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


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YES
We consider the system theBenchmark.

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

  0 : [] --> o 
  activate : [o] --> o 
  cons : [o * o] --> o 
  from : [o] --> o 
  minus : [o * o] --> o 
  n!6220!6220from : [o] --> o 
  n!6220!6220s : [o] --> o 
  n!6220!6220zWquot : [o * o] --> o 
  nil : [] --> o 
  quot : [o * o] --> o 
  s : [o] --> o 
  sel : [o * o] --> o 
  zWquot : [o * o] --> o 

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  sel(0, cons(X, Y)) => X 
  sel(s(X), cons(Y, Z)) => sel(X, activate(Z)) 
  minus(X, 0) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  quot(0, s(X)) => 0 
  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
  zWquot(X, nil) => nil 
  zWquot(nil, X) => nil 
  zWquot(cons(X, Y), cons(Z, U)) => cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  zWquot(X, Y) => n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) => zWquot(activate(X), activate(Y)) 
  activate(X) => X 

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] sel#(s(X), cons(Y, Z)) =#> sel#(X, activate(Z))   
    1] sel#(s(X), cons(Y, Z)) =#> activate#(Z)   
    2] minus#(s(X), s(Y)) =#> minus#(X, Y)   
    3] quot#(s(X), s(Y)) =#> s#(quot(minus(X, Y), s(Y)))   
    4] quot#(s(X), s(Y)) =#> quot#(minus(X, Y), s(Y))   
    5] quot#(s(X), s(Y)) =#> minus#(X, Y)   
    6] quot#(s(X), s(Y)) =#> s#(Y)   
    7] zWquot#(cons(X, Y), cons(Z, U)) =#> quot#(X, Z)   
    8] zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(Y)   
    9] zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(U)   
    10] activate#(n!6220!6220from(X)) =#> from#(activate(X))   
    11] activate#(n!6220!6220from(X)) =#> activate#(X)   
    12] activate#(n!6220!6220s(X)) =#> s#(activate(X))   
    13] activate#(n!6220!6220s(X)) =#> activate#(X)   
    14] activate#(n!6220!6220zWquot(X, Y)) =#> zWquot#(activate(X), activate(Y))   
    15] activate#(n!6220!6220zWquot(X, Y)) =#> activate#(X)   
    16] activate#(n!6220!6220zWquot(X, Y)) =#> activate#(Y)   

  Rules R_0:

    from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
    sel(0, cons(X, Y)) => X 
    sel(s(X), cons(Y, Z)) => sel(X, activate(Z)) 
    minus(X, 0) => 0 
    minus(s(X), s(Y)) => minus(X, Y) 
    quot(0, s(X)) => 0 
    quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
    zWquot(X, nil) => nil 
    zWquot(nil, X) => nil 
    zWquot(cons(X, Y), cons(Z, U)) => cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
    from(X) => n!6220!6220from(X) 
    s(X) => n!6220!6220s(X) 
    zWquot(X, Y) => n!6220!6220zWquot(X, Y) 
    activate(n!6220!6220from(X)) => from(activate(X)) 
    activate(n!6220!6220s(X)) => s(activate(X)) 
    activate(n!6220!6220zWquot(X, Y)) => zWquot(activate(X), activate(Y)) 
    activate(X) => X 

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

    * 0 : 0, 1 
    * 1 : 10, 11, 12, 13, 14, 15, 16 
    * 2 : 2 
    * 3 :  
    * 4 :  
    * 5 : 2 
    * 6 :  
    * 7 : 3, 4, 5, 6 
    * 8 : 10, 11, 12, 13, 14, 15, 16 
    * 9 : 10, 11, 12, 13, 14, 15, 16 
    * 10 :  
    * 11 : 10, 11, 12, 13, 14, 15, 16 
    * 12 :  
    * 13 : 10, 11, 12, 13, 14, 15, 16 
    * 14 : 7, 8, 9 
    * 15 : 10, 11, 12, 13, 14, 15, 16 
    * 16 : 10, 11, 12, 13, 14, 15, 16 

This graph has the following strongly connected components:

  P_1:

    sel#(s(X), cons(Y, Z)) =#> sel#(X, activate(Z))   

