Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

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( 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( s( X ) ) ) → s#( s( dbl( X ) ) )

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

active#( dbl( s( X ) ) ) → dbl#( X )

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

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

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

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

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

active#( terms( X ) ) → active#( X )

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

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

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

active#( recip( X ) ) → active#( X )

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

active#( sqr( X ) ) → active#( X )

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

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

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

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

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

active#( dbl( X ) ) → active#( X )

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

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

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

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

terms#( mark( X ) ) → terms#( X )

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

recip#( mark( X ) ) → recip#( X )

sqr#( mark( X ) ) → sqr#( X )

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

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

dbl#( mark( X ) ) → dbl#( X )

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

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

proper#( terms( X ) ) → terms#( proper( X ) )

proper#( terms( X ) ) → proper#( X )

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

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

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

proper#( recip( X ) ) → recip#( proper( X ) )

proper#( recip( X ) ) → proper#( X )

proper#( sqr( X ) ) → sqr#( proper( X ) )

proper#( sqr( X ) ) → proper#( X )

proper#( s( X ) ) → s#( proper( X ) )

proper#( s( X ) ) → proper#( X )

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

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

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

proper#( dbl( X ) ) → dbl#( proper( X ) )

proper#( dbl( X ) ) → proper#( X )

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

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

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

terms#( ok( X ) ) → terms#( X )

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

recip#( ok( X ) ) → recip#( X )

sqr#( ok( X ) ) → sqr#( X )

s#( ok( X ) ) → s#( X )

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

dbl#( ok( X ) ) → dbl#( X )

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

top#( mark( X ) ) → top#( proper( X ) )

top#( mark( X ) ) → proper#( X )

top#( ok( X ) ) → top#( active( X ) )

top#( ok( X ) ) → active#( X )

1.1: dependency graph processor

The dependency pairs are split into 11 component(s).

The 1st component contains the pair(s)

top#( ok( X ) ) → top#( active( X ) )

top#( mark( X ) ) → top#( proper( X ) )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[dbl (x₁) ] = x₁ + 2

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = x₁

[0] = 2

[nil] = 1

[cons (x₁, x₂) ] = x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = 1

[top# (x₁) ] = x₁

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

top#( ok( X ) ) → top#( active( X ) )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = x₁

[first (x₁, x₂) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 2

[nil] = 0

[cons (x₁, x₂) ] = x₁ + 2 x₂ + 2

[terms (x₁) ] = x₁

[recip (x₁) ] = 2 x₁ + 2

[sqr (x₁) ] = x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁ + 1

[top# (x₁) ] = 2 x₁

[ok (x₁) ] = 2 x₁ + 2

[top (x₁) ] = 0

[proper (x₁) ] = 3 x₁ + 2

[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: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

active#( terms( X ) ) → active#( X )

active#( recip( X ) ) → active#( X )

active#( sqr( X ) ) → active#( X )

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

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

active#( dbl( X ) ) → active#( X )

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

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

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[dbl (x₁) ] = 2 x₁ + 3

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = x₁

[0] = 3

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = 2 x₁ + 1

[sqr (x₁) ] = 2 x₁ + 3

[add (x₁, x₂) ] = 2 x₁ + 2 x₂

[s (x₁) ] = 0

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

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

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

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

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

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

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[dbl (x₁) ] = 3

[first (x₁, x₂) ] = x₁ + x₂

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + x₂ + 2

[terms (x₁) ] = 3

[recip (x₁) ] = 1

[sqr (x₁) ] = 3

[add (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = 2 x₁ + 2

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

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

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

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

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

1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[dbl (x₁) ] = 3

[first (x₁, x₂) ] = 3 x₁ + 2 x₂ + 3

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 3

[cons (x₁, x₂) ] = 2 x₁ + 3 x₂

[terms (x₁) ] = 0

[recip (x₁) ] = 3

[sqr (x₁) ] = 2

[add (x₁, x₂) ] = 2 x₁ + 2 x₂

[s (x₁) ] = 1

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

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

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

1.1.2.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[dbl (x₁) ] = 0

[first (x₁, x₂) ] = 0

[active (x₁) ] = x₁

[0] = 1

[nil] = 0

[cons (x₁, x₂) ] = 3 x₁ + 2

[terms (x₁) ] = 0

[recip (x₁) ] = 0

[sqr (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁ + x₂ + 2

[s (x₁) ] = 0

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.2.1.1.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

