2 ) ) → mark#( X1 )

mark#( add( X1 , X2 ) ) → mark#( X2 )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( dbl( X ) ) → mark#( X )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

mark#( first( X1 , X2 ) ) → mark#( X1 )

mark#( first( X1 , X2 ) ) → mark#( X2 )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 3 x₁ + 2 x₂ + 1

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 0

[nil] = 1

[cons (x₁, x₂) ] = x₁ + 3

[terms (x₁) ] = x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = x₁

[add (x₁, x₂) ] = x₁ + x₂ + 2

[s (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).

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

mark#( sqr( X ) ) → mark#( X )

mark#( add( X1 , X2 ) ) → active#( add( mark( X1 ) , mark( X2 ) ) )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[dbl (x₁) ] = 1

[first (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 1

[recip (x₁) ] = 0

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 0

[s (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).

mark#( terms( X ) ) → active#( terms( mark( X ) ) )

active#( terms( N ) ) → mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )

active#( sqr( s( X ) ) ) → mark#( s( add( sqr( X ) , dbl( X ) ) ) )

mark#( recip( X ) ) → mark#( X )

mark#( sqr( X ) ) → active#( sqr( mark( X ) ) )

mark#( sqr( X ) ) → mark#( X )

mark#( dbl( X ) ) → active#( dbl( mark( X ) ) )

mark#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 2

[active (x₁) ] = x₁

[mark# (x₁) ] = 3 x₁ + 2

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁ + 2

[add (x₁, x₂) ] = x₁

[s (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).

mark#( recip( X ) ) → mark#( X )

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁

[dbl (x₁) ] = 2 x₁

[first (x₁, x₂) ] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[recip (x₁) ] = 2 x₁ + 1

[terms (x₁) ] = 0

[sqr (x₁) ] = 2

[add (x₁, x₂) ] = 2 x₁

[s (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

1.1.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

terms#( active( X ) ) → terms#( X )

terms#( mark( X ) ) → terms#( X )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[terms# (x₁) ] = x₁

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 2

[recip (x₁) ] = 2

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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).

terms#( active( X ) ) → terms#( X )

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[terms# (x₁) ] = x₁

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[add (x₁, x₂) ] = 0

[s (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

1.1.2.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

cons#( X1 , mark( X2 ) ) → cons#( X1 , X2 )

cons#( mark( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( active( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( X1 , active( X2 ) ) → cons#( X1 , X2 )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁

[cons# (x₁, x₂) ] = 3 x₁ + 3 x₂

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 3

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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)

recip#( active( X ) ) → recip#( X )

recip#( mark( X ) ) → recip#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 2

[recip (x₁) ] = 2

[sqr (x₁) ] = 1

[recip# (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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).

recip#( active( X ) ) → recip#( X )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[recip# (x₁) ] = x₁

[add (x₁, x₂) ] = 0

[s (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

1.1.4.1.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

sqr#( active( X ) ) → sqr#( X )

sqr#( mark( X ) ) → sqr#( X )

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[sqr# (x₁) ] = x₁

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 2

[recip (x₁) ] = 2

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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).

sqr#( active( X ) ) → sqr#( X )

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[sqr# (x₁) ] = x₁

[mark (x₁) ] = 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[add (x₁, x₂) ] = 0

[s (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

1.1.5.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[s# (x₁) ] = x₁

[terms (x₁) ] = 2

[recip (x₁) ] = 2

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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).

s#( active( X ) ) → s#( X )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[s# (x₁) ] = x₁

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[add (x₁, x₂) ] = 0

[s (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

1.1.6.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

add#( X1 , mark( X2 ) ) → add#( X1 , X2 )

add#( mark( X1 ) , X2 ) → add#( X1 , X2 )

add#( active( X1 ) , X2 ) → add#( X1 , X2 )

add#( X1 , active( X2 ) ) → add#( X1 , X2 )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = x₁ + 1

[0] = 0

[add# (x₁, x₂) ] = 3 x₁ + 3 x₂

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 3

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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

1.1.7.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

dbl#( active( X ) ) → dbl#( X )

dbl#( mark( X ) ) → dbl#( X )

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 2

[recip (x₁) ] = 2

[sqr (x₁) ] = 1

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 0

[dbl# (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).

dbl#( active( X ) ) → dbl#( X )

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 0

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[add (x₁, x₂) ] = 0

[s (x₁) ] = 0

[dbl# (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).

none

1.1.8.1.1: P is empty

All dependency pairs have been removed.

