/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.

  active : [o] --> o 
  app : [o * o] --> o 
  cons : [o * o] --> o 
  from : [o] --> o 
  mark : [o] --> o 
  nil : [] --> o 
  prefix : [o] --> o 
  s : [o] --> o 
  zWadr : [o * o] --> o 

  active(app(nil, X)) => mark(X) 
  active(app(cons(X, Y), Z)) => mark(cons(X, app(Y, Z))) 
  active(from(X)) => mark(cons(X, from(s(X)))) 
  active(zWadr(nil, X)) => mark(nil) 
  active(zWadr(X, nil)) => mark(nil) 
  active(zWadr(cons(X, Y), cons(Z, U))) => mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active(prefix(X)) => mark(cons(nil, zWadr(X, prefix(X)))) 
  mark(app(X, Y)) => active(app(mark(X), mark(Y))) 
  mark(nil) => active(nil) 
  mark(cons(X, Y)) => active(cons(mark(X), Y)) 
  mark(from(X)) => active(from(mark(X))) 
  mark(s(X)) => active(s(mark(X))) 
  mark(zWadr(X, Y)) => active(zWadr(mark(X), mark(Y))) 
  mark(prefix(X)) => active(prefix(mark(X))) 
  app(mark(X), Y) => app(X, Y) 
  app(X, mark(Y)) => app(X, Y) 
  app(active(X), Y) => app(X, Y) 
  app(X, active(Y)) => app(X, Y) 
  cons(mark(X), Y) => cons(X, Y) 
  cons(X, mark(Y)) => cons(X, Y) 
  cons(active(X), Y) => cons(X, Y) 
  cons(X, active(Y)) => cons(X, Y) 
  from(mark(X)) => from(X) 
  from(active(X)) => from(X) 
  s(mark(X)) => s(X) 
  s(active(X)) => s(X) 
  zWadr(mark(X), Y) => zWadr(X, Y) 
  zWadr(X, mark(Y)) => zWadr(X, Y) 
  zWadr(active(X), Y) => zWadr(X, Y) 
  zWadr(X, active(Y)) => zWadr(X, Y) 
  prefix(mark(X)) => prefix(X) 
  prefix(active(X)) => prefix(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] active#(app(nil, X)) =#> mark#(X)   
    1] active#(app(cons(X, Y), Z)) =#> mark#(cons(X, app(Y, Z)))   
    2] active#(app(cons(X, Y), Z)) =#> cons#(X, app(Y, Z))   
    3] active#(app(cons(X, Y), Z)) =#> app#(Y, Z)   
    4] active#(from(X)) =#> mark#(cons(X, from(s(X))))   
    5] active#(from(X)) =#> cons#(X, from(s(X)))   
    6] active#(from(X)) =#> from#(s(X))   
    7] active#(from(X)) =#> s#(X)   
    8] active#(zWadr(nil, X)) =#> mark#(nil)   
    9] active#(zWadr(X, nil)) =#> mark#(nil)   
    10] active#(zWadr(cons(X, Y), cons(Z, U))) =#> mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U)))   
    11] active#(zWadr(cons(X, Y), cons(Z, U))) =#> cons#(app(Z, cons(X, nil)), zWadr(Y, U))   
    12] active#(zWadr(cons(X, Y), cons(Z, U))) =#> app#(Z, cons(X, nil))   
    13] active#(zWadr(cons(X, Y), cons(Z, U))) =#> cons#(X, nil)   
    14] active#(zWadr(cons(X, Y), cons(Z, U))) =#> zWadr#(Y, U)   
    15] active#(prefix(X)) =#> mark#(cons(nil, zWadr(X, prefix(X))))   
    16] active#(prefix(X)) =#> cons#(nil, zWadr(X, prefix(X)))   
    17] active#(prefix(X)) =#> zWadr#(X, prefix(X))   
    18] active#(prefix(X)) =#> prefix#(X)   
    19] mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
    20] mark#(app(X, Y)) =#> app#(mark(X), mark(Y))   
    21] mark#(app(X, Y)) =#> mark#(X)   
    22] mark#(app(X, Y)) =#> mark#(Y)   
    23] mark#(nil) =#> active#(nil)   
    24] mark#(cons(X, Y)) =#> active#(cons(mark(X), Y))   
    25] mark#(cons(X, Y)) =#> cons#(mark(X), Y)   
    26] mark#(cons(X, Y)) =#> mark#(X)   
    27] mark#(from(X)) =#> active#(from(mark(X)))   
    28] mark#(from(X)) =#> from#(mark(X))   
    29] mark#(from(X)) =#> mark#(X)   
    30] mark#(s(X)) =#> active#(s(mark(X)))   
    31] mark#(s(X)) =#> s#(mark(X))   
    32] mark#(s(X)) =#> mark#(X)   
    33] mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   
    34] mark#(zWadr(X, Y)) =#> zWadr#(mark(X), mark(Y))   
    35] mark#(zWadr(X, Y)) =#> mark#(X)   
    36] mark#(zWadr(X, Y)) =#> mark#(Y)   
    37] mark#(prefix(X)) =#> active#(prefix(mark(X)))   
    38] mark#(prefix(X)) =#> prefix#(mark(X))   
    39] mark#(prefix(X)) =#> mark#(X)   
    40] app#(mark(X), Y) =#> app#(X, Y)   
    41] app#(X, mark(Y)) =#> app#(X, Y)   
    42] app#(active(X), Y) =#> app#(X, Y)   
    43] app#(X, active(Y)) =#> app#(X, Y)   
    44] cons#(mark(X), Y) =#> cons#(X, Y)   
    45] cons#(X, mark(Y)) =#> cons#(X, Y)   
    46] cons#(active(X), Y) =#> cons#(X, Y)   
    47] cons#(X, active(Y)) =#> cons#(X, Y)   
    48] from#(mark(X)) =#> from#(X)   
    49] from#(active(X)) =#> from#(X)   
    50] s#(mark(X)) =#> s#(X)   
    51] s#(active(X)) =#> s#(X)   
    52] zWadr#(mark(X), Y) =#> zWadr#(X, Y)   
    53] zWadr#(X, mark(Y)) =#> zWadr#(X, Y)   
    54] zWadr#(active(X), Y) =#> zWadr#(X, Y)   
    55] zWadr#(X, active(Y)) =#> zWadr#(X, Y)   
    56] prefix#(mark(X)) =#> prefix#(X)   
    57] prefix#(active(X)) =#> prefix#(X)   