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

proper#( terms( X ) ) → proper#( X )

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

proper#( recip( X ) ) → proper#( X )

proper#( sqr( X ) ) → proper#( X )

proper#( s( X ) ) → proper#( X )

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

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

proper#( dbl( X ) ) → proper#( X )

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

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

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = x₁ + 1

[first (x₁, x₂) ] = 2 x₁ + 3 x₂ + 3

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + 3 x₂ + 1

[proper# (x₁) ] = 2 x₁

[terms (x₁) ] = 2 x₁

[recip (x₁) ] = 2 x₁ + 3

[sqr (x₁) ] = 2 x₁ + 3

[add (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[s (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

proper#( terms( X ) ) → proper#( X )

proper#( s( X ) ) → proper#( X )

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = 3 x₁

[first (x₁, x₂) ] = x₁ + 3 x₂ + 1

[active (x₁) ] = x₁

[0] = 1

[nil] = 0

[cons (x₁, x₂) ] = 3 x₁ + 3 x₂

[proper# (x₁) ] = x₁

[terms (x₁) ] = x₁

[recip (x₁) ] = 0

[sqr (x₁) ] = 0

[add (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = x₁ + 1

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

proper#( terms( X ) ) → proper#( X )

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2

[dbl (x₁) ] = 3 x₁ + 3

[first (x₁, x₂) ] = 3 x₁ + 3 x₂ + 2

[active (x₁) ] = x₁

[0] = 0

[nil] = 2

[cons (x₁, x₂) ] = 3 x₁ + 3 x₂ + 3

[proper# (x₁) ] = 2 x₁

[terms (x₁) ] = x₁ + 2

[recip (x₁) ] = 3 x₁ + 3

[sqr (x₁) ] = 3 x₁ + 3

[add (x₁, x₂) ] = 2 x₁ + 3 x₂ + 3

[s (x₁) ] = 2 x₁ + 3

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.3.1.1.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

terms#( ok( X ) ) → terms#( X )

terms#( mark( X ) ) → terms#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 3

[dbl (x₁) ] = 2 x₁

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = 2 x₁

[0] = 3

[nil] = 2

[terms# (x₁) ] = x₁

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 3 x₁ + 2 x₂

[s (x₁) ] = 1

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

terms#( ok( X ) ) → terms#( X )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[terms# (x₁) ] = 2 x₁

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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

1.1.4.1.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

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

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

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = x₁

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = x₁ + x₂

[cons# (x₁, x₂) ] = 3 x₁

[terms (x₁) ] = 2 x₁

[recip (x₁) ] = x₁

[sqr (x₁) ] = x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁

[ok (x₁) ] = 2 x₁ + 1

[top (x₁) ] = 0

[proper (x₁) ] = 3 x₁

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

one remains with the following pair(s).

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

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[dbl (x₁) ] = x₁

[first (x₁, x₂) ] = x₁ + x₂ + 1

[active (x₁) ] = x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = x₁

[cons# (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = x₁ + 1

[recip (x₁) ] = x₁

[sqr (x₁) ] = x₁

[add (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = 0

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.5.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

recip#( ok( X ) ) → recip#( X )

recip#( mark( X ) ) → recip#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 3

[dbl (x₁) ] = 2 x₁

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = 2 x₁

[0] = 3

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[recip# (x₁) ] = x₁

[add (x₁, x₂) ] = 3 x₁ + 2 x₂

[s (x₁) ] = 1

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

recip#( ok( X ) ) → recip#( X )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁

[recip# (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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

