We consider the system Applicative_first_order_05__#3.13.

  Alphabet:

    0 : [] --> a 
    cons : [d * e] --> e 
    edge : [a * a * b] --> b 
    empty : [] --> b 
    eq : [a * a] --> c 
    false : [] --> c 
    filter : [d -> c * e] --> e 
    filter2 : [c * d -> c * d * e] --> e 
    if!fac6220reach!fac62201 : [c * a * a * b * b] --> c 
    if!fac6220reach!fac62202 : [c * a * a * b * b] --> c 
    map : [d -> d * e] --> e 
    nil : [] --> e 
    or : [c * c] --> c 
    reach : [a * a * b * b] --> c 
    s : [a] --> a 
    true : [] --> c 
    union : [b * b] --> b 

  Rules:

    eq(0, 0) => true 
    eq(0, s(x)) => false 
    eq(s(x), 0) => false 
    eq(s(x), s(y)) => eq(x, y) 
    or(true, x) => true 
    or(false, x) => x 
    union(empty, x) => x 
    union(edge(x, y, z), u) => edge(x, y, union(z, u)) 
    reach(x, y, empty, z) => false 
    reach(x, y, edge(z, u, v), w) => if!fac6220reach!fac62201(eq(x, z), x, y, edge(z, u, v), w) 
    if!fac6220reach!fac62201(true, x, y, edge(z, u, v), w) => if!fac6220reach!fac62202(eq(y, u), x, y, edge(z, u, v), w) 
    if!fac6220reach!fac62201(false, x, y, edge(z, u, v), w) => reach(x, y, v, edge(z, u, w)) 
    if!fac6220reach!fac62202(true, x, y, edge(z, u, v), w) => true 
    if!fac6220reach!fac62202(false, x, y, edge(z, u, v), w) => or(reach(x, y, v, w), reach(u, y, union(v, w), empty)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(f, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

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

  Dependency Pairs P_0:

    0] eq#(s(X), s(Y)) =#> eq#(X, Y)   
    1] union#(edge(X, Y, Z), U) =#> union#(Z, U)   
    2] reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   
    3] reach#(X, Y, edge(Z, U, V), W) =#> eq#(X, Z)   
    4] if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W)   
    5] if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> eq#(Y, U)   
    6] if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W))   
    7] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> or#(reach(X, Y, V, W), reach(U, Y, union(V, W), empty))   
    8] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W)   
    9] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty)   
    10] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> union#(V, W)   
    11] map#(F, cons(X, Y)) =#> map#(F, Y)   
    12] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    13] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    14] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    eq(0, 0) => true 
    eq(0, s(X)) => false 
    eq(s(X), 0) => false 
    eq(s(X), s(Y)) => eq(X, Y) 
    or(true, X) => true 
    or(false, X) => X 
    union(empty, X) => X 
    union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) 
    reach(X, Y, empty, Z) => false 
    reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
    if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
    if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 
    if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true 
    if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 

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

We consider the dependency pair problem (P_0, R_0, static, 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 
    * 2 : 4, 5, 6 
    * 3 : 0 
    * 4 : 7, 8, 9, 10 
    * 5 : 0 
    * 6 : 2, 3 
    * 7 :  
    * 8 : 2, 3 
    * 9 : 2, 3 
    * 10 : 1 
    * 11 : 11 
    * 12 : 13, 14 
    * 13 : 12 
    * 14 : 12 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    union#(edge(X, Y, Z), U) =#> union#(Z, U)   

  P_3:

    reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   
    if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W)   
    if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W))   
    if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W)   
    if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty)   

  P_4:

    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_5:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, 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), (P_3, R_0, m, f), (P_4, R_0, m, f) and (P_5, R_0, m, f).

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

We consider the dependency pair problem (P_5, R_0, static, 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:

  filter#(F, cons(X, Y)) >? filter2#(F X, F, X, Y) 
  filter2#(true, F, X, Y) >? filter#(F, Y) 
  filter2#(false, F, X, Y) >? filter#(F, Y) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.1 + y1 
  edge = \y0y1y2.0 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.2y1 
  filter2 = \y0G1y2y3.1 + 2y3 
  filter2# = \y0G1y2y3.2y3 
  filter# = \G0y1.2y1 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.2y1 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  s = \y0.3 
  true = 0 
  union = \y0y1.2y1 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 2 + 2x2 > 2x2 = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = 2x2 >= 2x2 = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = 2x2 >= 2x2 = [[filter#(_F0, _x2)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = 2x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 2x3 >= 0 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 + 2x2 >= 1 + 2x2 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 2 + 2x2 >= 1 + 2x2 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 1 + 2x2 >= 1 + 2x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 1 + 2x2 >= 2x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, static, formative) by (P_6, R_0, static, formative), where P_6 consists of:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative), (P_3, R_0, static, formative), (P_4, R_0, static, formative) and (P_6, R_0, static, formative) is finite.