  Rules R_0:

    active(app(nil, X)) => mark(X) 
    active(app(cons(X, Y), Z)) => mark(cons(X, app(Y, Z))) 
    active(from(X)) => mark(cons(X, from(s(X)))) 
    active(zWadr(nil, X)) => mark(nil) 
    active(zWadr(X, nil)) => mark(nil) 
    active(zWadr(cons(X, Y), cons(Z, U))) => mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
    active(prefix(X)) => mark(cons(nil, zWadr(X, prefix(X)))) 
    mark(app(X, Y)) => active(app(mark(X), mark(Y))) 
    mark(nil) => active(nil) 
    mark(cons(X, Y)) => active(cons(mark(X), Y)) 
    mark(from(X)) => active(from(mark(X))) 
    mark(s(X)) => active(s(mark(X))) 
    mark(zWadr(X, Y)) => active(zWadr(mark(X), mark(Y))) 
    mark(prefix(X)) => active(prefix(mark(X))) 
    app(mark(X), Y) => app(X, Y) 
    app(X, mark(Y)) => app(X, Y) 
    app(active(X), Y) => app(X, Y) 
    app(X, active(Y)) => app(X, Y) 
    cons(mark(X), Y) => cons(X, Y) 
    cons(X, mark(Y)) => cons(X, Y) 
    cons(active(X), Y) => cons(X, Y) 
    cons(X, active(Y)) => cons(X, Y) 
    from(mark(X)) => from(X) 
    from(active(X)) => from(X) 
    s(mark(X)) => s(X) 
    s(active(X)) => s(X) 
    zWadr(mark(X), Y) => zWadr(X, Y) 
    zWadr(X, mark(Y)) => zWadr(X, Y) 
    zWadr(active(X), Y) => zWadr(X, Y) 
    zWadr(X, active(Y)) => zWadr(X, Y) 
    prefix(mark(X)) => prefix(X) 
    prefix(active(X)) => prefix(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 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 1 : 24, 25, 26 
    * 2 : 44, 46 
    * 3 : 40, 41, 42, 43 
    * 4 : 24, 25, 26 
    * 5 : 44, 46 
    * 6 :  
    * 7 : 50, 51 
    * 8 : 23 
    * 9 : 23 
    * 10 : 24, 25, 26 
    * 11 :  
    * 12 : 40, 42 
    * 13 : 44, 46 
    * 14 : 52, 53, 54, 55 
    * 15 : 24, 25, 26 
    * 16 :  
    * 17 : 52, 54 
    * 18 : 56, 57 
    * 19 : 0, 1, 2, 3 
    * 20 : 40, 41, 42, 43 
    * 21 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 22 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 23 :  
    * 24 :  
    * 25 : 44, 45, 46, 47 
    * 26 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 27 : 4, 5, 6, 7 
    * 28 : 48, 49 
    * 29 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 30 :  
    * 31 : 50, 51 
    * 32 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 33 : 8, 9, 10, 11, 12, 13, 14 
    * 34 : 52, 53, 54, 55 
    * 35 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 36 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 37 : 15, 16, 17, 18 
    * 38 : 56, 57 
    * 39 : 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 
    * 40 : 40, 41, 42, 43 
    * 41 : 40, 41, 42, 43 
    * 42 : 40, 41, 42, 43 
    * 43 : 40, 41, 42, 43 
    * 44 : 44, 45, 46, 47 
    * 45 : 44, 45, 46, 47 
    * 46 : 44, 45, 46, 47 
    * 47 : 44, 45, 46, 47 
    * 48 : 48, 49 
    * 49 : 48, 49 
    * 50 : 50, 51 
    * 51 : 50, 51 
    * 52 : 52, 53, 54, 55 
    * 53 : 52, 53, 54, 55 
    * 54 : 52, 53, 54, 55 
    * 55 : 52, 53, 54, 55 
    * 56 : 56, 57 
    * 57 : 56, 57 

