/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 
  ok : [o] --> o 
  prefix : [o] --> o 
  proper : [o] --> o 
  s : [o] --> o 
  top : [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)))) 
  active(app(X, Y)) => app(active(X), Y) 
  active(app(X, Y)) => app(X, active(Y)) 
  active(cons(X, Y)) => cons(active(X), Y) 
  active(from(X)) => from(active(X)) 
  active(s(X)) => s(active(X)) 
  active(zWadr(X, Y)) => zWadr(active(X), Y) 
  active(zWadr(X, Y)) => zWadr(X, active(Y)) 
  active(prefix(X)) => prefix(active(X)) 
  app(mark(X), Y) => mark(app(X, Y)) 
  app(X, mark(Y)) => mark(app(X, Y)) 
  cons(mark(X), Y) => mark(cons(X, Y)) 
  from(mark(X)) => mark(from(X)) 
  s(mark(X)) => mark(s(X)) 
  zWadr(mark(X), Y) => mark(zWadr(X, Y)) 
  zWadr(X, mark(Y)) => mark(zWadr(X, Y)) 
  prefix(mark(X)) => mark(prefix(X)) 
  proper(app(X, Y)) => app(proper(X), proper(Y)) 
  proper(nil) => ok(nil) 
  proper(cons(X, Y)) => cons(proper(X), proper(Y)) 
  proper(from(X)) => from(proper(X)) 
  proper(s(X)) => s(proper(X)) 
  proper(zWadr(X, Y)) => zWadr(proper(X), proper(Y)) 
  proper(prefix(X)) => prefix(proper(X)) 
  app(ok(X), ok(Y)) => ok(app(X, Y)) 
  cons(ok(X), ok(Y)) => ok(cons(X, Y)) 
  from(ok(X)) => ok(from(X)) 
  s(ok(X)) => ok(s(X)) 
  zWadr(ok(X), ok(Y)) => ok(zWadr(X, Y)) 
  prefix(ok(X)) => ok(prefix(X)) 
  top(mark(X)) => top(proper(X)) 
  top(ok(X)) => top(active(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(cons(X, Y), Z)) =#> cons#(X, app(Y, Z))   
    1] active#(app(cons(X, Y), Z)) =#> app#(Y, Z)   
    2] active#(from(X)) =#> cons#(X, from(s(X)))   
    3] active#(from(X)) =#> from#(s(X))   
    4] active#(from(X)) =#> s#(X)   
    5] active#(zWadr(cons(X, Y), cons(Z, U))) =#> cons#(app(Z, cons(X, nil)), zWadr(Y, U))   
    6] active#(zWadr(cons(X, Y), cons(Z, U))) =#> app#(Z, cons(X, nil))   
    7] active#(zWadr(cons(X, Y), cons(Z, U))) =#> cons#(X, nil)   
    8] active#(zWadr(cons(X, Y), cons(Z, U))) =#> zWadr#(Y, U)   
    9] active#(prefix(X)) =#> cons#(nil, zWadr(X, prefix(X)))   
    10] active#(prefix(X)) =#> zWadr#(X, prefix(X))   
    11] active#(prefix(X)) =#> prefix#(X)   
    12] active#(app(X, Y)) =#> app#(active(X), Y)   
    13] active#(app(X, Y)) =#> active#(X)   
    14] active#(app(X, Y)) =#> app#(X, active(Y))   
    15] active#(app(X, Y)) =#> active#(Y)   
    16] active#(cons(X, Y)) =#> cons#(active(X), Y)   
    17] active#(cons(X, Y)) =#> active#(X)   
    18] active#(from(X)) =#> from#(active(X))   
    19] active#(from(X)) =#> active#(X)   
    20] active#(s(X)) =#> s#(active(X))   
    21] active#(s(X)) =#> active#(X)   
    22] active#(zWadr(X, Y)) =#> zWadr#(active(X), Y)   
    23] active#(zWadr(X, Y)) =#> active#(X)   
    24] active#(zWadr(X, Y)) =#> zWadr#(X, active(Y))   
    25] active#(zWadr(X, Y)) =#> active#(Y)   
    26] active#(prefix(X)) =#> prefix#(active(X))   
    27] active#(prefix(X)) =#> active#(X)   
    28] app#(mark(X), Y) =#> app#(X, Y)   
    29] app#(X, mark(Y)) =#> app#(X, Y)   
    30] cons#(mark(X), Y) =#> cons#(X, Y)   
    31] from#(mark(X)) =#> from#(X)   
    32] s#(mark(X)) =#> s#(X)   
    33] zWadr#(mark(X), Y) =#> zWadr#(X, Y)   
    34] zWadr#(X, mark(Y)) =#> zWadr#(X, Y)   
    35] prefix#(mark(X)) =#> prefix#(X)   
    36] proper#(app(X, Y)) =#> app#(proper(X), proper(Y))   
    37] proper#(app(X, Y)) =#> proper#(X)   
    38] proper#(app(X, Y)) =#> proper#(Y)   
    39] proper#(cons(X, Y)) =#> cons#(proper(X), proper(Y))   
    40] proper#(cons(X, Y)) =#> proper#(X)   
    41] proper#(cons(X, Y)) =#> proper#(Y)   
    42] proper#(from(X)) =#> from#(proper(X))   
    43] proper#(from(X)) =#> proper#(X)   
    44] proper#(s(X)) =#> s#(proper(X))   
    45] proper#(s(X)) =#> proper#(X)   
    46] proper#(zWadr(X, Y)) =#> zWadr#(proper(X), proper(Y))   
    47] proper#(zWadr(X, Y)) =#> proper#(X)   
