Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( minus( 0 , Y ) ) → mark#( 0 )

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

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

active#( geq( X , 0 ) ) → mark#( true )

active#( geq( 0 , s( Y ) ) ) → mark#( false )

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

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

active#( div( 0 , s( Y ) ) ) → mark#( 0 )

active#( div( s( X ) , s( Y ) ) ) → mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )

active#( div( s( X ) , s( Y ) ) ) → if#( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 )

active#( div( s( X ) , s( Y ) ) ) → geq#( X , Y )

active#( div( s( X ) , s( Y ) ) ) → s#( div( minus( X , Y ) , s( Y ) ) )

active#( div( s( X ) , s( Y ) ) ) → div#( minus( X , Y ) , s( Y ) )

active#( div( s( X ) , s( Y ) ) ) → minus#( X , Y )

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

active#( if( true , X , Y ) ) → mark#( X )

active#( if( false , X , Y ) ) → mark#( Y )

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

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

mark#( 0 ) → active#( 0 )

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

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

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

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

mark#( true ) → active#( true )

mark#( false ) → active#( false )

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

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

mark#( div( X1 , X2 ) ) → mark#( X1 )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → if#( mark( X1 ) , X2 , X3 )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

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

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

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

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

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

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

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

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

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

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

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

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

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

if#( mark( X1 ) , X2 , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , mark( X2 ) , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , X2 , mark( X3 ) ) → if#( X1 , X2 , X3 )

if#( active( X1 ) , X2 , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , active( X2 ) , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , X2 , active( X3 ) ) → if#( X1 , X2 , X3 )

1.1: dependency graph processor

The dependency pairs are split into 6 component(s).

The 1st component contains the pair(s)

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

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

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

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

active#( div( s( X ) , s( Y ) ) ) → mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )

mark#( s( X ) ) → mark#( X )

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

active#( if( true , X , Y ) ) → mark#( X )

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

active#( if( false , X , Y ) ) → mark#( Y )

mark#( div( X1 , X2 ) ) → mark#( X1 )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 1

[true] = 0

[mark (x₁) ] = 0

[if (x₁, x₂, x₃) ] = 1

[active# (x₁) ] = x₁

[false] = 0

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[s (x₁) ] = 0

[0] = 0

[geq (x₁, x₂) ] = 1

[div (x₁, x₂) ] = 1

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

one remains with the following pair(s).

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

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

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

active#( div( s( X ) , s( Y ) ) ) → mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )

mark#( s( X ) ) → mark#( X )

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

active#( if( true , X , Y ) ) → mark#( X )

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

active#( if( false , X , Y ) ) → mark#( Y )

mark#( div( X1 , X2 ) ) → mark#( X1 )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂ + x₃

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[s (x₁) ] = x₁

[0] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 2 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).

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

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

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

active#( div( s( X ) , s( Y ) ) ) → mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )

mark#( s( X ) ) → mark#( X )

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

active#( if( true , X , Y ) ) → mark#( X )

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

active#( if( false , X , Y ) ) → mark#( Y )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[if (x₁, x₂, x₃) ] = x₁ + x₂ + 2 x₃

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[s (x₁) ] = 2 x₁ + 2

[0] = 0

[geq (x₁, x₂) ] = 0

[div (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).

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

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

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

active#( div( s( X ) , s( Y ) ) ) → mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )

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

active#( if( true , X , Y ) ) → mark#( X )

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

active#( if( false , X , Y ) ) → mark#( Y )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[if (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[s (x₁) ] = 2

[0] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 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#( minus( s( X ) , s( Y ) ) ) → mark#( minus( X , Y ) )

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

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

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

active#( if( true , X , Y ) ) → mark#( X )

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

active#( if( false , X , Y ) ) → mark#( Y )

mark#( if( X1 , X2 , X3 ) ) → active#( if( mark( X1 ) , X2 , X3 ) )

mark#( if( X1 , X2 , X3 ) ) → mark#( X1 )

1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[if (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 2 x₃ + 3

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁ + 1

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = 3 x₁ + 1

[s (x₁) ] = 0

[0] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 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).

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = x₁

[if (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2 x₃

[active# (x₁) ] = 2 x₁

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[s (x₁) ] = 2 x₁ + 2

[0] = 0

[geq (x₁, x₂) ] = 2 x₁

[div (x₁, x₂) ] = 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#( minus( s( X ) , s( Y ) ) ) → mark#( minus( X , Y ) )

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

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

1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 1

[true] = 1

[mark (x₁) ] = 2 x₁

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[active# (x₁) ] = 2 x₁ + 3

[false] = 0

[active (x₁) ] = 2 x₁

[mark# (x₁) ] = 3 x₁ + 2

[s (x₁) ] = 0

[0] = 0

[geq (x₁, x₂) ] = 2

[div (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

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

1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = x₁

[true] = 0

[mark (x₁) ] = x₁

[if (x₁, x₂, x₃) ] = x₁ + 2 x₂

[active# (x₁) ] = 2 x₁

[false] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁ + 2

[s (x₁) ] = x₁ + 1

[0] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 2 x₁ + x₂

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

one remains with the following pair(s).

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

1.1.1.1.1.1.1.1.1.1.1: dependency graph processor

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

The 2nd component contains the pair(s)

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

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

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

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

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = 2 x₁ + 3

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[false] = 0

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[s (x₁) ] = 0

[minus# (x₁, x₂) ] = 3 x₁ + x₂

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 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.