  P_2:

    minus#(s(X), s(Y)) =#> minus#(X, Y)   

  P_3:

    zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(Y)   
    zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(U)   
    activate#(n!6220!6220from(X)) =#> activate#(X)   
    activate#(n!6220!6220s(X)) =#> activate#(X)   
    activate#(n!6220!6220zWquot(X, Y)) =#> zWquot#(activate(X), activate(Y))   
    activate#(n!6220!6220zWquot(X, Y)) =#> activate#(X)   
    activate#(n!6220!6220zWquot(X, Y)) =#> activate#(Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

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

The formative rules of (P_3, R_0) are R_1 ::=

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  sel(0, cons(X, Y)) => X 
  sel(s(X), cons(Y, Z)) => sel(X, activate(Z)) 
  minus(X, 0) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  quot(0, s(X)) => 0 
  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
  zWquot(cons(X, Y), cons(Z, U)) => cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  zWquot(X, Y) => n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) => zWquot(activate(X), activate(Y)) 
  activate(X) => X 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_3, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_3, R_1) are:

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  minus(X, 0) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  quot(0, s(X)) => 0 
  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
  zWquot(cons(X, Y), cons(Z, U)) => cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  zWquot(X, Y) => n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) => zWquot(activate(X), activate(Y)) 
  activate(X) => X 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  zWquot#(cons(X, Y), cons(Z, U)) >? activate#(Y) 
  zWquot#(cons(X, Y), cons(Z, U)) >? activate#(U) 
  activate#(n!6220!6220from(X)) >? activate#(X) 
  activate#(n!6220!6220s(X)) >? activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >? zWquot#(activate(X), activate(Y)) 
  activate#(n!6220!6220zWquot(X, Y)) >? activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >? activate#(Y) 
  from(X) >= cons(X, n!6220!6220from(n!6220!6220s(X))) 
  minus(X, 0) >= 0 
  minus(s(X), s(Y)) >= minus(X, Y) 
  quot(0, s(X)) >= 0 
  quot(s(X), s(Y)) >= s(quot(minus(X, Y), s(Y))) 
  zWquot(cons(X, Y), cons(Z, U)) >= cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) >= n!6220!6220from(X) 
  s(X) >= n!6220!6220s(X) 
  zWquot(X, Y) >= n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) >= from(activate(X)) 
  activate(n!6220!6220s(X)) >= s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) >= zWquot(activate(X), activate(Y)) 
  activate(X) >= X 

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

We consider usable_rules with respect to the following argument filtering:

  cons(x_1,x_2) = cons(x_2) 

This leaves the following ordering requirements:

  zWquot#(cons(X, Y), cons(Z, U)) >= activate#(Y) 
  zWquot#(cons(X, Y), cons(Z, U)) >= activate#(U) 
  activate#(n!6220!6220from(X)) > activate#(X) 
  activate#(n!6220!6220s(X)) >= activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >= zWquot#(activate(X), activate(Y)) 
  activate#(n!6220!6220zWquot(X, Y)) >= activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >= activate#(Y) 
  from(X) >= cons(X, n!6220!6220from(n!6220!6220s(X))) 
  zWquot(cons(X, Y), cons(Z, U)) >= cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) >= n!6220!6220from(X) 
  s(X) >= n!6220!6220s(X) 
  zWquot(X, Y) >= n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) >= from(activate(X)) 
  activate(n!6220!6220s(X)) >= s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) >= zWquot(activate(X), activate(Y)) 
  activate(X) >= X 

The following interpretation satisfies the requirements:

  0 = 0 
  activate = \y0.y0 
  activate# = \y0.y0 
  cons = \y0y1.y1 
  from = \y0.2 + 2y0 
  minus = \y0y1.0 
  n!6220!6220from = \y0.2 + 2y0 
  n!6220!6220s = \y0.y0 
  n!6220!6220zWquot = \y0y1.2y0 + 2y1 
  quot = \y0y1.0 
  s = \y0.y0 
  zWquot = \y0y1.2y0 + 2y1 
  zWquot# = \y0y1.2y0 + 2y1 

Using this interpretation, the requirements translate to:

  [[zWquot#(cons(_x0, _x1), cons(_x2, _x3))]] = 2x1 + 2x3 >= x1 = [[activate#(_x1)]] 
  [[zWquot#(cons(_x0, _x1), cons(_x2, _x3))]] = 2x1 + 2x3 >= x3 = [[activate#(_x3)]] 
  [[activate#(n!6220!6220from(_x0))]] = 2 + 2x0 > x0 = [[activate#(_x0)]] 
  [[activate#(n!6220!6220s(_x0))]] = x0 >= x0 = [[activate#(_x0)]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[zWquot#(activate(_x0), activate(_x1))]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 2x0 + 2x1 >= x0 = [[activate#(_x0)]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 2x0 + 2x1 >= x1 = [[activate#(_x1)]] 
  [[from(_x0)]] = 2 + 2x0 >= 2 + 2x0 = [[cons(_x0, n!6220!6220from(n!6220!6220s(_x0)))]] 
  [[zWquot(cons(_x0, _x1), cons(_x2, _x3))]] = 2x1 + 2x3 >= 2x1 + 2x3 = [[cons(quot(_x0, _x2), n!6220!6220zWquot(activate(_x1), activate(_x3)))]] 
  [[from(_x0)]] = 2 + 2x0 >= 2 + 2x0 = [[n!6220!6220from(_x0)]] 
  [[s(_x0)]] = x0 >= x0 = [[n!6220!6220s(_x0)]] 
  [[zWquot(_x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[n!6220!6220zWquot(_x0, _x1)]] 
  [[activate(n!6220!6220from(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[from(activate(_x0))]] 
  [[activate(n!6220!6220s(_x0))]] = x0 >= x0 = [[s(activate(_x0))]] 
  [[activate(n!6220!6220zWquot(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[zWquot(activate(_x0), activate(_x1))]] 
  [[activate(_x0)]] = x0 >= x0 = [[_x0]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_1, minimal, formative) by (P_4, R_1, minimal, formative), where P_4 consists of:

  zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(Y)   
  zWquot#(cons(X, Y), cons(Z, U)) =#> activate#(U)   
  activate#(n!6220!6220s(X)) =#> activate#(X)   
  activate#(n!6220!6220zWquot(X, Y)) =#> zWquot#(activate(X), activate(Y))   
  activate#(n!6220!6220zWquot(X, Y)) =#> activate#(X)   
  activate#(n!6220!6220zWquot(X, Y)) =#> activate#(Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_1, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_4, R_1) are:

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  minus(X, 0) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  quot(0, s(X)) => 0 
  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
  zWquot(cons(X, Y), cons(Z, U)) => cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  zWquot(X, Y) => n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) => zWquot(activate(X), activate(Y)) 
  activate(X) => X 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  zWquot#(cons(X, Y), cons(Z, U)) >? activate#(Y) 
  zWquot#(cons(X, Y), cons(Z, U)) >? activate#(U) 
  activate#(n!6220!6220s(X)) >? activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >? zWquot#(activate(X), activate(Y)) 
  activate#(n!6220!6220zWquot(X, Y)) >? activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >? activate#(Y) 
  from(X) >= cons(X, n!6220!6220from(n!6220!6220s(X))) 
  minus(X, 0) >= 0 
  minus(s(X), s(Y)) >= minus(X, Y) 
  quot(0, s(X)) >= 0 
  quot(s(X), s(Y)) >= s(quot(minus(X, Y), s(Y))) 
  zWquot(cons(X, Y), cons(Z, U)) >= cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) >= n!6220!6220from(X) 
  s(X) >= n!6220!6220s(X) 
  zWquot(X, Y) >= n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) >= from(activate(X)) 
  activate(n!6220!6220s(X)) >= s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) >= zWquot(activate(X), activate(Y)) 
  activate(X) >= X 

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

We consider usable_rules with respect to the following argument filtering:

  cons(x_1,x_2) = cons(x_2) 