    48] proper#(zWadr(X, Y)) =#> proper#(Y)   
    49] proper#(prefix(X)) =#> prefix#(proper(X))   
    50] proper#(prefix(X)) =#> proper#(X)   
    51] app#(ok(X), ok(Y)) =#> app#(X, Y)   
    52] cons#(ok(X), ok(Y)) =#> cons#(X, Y)   
    53] from#(ok(X)) =#> from#(X)   
    54] s#(ok(X)) =#> s#(X)   
    55] zWadr#(ok(X), ok(Y)) =#> zWadr#(X, Y)   
    56] prefix#(ok(X)) =#> prefix#(X)   
    57] top#(mark(X)) =#> top#(proper(X))   
    58] top#(mark(X)) =#> proper#(X)   
    59] top#(ok(X)) =#> top#(active(X))   
    60] top#(ok(X)) =#> active#(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)))) 
    active(app(X, Y)) => app(active(X), Y) 
    active(app(X, Y)) => app(X, active(Y)) 
    active(cons(X, Y)) => cons(active(X), Y) 
    active(from(X)) => from(active(X)) 
    active(s(X)) => s(active(X)) 
    active(zWadr(X, Y)) => zWadr(active(X), Y) 
    active(zWadr(X, Y)) => zWadr(X, active(Y)) 
    active(prefix(X)) => prefix(active(X)) 
    app(mark(X), Y) => mark(app(X, Y)) 
    app(X, mark(Y)) => mark(app(X, Y)) 
    cons(mark(X), Y) => mark(cons(X, Y)) 
    from(mark(X)) => mark(from(X)) 
    s(mark(X)) => mark(s(X)) 
    zWadr(mark(X), Y) => mark(zWadr(X, Y)) 
    zWadr(X, mark(Y)) => mark(zWadr(X, Y)) 
    prefix(mark(X)) => mark(prefix(X)) 
    proper(app(X, Y)) => app(proper(X), proper(Y)) 
    proper(nil) => ok(nil) 
    proper(cons(X, Y)) => cons(proper(X), proper(Y)) 
    proper(from(X)) => from(proper(X)) 
    proper(s(X)) => s(proper(X)) 
    proper(zWadr(X, Y)) => zWadr(proper(X), proper(Y)) 
    proper(prefix(X)) => prefix(proper(X)) 
    app(ok(X), ok(Y)) => ok(app(X, Y)) 
    cons(ok(X), ok(Y)) => ok(cons(X, Y)) 
    from(ok(X)) => ok(from(X)) 
    s(ok(X)) => ok(s(X)) 
    zWadr(ok(X), ok(Y)) => ok(zWadr(X, Y)) 
    prefix(ok(X)) => ok(prefix(X)) 
    top(mark(X)) => top(proper(X)) 
    top(ok(X)) => top(active(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 : 30, 52 
    * 1 : 28, 29, 51 
    * 2 : 30, 52 
    * 3 : 31, 53 
    * 4 : 32, 54 
    * 5 : 30, 52 
    * 6 : 28, 29, 51 
    * 7 : 30 
    * 8 : 33, 34, 55 
    * 9 :  
    * 10 : 33, 34, 55 
    * 11 : 35, 56 
    * 12 : 28, 29, 51 
    * 13 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 14 : 28, 29, 51 
    * 15 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 16 : 30, 52 
    * 17 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 18 : 31, 53 
    * 19 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 20 : 32, 54 
    * 21 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 22 : 33, 34, 55 
    * 23 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 24 : 33, 34, 55 
    * 25 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 26 : 35, 56 
    * 27 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 
    * 28 : 28, 29, 51 
    * 29 : 28, 29, 51 
    * 30 : 30, 52 
    * 31 : 31, 53 
    * 32 : 32, 54 
    * 33 : 33, 34, 55 
    * 34 : 33, 34, 55 
    * 35 : 35, 56 
    * 36 : 28, 29, 51 
    * 37 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 38 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 39 : 30, 52 
    * 40 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 41 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 42 : 31, 53 
    * 43 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 44 : 32, 54 
    * 45 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 46 : 33, 34, 55 
    * 47 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 48 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 49 : 35, 56 
    * 50 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 51 : 28, 29, 51 
    * 52 : 30, 52 
    * 53 : 31, 53 
    * 54 : 32, 54 
    * 55 : 33, 34, 55 
    * 56 : 35, 56 
    * 57 : 57, 58, 59, 60 
    * 58 : 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 
    * 59 : 57, 58, 59, 60 
    * 60 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 

