We consider the system AotoYamada_05__010.

  Alphabet:

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

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

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

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

Argument functions:

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

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

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

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

With these choices, we have:

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

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

  14] @_{o -> o}(@_{o -> o -> o}(times, _|_), X) >= _|_  by (Bot) 

  15] @_{o -> o}(@_{o -> o -> o}(times, s(X)), Y) >= @_{o -> o}(@_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y)), Y)  because [16], by (Star) 
  16] @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= @_{o -> o}(@_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y)), Y)  because [17], by (Select) 
  17] @_{o -> o -> o}(times, s(X)) @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= @_{o -> o}(@_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y)), Y)  because [18] 
  18] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o}(@_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y)), Y)  because @_{o -> o -> o} > @_{o -> o}, [19] and [30], by (Copy) 
  19] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y))  because [20], by (Select) 
  20] times @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y))  because [21] 
  21] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= @_{o -> o -> o}(plus, @_{o -> o}(@_{o -> o -> o}(times, X), Y))  because times > @_{o -> o -> o}, [22] and [23], by (Copy) 
  22] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= plus  because times > plus, by (Copy) 
  23] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= @_{o -> o}(@_{o -> o -> o}(times, X), Y)  because [24], by (Select) 
  24] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o}(@_{o -> o -> o}(times, X), Y)  because @_{o -> o -> o} > @_{o -> o}, [25] and [30], by (Copy) 
  25] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o -> o}(times, X)  because @_{o -> o -> o} in Mul, [26] and [27], by (Stat) 
  26] times >= times  by (Fun) 
  27] s(X) > X  because [28], by definition 
  28] s*(X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= Y  because [31], by (Select) 
  31] @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= Y  because [32], by (Select) 
  32] Y >= Y  by (Meta) 

  33] @_{o -> o}(curry(F, X), Y) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because @_{o -> o} in Mul, [34] and [37], by (Fun) 
  34] curry(F, X) >= @_{o -> o -> o}(F, X)  because curry = @_{o -> o -> o}, curry in Mul, [35] and [36], by (Fun) 
  35] F >= F  by (Meta) 
  36] X >= X  by (Meta) 
  37] Y >= Y  by (Meta) 

  38] @_{o -> o}(map(F), _|_) >= _|_  by (Bot) 

  39] @_{o -> o}(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [40], by (Star) 
  40] @_{o -> o}*(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [41], by (Select) 
  41] map(F) @_{o -> o}*(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [42] 
  42] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because map > cons, [43] and [54], by (Copy) 
  43] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [44] and [46], by (Copy) 
  44] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= F  because [45], by (Select) 
  45] F >= F  by (Meta) 
  46] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= X  because [47], by (Select) 
  47] @_{o -> o}*(map(F), cons(X, Y)) >= X  because [48], by (Select) 
  48] map(F) @_{o -> o}*(map(F), cons(X, Y)) >= X  because [49] 
  49] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= X  because [50], by (Select) 
  50] @_{o -> o}*(map(F), cons(X, Y)) >= X  because [51], by (Select) 
  51] cons(X, Y) >= X  because [52], by (Star) 
  52] cons*(X, Y) >= X  because [53], by (Select) 
  53] X >= X  by (Meta) 
  54] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= @_{o -> o}(map(F), Y)  because [55], by (Select) 
  55] @_{o -> o}*(map(F), cons(X, Y)) >= @_{o -> o}(map(F), Y)  because @_{o -> o} in Mul, [56] and [58], by (Stat) 
  56] map(F) >= map(F)  because map in Mul and [57], by (Fun) 
  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) 

  61] inc > map(curry(plus, s(_|_)))  because [62], by definition 
  62] inc* >= map(curry(plus, s(_|_)))  because inc > map and [63], by (Copy) 
  63] inc* >= curry(plus, s(_|_))  because inc > curry, [64] and [65], by (Copy) 
  64] inc* >= plus  because inc > plus, by (Copy) 
  65] inc* >= s(_|_)  because inc > s and [66], by (Copy) 
  66] inc* >= _|_  by (Bot) 

  67] double >= map(curry(times, s(s(_|_))))  because [68], by (Star) 
  68] double* >= map(curry(times, s(s(_|_))))  because double > map and [69], by (Copy) 
  69] double* >= curry(times, s(s(_|_)))  because double > curry, [70] and [71], by (Copy) 
  70] double* >= times  because double > times, by (Copy) 
  71] double* >= s(s(_|_))  because double > s and [72], by (Copy) 
  72] double* >= s(_|_)  because double > s and [73], by (Copy) 
  73] double* >= _|_  by (Bot) 

We can thus remove the following rules:

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

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

Argument functions:

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

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

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

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

With these choices, we have:

  1] plus(s(X), Y) >= s(plus(X, Y))  because [2], by (Star) 
  2] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [3], by (Copy) 
  3] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [4] and [7], by (Stat) 
  4] s(X) > X  because [5], by definition 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] Y >= Y  by (Meta) 

  8] @_{o -> o}(@_{o -> o -> o}(times, _|_), X) >= _|_  by (Bot) 

