/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  a!6220!6220filter : [o * o * o] --> o 
  a!6220!6220nats : [o] --> o 
  a!6220!6220sieve : [o] --> o 
  a!6220!6220zprimes : [] --> o 
  cons : [o * o] --> o 
  filter : [o * o * o] --> o 
  mark : [o] --> o 
  nats : [o] --> o 
  s : [o] --> o 
  sieve : [o] --> o 
  zprimes : [] --> o 

  a!6220!6220filter(cons(X, Y), 0, Z) => cons(0, filter(Y, Z, Z)) 
  a!6220!6220filter(cons(X, Y), s(Z), U) => cons(mark(X), filter(Y, Z, U)) 
  a!6220!6220sieve(cons(0, X)) => cons(0, sieve(X)) 
  a!6220!6220sieve(cons(s(X), Y)) => cons(s(mark(X)), sieve(filter(Y, X, X))) 
  a!6220!6220nats(X) => cons(mark(X), nats(s(X))) 
  a!6220!6220zprimes => a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
  mark(filter(X, Y, Z)) => a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
  mark(sieve(X)) => a!6220!6220sieve(mark(X)) 
  mark(nats(X)) => a!6220!6220nats(mark(X)) 
  mark(zprimes) => a!6220!6220zprimes 
  mark(cons(X, Y)) => cons(mark(X), Y) 
  mark(0) => 0 
  mark(s(X)) => s(mark(X)) 
  a!6220!6220filter(X, Y, Z) => filter(X, Y, Z) 
  a!6220!6220sieve(X) => sieve(X) 
  a!6220!6220nats(X) => nats(X) 
  a!6220!6220zprimes => zprimes 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] a!6220!6220filter#(cons(X, Y), s(Z), U) =#> mark#(X)   
    1] a!6220!6220sieve#(cons(s(X), Y)) =#> mark#(X)   
    2] a!6220!6220nats#(X) =#> mark#(X)   
    3] a!6220!6220zprimes# =#> a!6220!6220sieve#(a!6220!6220nats(s(s(0))))   
    4] a!6220!6220zprimes# =#> a!6220!6220nats#(s(s(0)))   
    5] mark#(filter(X, Y, Z)) =#> a!6220!6220filter#(mark(X), mark(Y), mark(Z))   
    6] mark#(filter(X, Y, Z)) =#> mark#(X)   
    7] mark#(filter(X, Y, Z)) =#> mark#(Y)   
    8] mark#(filter(X, Y, Z)) =#> mark#(Z)   
    9] mark#(sieve(X)) =#> a!6220!6220sieve#(mark(X))   
    10] mark#(sieve(X)) =#> mark#(X)   
    11] mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
    12] mark#(nats(X)) =#> mark#(X)   
    13] mark#(zprimes) =#> a!6220!6220zprimes#   
    14] mark#(cons(X, Y)) =#> mark#(X)   
    15] mark#(s(X)) =#> mark#(X)   

  Rules R_0:

    a!6220!6220filter(cons(X, Y), 0, Z) => cons(0, filter(Y, Z, Z)) 
    a!6220!6220filter(cons(X, Y), s(Z), U) => cons(mark(X), filter(Y, Z, U)) 
    a!6220!6220sieve(cons(0, X)) => cons(0, sieve(X)) 
    a!6220!6220sieve(cons(s(X), Y)) => cons(s(mark(X)), sieve(filter(Y, X, X))) 
    a!6220!6220nats(X) => cons(mark(X), nats(s(X))) 
    a!6220!6220zprimes => a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
    mark(filter(X, Y, Z)) => a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
    mark(sieve(X)) => a!6220!6220sieve(mark(X)) 
    mark(nats(X)) => a!6220!6220nats(mark(X)) 
    mark(zprimes) => a!6220!6220zprimes 
    mark(cons(X, Y)) => cons(mark(X), Y) 
    mark(0) => 0 
    mark(s(X)) => s(mark(X)) 
    a!6220!6220filter(X, Y, Z) => filter(X, Y, Z) 
    a!6220!6220sieve(X) => sieve(X) 
    a!6220!6220nats(X) => nats(X) 
    a!6220!6220zprimes => zprimes 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220filter#(cons(X, Y), s(Z), U) >? mark#(X) 
  a!6220!6220sieve#(cons(s(X), Y)) >? mark#(X) 
  a!6220!6220nats#(X) >? mark#(X) 
  a!6220!6220zprimes# >? a!6220!6220sieve#(a!6220!6220nats(s(s(0)))) 
  a!6220!6220zprimes# >? a!6220!6220nats#(s(s(0))) 
  mark#(filter(X, Y, Z)) >? a!6220!6220filter#(mark(X), mark(Y), mark(Z)) 
  mark#(filter(X, Y, Z)) >? mark#(X) 
  mark#(filter(X, Y, Z)) >? mark#(Y) 
  mark#(filter(X, Y, Z)) >? mark#(Z) 
  mark#(sieve(X)) >? a!6220!6220sieve#(mark(X)) 
  mark#(sieve(X)) >? mark#(X) 
  mark#(nats(X)) >? a!6220!6220nats#(mark(X)) 
  mark#(nats(X)) >? mark#(X) 
  mark#(zprimes) >? a!6220!6220zprimes# 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220filter(cons(X, Y), 0, Z) >= cons(0, filter(Y, Z, Z)) 
  a!6220!6220filter(cons(X, Y), s(Z), U) >= cons(mark(X), filter(Y, Z, U)) 
  a!6220!6220sieve(cons(0, X)) >= cons(0, sieve(X)) 
  a!6220!6220sieve(cons(s(X), Y)) >= cons(s(mark(X)), sieve(filter(Y, X, X))) 
  a!6220!6220nats(X) >= cons(mark(X), nats(s(X))) 
  a!6220!6220zprimes >= a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
  mark(filter(X, Y, Z)) >= a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
  mark(sieve(X)) >= a!6220!6220sieve(mark(X)) 
  mark(nats(X)) >= a!6220!6220nats(mark(X)) 
  mark(zprimes) >= a!6220!6220zprimes 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  mark(0) >= 0 
  mark(s(X)) >= s(mark(X)) 
  a!6220!6220filter(X, Y, Z) >= filter(X, Y, Z) 
  a!6220!6220sieve(X) >= sieve(X) 
  a!6220!6220nats(X) >= nats(X) 
  a!6220!6220zprimes >= zprimes 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220filter = \y0y1y2.1 + y2 + 2y0 + 2y1 
  a!6220!6220filter# = \y0y1y2.1 + 2y0 
  a!6220!6220nats = \y0.2y0 
  a!6220!6220nats# = \y0.2y0 
  a!6220!6220sieve = \y0.2y0 
  a!6220!6220sieve# = \y0.y0 
  a!6220!6220zprimes = 0 
  a!6220!6220zprimes# = 0 
  cons = \y0y1.y0 
  filter = \y0y1y2.1 + y2 + 2y0 + 2y1 
  mark = \y0.y0 
  mark# = \y0.2y0 
  nats = \y0.2y0 
  s = \y0.2y0 
  sieve = \y0.2y0 
  zprimes = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220filter#(cons(_x0, _x1), s(_x2), _x3)]] = 1 + 2x0 > 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220sieve#(cons(s(_x0), _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220nats#(_x0)]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220zprimes#]] = 0 >= 0 = [[a!6220!6220sieve#(a!6220!6220nats(s(s(0))))]] 
  [[a!6220!6220zprimes#]] = 0 >= 0 = [[a!6220!6220nats#(s(s(0)))]] 
  [[mark#(filter(_x0, _x1, _x2))]] = 2 + 2x2 + 4x0 + 4x1 > 1 + 2x0 = [[a!6220!6220filter#(mark(_x0), mark(_x1), mark(_x2))]] 
  [[mark#(filter(_x0, _x1, _x2))]] = 2 + 2x2 + 4x0 + 4x1 > 2x0 = [[mark#(_x0)]] 
  [[mark#(filter(_x0, _x1, _x2))]] = 2 + 2x2 + 4x0 + 4x1 > 2x1 = [[mark#(_x1)]] 
  [[mark#(filter(_x0, _x1, _x2))]] = 2 + 2x2 + 4x0 + 4x1 > 2x2 = [[mark#(_x2)]] 
  [[mark#(sieve(_x0))]] = 4x0 >= x0 = [[a!6220!6220sieve#(mark(_x0))]] 