This graph has the following strongly connected components:

  P_1:

    active#(app(nil, X)) =#> mark#(X)   
    active#(app(cons(X, Y), Z)) =#> mark#(cons(X, app(Y, Z)))   
    active#(from(X)) =#> mark#(cons(X, from(s(X))))   
    active#(zWadr(cons(X, Y), cons(Z, U))) =#> mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U)))   
    active#(prefix(X)) =#> mark#(cons(nil, zWadr(X, prefix(X))))   
    mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
    mark#(app(X, Y)) =#> mark#(X)   
    mark#(app(X, Y)) =#> mark#(Y)   
    mark#(cons(X, Y)) =#> mark#(X)   
    mark#(from(X)) =#> active#(from(mark(X)))   
    mark#(from(X)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   
    mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   
    mark#(zWadr(X, Y)) =#> mark#(X)   
    mark#(zWadr(X, Y)) =#> mark#(Y)   
    mark#(prefix(X)) =#> active#(prefix(mark(X)))   
    mark#(prefix(X)) =#> mark#(X)   

  P_2:

    app#(mark(X), Y) =#> app#(X, Y)   
    app#(X, mark(Y)) =#> app#(X, Y)   
    app#(active(X), Y) =#> app#(X, Y)   
    app#(X, active(Y)) =#> app#(X, Y)   

  P_3:

    cons#(mark(X), Y) =#> cons#(X, Y)   
    cons#(X, mark(Y)) =#> cons#(X, Y)   
    cons#(active(X), Y) =#> cons#(X, Y)   
    cons#(X, active(Y)) =#> cons#(X, Y)   

  P_4:

    from#(mark(X)) =#> from#(X)   
    from#(active(X)) =#> from#(X)   

  P_5:

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

  P_6:

    zWadr#(mark(X), Y) =#> zWadr#(X, Y)   
    zWadr#(X, mark(Y)) =#> zWadr#(X, Y)   
    zWadr#(active(X), Y) =#> zWadr#(X, Y)   
    zWadr#(X, active(Y)) =#> zWadr#(X, Y)   

  P_7:

    prefix#(mark(X)) =#> prefix#(X)   
    prefix#(active(X)) =#> prefix#(X)   

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), (P_3, R_0, m, f), (P_4, R_0, m, f), (P_5, R_0, m, f), (P_6, R_0, m, f) and (P_7, 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), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative), (P_6, R_0, minimal, formative) and (P_7, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(prefix#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(prefix#(mark(X))) = mark(X) |> X = nu(prefix#(X)) 
  nu(prefix#(active(X))) = active(X) |> X = nu(prefix#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_7, 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 each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (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(zWadr#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(zWadr#(mark(X), Y)) = mark(X) |> X = nu(zWadr#(X, Y)) 
  nu(zWadr#(X, mark(Y))) = X = X = nu(zWadr#(X, Y)) 
  nu(zWadr#(active(X), Y)) = active(X) |> X = nu(zWadr#(X, Y)) 
  nu(zWadr#(X, active(Y))) = X = X = nu(zWadr#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_6, R_0, minimal, f) by (P_8, R_0, minimal, f), where P_8 contains:

  zWadr#(X, mark(Y)) =#> zWadr#(X, Y)   
  zWadr#(X, active(Y)) =#> zWadr#(X, Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_8, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_8, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(zWadr#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(zWadr#(X, mark(Y))) = mark(Y) |> Y = nu(zWadr#(X, Y)) 
  nu(zWadr#(X, active(Y))) = active(Y) |> Y = nu(zWadr#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_8, 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 each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative) and (P_5, R_0, minimal, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(s#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_5, 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 each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative) and (P_4, R_0, minimal, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(from#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(from#(mark(X))) = mark(X) |> X = nu(from#(X)) 
  nu(from#(active(X))) = active(X) |> X = nu(from#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_4, 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 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).

We apply the subterm criterion with the following projection function:

  nu(cons#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(cons#(mark(X), Y)) = mark(X) |> X = nu(cons#(X, Y)) 
  nu(cons#(X, mark(Y))) = X = X = nu(cons#(X, Y)) 
  nu(cons#(active(X), Y)) = active(X) |> X = nu(cons#(X, Y)) 
  nu(cons#(X, active(Y))) = X = X = nu(cons#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_3, R_0, minimal, f) by (P_9, R_0, minimal, f), where P_9 contains:

  cons#(X, mark(Y)) =#> cons#(X, Y)   
  cons#(X, active(Y)) =#> cons#(X, Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_9, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_9, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(cons#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(cons#(X, mark(Y))) = mark(Y) |> Y = nu(cons#(X, Y)) 
  nu(cons#(X, active(Y))) = active(Y) |> Y = nu(cons#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_9, 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 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(app#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(app#(mark(X), Y)) = mark(X) |> X = nu(app#(X, Y)) 
  nu(app#(X, mark(Y))) = X = X = nu(app#(X, Y)) 
  nu(app#(active(X), Y)) = active(X) |> X = nu(app#(X, Y)) 
  nu(app#(X, active(Y))) = X = X = nu(app#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by (P_10, R_0, minimal, f), where P_10 contains:

  app#(X, mark(Y)) =#> app#(X, Y)   
  app#(X, active(Y)) =#> app#(X, Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_10, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_10, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(app#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(app#(X, mark(Y))) = mark(Y) |> Y = nu(app#(X, Y)) 
  nu(app#(X, active(Y))) = active(Y) |> Y = nu(app#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_10, 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 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:

  active#(app(nil, X)) >? mark#(X) 
  active#(app(cons(X, Y), Z)) >? mark#(cons(X, app(Y, Z))) 
  active#(from(X)) >? mark#(cons(X, from(s(X)))) 
  active#(zWadr(cons(X, Y), cons(Z, U))) >? mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active#(prefix(X)) >? mark#(cons(nil, zWadr(X, prefix(X)))) 
  mark#(app(X, Y)) >? active#(app(mark(X), mark(Y))) 
  mark#(app(X, Y)) >? mark#(X) 
  mark#(app(X, Y)) >? mark#(Y) 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(from(X)) >? active#(from(mark(X))) 
  mark#(from(X)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? active#(zWadr(mark(X), mark(Y))) 
  mark#(zWadr(X, Y)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? mark#(Y) 
  mark#(prefix(X)) >? active#(prefix(mark(X))) 
  mark#(prefix(X)) >? mark#(X) 
  active(app(nil, X)) >= mark(X) 
  active(app(cons(X, Y), Z)) >= mark(cons(X, app(Y, Z))) 
  active(from(X)) >= mark(cons(X, from(s(X)))) 
  active(zWadr(nil, X)) >= mark(nil) 
  active(zWadr(X, nil)) >= mark(nil) 
  active(zWadr(cons(X, Y), cons(Z, U))) >= mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active(prefix(X)) >= mark(cons(nil, zWadr(X, prefix(X)))) 
  mark(app(X, Y)) >= active(app(mark(X), mark(Y))) 
  mark(nil) >= active(nil) 
  mark(cons(X, Y)) >= active(cons(mark(X), Y)) 
  mark(from(X)) >= active(from(mark(X))) 
  mark(s(X)) >= active(s(mark(X))) 
  mark(zWadr(X, Y)) >= active(zWadr(mark(X), mark(Y))) 
  mark(prefix(X)) >= active(prefix(mark(X))) 
  app(mark(X), Y) >= app(X, Y) 
  app(X, mark(Y)) >= app(X, Y) 
  app(active(X), Y) >= app(X, Y) 
  app(X, active(Y)) >= app(X, Y) 
  cons(mark(X), Y) >= cons(X, Y) 
  cons(X, mark(Y)) >= cons(X, Y) 
  cons(active(X), Y) >= cons(X, Y) 
  cons(X, active(Y)) >= cons(X, Y) 
  from(mark(X)) >= from(X) 
  from(active(X)) >= from(X) 
  s(mark(X)) >= s(X) 
  s(active(X)) >= s(X) 
  zWadr(mark(X), Y) >= zWadr(X, Y) 
  zWadr(X, mark(Y)) >= zWadr(X, Y) 
  zWadr(active(X), Y) >= zWadr(X, Y) 
  zWadr(X, active(Y)) >= zWadr(X, Y) 
  prefix(mark(X)) >= prefix(X) 
  prefix(active(X)) >= prefix(X) 

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

The following interpretation satisfies the requirements:

  active = \y0.y0 
  active# = \y0.y0 
  app = \y0y1.y0 + 2y1 
  cons = \y0y1.y0 
  from = \y0.1 + y0 
  mark = \y0.y0 
  mark# = \y0.y0 
  nil = 0 
  prefix = \y0.1 + 2y0 
  s = \y0.y0 
  zWadr = \y0y1.y1 + 2y0 

Using this interpretation, the requirements translate to:

  [[active#(app(nil, _x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[active#(app(cons(_x0, _x1), _x2))]] = x0 + 2x2 >= x0 = [[mark#(cons(_x0, app(_x1, _x2)))]] 
  [[active#(from(_x0))]] = 1 + x0 > x0 = [[mark#(cons(_x0, from(s(_x0))))]] 
  [[active#(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = x2 + 2x0 >= x2 + 2x0 = [[mark#(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[active#(prefix(_x0))]] = 1 + 2x0 > 0 = [[mark#(cons(nil, zWadr(_x0, prefix(_x0))))]] 
  [[mark#(app(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[active#(app(mark(_x0), mark(_x1)))]] 
  [[mark#(app(_x0, _x1))]] = x0 + 2x1 >= x0 = [[mark#(_x0)]] 
  [[mark#(app(_x0, _x1))]] = x0 + 2x1 >= x1 = [[mark#(_x1)]] 
  [[mark#(cons(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(from(_x0))]] = 1 + x0 >= 1 + x0 = [[active#(from(mark(_x0)))]] 
  [[mark#(from(_x0))]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[active#(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark#(zWadr(_x0, _x1))]] = x1 + 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = x1 + 2x0 >= x1 = [[mark#(_x1)]] 
  [[mark#(prefix(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[active#(prefix(mark(_x0)))]] 
  [[mark#(prefix(_x0))]] = 1 + 2x0 > x0 = [[mark#(_x0)]] 
  [[active(app(nil, _x0))]] = 2x0 >= x0 = [[mark(_x0)]] 
  [[active(app(cons(_x0, _x1), _x2))]] = x0 + 2x2 >= x0 = [[mark(cons(_x0, app(_x1, _x2)))]] 
  [[active(from(_x0))]] = 1 + x0 >= x0 = [[mark(cons(_x0, from(s(_x0))))]] 
  [[active(zWadr(nil, _x0))]] = x0 >= 0 = [[mark(nil)]] 
  [[active(zWadr(_x0, nil))]] = 2x0 >= 0 = [[mark(nil)]] 
  [[active(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = x2 + 2x0 >= x2 + 2x0 = [[mark(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[active(prefix(_x0))]] = 1 + 2x0 >= 0 = [[mark(cons(nil, zWadr(_x0, prefix(_x0))))]] 
  [[mark(app(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[active(app(mark(_x0), mark(_x1)))]] 
  [[mark(nil)]] = 0 >= 0 = [[active(nil)]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[active(cons(mark(_x0), _x1))]] 
  [[mark(from(_x0))]] = 1 + x0 >= 1 + x0 = [[active(from(mark(_x0)))]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[active(s(mark(_x0)))]] 
  [[mark(zWadr(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[active(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark(prefix(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[active(prefix(mark(_x0)))]] 
  [[app(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[app(_x0, _x1)]] 
  [[app(_x0, mark(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[app(_x0, _x1)]] 
  [[app(active(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[app(_x0, _x1)]] 
  [[app(_x0, active(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[app(_x0, _x1)]] 
  [[cons(mark(_x0), _x1)]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, mark(_x1))]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(active(_x0), _x1)]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, active(_x1))]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[from(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[from(_x0)]] 
  [[from(active(_x0))]] = 1 + x0 >= 1 + x0 = [[from(_x0)]] 
  [[s(mark(_x0))]] = x0 >= x0 = [[s(_x0)]] 
  [[s(active(_x0))]] = x0 >= x0 = [[s(_x0)]] 
  [[zWadr(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, mark(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[zWadr(_x0, _x1)]] 
  [[zWadr(active(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, active(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[zWadr(_x0, _x1)]] 
  [[prefix(mark(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[prefix(_x0)]] 
  [[prefix(active(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[prefix(_x0)]] 

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_11, R_0, minimal, formative), where P_11 consists of:

  active#(app(nil, X)) =#> mark#(X)   
  active#(app(cons(X, Y), Z)) =#> mark#(cons(X, app(Y, Z)))   
  active#(zWadr(cons(X, Y), cons(Z, U))) =#> mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U)))   
  mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
  mark#(app(X, Y)) =#> mark#(X)   
  mark#(app(X, Y)) =#> mark#(Y)   
  mark#(cons(X, Y)) =#> mark#(X)   
  mark#(from(X)) =#> active#(from(mark(X)))   
  mark#(s(X)) =#> mark#(X)   
  mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   
  mark#(zWadr(X, Y)) =#> mark#(X)   
  mark#(zWadr(X, Y)) =#> mark#(Y)   
  mark#(prefix(X)) =#> active#(prefix(mark(X)))   

Thus, the original system is terminating if (P_11, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_11, 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 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 1 : 6 
    * 2 : 6 
    * 3 : 0, 1 
    * 4 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 5 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 6 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 7 :  
    * 8 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 9 : 2 
    * 10 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 11 : 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 
    * 12 :  

This graph has the following strongly connected components:

  P_12:

    active#(app(nil, X)) =#> mark#(X)   
    active#(app(cons(X, Y), Z)) =#> mark#(cons(X, app(Y, Z)))   
    active#(zWadr(cons(X, Y), cons(Z, U))) =#> mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U)))   
    mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
    mark#(app(X, Y)) =#> mark#(X)   
    mark#(app(X, Y)) =#> mark#(Y)   
    mark#(cons(X, Y)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   
    mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   
    mark#(zWadr(X, Y)) =#> mark#(X)   
    mark#(zWadr(X, Y)) =#> mark#(Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_11, R_0, m, f) by (P_12, R_0, m, f).

Thus, the original system is terminating if (P_12, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_12, R_0, minimal, formative).