We consider the dependency pair problem (P_6, R_0, static, 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 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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

We consider the dependency pair problem (P_4, R_0, static, 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:

  map#(F, cons(X, Y)) >? map#(F, Y) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.2 + y0 + y1 
  edge = \y0y1y2.0 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.y1 
  filter2 = \y0G1y2y3.2 + y2 + y3 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.2y1 + 2y1G0(y1) 
  map# = \G0y1.y1 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  s = \y0.3 
  true = 0 
  union = \y0y1.y1 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 2 + x1 + x2 > x2 = [[map#(_F0, _x2)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = x3 >= 0 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 4 + 2x1 + 2x2 + 2x1F0(2 + x1 + x2) + 2x2F0(2 + x1 + x2) + 4F0(2 + x1 + x2) >= 2 + 2x2 + F0(x1) + 2x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 2 + x1 + x2 >= 2 + x1 + x2 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + x1 + x2 >= 2 + x1 + x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + x1 + x2 >= x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_4, R_0) by ({}, R_0).  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, static, formative), (P_2, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

We consider the dependency pair problem (P_3, R_0, static, 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:

  reach#(X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, W) 
  if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) >? reach#(U, Y, union(V, W), empty) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.0 
  edge = \y0y1y2.2 + y2 
  empty = 2 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.0 
  filter2 = \y0G1y2y3.0 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62201# = \y0y1y2y3y4.2 + y3 + y4 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202# = \y0y1y2y3y4.2 + y3 + y4 
  map = \G0y1.2 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  reach# = \y0y1y2y3.2 + y2 + y3 
  s = \y0.3 
  true = 0 
  union = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[reach#(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 4 + x4 + x5 >= 4 + x4 + x5 = [[if!fac6220reach!fac62201#(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201#(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 4 + x4 + x5 >= 4 + x4 + x5 = [[if!fac6220reach!fac62202#(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 4 + x4 + x5 >= 4 + x4 + x5 = [[reach#(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 4 + x4 + x5 > 2 + x4 + x5 = [[reach#(_x0, _x1, _x4, _x5)]] 
  [[if!fac6220reach!fac62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 4 + x4 + x5 >= 4 + x4 + x5 = [[reach#(_x3, _x1, union(_x4, _x5), empty)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = 2 + x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 2 + x2 + x3 >= 2 + x2 + x3 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 2 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 >= 0 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 0 >= 0 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 0 >= 0 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, static, formative) by (P_7, R_0, static, formative), where P_7 consists of:

  reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   
  if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W)   
  if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W))   
  if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_7, R_0, static, formative) is finite.

We consider the dependency pair problem (P_7, R_0, static, 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:

  reach#(X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) >? reach#(U, Y, union(V, W), empty) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.0 
  edge = \y0y1y2.1 + y2 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.0 
  filter2 = \y0G1y2y3.0 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62201# = \y0y1y2y3y4.y3 + y4 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202# = \y0y1y2y3y4.y3 + y4 
  map = \G0y1.0 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  reach# = \y0y1y2y3.y2 + y3 
  s = \y0.3 
  true = 0 
  union = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[reach#(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[if!fac6220reach!fac62201#(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201#(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[if!fac6220reach!fac62202#(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[reach#(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 > x4 + x5 = [[reach#(_x3, _x1, union(_x4, _x5), empty)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 1 + x2 + x3 >= 1 + x2 + x3 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 0 >= 0 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 0 >= 0 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_7, R_0, static, formative) by (P_8, R_0, static, formative), where P_8 consists of:

  reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   
  if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W)   
  if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W))   

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

We consider the dependency pair problem (P_8, R_0, static, 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, 2 
    * 1 :  
    * 2 : 0 

This graph has the following strongly connected components:

  P_9:

    reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   
    if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W))   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_8, R_0, m, f) by (P_9, R_0, m, f).