1.1.6.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

sqr#( ok( X ) ) → sqr#( X )

sqr#( mark( X ) ) → sqr#( X )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[sqr# (x₁) ] = x₁

[mark (x₁) ] = x₁ + 3

[dbl (x₁) ] = 2 x₁

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = 2 x₁

[0] = 3

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 3 x₁ + 2 x₂

[s (x₁) ] = 1

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

sqr#( ok( X ) ) → sqr#( X )

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[sqr# (x₁) ] = 2 x₁

[mark (x₁) ] = 0

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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

1.1.7.1.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

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

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

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

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[dbl (x₁) ] = x₁

[first (x₁, x₂) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 3

[add# (x₁, x₂) ] = x₁

[nil] = 3

[cons (x₁, x₂) ] = x₁

[terms (x₁) ] = x₁

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁

[ok (x₁) ] = 2 x₁ + 1

[top (x₁) ] = 0

[proper (x₁) ] = 3 x₁

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

one remains with the following pair(s).

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

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

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + x₂

[active (x₁) ] = 2 x₁

[0] = 2

[add# (x₁, x₂) ] = x₁ + x₂

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = x₁ + 1

[add (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[s (x₁) ] = 0

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.8.1.1: P is empty

All dependency pairs have been removed.

The 9th component contains the pair(s)

dbl#( ok( X ) ) → dbl#( X )

dbl#( mark( X ) ) → dbl#( X )

1.1.9: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 3

[dbl (x₁) ] = 2 x₁

[first (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = 2 x₁

[0] = 3

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 3 x₁ + 2 x₂

[s (x₁) ] = 1

[ok (x₁) ] = x₁

[dbl# (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

dbl#( ok( X ) ) → dbl#( X )

1.1.9.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 1

[dbl# (x₁) ] = 2 x₁

[top (x₁) ] = 0

[proper (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

1.1.9.1.1: P is empty

All dependency pairs have been removed.

The 10th component contains the pair(s)

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

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

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

1.1.10: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[dbl (x₁) ] = x₁

[first (x₁, x₂) ] = x₁

[active (x₁) ] = 2 x₁

[first# (x₁, x₂) ] = x₁

[0] = 3

[nil] = 3

[cons (x₁, x₂) ] = x₁

[terms (x₁) ] = x₁

[recip (x₁) ] = x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁

[ok (x₁) ] = 2 x₁ + 1

[top (x₁) ] = 0

[proper (x₁) ] = 3 x₁

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

one remains with the following pair(s).

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

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

1.1.10.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = x₁ + x₂

[active (x₁) ] = 2 x₁

[first# (x₁, x₂) ] = x₁ + x₂

[0] = 2

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁ + 2

[terms (x₁) ] = 2 x₁ + 3

[recip (x₁) ] = x₁

[sqr (x₁) ] = x₁ + 1

[add (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[s (x₁) ] = 0

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.10.1.1: P is empty

All dependency pairs have been removed.

The 11th component contains the pair(s)

s#( ok( X ) ) → s#( X )

1.1.11: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[dbl (x₁) ] = 2 x₁ + 2

[first (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[s# (x₁) ] = 2 x₁

[terms (x₁) ] = 3 x₁ + 2

[recip (x₁) ] = 2 x₁

[sqr (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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

1.1.11.1: P is empty

All dependency pairs have been removed.