The formative rules of (P_12, R_0) are R_1 ::=

  active(app(nil, X)) => mark(X) 
  active(app(cons(X, Y), Z)) => mark(cons(X, app(Y, Z))) 
  active(from(X)) => mark(cons(X, from(s(X)))) 
  active(zWadr(nil, X)) => mark(nil) 
  active(zWadr(X, nil)) => mark(nil) 
  active(zWadr(cons(X, Y), cons(Z, U))) => mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active(prefix(X)) => mark(cons(nil, zWadr(X, prefix(X)))) 
  mark(app(X, Y)) => active(app(mark(X), mark(Y))) 
  mark(nil) => active(nil) 
  mark(cons(X, Y)) => active(cons(mark(X), Y)) 
  mark(from(X)) => active(from(mark(X))) 
  mark(s(X)) => active(s(mark(X))) 
  mark(zWadr(X, Y)) => active(zWadr(mark(X), mark(Y))) 
  mark(prefix(X)) => active(prefix(mark(X))) 
  app(mark(X), Y) => app(X, Y) 
  app(X, mark(Y)) => app(X, Y) 
  app(active(X), Y) => app(X, Y) 
  app(X, active(Y)) => app(X, Y) 
  cons(mark(X), Y) => cons(X, Y) 
  cons(X, mark(Y)) => cons(X, Y) 
  cons(active(X), Y) => cons(X, Y) 
  cons(X, active(Y)) => cons(X, Y) 
  s(mark(X)) => s(X) 
  s(active(X)) => s(X) 
  zWadr(mark(X), Y) => zWadr(X, Y) 
  zWadr(X, mark(Y)) => zWadr(X, Y) 
  zWadr(active(X), Y) => zWadr(X, Y) 
  zWadr(X, active(Y)) => zWadr(X, Y) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_12, R_0, minimal, formative) by (P_12, R_1, minimal, formative).

Thus, the original system is terminating if (P_12, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_12, R_1, 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:

  active#(app(nil, X)) >? mark#(X) 
  active#(app(cons(X, Y), Z)) >? mark#(cons(X, app(Y, Z))) 
  active#(zWadr(cons(X, Y), cons(Z, U))) >? mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  mark#(app(X, Y)) >? active#(app(mark(X), mark(Y))) 
  mark#(app(X, Y)) >? mark#(X) 
  mark#(app(X, Y)) >? mark#(Y) 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? active#(zWadr(mark(X), mark(Y))) 
  mark#(zWadr(X, Y)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? mark#(Y) 
  active(app(nil, X)) >= mark(X) 
  active(app(cons(X, Y), Z)) >= mark(cons(X, app(Y, Z))) 
  active(from(X)) >= mark(cons(X, from(s(X)))) 
  active(zWadr(nil, X)) >= mark(nil) 
  active(zWadr(X, nil)) >= mark(nil) 
  active(zWadr(cons(X, Y), cons(Z, U))) >= mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active(prefix(X)) >= mark(cons(nil, zWadr(X, prefix(X)))) 
  mark(app(X, Y)) >= active(app(mark(X), mark(Y))) 
  mark(nil) >= active(nil) 
  mark(cons(X, Y)) >= active(cons(mark(X), Y)) 
  mark(from(X)) >= active(from(mark(X))) 
  mark(s(X)) >= active(s(mark(X))) 
  mark(zWadr(X, Y)) >= active(zWadr(mark(X), mark(Y))) 
  mark(prefix(X)) >= active(prefix(mark(X))) 
  app(mark(X), Y) >= app(X, Y) 
  app(X, mark(Y)) >= app(X, Y) 
  app(active(X), Y) >= app(X, Y) 
  app(X, active(Y)) >= app(X, Y) 
  cons(mark(X), Y) >= cons(X, Y) 
  cons(X, mark(Y)) >= cons(X, Y) 
  cons(active(X), Y) >= cons(X, Y) 
  cons(X, active(Y)) >= cons(X, Y) 
  s(mark(X)) >= s(X) 
  s(active(X)) >= s(X) 
  zWadr(mark(X), Y) >= zWadr(X, Y) 
  zWadr(X, mark(Y)) >= zWadr(X, Y) 
  zWadr(active(X), Y) >= zWadr(X, Y) 
  zWadr(X, active(Y)) >= zWadr(X, Y) 

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

The following interpretation satisfies the requirements:

  active = \y0.y0 
  active# = \y0.y0 
  app = \y0y1.y0 + y1 
  cons = \y0y1.y0 
  from = \y0.y0 
  mark = \y0.y0 
  mark# = \y0.y0 
  nil = 2 
  prefix = \y0.2 
  s = \y0.2y0 
  zWadr = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[active#(app(nil, _x0))]] = 2 + x0 > x0 = [[mark#(_x0)]] 
  [[active#(app(cons(_x0, _x1), _x2))]] = x0 + x2 >= x0 = [[mark#(cons(_x0, app(_x1, _x2)))]] 
  [[active#(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = x0 + x2 >= x0 + x2 = [[mark#(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[mark#(app(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[active#(app(mark(_x0), mark(_x1)))]] 
  [[mark#(app(_x0, _x1))]] = x0 + x1 >= x0 = [[mark#(_x0)]] 
  [[mark#(app(_x0, _x1))]] = x0 + x1 >= x1 = [[mark#(_x1)]] 
  [[mark#(cons(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[active#(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark#(zWadr(_x0, _x1))]] = x0 + x1 >= x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = x0 + x1 >= x1 = [[mark#(_x1)]] 
  [[active(app(nil, _x0))]] = 2 + x0 >= x0 = [[mark(_x0)]] 
  [[active(app(cons(_x0, _x1), _x2))]] = x0 + x2 >= x0 = [[mark(cons(_x0, app(_x1, _x2)))]] 
  [[active(from(_x0))]] = x0 >= x0 = [[mark(cons(_x0, from(s(_x0))))]] 
  [[active(zWadr(nil, _x0))]] = 2 + x0 >= 2 = [[mark(nil)]] 
  [[active(zWadr(_x0, nil))]] = 2 + x0 >= 2 = [[mark(nil)]] 
  [[active(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = x0 + x2 >= x0 + x2 = [[mark(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[active(prefix(_x0))]] = 2 >= 2 = [[mark(cons(nil, zWadr(_x0, prefix(_x0))))]] 
  [[mark(app(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[active(app(mark(_x0), mark(_x1)))]] 
  [[mark(nil)]] = 2 >= 2 = [[active(nil)]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[active(cons(mark(_x0), _x1))]] 
  [[mark(from(_x0))]] = x0 >= x0 = [[active(from(mark(_x0)))]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[active(s(mark(_x0)))]] 
  [[mark(zWadr(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[active(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark(prefix(_x0))]] = 2 >= 2 = [[active(prefix(mark(_x0)))]] 
  [[app(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[app(_x0, _x1)]] 
  [[app(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[app(_x0, _x1)]] 
  [[app(active(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[app(_x0, _x1)]] 
  [[app(_x0, active(_x1))]] = x0 + x1 >= x0 + x1 = [[app(_x0, _x1)]] 
  [[cons(mark(_x0), _x1)]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, mark(_x1))]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(active(_x0), _x1)]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, active(_x1))]] = x0 >= x0 = [[cons(_x0, _x1)]] 
  [[s(mark(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] 
  [[s(active(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] 
  [[zWadr(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(active(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, active(_x1))]] = x0 + x1 >= x0 + x1 = [[zWadr(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_12, R_1, minimal, formative) by (P_13, R_1, minimal, formative), where P_13 consists of:

  active#(app(cons(X, Y), Z)) =#> mark#(cons(X, app(Y, Z)))   
  active#(zWadr(cons(X, Y), cons(Z, U))) =#> mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U)))   
  mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
  mark#(app(X, Y)) =#> mark#(X)   
  mark#(app(X, Y)) =#> mark#(Y)   
  mark#(cons(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   
  mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   
  mark#(zWadr(X, Y)) =#> mark#(X)   
  mark#(zWadr(X, Y)) =#> mark#(Y)   

Thus, the original system is terminating if (P_13, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_13, R_1, 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:

  active#(app(cons(X, Y), Z)) >? mark#(cons(X, app(Y, Z))) 
  active#(zWadr(cons(X, Y), cons(Z, U))) >? mark#(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  mark#(app(X, Y)) >? active#(app(mark(X), mark(Y))) 
  mark#(app(X, Y)) >? mark#(X) 
  mark#(app(X, Y)) >? mark#(Y) 
  mark#(cons(X, Y)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? active#(zWadr(mark(X), mark(Y))) 
  mark#(zWadr(X, Y)) >? mark#(X) 
  mark#(zWadr(X, Y)) >? mark#(Y) 
  active(app(nil, X)) >= mark(X) 
  active(app(cons(X, Y), Z)) >= mark(cons(X, app(Y, Z))) 
  active(from(X)) >= mark(cons(X, from(s(X)))) 
  active(zWadr(nil, X)) >= mark(nil) 
  active(zWadr(X, nil)) >= mark(nil) 
  active(zWadr(cons(X, Y), cons(Z, U))) >= mark(cons(app(Z, cons(X, nil)), zWadr(Y, U))) 
  active(prefix(X)) >= mark(cons(nil, zWadr(X, prefix(X)))) 
  mark(app(X, Y)) >= active(app(mark(X), mark(Y))) 
  mark(nil) >= active(nil) 
  mark(cons(X, Y)) >= active(cons(mark(X), Y)) 
  mark(from(X)) >= active(from(mark(X))) 
  mark(s(X)) >= active(s(mark(X))) 
  mark(zWadr(X, Y)) >= active(zWadr(mark(X), mark(Y))) 
  mark(prefix(X)) >= active(prefix(mark(X))) 
  app(mark(X), Y) >= app(X, Y) 
  app(X, mark(Y)) >= app(X, Y) 
  app(active(X), Y) >= app(X, Y) 
  app(X, active(Y)) >= app(X, Y) 
  cons(mark(X), Y) >= cons(X, Y) 
  cons(X, mark(Y)) >= cons(X, Y) 
  cons(active(X), Y) >= cons(X, Y) 
  cons(X, active(Y)) >= cons(X, Y) 
  s(mark(X)) >= s(X) 
  s(active(X)) >= s(X) 
  zWadr(mark(X), Y) >= zWadr(X, Y) 
  zWadr(X, mark(Y)) >= zWadr(X, Y) 
  zWadr(active(X), Y) >= zWadr(X, Y) 
  zWadr(X, active(Y)) >= zWadr(X, Y) 