This leaves the following ordering requirements:

  zWquot#(cons(X, Y), cons(Z, U)) >= activate#(Y) 
  zWquot#(cons(X, Y), cons(Z, U)) > activate#(U) 
  activate#(n!6220!6220s(X)) >= activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >= zWquot#(activate(X), activate(Y)) 
  activate#(n!6220!6220zWquot(X, Y)) >= activate#(X) 
  activate#(n!6220!6220zWquot(X, Y)) >= activate#(Y) 
  from(X) >= cons(X, n!6220!6220from(n!6220!6220s(X))) 
  zWquot(cons(X, Y), cons(Z, U)) >= cons(quot(X, Z), n!6220!6220zWquot(activate(Y), activate(U))) 
  from(X) >= n!6220!6220from(X) 
  s(X) >= n!6220!6220s(X) 
  zWquot(X, Y) >= n!6220!6220zWquot(X, Y) 
  activate(n!6220!6220from(X)) >= from(activate(X)) 
  activate(n!6220!6220s(X)) >= s(activate(X)) 
  activate(n!6220!6220zWquot(X, Y)) >= zWquot(activate(X), activate(Y)) 
  activate(X) >= X 

The following interpretation satisfies the requirements:

  0 = 0 
  activate = \y0.y0 
  activate# = \y0.y0 
  cons = \y0y1.y1 
  from = \y0.0 
  minus = \y0y1.0 
  n!6220!6220from = \y0.0 
  n!6220!6220s = \y0.2y0 
  n!6220!6220zWquot = \y0y1.1 + y1 + 2y0 
  quot = \y0y1.0 
  s = \y0.2y0 
  zWquot = \y0y1.1 + y1 + 2y0 
  zWquot# = \y0y1.1 + y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[zWquot#(cons(_x0, _x1), cons(_x2, _x3))]] = 1 + x3 + 2x1 > x1 = [[activate#(_x1)]] 
  [[zWquot#(cons(_x0, _x1), cons(_x2, _x3))]] = 1 + x3 + 2x1 > x3 = [[activate#(_x3)]] 
  [[activate#(n!6220!6220s(_x0))]] = 2x0 >= x0 = [[activate#(_x0)]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[zWquot#(activate(_x0), activate(_x1))]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 1 + x1 + 2x0 > x0 = [[activate#(_x0)]] 
  [[activate#(n!6220!6220zWquot(_x0, _x1))]] = 1 + x1 + 2x0 > x1 = [[activate#(_x1)]] 
  [[from(_x0)]] = 0 >= 0 = [[cons(_x0, n!6220!6220from(n!6220!6220s(_x0)))]] 
  [[zWquot(cons(_x0, _x1), cons(_x2, _x3))]] = 1 + x3 + 2x1 >= 1 + x3 + 2x1 = [[cons(quot(_x0, _x2), n!6220!6220zWquot(activate(_x1), activate(_x3)))]] 
  [[from(_x0)]] = 0 >= 0 = [[n!6220!6220from(_x0)]] 
  [[s(_x0)]] = 2x0 >= 2x0 = [[n!6220!6220s(_x0)]] 
  [[zWquot(_x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[n!6220!6220zWquot(_x0, _x1)]] 
  [[activate(n!6220!6220from(_x0))]] = 0 >= 0 = [[from(activate(_x0))]] 
  [[activate(n!6220!6220s(_x0))]] = 2x0 >= 2x0 = [[s(activate(_x0))]] 
  [[activate(n!6220!6220zWquot(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[zWquot(activate(_x0), activate(_x1))]] 
  [[activate(_x0)]] = x0 >= x0 = [[_x0]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_1, minimal, formative) by (P_5, R_1, minimal, formative), where P_5 consists of:

  activate#(n!6220!6220s(X)) =#> activate#(X)   
  activate#(n!6220!6220zWquot(X, Y)) =#> zWquot#(activate(X), activate(Y))   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_5, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_1, 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 : 0, 1 
    * 1 :  

This graph has the following strongly connected components:

  P_6:

    activate#(n!6220!6220s(X)) =#> activate#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_5, R_1, m, f) by (P_6, R_1, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_6, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(activate#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(activate#(n!6220!6220s(X))) = n!6220!6220s(X) |> X = nu(activate#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_6, R_1, minimal, f) by ({}, R_1, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(minus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(minus#(s(X), s(Y))) = s(X) |> X = nu(minus#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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 apply the subterm criterion with the following projection function:

  nu(sel#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(sel#(s(X), cons(Y, Z))) = s(X) |> X = nu(sel#(X, activate(Z))) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, 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.