The 9th component contains the pair(s)

first#( X1 , mark( X2 ) ) → first#( X1 , X2 )

first#( mark( X1 ) , X2 ) → first#( X1 , X2 )

first#( active( X1 ) , X2 ) → first#( X1 , X2 )

first#( X1 , active( X2 ) ) → first#( X1 , X2 )

1.1.9: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = x₁ + 1

[first# (x₁, x₂) ] = 3 x₁ + 3 x₂

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = 3

[add (x₁, x₂) ] = 2 x₁ + 2

[s (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

1.1.9.1: P is empty

All dependency pairs have been removed.

active#( terms( N ) )	→	mark#( cons( recip( sqr( N ) ) , terms( s( N ) ) ) )
active#( terms( N ) )	→	cons#( recip( sqr( N ) ) , terms( s( N ) ) )
active#( terms( N ) )	→	recip#( sqr( N ) )
active#( terms( N ) )	→	sqr#( N )
active#( terms( N ) )	→	terms#( s( N ) )
active#( terms( N ) )	→	s#( N )
active#( sqr( 0 ) )	→	mark#( 0 )
active#( sqr( s( X ) ) )	→	mark#( s( add( sqr( X ) , dbl( X ) ) ) )
active#( sqr( s( X ) ) )	→	s#( add( sqr( X ) , dbl( X ) ) )
active#( sqr( s( X ) ) )	→	add#( sqr( X ) , dbl( X ) )
active#( sqr( s( X ) ) )	→	sqr#( X )
active#( sqr( s( X ) ) )	→	dbl#( X )
active#( dbl( 0 ) )	→	mark#( 0 )
active#( dbl( s( X ) ) )	→	mark#( s( s( dbl( X ) ) ) )
active#( dbl( s( X ) ) )	→	s#( s( dbl( X ) ) )
active#( dbl( s( X ) ) )	→	s#( dbl( X ) )
active#( dbl( s( X ) ) )	→	dbl#( X )
active#( add( 0 , X ) )	→	mark#( X )
active#( add( s( X ) , Y ) )	→	mark#( s( add( X , Y ) ) )
active#( add( s( X ) , Y ) )	→	s#( add( X , Y ) )
active#( add( s( X ) , Y ) )	→	add#( X , Y )
active#( first( 0 , X ) )	→	mark#( nil )
active#( first( s( X ) , cons( Y , Z ) ) )	→	mark#( cons( Y , first( X , Z ) ) )
active#( first( s( X ) , cons( Y , Z ) ) )	→	cons#( Y , first( X , Z ) )
active#( first( s( X ) , cons( Y , Z ) ) )	→	first#( X , Z )
mark#( terms( X ) )	→	active#( terms( mark( X ) ) )
mark#( terms( X ) )	→	terms#( mark( X ) )
mark#( terms( X ) )	→	mark#( X )
mark#( cons( X1 , X2 ) )	→	active#( cons( mark( X1 ) , X2 ) )
mark#( cons( X1 , X2 ) )	→	cons#( mark( X1 ) , X2 )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( recip( X ) )	→	active#( recip( mark( X ) ) )
mark#( recip( X ) )	→	recip#( mark( X ) )
mark#( recip( X ) )	→	mark#( X )
mark#( sqr( X ) )	→	active#( sqr( mark( X ) ) )
mark#( sqr( X ) )	→	sqr#( mark( X ) )
mark#( sqr( X ) )	→	mark#( X )
mark#( s( X ) )	→	active#( s( X ) )
mark#( s( X ) )	→	s#( X )
mark#( 0 )	→	active#( 0 )
mark#( add( X1 , X2 ) )	→	active#( add( mark( X1 ) , mark( X2 ) ) )
mark#( add( X1 , X2 ) )	→	add#( mark( X1 ) , mark( X2 ) )
mark#( add( X1 , X2 ) )	→	mark#( X1 )
mark#( add( X1 , X2 ) )	→	mark#( X2 )
mark#( dbl( X ) )	→	active#( dbl( mark( X ) ) )
mark#( dbl( X ) )	→	dbl#( mark( X ) )
mark#( dbl( X ) )	→	mark#( X )
mark#( first( X1 , X2 ) )	→	active#( first( mark( X1 ) , mark( X2 ) ) )
mark#( first( X1 , X2 ) )	→	first#( mark( X1 ) , mark( X2 ) )
mark#( first( X1 , X2 ) )	→	mark#( X1 )
mark#( first( X1 , X2 ) )	→	mark#( X2 )
mark#( nil )	→	active#( nil )
terms#( mark( X ) )	→	terms#( X )
terms#( active( X ) )	→	terms#( X )
cons#( mark( X1 ) , X2 )	→	cons#( X1 , X2 )
cons#( X1 , mark( X2 ) )	→	cons#( X1 , X2 )
cons#( active( X1 ) , X2 )	→	cons#( X1 , X2 )
cons#( X1 , active( X2 ) )	→	cons#( X1 , X2 )
recip#( mark( X ) )	→	recip#( X )
recip#( active( X ) )	→	recip#( X )
sqr#( mark( X ) )	→	sqr#( X )
sqr#( active( X ) )	→	sqr#( X )
s#( mark( X ) )	→	s#( X )
s#( active( X ) )	→	s#( X )
add#( mark( X1 ) , X2 )	→	add#( X1 , X2 )
add#( X1 , mark( X2 ) )	→	add#( X1 , X2 )
add#( active( X1 ) , X2 )	→	add#( X1 , X2 )
add#( X1 , active( X2 ) )	→	add#( X1 , X2 )
dbl#( mark( X ) )	→	dbl#( X )
dbl#( active( X ) )	→	dbl#( X )
first#( mark( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , mark( X2 ) )	→	first#( X1 , X2 )
first#( active( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , active( X2 ) )	→	first#( X1 , X2 )

