We consider the system Applicative_05__Ex9Maps.

  Alphabet:

    cons : [d * e] --> e 
    g : [] --> b 
    map!fac62201 : [d -> d * e] --> e 
    map!fac62202 : [d -> a -> d * a * e] --> e 
    map!fac62203 : [b -> d -> c -> d * b * c * e] --> e 

  Rules:

    map!fac62201(f, cons(x, y)) => cons(f x, map!fac62201(f, y)) 
    map!fac62202(f, x, cons(y, z)) => cons(f y x, map!fac62202(f, x, z)) 
    map!fac62203(f, g, x, cons(y, z)) => cons(f g y x, map!fac62203(f, g, x, z)) 

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]):

  map!fac62201(F, cons(X, Y)) >? cons(F X, map!fac62201(F, Y)) 
  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 
  map!fac62203(F, g, X, cons(Y, Z)) >? cons(F g Y X, map!fac62203(F, g, X, Z)) 

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

We choose Lex = {} and Mul = {@_{o -> o -> o -> o}, @_{o -> o -> o}, @_{o -> o}, cons, g, map!fac62201, map!fac62202, map!fac62203}, and the following precedence: map!fac62202 > map!fac62203 > @_{o -> o -> o} > @_{o -> o} = map!fac62201 > g > @_{o -> o -> o -> o} > cons

With these choices, we have:

  1] map!fac62201(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because [2], by (Star) 
  2] map!fac62201*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because map!fac62201 > cons, [3] and [8], by (Copy) 
  3] map!fac62201*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201 = @_{o -> o}, map!fac62201 in Mul, [4] and [5], by (Stat) 
  4] F >= F  by (Meta) 
  5] cons(X, Y) > X  because [6], by definition 
  6] cons*(X, Y) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] map!fac62201*(F, cons(X, Y)) >= map!fac62201(F, Y)  because map!fac62201 in Mul, [4] and [9], by (Stat) 
  9] cons(X, Y) > Y  because [10], by definition 
  10] cons*(X, Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

  12] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [13], by (Star) 
  13] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [14] and [24], by (Copy) 
  14] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [15] and [22], by (Copy) 
  15] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 > @_{o -> o -> o}, [16] and [18], by (Copy) 
  16] map!fac62202*(F, X, cons(Y, Z)) >= F  because [17], by (Select) 
  17] F >= F  by (Meta) 
  18] map!fac62202*(F, X, cons(Y, Z)) >= Y  because [19], by (Select) 
  19] cons(Y, Z) >= Y  because [20], by (Star) 
  20] cons*(Y, Z) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] map!fac62202*(F, X, cons(Y, Z)) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [25], [26] and [27], by (Stat) 
  25] F >= F  by (Meta) 
  26] X >= X  by (Meta) 
  27] cons(Y, Z) > Z  because [28], by definition 
  28] cons*(Y, Z) >= Z  because [29], by (Select) 
  29] Z >= Z  by (Meta) 

  30] map!fac62203(F, g, X, cons(Y, Z)) > cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X), map!fac62203(F, g, X, Z))  because [31], by definition 
  31] map!fac62203*(F, g, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X), map!fac62203(F, g, X, Z))  because map!fac62203 > cons, [32] and [45], by (Copy) 
  32] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y), X)  because map!fac62203 > @_{o -> o}, [33] and [43], by (Copy) 
  33] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, g), Y)  because map!fac62203 > @_{o -> o -> o}, [34] and [39], by (Copy) 
  34] map!fac62203*(F, g, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, g)  because map!fac62203 > @_{o -> o -> o -> o}, [35] and [37], by (Copy) 
  35] map!fac62203*(F, g, X, cons(Y, Z)) >= F  because [36], by (Select) 
  36] F >= F  by (Meta) 
  37] map!fac62203*(F, g, X, cons(Y, Z)) >= g  because [38], by (Select) 
  38] g >= g  by (Fun) 
  39] map!fac62203*(F, g, X, cons(Y, Z)) >= Y  because [40], by (Select) 
  40] cons(Y, Z) >= Y  because [41], by (Star) 
  41] cons*(Y, Z) >= Y  because [42], by (Select) 
  42] Y >= Y  by (Meta) 
  43] map!fac62203*(F, g, X, cons(Y, Z)) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 
  45] map!fac62203*(F, g, X, cons(Y, Z)) >= map!fac62203(F, g, X, Z)  because map!fac62203 in Mul, [46], [47], [48] and [49], by (Stat) 
  46] F >= F  by (Meta) 
  47] g >= g  by (Fun) 
  48] X >= X  by (Meta) 
  49] cons(Y, Z) > Z  because [50], by definition 
  50] cons*(Y, Z) >= Z  because [51], by (Select) 
  51] Z >= Z  by (Meta) 

