Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

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

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

active#( from( X ) ) → cons#( X , from( s( X ) ) )

active#( from( X ) ) → from#( s( X ) )

active#( from( X ) ) → s#( X )

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

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

active#( len( cons( X , Z ) ) ) → s#( len( Z ) )

active#( len( cons( X , Z ) ) ) → len#( Z )

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

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

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

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

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

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

active#( from( X ) ) → from#( active( X ) )

active#( from( 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#( len( X ) ) → len#( active( X ) )

active#( len( X ) ) → active#( X )

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

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

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

from#( mark( X ) ) → from#( X )

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

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

len#( mark( X ) ) → len#( X )

proper#( s( X ) ) → s#( proper( X ) )

proper#( s( 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#( fst( X1 , X2 ) ) → fst#( proper( X1 ) , proper( X2 ) )

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

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

proper#( from( X ) ) → from#( proper( X ) )

proper#( from( 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#( len( X ) ) → len#( proper( X ) )

proper#( len( X ) ) → proper#( X )

s#( ok( X ) ) → s#( X )

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

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

from#( ok( X ) ) → from#( X )

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

len#( ok( X ) ) → len#( X )

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 9 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

[from (x₁) ] = x₁ + 2

[fst (x₁, x₂) ] = x₁ + x₂ + 3

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[nil] = 2

[cons (x₁, x₂) ] = x₁ + 1

[len (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[s (x₁) ] = 0

[top# (x₁) ] = x₁

[ok (x₁) ] = x₁

[top (x₁) ] = 2 x₁

[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

[from (x₁) ] = x₁

[fst (x₁, x₂) ] = x₁ + x₂ + 2

[mark (x₁) ] = 0

[active (x₁) ] = x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 3 x₁ + 3

[len (x₁) ] = x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁ + 2

[top# (x₁) ] = 3 x₁

[ok (x₁) ] = 2 x₁ + 3

[top (x₁) ] = 0

[proper (x₁) ] = 3 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).

none

1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

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

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

active#( from( X ) ) → active#( X )

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

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

active#( len( X ) ) → active#( X )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁ + 2 x₂

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = x₁ + 2

[len (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁ + 2 x₂

[s (x₁) ] = 0

[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).

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

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

active#( from( X ) ) → active#( X )

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

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

active#( len( X ) ) → active#( X )

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁ + 2 x₂

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 2 x₁

[0] = 3

[nil] = 0

[cons (x₁, x₂) ] = 0

[len (x₁) ] = 2 x₁

[add (x₁, x₂) ] = x₁ + 2 x₂ + 1

[s (x₁) ] = 2

[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#( fst( X1 , X2 ) ) → active#( X1 )

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

active#( from( X ) ) → active#( X )

active#( len( X ) ) → active#( X )

1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 2

[fst (x₁, x₂) ] = x₁ + 2 x₂ + 2

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 1

[nil] = 2

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = x₁ + 1

[add (x₁, x₂) ] = 2 x₁ + 3

[s (x₁) ] = 2 x₁

[ok (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

1.1.2.1.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

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

proper#( s( X ) ) → proper#( X )

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

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

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

proper#( from( X ) ) → proper#( X )

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

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

proper#( len( X ) ) → proper#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁ + x₂

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 1

[nil] = 1

[cons (x₁, x₂) ] = x₁ + x₂

[len (x₁) ] = x₁

[proper# (x₁) ] = 2 x₁

[add (x₁, x₂) ] = x₁ + 3 x₂ + 1

[s (x₁) ] = 2 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).

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁ + x₂

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 2

[nil] = 3

[cons (x₁, x₂) ] = 2 x₁ + x₂

[len (x₁) ] = x₁ + 1

[proper# (x₁) ] = x₁

[add (x₁, x₂) ] = 0

[s (x₁) ] = x₁

[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).

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[fst (x₁, x₂) ] = x₁ + x₂ + 1

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = 2 x₁ + x₂

[len (x₁) ] = 0

[proper# (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 1

[s (x₁) ] = 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).