[mark (x₁) ]	=	0
[active# (x₁) ]	=	x₁
[dbl (x₁) ]	=	1
[first (x₁, x₂) ]	=	1
[active (x₁) ]	=	0
[mark# (x₁) ]	=	1
[0]	=	0
[nil]	=	0
[cons (x₁, x₂) ]	=	1
[terms (x₁) ]	=	1
[recip (x₁) ]	=	1
[sqr (x₁) ]	=	1
[add (x₁, x₂) ]	=	1
[s (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	2 x₁
[dbl (x₁) ]	=	2 x₁
[first (x₁, x₂) ]	=	0
[active (x₁) ]	=	x₁
[mark# (x₁) ]	=	2 x₁
[0]	=	0
[nil]	=	0
[cons (x₁, x₂) ]	=	0
[recip (x₁) ]	=	2 x₁ + 1
[terms (x₁) ]	=	0
[sqr (x₁) ]	=	2
[add (x₁, x₂) ]	=	2 x₁
[s (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[sqr# (x₁) ]	=	x₁
[mark (x₁) ]	=	2 x₁ + 1
[dbl (x₁) ]	=	2 x₁ + 2
[first (x₁, x₂) ]	=	x₁ + 1
[active (x₁) ]	=	x₁
[0]	=	0
[nil]	=	0
[cons (x₁, x₂) ]	=	0
[terms (x₁) ]	=	2
[recip (x₁) ]	=	2
[sqr (x₁) ]	=	1
[add (x₁, x₂) ]	=	2 x₁ + 2
[s (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n