This graph has the following strongly connected components:

  P_1:

    active#(app(X, Y)) =#> active#(X)   
    active#(app(X, Y)) =#> active#(Y)   
    active#(cons(X, Y)) =#> active#(X)   
    active#(from(X)) =#> active#(X)   
    active#(s(X)) =#> active#(X)   
    active#(zWadr(X, Y)) =#> active#(X)   
    active#(zWadr(X, Y)) =#> active#(Y)   
    active#(prefix(X)) =#> active#(X)   

  P_2:

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

  P_3:

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

  P_4:

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

  P_5:

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

  P_6:

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

  P_7:

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

  P_8:

    proper#(app(X, Y)) =#> proper#(X)   
    proper#(app(X, Y)) =#> proper#(Y)   
    proper#(cons(X, Y)) =#> proper#(X)   
    proper#(cons(X, Y)) =#> proper#(Y)   
    proper#(from(X)) =#> proper#(X)   
    proper#(s(X)) =#> proper#(X)   
    proper#(zWadr(X, Y)) =#> proper#(X)   
    proper#(zWadr(X, Y)) =#> proper#(Y)   
    proper#(prefix(X)) =#> proper#(X)   

  P_9:

    top#(mark(X)) =#> top#(proper(X))   
    top#(ok(X)) =#> top#(active(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), (P_7, R_0, m, f), (P_8, R_0, m, f) and (P_9, 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), (P_7, R_0, minimal, formative), (P_8, 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).