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

We consider the dependency pair problem (P_9, R_0, static, 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:

  reach#(X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, edge(Z, U, W)) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.0 
  edge = \y0y1y2.2 + y2 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.0 
  filter2 = \y0G1y2y3.0 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62201# = \y0y1y2y3y4.y3 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.0 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  reach# = \y0y1y2y3.y2 
  s = \y0.3 
  true = 0 
  union = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[reach#(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2 + x4 >= 2 + x4 = [[if!fac6220reach!fac62201#(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2 + x4 > x4 = [[reach#(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 2 + x2 + x3 >= 2 + x2 + x3 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 0 >= 0 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 0 >= 0 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_9, R_0, static, formative) by (P_10, R_0, static, formative), where P_10 consists of:

  reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_10, R_0, static, formative) is finite.

We consider the dependency pair problem (P_10, R_0, static, 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 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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

We consider the dependency pair problem (P_2, R_0, static, 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:

  union#(edge(X, Y, Z), U) >? union#(Z, U) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.0 
  edge = \y0y1y2.1 + y2 
  empty = 0 
  eq = \y0y1.0 
  false = 0 
  filter = \G0y1.0 
  filter2 = \y0G1y2y3.0 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.0 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.0 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.0 
  s = \y0.3 
  true = 0 
  union = \y0y1.y0 + y1 
  union# = \y0y1.y0 

Using this interpretation, the requirements translate to:

  [[union#(edge(_x0, _x1, _x2), _x3)]] = 1 + x2 > x2 = [[union#(_x2, _x3)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 1 + x2 + x3 >= 1 + x2 + x3 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 0 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 0 >= 0 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 0 >= 0 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  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, static, formative) is finite.

We consider the dependency pair problem (P_1, R_0, static, 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:

  eq#(s(X), s(Y)) >? eq#(X, Y) 
  eq(0, 0) >= true 
  eq(0, s(X)) >= false 
  eq(s(X), 0) >= false 
  eq(s(X), s(Y)) >= eq(X, Y) 
  or(true, X) >= true 
  or(false, X) >= X 
  union(empty, X) >= X 
  union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) 
  reach(X, Y, empty, Z) >= false 
  reach(X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) >= if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 
  if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) >= reach(X, Y, V, edge(Z, U, W)) 
  if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) >= true 
  if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) >= or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.0 
  edge = \y0y1y2.0 
  empty = 0 
  eq = \y0y1.0 
  eq# = \y0y1.y1 
  false = 0 
  filter = \G0y1.0 
  filter2 = \y0G1y2y3.0 
  if!fac6220reach!fac62201 = \y0y1y2y3y4.2y4 
  if!fac6220reach!fac62202 = \y0y1y2y3y4.0 
  map = \G0y1.0 
  nil = 0 
  or = \y0y1.y1 
  reach = \y0y1y2y3.2y3 
  s = \y0.3 + y0 
  true = 0 
  union = \y0y1.2y1 

Using this interpretation, the requirements translate to:

  [[eq#(s(_x0), s(_x1))]] = 3 + x1 > x1 = [[eq#(_x0, _x1)]] 
  [[eq(0, 0)]] = 0 >= 0 = [[true]] 
  [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] 
  [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] 
  [[or(true, _x0)]] = x0 >= 0 = [[true]] 
  [[or(false, _x0)]] = x0 >= x0 = [[_x0]] 
  [[union(empty, _x0)]] = 2x0 >= x0 = [[_x0]] 
  [[union(edge(_x0, _x1, _x2), _x3)]] = 2x3 >= 0 = [[edge(_x0, _x1, union(_x2, _x3))]] 
  [[reach(_x0, _x1, empty, _x2)]] = 2x2 >= 0 = [[false]] 
  [[reach(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2x5 >= 2x5 = [[if!fac6220reach!fac62201(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2x5 >= 0 = [[if!fac6220reach!fac62202(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] 
  [[if!fac6220reach!fac62201(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 2x5 >= 0 = [[reach(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] 
  [[if!fac6220reach!fac62202(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[true]] 
  [[if!fac6220reach!fac62202(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 0 >= 0 = [[or(reach(_x0, _x1, _x4, _x5), reach(_x3, _x1, union(_x4, _x5), empty))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 0 >= 0 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 0 >= 0 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 0 >= 0 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_0) by ({}, R_0).  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.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[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.