The 3rd component contains the pair(s)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = 2 x₁ + 3

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[false] = 0

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[s (x₁) ] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = x₁

[s# (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: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

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

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

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

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

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = 2 x₁ + 3

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[false] = 0

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[s (x₁) ] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 0

[geq# (x₁, x₂) ] = 3 x₁ + x₂

[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: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

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

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

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

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

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = 2 x₁ + 3

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[false] = 0

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[s (x₁) ] = 0

[div# (x₁, x₂) ] = 3 x₁ + x₂

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 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: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

if#( X1 , mark( X2 ) , X3 ) → if#( X1 , X2 , X3 )

if#( mark( X1 ) , X2 , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , X2 , mark( X3 ) ) → if#( X1 , X2 , X3 )

if#( active( X1 ) , X2 , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , active( X2 ) , X3 ) → if#( X1 , X2 , X3 )

if#( X1 , X2 , active( X3 ) ) → if#( X1 , X2 , X3 )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[minus (x₁, x₂) ] = 0

[true] = 0

[mark (x₁) ] = 2 x₁ + 3

[if (x₁, x₂, x₃) ] = 2 x₁ + x₂

[false] = 0

[if# (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3 x₃

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[s (x₁) ] = 0

[geq (x₁, x₂) ] = 0

[div (x₁, x₂) ] = 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: P is empty

All dependency pairs have been removed.

active#( minus( 0 , Y ) )	→	mark#( 0 )
active#( minus( s( X ) , s( Y ) ) )	→	mark#( minus( X , Y ) )
active#( minus( s( X ) , s( Y ) ) )	→	minus#( X , Y )
active#( geq( X , 0 ) )	→	mark#( true )
active#( geq( 0 , s( Y ) ) )	→	mark#( false )
active#( geq( s( X ) , s( Y ) ) )	→	mark#( geq( X , Y ) )
active#( geq( s( X ) , s( Y ) ) )	→	geq#( X , Y )
active#( div( 0 , s( Y ) ) )	→	mark#( 0 )
active#( div( s( X ) , s( Y ) ) )	→	mark#( if( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 ) )
active#( div( s( X ) , s( Y ) ) )	→	if#( geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 )
active#( div( s( X ) , s( Y ) ) )	→	geq#( X , Y )
active#( div( s( X ) , s( Y ) ) )	→	s#( div( minus( X , Y ) , s( Y ) ) )
active#( div( s( X ) , s( Y ) ) )	→	div#( minus( X , Y ) , s( Y ) )
active#( div( s( X ) , s( Y ) ) )	→	minus#( X , Y )
active#( div( s( X ) , s( Y ) ) )	→	s#( Y )
active#( if( true , X , Y ) )	→	mark#( X )
active#( if( false , X , Y ) )	→	mark#( Y )
mark#( minus( X1 , X2 ) )	→	active#( minus( X1 , X2 ) )
mark#( minus( X1 , X2 ) )	→	minus#( X1 , X2 )
mark#( 0 )	→	active#( 0 )
mark#( s( X ) )	→	active#( s( mark( X ) ) )
mark#( s( X ) )	→	s#( mark( X ) )
mark#( s( X ) )	→	mark#( X )
mark#( geq( X1 , X2 ) )	→	active#( geq( X1 , X2 ) )
mark#( geq( X1 , X2 ) )	→	geq#( X1 , X2 )
mark#( true )	→	active#( true )
mark#( false )	→	active#( false )
mark#( div( X1 , X2 ) )	→	active#( div( mark( X1 ) , X2 ) )
mark#( div( X1 , X2 ) )	→	div#( mark( X1 ) , X2 )
mark#( div( X1 , X2 ) )	→	mark#( X1 )
mark#( if( X1 , X2 , X3 ) )	→	active#( if( mark( X1 ) , X2 , X3 ) )
mark#( if( X1 , X2 , X3 ) )	→	if#( mark( X1 ) , X2 , X3 )
mark#( if( X1 , X2 , X3 ) )	→	mark#( X1 )
minus#( mark( X1 ) , X2 )	→	minus#( X1 , X2 )
minus#( X1 , mark( X2 ) )	→	minus#( X1 , X2 )
minus#( active( X1 ) , X2 )	→	minus#( X1 , X2 )
minus#( X1 , active( X2 ) )	→	minus#( X1 , X2 )
s#( mark( X ) )	→	s#( X )
s#( active( X ) )	→	s#( X )
geq#( mark( X1 ) , X2 )	→	geq#( X1 , X2 )
geq#( X1 , mark( X2 ) )	→	geq#( X1 , X2 )
geq#( active( X1 ) , X2 )	→	geq#( X1 , X2 )
geq#( X1 , active( X2 ) )	→	geq#( X1 , X2 )
div#( mark( X1 ) , X2 )	→	div#( X1 , X2 )
div#( X1 , mark( X2 ) )	→	div#( X1 , X2 )
div#( active( X1 ) , X2 )	→	div#( X1 , X2 )
div#( X1 , active( X2 ) )	→	div#( X1 , X2 )
if#( mark( X1 ) , X2 , X3 )	→	if#( X1 , X2 , X3 )
if#( X1 , mark( X2 ) , X3 )	→	if#( X1 , X2 , X3 )
if#( X1 , X2 , mark( X3 ) )	→	if#( X1 , X2 , X3 )
if#( active( X1 ) , X2 , X3 )	→	if#( X1 , X2 , X3 )
if#( X1 , active( X2 ) , X3 )	→	if#( X1 , X2 , X3 )
if#( X1 , X2 , active( X3 ) )	→	if#( X1 , X2 , X3 )

[minus (x₁, x₂) ]	=	1
[true]	=	0
[mark (x₁) ]	=	0
[if (x₁, x₂, x₃) ]	=	1
[active# (x₁) ]	=	x₁
[false]	=	0
[active (x₁) ]	=	0
[mark# (x₁) ]	=	1
[s (x₁) ]	=	0
[0]	=	0
[geq (x₁, x₂) ]	=	1
[div (x₁, x₂) ]	=	1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n