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

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

proper#( from( X ) ) → proper#( X )

1.1.3.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = x₁ + x₂ + 1

[len (x₁) ] = 0

[proper# (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = 2 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#( from( X ) ) → proper#( X )

1.1.3.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 1

[fst (x₁, x₂) ] = x₁ + 3 x₂

[mark (x₁) ] = 0

[active (x₁) ] = x₁

[0] = 1

[nil] = 1

[cons (x₁, x₂) ] = 2 x₁ + 1

[len (x₁) ] = 0

[proper# (x₁) ] = x₁

[add (x₁, x₂) ] = 3 x₁

[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

1.1.3.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

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

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

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 1

[fst (x₁, x₂) ] = x₁

[mark (x₁) ] = 0

[active (x₁) ] = x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁ + 1

[len (x₁) ] = 2 x₁

[cons# (x₁, x₂) ] = x₁

[add (x₁, x₂) ] = 2 x₁

[s (x₁) ] = x₁ + 2

[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).

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

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 2

[fst (x₁, x₂) ] = x₁ + 2 x₂ + 2

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 1

[cons (x₁, x₂) ] = 2 x₁ + 2

[len (x₁) ] = 2 x₁

[cons# (x₁, x₂) ] = 2 x₁

[add (x₁, x₂) ] = 2 x₁ + 2 x₂ + 3

[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

1.1.4.1.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

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

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

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

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 3

[fst (x₁, x₂) ] = x₁ + 2 x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = x₁ + 1

[len (x₁) ] = 2 x₁

[fst# (x₁, x₂) ] = x₁

[add (x₁, x₂) ] = x₁ + 2 x₂ + 1

[s (x₁) ] = 0

[ok (x₁) ] = x₁

[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).

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

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

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[fst (x₁, x₂) ] = x₁ + x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = x₁

[fst# (x₁, x₂) ] = 2 x₁

[add (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = 0

[ok (x₁) ] = 2 x₁

[top (x₁) ] = 0

[proper (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).

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

1.1.5.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 2

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[len (x₁) ] = x₁

[fst# (x₁, x₂) ] = 3 x₁ + 3 x₂

[add (x₁, x₂) ] = 2 x₁

[s (x₁) ] = 3 x₁ + 2

[ok (x₁) ] = 2 x₁ + 2

[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).

none

1.1.5.1.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

from#( ok( X ) ) → from#( X )

from#( mark( X ) ) → from#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = x₁ + 1

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 3

[from# (x₁) ] = x₁

[nil] = 2

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 3 x₁ + x₂ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 2

[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).

from#( mark( X ) ) → from#( X )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 2

[fst (x₁, x₂) ] = x₁ + 2 x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 3

[from# (x₁) ] = x₁

[nil] = 3

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = 2 x₁ + 2

[add (x₁, x₂) ] = x₁ + 2 x₂

[s (x₁) ] = 2

[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

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

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 3

[fst (x₁, x₂) ] = x₁ + 2 x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 2

[add# (x₁, x₂) ] = x₁

[nil] = 2

[cons (x₁, x₂) ] = x₁ + 1

[len (x₁) ] = 2 x₁

[add (x₁, x₂) ] = x₁ + 2 x₂ + 1

[s (x₁) ] = 0

[ok (x₁) ] = x₁

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

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[fst (x₁, x₂) ] = x₁ + x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[add# (x₁, x₂) ] = 2 x₁

[nil] = 0

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = x₁

[add (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = 0

[ok (x₁) ] = 2 x₁

[top (x₁) ] = 0

[proper (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).

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

1.1.7.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = 2 x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 2

[add# (x₁, x₂) ] = 3 x₁ + 3 x₂

[nil] = 2

[cons (x₁, x₂) ] = 2 x₁

[len (x₁) ] = x₁

[add (x₁, x₂) ] = 2 x₁

[s (x₁) ] = 3 x₁ + 2

[ok (x₁) ] = 2 x₁ + 2

[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).

none

1.1.7.1.1.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

len#( ok( X ) ) → len#( X )

len#( mark( X ) ) → len#( X )

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[fst (x₁, x₂) ] = x₁ + 1

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 3

[len# (x₁) ] = x₁

[nil] = 2

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = 2 x₁

[add (x₁, x₂) ] = 3 x₁ + x₂ + 2

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁ + 2

[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).

len#( mark( X ) ) → len#( X )

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 2

[fst (x₁, x₂) ] = x₁ + 2 x₂

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 3

[len# (x₁) ] = x₁

[nil] = 3

[cons (x₁, x₂) ] = x₁

[len (x₁) ] = 2 x₁ + 2

[add (x₁, x₂) ] = x₁ + 2 x₂

[s (x₁) ] = 2

[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

1.1.8.1.1: P is empty

All dependency pairs have been removed.