  [[mark#(sieve(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(nats(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220nats#(mark(_x0))]] 
  [[mark#(nats(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(zprimes)]] = 0 >= 0 = [[a!6220!6220zprimes#]] 
  [[mark#(cons(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220filter(cons(_x0, _x1), 0, _x2)]] = 1 + x2 + 2x0 >= 0 = [[cons(0, filter(_x1, _x2, _x2))]] 
  [[a!6220!6220filter(cons(_x0, _x1), s(_x2), _x3)]] = 1 + x3 + 2x0 + 4x2 >= x0 = [[cons(mark(_x0), filter(_x1, _x2, _x3))]] 
  [[a!6220!6220sieve(cons(0, _x0))]] = 0 >= 0 = [[cons(0, sieve(_x0))]] 
  [[a!6220!6220sieve(cons(s(_x0), _x1))]] = 4x0 >= 2x0 = [[cons(s(mark(_x0)), sieve(filter(_x1, _x0, _x0)))]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= x0 = [[cons(mark(_x0), nats(s(_x0)))]] 
  [[a!6220!6220zprimes]] = 0 >= 0 = [[a!6220!6220sieve(a!6220!6220nats(s(s(0))))]] 
  [[mark(filter(_x0, _x1, _x2))]] = 1 + x2 + 2x0 + 2x1 >= 1 + x2 + 2x0 + 2x1 = [[a!6220!6220filter(mark(_x0), mark(_x1), mark(_x2))]] 
  [[mark(sieve(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220sieve(mark(_x0))]] 
  [[mark(nats(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220nats(mark(_x0))]] 
  [[mark(zprimes)]] = 0 >= 0 = [[a!6220!6220zprimes]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[cons(mark(_x0), _x1)]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[a!6220!6220filter(_x0, _x1, _x2)]] = 1 + x2 + 2x0 + 2x1 >= 1 + x2 + 2x0 + 2x1 = [[filter(_x0, _x1, _x2)]] 
  [[a!6220!6220sieve(_x0)]] = 2x0 >= 2x0 = [[sieve(_x0)]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= 2x0 = [[nats(_x0)]] 
  [[a!6220!6220zprimes]] = 0 >= 0 = [[zprimes]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  a!6220!6220sieve#(cons(s(X), Y)) =#> mark#(X)   
  a!6220!6220nats#(X) =#> mark#(X)   
  a!6220!6220zprimes# =#> a!6220!6220sieve#(a!6220!6220nats(s(s(0))))   
  a!6220!6220zprimes# =#> a!6220!6220nats#(s(s(0)))   
  mark#(sieve(X)) =#> a!6220!6220sieve#(mark(X))   
  mark#(sieve(X)) =#> mark#(X)   
  mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
  mark#(nats(X)) =#> mark#(X)   
  mark#(zprimes) =#> a!6220!6220zprimes#   
  mark#(cons(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220sieve#(cons(s(X), Y)) >? mark#(X) 
  a!6220!6220nats#(X) >? mark#(X) 
  a!6220!6220zprimes# >? a!6220!6220sieve#(a!6220!6220nats(s(s(0)))) 
  a!6220!6220zprimes# >? a!6220!6220nats#(s(s(0))) 
  mark#(sieve(X)) >? a!6220!6220sieve#(mark(X)) 
  mark#(sieve(X)) >? mark#(X) 
  mark#(nats(X)) >? a!6220!6220nats#(mark(X)) 
  mark#(nats(X)) >? mark#(X) 
  mark#(zprimes) >? a!6220!6220zprimes# 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220filter(cons(X, Y), 0, Z) >= cons(0, filter(Y, Z, Z)) 
  a!6220!6220filter(cons(X, Y), s(Z), U) >= cons(mark(X), filter(Y, Z, U)) 
  a!6220!6220sieve(cons(0, X)) >= cons(0, sieve(X)) 
  a!6220!6220sieve(cons(s(X), Y)) >= cons(s(mark(X)), sieve(filter(Y, X, X))) 
  a!6220!6220nats(X) >= cons(mark(X), nats(s(X))) 
  a!6220!6220zprimes >= a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
  mark(filter(X, Y, Z)) >= a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
  mark(sieve(X)) >= a!6220!6220sieve(mark(X)) 
  mark(nats(X)) >= a!6220!6220nats(mark(X)) 
  mark(zprimes) >= a!6220!6220zprimes 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  mark(0) >= 0 
  mark(s(X)) >= s(mark(X)) 
  a!6220!6220filter(X, Y, Z) >= filter(X, Y, Z) 
  a!6220!6220sieve(X) >= sieve(X) 
  a!6220!6220nats(X) >= nats(X) 
  a!6220!6220zprimes >= zprimes 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220filter = \y0y1y2.2 + 2y0 + 2y1 + 2y2 
  a!6220!6220nats = \y0.2y0 
  a!6220!6220nats# = \y0.2y0 
  a!6220!6220sieve = \y0.2y0 
  a!6220!6220sieve# = \y0.2y0 
  a!6220!6220zprimes = 2 
  a!6220!6220zprimes# = 1 
  cons = \y0y1.y0 
  filter = \y0y1y2.2 + 2y0 + 2y1 + 2y2 
  mark = \y0.y0 
  mark# = \y0.y0 
  nats = \y0.2y0 
  s = \y0.y0 
  sieve = \y0.2y0 
  zprimes = 2 