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( 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( s( X ) ) )	→	s#( s( dbl( X ) ) )
active#( dbl( s( X ) ) )	→	s#( dbl( X ) )
active#( dbl( s( X ) ) )	→	dbl#( X )
active#( add( s( X ) , Y ) )	→	s#( add( X , Y ) )
active#( add( s( X ) , Y ) )	→	add#( X , Y )
active#( first( s( X ) , cons( Y , Z ) ) )	→	cons#( Y , first( X , Z ) )
active#( first( s( X ) , cons( Y , Z ) ) )	→	first#( X , Z )
active#( terms( X ) )	→	terms#( active( X ) )
active#( terms( X ) )	→	active#( X )
active#( cons( X1 , X2 ) )	→	cons#( active( X1 ) , X2 )
active#( cons( X1 , X2 ) )	→	active#( X1 )
active#( recip( X ) )	→	recip#( active( X ) )
active#( recip( X ) )	→	active#( X )
active#( sqr( X ) )	→	sqr#( active( X ) )
active#( sqr( X ) )	→	active#( X )
active#( add( X1 , X2 ) )	→	add#( active( X1 ) , X2 )
active#( add( X1 , X2 ) )	→	active#( X1 )
active#( add( X1 , X2 ) )	→	add#( X1 , active( X2 ) )
active#( add( X1 , X2 ) )	→	active#( X2 )
active#( dbl( X ) )	→	dbl#( active( X ) )
active#( dbl( X ) )	→	active#( X )
active#( first( X1 , X2 ) )	→	first#( active( X1 ) , X2 )
active#( first( X1 , X2 ) )	→	active#( X1 )
active#( first( X1 , X2 ) )	→	first#( X1 , active( X2 ) )
active#( first( X1 , X2 ) )	→	active#( X2 )
terms#( mark( X ) )	→	terms#( X )
cons#( mark( X1 ) , X2 )	→	cons#( X1 , X2 )
recip#( mark( X ) )	→	recip#( X )
sqr#( mark( X ) )	→	sqr#( X )
add#( mark( X1 ) , X2 )	→	add#( X1 , X2 )
add#( X1 , mark( X2 ) )	→	add#( X1 , X2 )
dbl#( mark( X ) )	→	dbl#( X )
first#( mark( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , mark( X2 ) )	→	first#( X1 , X2 )
proper#( terms( X ) )	→	terms#( proper( X ) )
proper#( terms( X ) )	→	proper#( X )
proper#( cons( X1 , X2 ) )	→	cons#( proper( X1 ) , proper( X2 ) )
proper#( cons( X1 , X2 ) )	→	proper#( X1 )
proper#( cons( X1 , X2 ) )	→	proper#( X2 )
proper#( recip( X ) )	→	recip#( proper( X ) )
proper#( recip( X ) )	→	proper#( X )
proper#( sqr( X ) )	→	sqr#( proper( X ) )
proper#( sqr( X ) )	→	proper#( X )
proper#( s( X ) )	→	s#( proper( X ) )
proper#( s( X ) )	→	proper#( X )
proper#( add( X1 , X2 ) )	→	add#( proper( X1 ) , proper( X2 ) )
proper#( add( X1 , X2 ) )	→	proper#( X1 )
proper#( add( X1 , X2 ) )	→	proper#( X2 )
proper#( dbl( X ) )	→	dbl#( proper( X ) )
proper#( dbl( X ) )	→	proper#( X )
proper#( first( X1 , X2 ) )	→	first#( proper( X1 ) , proper( X2 ) )
proper#( first( X1 , X2 ) )	→	proper#( X1 )
proper#( first( X1 , X2 ) )	→	proper#( X2 )
terms#( ok( X ) )	→	terms#( X )
cons#( ok( X1 ) , ok( X2 ) )	→	cons#( X1 , X2 )
recip#( ok( X ) )	→	recip#( X )
sqr#( ok( X ) )	→	sqr#( X )
s#( ok( X ) )	→	s#( X )
add#( ok( X1 ) , ok( X2 ) )	→	add#( X1 , X2 )
dbl#( ok( X ) )	→	dbl#( X )
first#( ok( X1 ) , ok( X2 ) )	→	first#( X1 , X2 )
top#( mark( X ) )	→	top#( proper( X ) )
top#( mark( X ) )	→	proper#( X )
top#( ok( X ) )	→	top#( active( X ) )
top#( ok( X ) )	→	active#( X )