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

The following interpretation satisfies the requirements:

  active = \y0.y0 
  active# = \y0.y0 
  app = \y0y1.y1 + 2y0 
  cons = \y0y1.1 + y0 
  from = \y0.1 + y0 
  mark = \y0.y0 
  mark# = \y0.y0 
  nil = 0 
  prefix = \y0.1 
  s = \y0.y0 
  zWadr = \y0y1.2 + y0 + 2y1 

Using this interpretation, the requirements translate to:

  [[active#(app(cons(_x0, _x1), _x2))]] = 2 + x2 + 2x0 > 1 + x0 = [[mark#(cons(_x0, app(_x1, _x2)))]] 
  [[active#(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = 5 + x0 + 2x2 > 2 + x0 + 2x2 = [[mark#(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[mark#(app(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[active#(app(mark(_x0), mark(_x1)))]] 
  [[mark#(app(_x0, _x1))]] = x1 + 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(app(_x0, _x1))]] = x1 + 2x0 >= x1 = [[mark#(_x1)]] 
  [[mark#(cons(_x0, _x1))]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[active#(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark#(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 > x0 = [[mark#(_x0)]] 
  [[mark#(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 > x1 = [[mark#(_x1)]] 
  [[active(app(nil, _x0))]] = x0 >= x0 = [[mark(_x0)]] 
  [[active(app(cons(_x0, _x1), _x2))]] = 2 + x2 + 2x0 >= 1 + x0 = [[mark(cons(_x0, app(_x1, _x2)))]] 
  [[active(from(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(cons(_x0, from(s(_x0))))]] 
  [[active(zWadr(nil, _x0))]] = 2 + 2x0 >= 0 = [[mark(nil)]] 
  [[active(zWadr(_x0, nil))]] = 2 + x0 >= 0 = [[mark(nil)]] 
  [[active(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = 5 + x0 + 2x2 >= 2 + x0 + 2x2 = [[mark(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[active(prefix(_x0))]] = 1 >= 1 = [[mark(cons(nil, zWadr(_x0, prefix(_x0))))]] 
  [[mark(app(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[active(app(mark(_x0), mark(_x1)))]] 
  [[mark(nil)]] = 0 >= 0 = [[active(nil)]] 
  [[mark(cons(_x0, _x1))]] = 1 + x0 >= 1 + x0 = [[active(cons(mark(_x0), _x1))]] 
  [[mark(from(_x0))]] = 1 + x0 >= 1 + x0 = [[active(from(mark(_x0)))]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[active(s(mark(_x0)))]] 
  [[mark(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[active(zWadr(mark(_x0), mark(_x1)))]] 
  [[mark(prefix(_x0))]] = 1 >= 1 = [[active(prefix(mark(_x0)))]] 
  [[app(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[app(_x0, _x1)]] 
  [[app(_x0, mark(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[app(_x0, _x1)]] 
  [[app(active(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[app(_x0, _x1)]] 
  [[app(_x0, active(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[app(_x0, _x1)]] 
  [[cons(mark(_x0), _x1)]] = 1 + x0 >= 1 + x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, mark(_x1))]] = 1 + x0 >= 1 + x0 = [[cons(_x0, _x1)]] 
  [[cons(active(_x0), _x1)]] = 1 + x0 >= 1 + x0 = [[cons(_x0, _x1)]] 
  [[cons(_x0, active(_x1))]] = 1 + x0 >= 1 + x0 = [[cons(_x0, _x1)]] 
  [[s(mark(_x0))]] = x0 >= x0 = [[s(_x0)]] 
  [[s(active(_x0))]] = x0 >= x0 = [[s(_x0)]] 
  [[zWadr(mark(_x0), _x1)]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, mark(_x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(active(_x0), _x1)]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(_x0, _x1)]] 
  [[zWadr(_x0, active(_x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_13, R_1, minimal, formative) by (P_14, R_1, minimal, formative), where P_14 consists of:

  mark#(app(X, Y)) =#> active#(app(mark(X), mark(Y)))   
  mark#(app(X, Y)) =#> mark#(X)   
  mark#(app(X, Y)) =#> mark#(Y)   
  mark#(s(X)) =#> mark#(X)   
  mark#(zWadr(X, Y)) =#> active#(zWadr(mark(X), mark(Y)))   

Thus, the original system is terminating if (P_14, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_14, 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 :  
    * 1 : 0, 1, 2, 3, 4 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 0, 1, 2, 3, 4 
    * 4 :  

This graph has the following strongly connected components:

  P_15:

    mark#(app(X, Y)) =#> mark#(X)   
    mark#(app(X, Y)) =#> mark#(Y)   
    mark#(s(X)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_14, R_1, m, f) by (P_15, R_1, m, f).

Thus, the original system is terminating if (P_15, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_15, R_1, 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#(app(X, Y))) = app(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(app(X, Y))) = app(X, Y) |> Y = nu(mark#(Y)) 
  nu(mark#(s(X))) = s(X) |> X = nu(mark#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_15, R_1, minimal, f) by ({}, R_1, 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.