Using this interpretation, the requirements translate to:

  [[a!6220!6220sieve#(cons(s(_x0), _x1))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[a!6220!6220nats#(_x0)]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[a!6220!6220zprimes#]] = 1 > 0 = [[a!6220!6220sieve#(a!6220!6220nats(s(s(0))))]] 
  [[a!6220!6220zprimes#]] = 1 > 0 = [[a!6220!6220nats#(s(s(0)))]] 
  [[mark#(sieve(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220sieve#(mark(_x0))]] 
  [[mark#(sieve(_x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(nats(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220nats#(mark(_x0))]] 
  [[mark#(nats(_x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(zprimes)]] = 2 > 1 = [[a!6220!6220zprimes#]] 
  [[mark#(cons(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[a!6220!6220filter(cons(_x0, _x1), 0, _x2)]] = 2 + 2x0 + 2x2 >= 0 = [[cons(0, filter(_x1, _x2, _x2))]] 
  [[a!6220!6220filter(cons(_x0, _x1), s(_x2), _x3)]] = 2 + 2x0 + 2x2 + 2x3 >= x0 = [[cons(mark(_x0), filter(_x1, _x2, _x3))]] 
  [[a!6220!6220sieve(cons(0, _x0))]] = 0 >= 0 = [[cons(0, sieve(_x0))]] 
  [[a!6220!6220sieve(cons(s(_x0), _x1))]] = 2x0 >= x0 = [[cons(s(mark(_x0)), sieve(filter(_x1, _x0, _x0)))]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= x0 = [[cons(mark(_x0), nats(s(_x0)))]] 
  [[a!6220!6220zprimes]] = 2 >= 0 = [[a!6220!6220sieve(a!6220!6220nats(s(s(0))))]] 
  [[mark(filter(_x0, _x1, _x2))]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[a!6220!6220filter(mark(_x0), mark(_x1), mark(_x2))]] 
  [[mark(sieve(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220sieve(mark(_x0))]] 
  [[mark(nats(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220nats(mark(_x0))]] 
  [[mark(zprimes)]] = 2 >= 2 = [[a!6220!6220zprimes]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[cons(mark(_x0), _x1)]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[a!6220!6220filter(_x0, _x1, _x2)]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[filter(_x0, _x1, _x2)]] 
  [[a!6220!6220sieve(_x0)]] = 2x0 >= 2x0 = [[sieve(_x0)]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= 2x0 = [[nats(_x0)]] 
  [[a!6220!6220zprimes]] = 2 >= 2 = [[zprimes]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_2, R_0, minimal, formative), where P_2 consists of:

  a!6220!6220sieve#(cons(s(X), Y)) =#> mark#(X)   
  a!6220!6220nats#(X) =#> mark#(X)   
  mark#(sieve(X)) =#> a!6220!6220sieve#(mark(X))   
  mark#(sieve(X)) =#> mark#(X)   
  mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
  mark#(nats(X)) =#> mark#(X)   
  mark#(cons(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220sieve#(cons(s(X), Y)) >? mark#(X) 
  a!6220!6220nats#(X) >? mark#(X) 
  mark#(sieve(X)) >? a!6220!6220sieve#(mark(X)) 
  mark#(sieve(X)) >? mark#(X) 
  mark#(nats(X)) >? a!6220!6220nats#(mark(X)) 
  mark#(nats(X)) >? mark#(X) 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220filter(cons(X, Y), 0, Z) >= cons(0, filter(Y, Z, Z)) 
  a!6220!6220filter(cons(X, Y), s(Z), U) >= cons(mark(X), filter(Y, Z, U)) 
  a!6220!6220sieve(cons(0, X)) >= cons(0, sieve(X)) 
  a!6220!6220sieve(cons(s(X), Y)) >= cons(s(mark(X)), sieve(filter(Y, X, X))) 
  a!6220!6220nats(X) >= cons(mark(X), nats(s(X))) 
  a!6220!6220zprimes >= a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
  mark(filter(X, Y, Z)) >= a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
  mark(sieve(X)) >= a!6220!6220sieve(mark(X)) 
  mark(nats(X)) >= a!6220!6220nats(mark(X)) 
  mark(zprimes) >= a!6220!6220zprimes 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  mark(0) >= 0 
  mark(s(X)) >= s(mark(X)) 
  a!6220!6220filter(X, Y, Z) >= filter(X, Y, Z) 
  a!6220!6220sieve(X) >= sieve(X) 
  a!6220!6220nats(X) >= nats(X) 
  a!6220!6220zprimes >= zprimes 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220filter = \y0y1y2.y0 + 2y2 
  a!6220!6220nats = \y0.2y0 
  a!6220!6220nats# = \y0.2y0 
  a!6220!6220sieve = \y0.1 + 2y0 
  a!6220!6220sieve# = \y0.y0 
  a!6220!6220zprimes = 2 
  cons = \y0y1.y0 
  filter = \y0y1y2.y0 + 2y2 
  mark = \y0.y0 
  mark# = \y0.2y0 
  nats = \y0.2y0 
  s = \y0.2y0 
  sieve = \y0.1 + 2y0 
  zprimes = 2 

Using this interpretation, the requirements translate to:

  [[a!6220!6220sieve#(cons(s(_x0), _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220nats#(_x0)]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(sieve(_x0))]] = 2 + 4x0 > x0 = [[a!6220!6220sieve#(mark(_x0))]] 
  [[mark#(sieve(_x0))]] = 2 + 4x0 > 2x0 = [[mark#(_x0)]] 
  [[mark#(nats(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220nats#(mark(_x0))]] 
  [[mark#(nats(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(cons(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220filter(cons(_x0, _x1), 0, _x2)]] = x0 + 2x2 >= 0 = [[cons(0, filter(_x1, _x2, _x2))]] 
  [[a!6220!6220filter(cons(_x0, _x1), s(_x2), _x3)]] = x0 + 2x3 >= x0 = [[cons(mark(_x0), filter(_x1, _x2, _x3))]] 
  [[a!6220!6220sieve(cons(0, _x0))]] = 1 >= 0 = [[cons(0, sieve(_x0))]] 
  [[a!6220!6220sieve(cons(s(_x0), _x1))]] = 1 + 4x0 >= 2x0 = [[cons(s(mark(_x0)), sieve(filter(_x1, _x0, _x0)))]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= x0 = [[cons(mark(_x0), nats(s(_x0)))]] 
  [[a!6220!6220zprimes]] = 2 >= 1 = [[a!6220!6220sieve(a!6220!6220nats(s(s(0))))]] 
  [[mark(filter(_x0, _x1, _x2))]] = x0 + 2x2 >= x0 + 2x2 = [[a!6220!6220filter(mark(_x0), mark(_x1), mark(_x2))]] 
  [[mark(sieve(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[a!6220!6220sieve(mark(_x0))]] 
  [[mark(nats(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220nats(mark(_x0))]] 
  [[mark(zprimes)]] = 2 >= 2 = [[a!6220!6220zprimes]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[cons(mark(_x0), _x1)]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[a!6220!6220filter(_x0, _x1, _x2)]] = x0 + 2x2 >= x0 + 2x2 = [[filter(_x0, _x1, _x2)]] 
  [[a!6220!6220sieve(_x0)]] = 1 + 2x0 >= 1 + 2x0 = [[sieve(_x0)]] 
  [[a!6220!6220nats(_x0)]] = 2x0 >= 2x0 = [[nats(_x0)]] 
  [[a!6220!6220zprimes]] = 2 >= 2 = [[zprimes]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_0, minimal, formative) by (P_3, R_0, minimal, formative), where P_3 consists of:

  a!6220!6220sieve#(cons(s(X), Y)) =#> mark#(X)   
  a!6220!6220nats#(X) =#> mark#(X)   
  mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
  mark#(nats(X)) =#> mark#(X)   
  mark#(cons(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 2, 3, 4, 5 
    * 1 : 2, 3, 4, 5 
    * 2 : 1 
    * 3 : 2, 3, 4, 5 
    * 4 : 2, 3, 4, 5 
    * 5 : 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_4:

    a!6220!6220nats#(X) =#> mark#(X)   
    mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
    mark#(nats(X)) =#> mark#(X)   
    mark#(cons(X, Y)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_0, m, f) by (P_4, R_0, m, f).

