We consider the system AotoYamada_05__021.

  Alphabet:

    0 : [] --> a 
    cons : [a * b] --> b 
    double : [b] --> b 
    inc : [b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 
    plus : [a] --> a -> a 
    s : [a] --> a 
    times : [a] --> a -> a 

  Rules:

    plus(0) x => x 
    plus(s(x)) y => s(plus(x) y) 
    times(0) x => 0 
    times(s(x)) y => plus(times(x) y) y 
    inc(x) => map(plus(s(0)), x) 
    double(x) => map(times(s(s(0))), x) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(0) X >? X 
  plus(s(X)) Y >? s(plus(X) Y) 
  times(0) X >? 0 
  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, double, inc, map, plus, s, times}, and the following precedence: double > inc > times > plus > map > @_{o -> o} > cons > s

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(plus(_|_), X) > X 
  @_{o -> o}(plus(s(X)), Y) > s(@_{o -> o}(plus(X), Y)) 
  @_{o -> o}(times(_|_), X) > _|_ 
  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) > map(times(s(s(_|_))), X) 
  map(F, _|_) >= _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] @_{o -> o}(plus(_|_), X) > X  because [2], by definition 
  2] @_{o -> o}*(plus(_|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(plus(s(X)), Y) > s(@_{o -> o}(plus(X), Y))  because [5], by definition 
  5] @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because [6], by (Select) 
  6] plus(s(X)) @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because [7] 
  7] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= s(@_{o -> o}(plus(X), Y))  because plus > s and [8], by (Copy) 
  8] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= @_{o -> o}(plus(X), Y)  because plus > @_{o -> o}, [9] and [13], by (Copy) 
  9] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= plus(X)  because plus in Mul and [10], by (Stat) 
  10] s(X) > X  because [11], by definition 
  11] s*(X) >= X  because [12], by (Select) 
  12] X >= X  by (Meta) 
  13] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= Y  because [14], by (Select) 
  14] @_{o -> o}*(plus(s(X)), Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

  16] @_{o -> o}(times(_|_), X) > _|_  because [17], by definition 
  17] @_{o -> o}*(times(_|_), X) >= _|_  by (Bot) 

  18] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [19], by (Star) 
  19] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [20], by (Select) 
  20] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [21] 
  21] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [22] and [28], by (Copy) 
  22] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [23], by (Copy) 
  23] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because times > @_{o -> o}, [24] and [28], by (Copy) 
  24] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X)  because times in Mul and [25], by (Stat) 
  25] s(X) > X  because [26], by definition 
  26] s*(X) >= X  because [27], by (Select) 
  27] X >= X  by (Meta) 
  28] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [29], by (Select) 
  29] @_{o -> o}*(times(s(X)), Y) >= Y  because [30], by (Select) 
  30] Y >= Y  by (Meta) 

  31] inc(X) >= map(plus(s(_|_)), X)  because [32], by (Star) 
  32] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [33] and [36], by (Copy) 
  33] inc*(X) >= plus(s(_|_))  because inc > plus and [34], by (Copy) 
  34] inc*(X) >= s(_|_)  because inc > s and [35], by (Copy) 
  35] inc*(X) >= _|_  by (Bot) 
  36] inc*(X) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 

  38] double(X) > map(times(s(s(_|_))), X)  because [39], by definition 
  39] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [40] and [44], by (Copy) 
  40] double*(X) >= times(s(s(_|_)))  because double > times and [41], by (Copy) 
  41] double*(X) >= s(s(_|_))  because double > s and [42], by (Copy) 
  42] double*(X) >= s(_|_)  because double > s and [43], by (Copy) 
  43] double*(X) >= _|_  by (Bot) 
  44] double*(X) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 

  46] map(F, _|_) >= _|_  by (Bot) 

  47] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [48], by (Star) 
  48] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [49] and [56], by (Copy) 
  49] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [50] and [52], by (Copy) 
  50] map*(F, cons(X, Y)) >= F  because [51], by (Select) 
  51] F >= F  by (Meta) 
  52] map*(F, cons(X, Y)) >= X  because [53], by (Select) 
  53] cons(X, Y) >= X  because [54], by (Star) 
  54] cons*(X, Y) >= X  because [55], by (Select) 
  55] X >= X  by (Meta) 
  56] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [57] and [58], by (Stat) 
  57] F >= F  by (Meta) 
  58] cons(X, Y) > Y  because [59], by definition 
  59] cons*(X, Y) >= Y  because [60], by (Select) 
  60] Y >= Y  by (Meta) 