[mark (x₁) ]	=	x₁ + 1
[dbl (x₁) ]	=	x₁ + 2
[first (x₁, x₂) ]	=	x₁ + 2 x₂
[active (x₁) ]	=	x₁
[0]	=	2
[nil]	=	1
[cons (x₁, x₂) ]	=	x₁
[terms (x₁) ]	=	3 x₁ + 2
[recip (x₁) ]	=	x₁
[sqr (x₁) ]	=	2 x₁
[add (x₁, x₂) ]	=	2 x₁ + x₂
[s (x₁) ]	=	1
[top# (x₁) ]	=	x₁
[ok (x₁) ]	=	x₁
[top (x₁) ]	=	0
[proper (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	0
[active# (x₁) ]	=	2 x₁
[dbl (x₁) ]	=	2 x₁ + 3
[first (x₁, x₂) ]	=	x₁ + 2 x₂
[active (x₁) ]	=	x₁
[0]	=	3
[nil]	=	0
[cons (x₁, x₂) ]	=	2 x₁
[terms (x₁) ]	=	2 x₁ + 3
[recip (x₁) ]	=	2 x₁ + 1
[sqr (x₁) ]	=	2 x₁ + 3
[add (x₁, x₂) ]	=	2 x₁ + 2 x₂
[s (x₁) ]	=	0
[ok (x₁) ]	=	0
[top (x₁) ]	=	0
[proper (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	0
[dbl (x₁) ]	=	3 x₁
[first (x₁, x₂) ]	=	x₁ + 3 x₂ + 1
[active (x₁) ]	=	x₁
[0]	=	1
[nil]	=	0
[cons (x₁, x₂) ]	=	3 x₁ + 3 x₂
[proper# (x₁) ]	=	x₁
[terms (x₁) ]	=	x₁
[recip (x₁) ]	=	0
[sqr (x₁) ]	=	0
[add (x₁, x₂) ]	=	2 x₁ + x₂
[s (x₁) ]	=	x₁ + 1
[ok (x₁) ]	=	0
[top (x₁) ]	=	0
[proper (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	2
[dbl (x₁) ]	=	3 x₁ + 3
[first (x₁, x₂) ]	=	3 x₁ + 3 x₂ + 2
[active (x₁) ]	=	x₁
[0]	=	0
[nil]	=	2
[cons (x₁, x₂) ]	=	3 x₁ + 3 x₂ + 3
[proper# (x₁) ]	=	2 x₁
[terms (x₁) ]	=	x₁ + 2
[recip (x₁) ]	=	3 x₁ + 3
[sqr (x₁) ]	=	3 x₁ + 3
[add (x₁, x₂) ]	=	2 x₁ + 3 x₂ + 3
[s (x₁) ]	=	2 x₁ + 3
[ok (x₁) ]	=	0
[top (x₁) ]	=	0
[proper (x₁) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	x₁ + 3
[dbl (x₁) ]	=	2 x₁
[first (x₁, x₂) ]	=	x₁ + 2 x₂
[active (x₁) ]	=	2 x₁
[0]	=	3
[nil]	=	2
[terms# (x₁) ]	=	x₁
[cons (x₁, x₂) ]	=	2 x₁ + 2
[terms (x₁) ]	=	2 x₁ + 3
[recip (x₁) ]	=	x₁
[sqr (x₁) ]	=	2 x₁
[add (x₁, x₂) ]	=	3 x₁ + 2 x₂
[s (x₁) ]	=	1
[ok (x₁) ]	=	x₁
[top (x₁) ]	=	0
[proper (x₁) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[sqr# (x₁) ]	=	x₁
[mark (x₁) ]	=	x₁ + 3
[dbl (x₁) ]	=	2 x₁
[first (x₁, x₂) ]	=	x₁ + 2 x₂
[active (x₁) ]	=	2 x₁
[0]	=	3
[nil]	=	2
[cons (x₁, x₂) ]	=	2 x₁ + 2
[terms (x₁) ]	=	2 x₁ + 3
[recip (x₁) ]	=	x₁
[sqr (x₁) ]	=	2 x₁
[add (x₁, x₂) ]	=	3 x₁ + 2 x₂
[s (x₁) ]	=	1
[ok (x₁) ]	=	x₁
[top (x₁) ]	=	0
[proper (x₁) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n