Thus, the original system is terminating if (P_4, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220nats#(X) >? mark#(X) 
  mark#(nats(X)) >? a!6220!6220nats#(mark(X)) 
  mark#(nats(X)) >? mark#(X) 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220filter(cons(X, Y), 0, Z) >= cons(0, filter(Y, Z, Z)) 
  a!6220!6220filter(cons(X, Y), s(Z), U) >= cons(mark(X), filter(Y, Z, U)) 
  a!6220!6220sieve(cons(0, X)) >= cons(0, sieve(X)) 
  a!6220!6220sieve(cons(s(X), Y)) >= cons(s(mark(X)), sieve(filter(Y, X, X))) 
  a!6220!6220nats(X) >= cons(mark(X), nats(s(X))) 
  a!6220!6220zprimes >= a!6220!6220sieve(a!6220!6220nats(s(s(0)))) 
  mark(filter(X, Y, Z)) >= a!6220!6220filter(mark(X), mark(Y), mark(Z)) 
  mark(sieve(X)) >= a!6220!6220sieve(mark(X)) 
  mark(nats(X)) >= a!6220!6220nats(mark(X)) 
  mark(zprimes) >= a!6220!6220zprimes 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  mark(0) >= 0 
  mark(s(X)) >= s(mark(X)) 
  a!6220!6220filter(X, Y, Z) >= filter(X, Y, Z) 
  a!6220!6220sieve(X) >= sieve(X) 
  a!6220!6220nats(X) >= nats(X) 
  a!6220!6220zprimes >= zprimes 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220filter = \y0y1y2.2y0 
  a!6220!6220nats = \y0.1 + y0 
  a!6220!6220nats# = \y0.1 + y0 
  a!6220!6220sieve = \y0.2y0 
  a!6220!6220zprimes = 2 
  cons = \y0y1.1 + y0 
  filter = \y0y1y2.2y0 
  mark = \y0.y0 
  mark# = \y0.y0 
  nats = \y0.1 + y0 
  s = \y0.y0 
  sieve = \y0.2y0 
  zprimes = 2 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats#(_x0)]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[mark#(nats(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220nats#(mark(_x0))]] 
  [[mark#(nats(_x0))]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[mark#(cons(_x0, _x1))]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[a!6220!6220filter(cons(_x0, _x1), 0, _x2)]] = 2 + 2x0 >= 1 = [[cons(0, filter(_x1, _x2, _x2))]] 
  [[a!6220!6220filter(cons(_x0, _x1), s(_x2), _x3)]] = 2 + 2x0 >= 1 + x0 = [[cons(mark(_x0), filter(_x1, _x2, _x3))]] 
  [[a!6220!6220sieve(cons(0, _x0))]] = 2 >= 1 = [[cons(0, sieve(_x0))]] 
  [[a!6220!6220sieve(cons(s(_x0), _x1))]] = 2 + 2x0 >= 1 + x0 = [[cons(s(mark(_x0)), sieve(filter(_x1, _x0, _x0)))]] 
  [[a!6220!6220nats(_x0)]] = 1 + x0 >= 1 + x0 = [[cons(mark(_x0), nats(s(_x0)))]] 
  [[a!6220!6220zprimes]] = 2 >= 2 = [[a!6220!6220sieve(a!6220!6220nats(s(s(0))))]] 
  [[mark(filter(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[a!6220!6220filter(mark(_x0), mark(_x1), mark(_x2))]] 
  [[mark(sieve(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220sieve(mark(_x0))]] 
  [[mark(nats(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220nats(mark(_x0))]] 
  [[mark(zprimes)]] = 2 >= 2 = [[a!6220!6220zprimes]] 
  [[mark(cons(_x0, _x1))]] = 1 + x0 >= 1 + x0 = [[cons(mark(_x0), _x1)]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[a!6220!6220filter(_x0, _x1, _x2)]] = 2x0 >= 2x0 = [[filter(_x0, _x1, _x2)]] 
  [[a!6220!6220sieve(_x0)]] = 2x0 >= 2x0 = [[sieve(_x0)]] 
  [[a!6220!6220nats(_x0)]] = 1 + x0 >= 1 + x0 = [[nats(_x0)]] 
  [[a!6220!6220zprimes]] = 2 >= 2 = [[zprimes]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_0, minimal, formative) by (P_5, R_0, minimal, formative), where P_5 consists of:

  mark#(nats(X)) =#> a!6220!6220nats#(mark(X))   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if (P_5, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 : 0, 1 

This graph has the following strongly connected components:

  P_6:

    mark#(s(X)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_5, R_0, m, f) by (P_6, R_0, m, f).

Thus, the original system is terminating if (P_6, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(mark#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(mark#(s(X))) = s(X) |> X = nu(mark#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_6, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.