  9] @_{o -> o}(@_{o -> o -> o}(times, s(X)), Y) > plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because [10], by definition 
  10] @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because [11], by (Select) 
  11] @_{o -> o -> o}(times, s(X)) @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because [12] 
  12] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because [13], by (Select) 
  13] times @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because [14] 
  14] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)), @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= plus(@_{o -> o}(@_{o -> o -> o}(times, X), Y), Y)  because times > plus, [15] and [25], by (Copy) 
  15] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)), @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= @_{o -> o}(@_{o -> o -> o}(times, X), Y)  because [16], by (Select) 
  16] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= @_{o -> o}(@_{o -> o -> o}(times, X), Y)  because [17], by (Select) 
  17] @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= @_{o -> o}(@_{o -> o -> o}(times, X), Y)  because @_{o -> o} in Mul, [18] and [24], by (Stat) 
  18] @_{o -> o -> o}(times, s(X)) > @_{o -> o -> o}(times, X)  because [19], by definition 
  19] @_{o -> o -> o}*(times, s(X)) >= @_{o -> o -> o}(times, X)  because @_{o -> o -> o} in Mul, [20] and [21], by (Stat) 
  20] times >= times  by (Fun) 
  21] s(X) > X  because [22], by definition 
  22] s*(X) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] Y >= Y  by (Meta) 
  25] times*(@_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)), @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y))) >= Y  because [26], by (Select) 
  26] @_{o -> o -> o}*(times, s(X), @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y)) >= Y  because [27], by (Select) 
  27] @_{o -> o}*(@_{o -> o -> o}(times, s(X)), Y) >= Y  because [24], by (Select) 

  28] @_{o -> o}(curry(F, X), Y) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [29], by (Star) 
  29] @_{o -> o}*(curry(F, X), Y) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [30], by (Select) 
  30] curry(F, X) @_{o -> o}*(curry(F, X), Y) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [31] 
  31] curry*(F, X, @_{o -> o}*(curry(F, X), Y)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because curry > @_{o -> o}, [32] and [37], by (Copy) 
  32] curry*(F, X, @_{o -> o}*(curry(F, X), Y)) >= @_{o -> o -> o}(F, X)  because curry > @_{o -> o -> o}, [33] and [35], by (Copy) 
  33] curry*(F, X, @_{o -> o}*(curry(F, X), Y)) >= F  because [34], by (Select) 
  34] F >= F  by (Meta) 
  35] curry*(F, X, @_{o -> o}*(curry(F, X), Y)) >= X  because [36], by (Select) 
  36] X >= X  by (Meta) 
  37] curry*(F, X, @_{o -> o}*(curry(F, X), Y)) >= Y  because [38], by (Select) 
  38] @_{o -> o}*(curry(F, X), Y) >= Y  because [39], by (Select) 
  39] Y >= Y  by (Meta) 

  40] @_{o -> o}(map(F), _|_) >= _|_  by (Bot) 

  41] @_{o -> o}(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [42], by (Star) 
  42] @_{o -> o}*(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [43], by (Select) 
  43] map(F) @_{o -> o}*(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because [44] 
  44] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y))  because map > cons, [45] and [53], by (Copy) 
  45] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [46] and [48], by (Copy) 
  46] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= F  because [47], by (Select) 
  47] F >= F  by (Meta) 
  48] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= X  because [49], by (Select) 
  49] @_{o -> o}*(map(F), cons(X, Y)) >= X  because [50], by (Select) 
  50] cons(X, Y) >= X  because [51], by (Star) 
  51] cons*(X, Y) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 
  53] map*(F, @_{o -> o}*(map(F), cons(X, Y))) >= @_{o -> o}(map(F), Y)  because [54], by (Select) 
  54] @_{o -> o}*(map(F), cons(X, Y)) >= @_{o -> o}(map(F), Y)  because @_{o -> o} in Mul, [55] and [57], by (Stat) 
  55] map(F) >= map(F)  because map in Mul and [56], by (Fun) 
  56] F >= F  by (Meta) 
  57] cons(X, Y) > Y  because [58], by definition 
  58] cons*(X, Y) >= Y  because [59], by (Select) 
  59] Y >= Y  by (Meta) 

  60] double > map(curry(times, s(s(_|_))))  because [61], by definition 
  61] double* >= map(curry(times, s(s(_|_))))  because double > map and [62], by (Copy) 
  62] double* >= curry(times, s(s(_|_)))  because double > curry, [63] and [64], by (Copy) 
  63] double* >= times  because double > times, by (Copy) 
  64] double* >= s(s(_|_))  because double > s and [65], by (Copy) 
  65] double* >= s(_|_)  because double > s and [66], by (Copy) 
  66] double* >= _|_  by (Bot) 

We can thus remove the following rules:

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

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

The following interpretation satisfies the requirements:

  0 = 0 
  cons = \y0y1.3 + y0 + y1 
  curry = \G0y1y2.3 + y1 + y2 + G0(y1,y2) 
  map = \G0y1.3 + 3y1 + G0(y1) + 2y1G0(y1) 
  nil = 0 
  plus = \y0y1.y1 + 3y0 
  s = \y0.3 + y0 
  times = \y0y1.3 + y1 + 3y0 

Using this interpretation, the requirements translate to:

  [[plus(s(_x0), _x1)]] = 9 + x1 + 3x0 > 3 + x1 + 3x0 = [[s(plus(_x0, _x1))]] 
  [[times(0, _x0)]] = 3 + x0 > 0 = [[0]] 
  [[curry(_F0, _x1, _x2)]] = 3 + x1 + x2 + F0(x1,x2) > x1 + x2 + F0(x1,x2) = [[_F0 _x1 _x2]] 
  [[map(_F0, nil)]] = 3 + F0(0) > 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 12 + 3x1 + 3x2 + 2x1F0(3 + x1 + x2) + 2x2F0(3 + x1 + x2) + 7F0(3 + x1 + x2) > 6 + x1 + 3x2 + F0(x1) + F0(x2) + 2x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 

We can thus remove the following rules:

  plus(s(X), Y) => s(plus(X, Y)) 
  times(0, X) => 0 
  curry(F, X, Y) => F X Y 
  map(F, nil) => nil 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 

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.