The 9th component contains the pair(s)

s#( ok( X ) ) → s#( X )

1.1.9: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[fst (x₁, x₂) ] = x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[0] = 0

[nil] = 0

[cons (x₁, x₂) ] = x₁

[s# (x₁) ] = 2 x₁

[len (x₁) ] = x₁

[add (x₁, x₂) ] = x₁

[s (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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

1.1.9.1: P is empty

All dependency pairs have been removed.

active#( fst( s( X ) , cons( Y , Z ) ) )	→	cons#( Y , fst( X , Z ) )
active#( fst( s( X ) , cons( Y , Z ) ) )	→	fst#( X , Z )
active#( from( X ) )	→	cons#( X , from( s( X ) ) )
active#( from( X ) )	→	from#( s( X ) )
active#( from( X ) )	→	s#( X )
active#( add( s( X ) , Y ) )	→	s#( add( X , Y ) )
active#( add( s( X ) , Y ) )	→	add#( X , Y )
active#( len( cons( X , Z ) ) )	→	s#( len( Z ) )
active#( len( cons( X , Z ) ) )	→	len#( Z )
active#( cons( X1 , X2 ) )	→	cons#( active( X1 ) , X2 )
active#( cons( X1 , X2 ) )	→	active#( X1 )
active#( fst( X1 , X2 ) )	→	fst#( active( X1 ) , X2 )
active#( fst( X1 , X2 ) )	→	active#( X1 )
active#( fst( X1 , X2 ) )	→	fst#( X1 , active( X2 ) )
active#( fst( X1 , X2 ) )	→	active#( X2 )
active#( from( X ) )	→	from#( active( X ) )
active#( from( 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#( len( X ) )	→	len#( active( X ) )
active#( len( X ) )	→	active#( X )
cons#( mark( X1 ) , X2 )	→	cons#( X1 , X2 )
fst#( mark( X1 ) , X2 )	→	fst#( X1 , X2 )
fst#( X1 , mark( X2 ) )	→	fst#( X1 , X2 )
from#( mark( X ) )	→	from#( X )
add#( mark( X1 ) , X2 )	→	add#( X1 , X2 )
add#( X1 , mark( X2 ) )	→	add#( X1 , X2 )
len#( mark( X ) )	→	len#( X )
proper#( s( X ) )	→	s#( proper( X ) )
proper#( s( 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#( fst( X1 , X2 ) )	→	fst#( proper( X1 ) , proper( X2 ) )
proper#( fst( X1 , X2 ) )	→	proper#( X1 )
proper#( fst( X1 , X2 ) )	→	proper#( X2 )
proper#( from( X ) )	→	from#( proper( X ) )
proper#( from( 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#( len( X ) )	→	len#( proper( X ) )
proper#( len( X ) )	→	proper#( X )
s#( ok( X ) )	→	s#( X )
cons#( ok( X1 ) , ok( X2 ) )	→	cons#( X1 , X2 )
fst#( ok( X1 ) , ok( X2 ) )	→	fst#( X1 , X2 )
from#( ok( X ) )	→	from#( X )
add#( ok( X1 ) , ok( X2 ) )	→	add#( X1 , X2 )
len#( ok( X ) )	→	len#( X )
top#( mark( X ) )	→	top#( proper( X ) )
top#( mark( X ) )	→	proper#( X )
top#( ok( X ) )	→	top#( active( X ) )
top#( ok( X ) )	→	active#( X )

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

[from (x₁) ]	=	2 x₁ + 1
[fst (x₁, x₂) ]	=	x₁
[mark (x₁) ]	=	0
[active (x₁) ]	=	x₁
[0]	=	2
[nil]	=	2
[cons (x₁, x₂) ]	=	2 x₁ + 1
[len (x₁) ]	=	2 x₁
[cons# (x₁, x₂) ]	=	x₁
[add (x₁, x₂) ]	=	2 x₁
[s (x₁) ]	=	x₁ + 2
[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

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

[from (x₁) ]	=	2 x₁ + 2
[fst (x₁, x₂) ]	=	x₁ + 2 x₂
[mark (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	2 x₁
[0]	=	3
[from# (x₁) ]	=	x₁
[nil]	=	3
[cons (x₁, x₂) ]	=	x₁
[len (x₁) ]	=	2 x₁ + 2
[add (x₁, x₂) ]	=	x₁ + 2 x₂
[s (x₁) ]	=	2
[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