We can thus remove the following rules:

  plus(0) X => X 
  plus(s(X)) Y => s(plus(X) Y) 
  times(0) X => 0 
  double(X) => map(times(s(s(0))), X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  times(s(X), Y) >? plus(times(X, Y)) Y 
  inc(X) >? map(plus(s(0)), X) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, inc, map, plus, s, times}, and the following precedence: times > inc > map > @_{o -> o} > cons > plus > s

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  times(s(X), Y) > @_{o -> o}(plus(times(X, Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  map(F, _|_) > _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] times(s(X), Y) > @_{o -> o}(plus(times(X, Y)), Y)  because [2], by definition 
  2] times*(s(X), Y) >= @_{o -> o}(plus(times(X, Y)), Y)  because times > @_{o -> o}, [3] and [9], by (Copy) 
  3] times*(s(X), Y) >= plus(times(X, Y))  because times > plus and [4], by (Copy) 
  4] times*(s(X), Y) >= times(X, Y)  because times in Mul, [5] and [8], by (Stat) 
  5] s(X) > X  because [6], by definition 
  6] s*(X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Y >= Y  by (Meta) 
  9] times*(s(X), Y) >= Y  because [8], by (Select) 

  10] inc(X) >= map(plus(s(_|_)), X)  because [11], by (Star) 
  11] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [12] and [15], by (Copy) 
  12] inc*(X) >= plus(s(_|_))  because inc > plus and [13], by (Copy) 
  13] inc*(X) >= s(_|_)  because inc > s and [14], by (Copy) 
  14] inc*(X) >= _|_  by (Bot) 
  15] inc*(X) >= X  because [16], by (Select) 
  16] X >= X  by (Meta) 

  17] map(F, _|_) > _|_  because [18], by definition 
  18] map*(F, _|_) >= _|_  by (Bot) 

  19] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [20], by (Star) 
  20] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [21] and [28], by (Copy) 
  21] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [22] and [24], by (Copy) 
  22] map*(F, cons(X, Y)) >= F  because [23], by (Select) 
  23] F >= F  by (Meta) 
  24] map*(F, cons(X, Y)) >= X  because [25], by (Select) 
  25] cons(X, Y) >= X  because [26], by (Star) 
  26] cons*(X, Y) >= X  because [27], by (Select) 
  27] X >= X  by (Meta) 
  28] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [29] and [30], by (Stat) 
  29] F >= F  by (Meta) 
  30] cons(X, Y) > Y  because [31], by definition 
  31] cons*(X, Y) >= Y  because [32], by (Select) 
  32] Y >= Y  by (Meta) 

We can thus remove the following rules:

  times(s(X), Y) => plus(times(X, Y)) Y 
  map(F, nil) => nil 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  inc(X) >? map(plus(s(0)), X) 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[s(x_1)]] = x_1 

We choose Lex = {} and Mul = {@_{o -> o}, cons, inc, map, plus}, and the following precedence: inc > plus > map > @_{o -> o} > cons

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  inc(X) >= map(plus(_|_), X) 
  map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] inc(X) >= map(plus(_|_), X)  because [2], by (Star) 
  2] inc*(X) >= map(plus(_|_), X)  because inc > map, [3] and [5], by (Copy) 
  3] inc*(X) >= plus(_|_)  because inc > plus and [4], by (Copy) 
  4] inc*(X) >= _|_  by (Bot) 
  5] inc*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [8], by definition 
  8] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [9] and [16], by (Copy) 
  9] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [10] and [12], by (Copy) 
  10] map*(F, cons(X, Y)) >= F  because [11], by (Select) 
  11] F >= F  by (Meta) 
  12] map*(F, cons(X, Y)) >= X  because [13], by (Select) 
  13] cons(X, Y) >= X  because [14], by (Star) 
  14] cons*(X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [17] and [18], by (Stat) 
  17] F >= F  by (Meta) 
  18] cons(X, Y) > Y  because [19], by definition 
  19] cons*(X, Y) >= Y  because [20], by (Select) 
  20] Y >= Y  by (Meta) 

We can thus remove the following rules:

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  inc(X) >? map(plus(s(0)), X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[s(x_1)]] = x_1 

We choose Lex = {} and Mul = {inc, map, plus}, and the following precedence: inc > plus > map

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  inc(X) > map(plus(_|_), X) 

With these choices, we have:

  1] inc(X) > map(plus(_|_), X)  because [2], by definition 
  2] inc*(X) >= map(plus(_|_), X)  because inc > map, [3] and [5], by (Copy) 
  3] inc*(X) >= plus(_|_)  because inc > plus and [4], by (Copy) 
  4] inc*(X) >= _|_  by (Bot) 
  5] inc*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

We can thus remove the following rules:

  inc(X) => map(plus(s(0)), X) 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.