Termination proof

1: switching to dependency pairs

eq#( s( x ) , s( y ) )	→	eq#( x , y )
union#( edge( x , y , i ) , h )	→	union#( i , h )
reach#( x , y , edge( u , v , i ) , h )	→	if_reach_1#( eq( x , u ) , x , y , edge( u , v , i ) , h )
reach#( x , y , edge( u , v , i ) , h )	→	eq#( x , u )
if_reach_1#( true , x , y , edge( u , v , i ) , h )	→	if_reach_2#( eq( y , v ) , x , y , edge( u , v , i ) , h )
if_reach_1#( true , x , y , edge( u , v , i ) , h )	→	eq#( y , v )
if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	or#( reach( x , y , i , h ) , reach( v , y , union( i , h ) , empty ) )
if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , h )
if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	reach#( v , y , union( i , h ) , empty )
if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	union#( i , h )
if_reach_1#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , edge( u , v , h ) )

1.1: dependency graph processor

The dependency pairs are split into 3 component(s).

• The 1st component contains the pair(s)

  reach#( x , y , edge( u , v , i ) , h )	→	if_reach_1#( eq( x , u ) , x , y , edge( u , v , i ) , h )
  if_reach_1#( true , x , y , edge( u , v , i ) , h )	→	if_reach_2#( eq( y , v ) , x , y , edge( u , v , i ) , h )
  if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , h )
  if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	reach#( v , y , union( i , h ) , empty )
  if_reach_1#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , edge( u , v , h ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [union (x1, x2) ]	=	x1 + x2
  [if_reach_2 (x1, ..., x5) ]	=	0
  [if_reach_1 (x1, ..., x5) ]	=	0
  [or (x1, x2) ]	=	2 x1
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [empty]	=	1
  [reach (x1, ..., x4) ]	=	0
  [true]	=	0
  [edge (x1, x2, x3) ]	=	x1 + 1
  [if_reach_2# (x1, ..., x5) ]	=	x1 + x2
  [false]	=	0
  [reach# (x1, ..., x4) ]	=	x1 + x2
  [s (x1) ]	=	0
  [if_reach_1# (x1, ..., x5) ]	=	x1 + x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  reach#( x , y , edge( u , v , i ) , h )	→	if_reach_1#( eq( x , u ) , x , y , edge( u , v , i ) , h )
  if_reach_1#( true , x , y , edge( u , v , i ) , h )	→	if_reach_2#( eq( y , v ) , x , y , edge( u , v , i ) , h )
  if_reach_2#( false , x , y , edge( u , v , i ) , h )	→	reach#( v , y , union( i , h ) , empty )
  if_reach_1#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , edge( u , v , h ) )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [union (x1, x2) ]	=	x1 + x2
  [if_reach_2 (x1, ..., x5) ]	=	0
  [if_reach_1 (x1, ..., x5) ]	=	0
  [or (x1, x2) ]	=	2 x1
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [empty]	=	0
  [reach (x1, ..., x4) ]	=	0
  [true]	=	0
  [edge (x1, x2, x3) ]	=	x1 + 1
  [if_reach_2# (x1, ..., x5) ]	=	x1 + x2
  [false]	=	0
  [reach# (x1, ..., x4) ]	=	x1 + x2
  [s (x1) ]	=	0
  [if_reach_1# (x1, ..., x5) ]	=	x1 + x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  reach#( x , y , edge( u , v , i ) , h )	→	if_reach_1#( eq( x , u ) , x , y , edge( u , v , i ) , h )
  if_reach_1#( true , x , y , edge( u , v , i ) , h )	→	if_reach_2#( eq( y , v ) , x , y , edge( u , v , i ) , h )
  if_reach_1#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , edge( u , v , h ) )

  1.1.1.1.1: dependency graph processor

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

  • The 1st component contains the pair(s)

    if_reach_1#( false , x , y , edge( u , v , i ) , h )	→	reach#( x , y , i , edge( u , v , h ) )
    reach#( x , y , edge( u , v , i ) , h )	→	if_reach_1#( eq( x , u ) , x , y , edge( u , v , i ) , h )

    1.1.1.1.1.1: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [union (x1, x2) ]	=	x1 + 2 x2
    [if_reach_2 (x1, ..., x5) ]	=	0
    [if_reach_1 (x1, ..., x5) ]	=	0
    [or (x1, x2) ]	=	2 x1
    [eq (x1, x2) ]	=	1
    [0]	=	0
    [empty]	=	0
    [reach (x1, ..., x4) ]	=	0
    [true]	=	0
    [edge (x1, x2, x3) ]	=	x1 + 1
    [false]	=	0
    [reach# (x1, ..., x4) ]	=	2 x1 + 1
    [s (x1) ]	=	0
    [if_reach_1# (x1, ..., x5) ]	=	2 x1
    [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

    one remains with the following pair(s).

    none

    1.1.1.1.1.1.1: P is empty

    All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  eq#( s( x ) , s( y ) )

  →

  eq#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [union (x1, x2) ]	=	2 x1 + 2 x2
  [if_reach_2 (x1, ..., x5) ]	=	0
  [if_reach_1 (x1, ..., x5) ]	=	0
  [or (x1, x2) ]	=	2 x1
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [eq# (x1, x2) ]	=	x1
  [empty]	=	0
  [reach (x1, ..., x4) ]	=	0
  [true]	=	0
  [edge (x1, x2, x3) ]	=	x1
  [false]	=	0
  [s (x1) ]	=	2 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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)

  union#( edge( x , y , i ) , h )

  →

  union#( i , h )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [union (x1, x2) ]	=	3 x1 + x2
  [if_reach_2 (x1, ..., x5) ]	=	0
  [if_reach_1 (x1, ..., x5) ]	=	0
  [or (x1, x2) ]	=	2 x1
  [eq (x1, x2) ]	=	3 x1
  [0]	=	0
  [empty]	=	0
  [reach (x1, ..., x4) ]	=	0
  [union# (x1, x2) ]	=	2 x1
  [true]	=	0
  [edge (x1, x2, x3) ]	=	2 x1 + 3 x2 + x3 + 2
  [false]	=	0
  [s (x1) ]	=	2 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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.