The formative rules of (P_9, 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)))) 
  active(app(X, Y)) => app(active(X), Y) 
  active(app(X, Y)) => app(X, active(Y)) 
  active(cons(X, Y)) => cons(active(X), Y) 
  active(from(X)) => from(active(X)) 
  active(s(X)) => s(active(X)) 
  active(zWadr(X, Y)) => zWadr(active(X), Y) 
  active(zWadr(X, Y)) => zWadr(X, active(Y)) 
  active(prefix(X)) => prefix(active(X)) 
  app(mark(X), Y) => mark(app(X, Y)) 
  app(X, mark(Y)) => mark(app(X, Y)) 
  cons(mark(X), Y) => mark(cons(X, Y)) 
  from(mark(X)) => mark(from(X)) 
  s(mark(X)) => mark(s(X)) 
  zWadr(mark(X), Y) => mark(zWadr(X, Y)) 
  zWadr(X, mark(Y)) => mark(zWadr(X, Y)) 
  prefix(mark(X)) => mark(prefix(X)) 
  proper(app(X, Y)) => app(proper(X), proper(Y)) 
  proper(nil) => ok(nil) 
  proper(cons(X, Y)) => cons(proper(X), proper(Y)) 
  proper(from(X)) => from(proper(X)) 
  proper(s(X)) => s(proper(X)) 
  proper(zWadr(X, Y)) => zWadr(proper(X), proper(Y)) 
  proper(prefix(X)) => prefix(proper(X)) 
  app(ok(X), ok(Y)) => ok(app(X, Y)) 
  cons(ok(X), ok(Y)) => ok(cons(X, Y)) 
  from(ok(X)) => ok(from(X)) 
  s(ok(X)) => ok(s(X)) 
  zWadr(ok(X), ok(Y)) => ok(zWadr(X, Y)) 
  prefix(ok(X)) => ok(prefix(X)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_9, R_0, minimal, formative) by (P_9, 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), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative), (P_6, R_0, minimal, formative), (P_7, R_0, minimal, formative), (P_8, R_0, minimal, formative) and (P_9, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_9, 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:

  top#(mark(X)) >? top#(proper(X)) 
  top#(ok(X)) >? top#(active(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)))) 
  active(app(X, Y)) >= app(active(X), Y) 
  active(app(X, Y)) >= app(X, active(Y)) 
  active(cons(X, Y)) >= cons(active(X), Y) 
  active(from(X)) >= from(active(X)) 
  active(s(X)) >= s(active(X)) 
  active(zWadr(X, Y)) >= zWadr(active(X), Y) 
  active(zWadr(X, Y)) >= zWadr(X, active(Y)) 
  active(prefix(X)) >= prefix(active(X)) 
  app(mark(X), Y) >= mark(app(X, Y)) 
  app(X, mark(Y)) >= mark(app(X, Y)) 
  cons(mark(X), Y) >= mark(cons(X, Y)) 
  from(mark(X)) >= mark(from(X)) 
  s(mark(X)) >= mark(s(X)) 
  zWadr(mark(X), Y) >= mark(zWadr(X, Y)) 
  zWadr(X, mark(Y)) >= mark(zWadr(X, Y)) 
  prefix(mark(X)) >= mark(prefix(X)) 
  proper(app(X, Y)) >= app(proper(X), proper(Y)) 
  proper(nil) >= ok(nil) 
  proper(cons(X, Y)) >= cons(proper(X), proper(Y)) 
  proper(from(X)) >= from(proper(X)) 
  proper(s(X)) >= s(proper(X)) 
  proper(zWadr(X, Y)) >= zWadr(proper(X), proper(Y)) 
  proper(prefix(X)) >= prefix(proper(X)) 
  app(ok(X), ok(Y)) >= ok(app(X, Y)) 
  cons(ok(X), ok(Y)) >= ok(cons(X, Y)) 
  from(ok(X)) >= ok(from(X)) 
  s(ok(X)) >= ok(s(X)) 
  zWadr(ok(X), ok(Y)) >= ok(zWadr(X, Y)) 
  prefix(ok(X)) >= ok(prefix(X)) 

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

The following interpretation satisfies the requirements:

  active = \y0.y0 
  app = \y0y1.y1 + 2y0 
  cons = \y0y1.1 + y0 
  from = \y0.2 + 2y0 
  mark = \y0.1 + y0 
  nil = 1 
  ok = \y0.y0 
  prefix = \y0.3 + 2y0 
  proper = \y0.y0 
  s = \y0.y0 
  top# = \y0.y0 
  zWadr = \y0y1.2 + y0 + 2y1 

Using this interpretation, the requirements translate to:

  [[top#(mark(_x0))]] = 1 + x0 > x0 = [[top#(proper(_x0))]] 
  [[top#(ok(_x0))]] = x0 >= x0 = [[top#(active(_x0))]] 
  [[active(app(nil, _x0))]] = 2 + x0 >= 1 + x0 = [[mark(_x0)]] 
  [[active(app(cons(_x0, _x1), _x2))]] = 2 + x2 + 2x0 >= 2 + x0 = [[mark(cons(_x0, app(_x1, _x2)))]] 
  [[active(from(_x0))]] = 2 + 2x0 >= 2 + x0 = [[mark(cons(_x0, from(s(_x0))))]] 
  [[active(zWadr(nil, _x0))]] = 3 + 2x0 >= 2 = [[mark(nil)]] 
  [[active(zWadr(_x0, nil))]] = 4 + x0 >= 2 = [[mark(nil)]] 
  [[active(zWadr(cons(_x0, _x1), cons(_x2, _x3)))]] = 5 + x0 + 2x2 >= 3 + x0 + 2x2 = [[mark(cons(app(_x2, cons(_x0, nil)), zWadr(_x1, _x3)))]] 
  [[active(prefix(_x0))]] = 3 + 2x0 >= 3 = [[mark(cons(nil, zWadr(_x0, prefix(_x0))))]] 
  [[active(app(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[app(active(_x0), _x1)]] 
  [[active(app(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[app(_x0, active(_x1))]] 
  [[active(cons(_x0, _x1))]] = 1 + x0 >= 1 + x0 = [[cons(active(_x0), _x1)]] 
  [[active(from(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[from(active(_x0))]] 
  [[active(s(_x0))]] = x0 >= x0 = [[s(active(_x0))]] 
  [[active(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(active(_x0), _x1)]] 
  [[active(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(_x0, active(_x1))]] 
  [[active(prefix(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[prefix(active(_x0))]] 
  [[app(mark(_x0), _x1)]] = 2 + x1 + 2x0 >= 1 + x1 + 2x0 = [[mark(app(_x0, _x1))]] 
  [[app(_x0, mark(_x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[mark(app(_x0, _x1))]] 
  [[cons(mark(_x0), _x1)]] = 2 + x0 >= 2 + x0 = [[mark(cons(_x0, _x1))]] 
  [[from(mark(_x0))]] = 4 + 2x0 >= 3 + 2x0 = [[mark(from(_x0))]] 
  [[s(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(s(_x0))]] 
  [[zWadr(mark(_x0), _x1)]] = 3 + x0 + 2x1 >= 3 + x0 + 2x1 = [[mark(zWadr(_x0, _x1))]] 
  [[zWadr(_x0, mark(_x1))]] = 4 + x0 + 2x1 >= 3 + x0 + 2x1 = [[mark(zWadr(_x0, _x1))]] 
  [[prefix(mark(_x0))]] = 5 + 2x0 >= 4 + 2x0 = [[mark(prefix(_x0))]] 
  [[proper(app(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[app(proper(_x0), proper(_x1))]] 
  [[proper(nil)]] = 1 >= 1 = [[ok(nil)]] 
  [[proper(cons(_x0, _x1))]] = 1 + x0 >= 1 + x0 = [[cons(proper(_x0), proper(_x1))]] 
  [[proper(from(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[from(proper(_x0))]] 
  [[proper(s(_x0))]] = x0 >= x0 = [[s(proper(_x0))]] 
  [[proper(zWadr(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[zWadr(proper(_x0), proper(_x1))]] 
  [[proper(prefix(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[prefix(proper(_x0))]] 
  [[app(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(app(_x0, _x1))]] 
  [[cons(ok(_x0), ok(_x1))]] = 1 + x0 >= 1 + x0 = [[ok(cons(_x0, _x1))]] 
  [[from(ok(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[ok(from(_x0))]] 
  [[s(ok(_x0))]] = x0 >= x0 = [[ok(s(_x0))]] 
  [[zWadr(ok(_x0), ok(_x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[ok(zWadr(_x0, _x1))]] 
  [[prefix(ok(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[ok(prefix(_x0))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_9, R_1, minimal, formative) by (P_10, R_1, minimal, formative), where P_10 consists of:

  top#(ok(X)) =#> top#(active(X))   

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), (P_7, R_0, minimal, formative), (P_8, R_0, minimal, formative) and (P_10, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_10, R_1, minimal, formative).

The formative rules of (P_10, R_1) are R_2 ::=

  active(app(X, Y)) => app(active(X), Y) 
  active(app(X, Y)) => app(X, active(Y)) 
  active(cons(X, Y)) => cons(active(X), Y) 
  active(from(X)) => from(active(X)) 
  active(s(X)) => s(active(X)) 
  active(zWadr(X, Y)) => zWadr(active(X), Y) 
  active(zWadr(X, Y)) => zWadr(X, active(Y)) 
  active(prefix(X)) => prefix(active(X)) 
  proper(app(X, Y)) => app(proper(X), proper(Y)) 
  proper(nil) => ok(nil) 
  proper(cons(X, Y)) => cons(proper(X), proper(Y)) 
  proper(from(X)) => from(proper(X)) 
  proper(s(X)) => s(proper(X)) 
  proper(zWadr(X, Y)) => zWadr(proper(X), proper(Y)) 
  proper(prefix(X)) => prefix(proper(X)) 
  app(ok(X), ok(Y)) => ok(app(X, Y)) 
  cons(ok(X), ok(Y)) => ok(cons(X, Y)) 
  from(ok(X)) => ok(from(X)) 
  s(ok(X)) => ok(s(X)) 
  zWadr(ok(X), ok(Y)) => ok(zWadr(X, Y)) 
  prefix(ok(X)) => ok(prefix(X)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_10, R_1, minimal, formative) by (P_10, R_2, minimal, formative).

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), (P_7, R_0, minimal, formative), (P_8, R_0, minimal, formative) and (P_10, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_10, R_2, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_10, R_2) are:

  active(app(X, Y)) => app(active(X), Y) 
  active(app(X, Y)) => app(X, active(Y)) 
  active(cons(X, Y)) => cons(active(X), Y) 
  active(from(X)) => from(active(X)) 
  active(s(X)) => s(active(X)) 
  active(zWadr(X, Y)) => zWadr(active(X), Y) 
  active(zWadr(X, Y)) => zWadr(X, active(Y)) 
  active(prefix(X)) => prefix(active(X)) 
  app(ok(X), ok(Y)) => ok(app(X, Y)) 
  cons(ok(X), ok(Y)) => ok(cons(X, Y)) 
  from(ok(X)) => ok(from(X)) 
  s(ok(X)) => ok(s(X)) 
  zWadr(ok(X), ok(Y)) => ok(zWadr(X, Y)) 
  prefix(ok(X)) => ok(prefix(X)) 

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

  top#(ok(X)) >? top#(active(X)) 
  active(app(X, Y)) >= app(active(X), Y) 
  active(app(X, Y)) >= app(X, active(Y)) 
  active(cons(X, Y)) >= cons(active(X), Y) 
  active(from(X)) >= from(active(X)) 
  active(s(X)) >= s(active(X)) 
  active(zWadr(X, Y)) >= zWadr(active(X), Y) 
  active(zWadr(X, Y)) >= zWadr(X, active(Y)) 
  active(prefix(X)) >= prefix(active(X)) 
  app(ok(X), ok(Y)) >= ok(app(X, Y)) 
  cons(ok(X), ok(Y)) >= ok(cons(X, Y)) 
  from(ok(X)) >= ok(from(X)) 
  s(ok(X)) >= ok(s(X)) 
  zWadr(ok(X), ok(Y)) >= ok(zWadr(X, Y)) 
  prefix(ok(X)) >= ok(prefix(X)) 

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

The following interpretation satisfies the requirements:

  active = \y0.y0 
  app = \y0y1.1 + 2y0 
  cons = \y0y1.y1 
  from = \y0.2y0 
  ok = \y0.1 + y0 
  prefix = \y0.y0 
  s = \y0.1 + y0 
  top# = \y0.y0 
  zWadr = \y0y1.2y0 

Using this interpretation, the requirements translate to:

  [[top#(ok(_x0))]] = 1 + x0 > x0 = [[top#(active(_x0))]] 
  [[active(app(_x0, _x1))]] = 1 + 2x0 >= 1 + 2x0 = [[app(active(_x0), _x1)]] 
  [[active(app(_x0, _x1))]] = 1 + 2x0 >= 1 + 2x0 = [[app(_x0, active(_x1))]] 
  [[active(cons(_x0, _x1))]] = x1 >= x1 = [[cons(active(_x0), _x1)]] 
  [[active(from(_x0))]] = 2x0 >= 2x0 = [[from(active(_x0))]] 
  [[active(s(_x0))]] = 1 + x0 >= 1 + x0 = [[s(active(_x0))]] 
  [[active(zWadr(_x0, _x1))]] = 2x0 >= 2x0 = [[zWadr(active(_x0), _x1)]] 
  [[active(zWadr(_x0, _x1))]] = 2x0 >= 2x0 = [[zWadr(_x0, active(_x1))]] 
  [[active(prefix(_x0))]] = x0 >= x0 = [[prefix(active(_x0))]] 
  [[app(ok(_x0), ok(_x1))]] = 3 + 2x0 >= 2 + 2x0 = [[ok(app(_x0, _x1))]] 
  [[cons(ok(_x0), ok(_x1))]] = 1 + x1 >= 1 + x1 = [[ok(cons(_x0, _x1))]] 
  [[from(ok(_x0))]] = 2 + 2x0 >= 1 + 2x0 = [[ok(from(_x0))]] 
  [[s(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(s(_x0))]] 
  [[zWadr(ok(_x0), ok(_x1))]] = 2 + 2x0 >= 1 + 2x0 = [[ok(zWadr(_x0, _x1))]] 
  [[prefix(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(prefix(_x0))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_10, R_2) by ({}, R_2).  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), (P_6, R_0, minimal, formative), (P_7, 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(proper#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(proper#(app(X, Y))) = app(X, Y) |> X = nu(proper#(X)) 
  nu(proper#(app(X, Y))) = app(X, Y) |> Y = nu(proper#(Y)) 
  nu(proper#(cons(X, Y))) = cons(X, Y) |> X = nu(proper#(X)) 
  nu(proper#(cons(X, Y))) = cons(X, Y) |> Y = nu(proper#(Y)) 
  nu(proper#(from(X))) = from(X) |> X = nu(proper#(X)) 
  nu(proper#(s(X))) = s(X) |> X = nu(proper#(X)) 
  nu(proper#(zWadr(X, Y))) = zWadr(X, Y) |> X = nu(proper#(X)) 
  nu(proper#(zWadr(X, Y))) = zWadr(X, Y) |> Y = nu(proper#(Y)) 
  nu(proper#(prefix(X))) = prefix(X) |> X = nu(proper#(X)) 

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), (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#(ok(X))) = ok(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#(ok(X), ok(Y))) = ok(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_11, R_0, minimal, f), where P_11 contains:

  zWadr#(X, mark(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_11, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_11, 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)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_11, 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#(ok(X))) = ok(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#(ok(X))) = ok(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#(ok(X), ok(Y))) = ok(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 ({}, 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#(ok(X), ok(Y))) = ok(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_12, R_0, minimal, f), where P_12 contains:

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

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_12, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_12, 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)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_12, 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(active#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(active#(app(X, Y))) = app(X, Y) |> X = nu(active#(X)) 
  nu(active#(app(X, Y))) = app(X, Y) |> Y = nu(active#(Y)) 
  nu(active#(cons(X, Y))) = cons(X, Y) |> X = nu(active#(X)) 
  nu(active#(from(X))) = from(X) |> X = nu(active#(X)) 
  nu(active#(s(X))) = s(X) |> X = nu(active#(X)) 
  nu(active#(zWadr(X, Y))) = zWadr(X, Y) |> X = nu(active#(X)) 
  nu(active#(zWadr(X, Y))) = zWadr(X, Y) |> Y = nu(active#(Y)) 
  nu(active#(prefix(X))) = prefix(X) |> X = nu(active#(X)) 

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.