We can thus remove the following rules:

  map!fac62203(F, g, X, cons(Y, Z)) => cons(F g Y X, map!fac62203(F, g, X, Z)) 

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

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

  map!fac62201(F, cons(X, Y)) >? cons(F X, map!fac62201(F, Y)) 
  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 

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

We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, map!fac62201, map!fac62202}, and the following precedence: map!fac62202 > @_{o -> o -> o} > map!fac62201 > cons > @_{o -> o}

With these choices, we have:

  1] map!fac62201(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because [2], by definition 
  2] map!fac62201*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because map!fac62201 > cons, [3] and [10], by (Copy) 
  3] map!fac62201*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201 > @_{o -> o}, [4] and [6], by (Copy) 
  4] map!fac62201*(F, cons(X, Y)) >= F  because [5], by (Select) 
  5] F >= F  by (Meta) 
  6] map!fac62201*(F, cons(X, Y)) >= X  because [7], by (Select) 
  7] cons(X, Y) >= X  because [8], by (Star) 
  8] cons*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] map!fac62201*(F, cons(X, Y)) >= map!fac62201(F, Y)  because map!fac62201 in Mul, [11] and [12], by (Stat) 
  11] F >= F  by (Meta) 
  12] cons(X, Y) > Y  because [13], by definition 
  13] cons*(X, Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [16], by (Star) 
  16] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [17] and [27], by (Copy) 
  17] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [18] and [25], by (Copy) 
  18] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 > @_{o -> o -> o}, [19] and [21], by (Copy) 
  19] map!fac62202*(F, X, cons(Y, Z)) >= F  because [20], by (Select) 
  20] F >= F  by (Meta) 
  21] map!fac62202*(F, X, cons(Y, Z)) >= Y  because [22], by (Select) 
  22] cons(Y, Z) >= Y  because [23], by (Star) 
  23] cons*(Y, Z) >= Y  because [24], by (Select) 
  24] Y >= Y  by (Meta) 
  25] map!fac62202*(F, X, cons(Y, Z)) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 
  27] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [28], [29] and [30], by (Stat) 
  28] F >= F  by (Meta) 
  29] X >= X  by (Meta) 
  30] cons(Y, Z) > Z  because [31], by definition 
  31] cons*(Y, Z) >= Z  because [32], by (Select) 
  32] Z >= Z  by (Meta) 

We can thus remove the following rules:

  map!fac62201(F, cons(X, Y)) => cons(F X, map!fac62201(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]):

  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 

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

We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, map!fac62202}, and the following precedence: @_{o -> o -> o} = map!fac62202 > @_{o -> o} > cons

With these choices, we have:

  1] map!fac62202(F, X, cons(Y, Z)) > cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [2], by definition 
  2] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [3] and [11], by (Copy) 
  3] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [4] and [9], by (Copy) 
  4] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 = @_{o -> o -> o}, map!fac62202 in Mul, [5] and [6], by (Stat) 
  5] F >= F  by (Meta) 
  6] cons(Y, Z) >= Y  because [7], by (Star) 
  7] cons*(Y, Z) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 
  9] map!fac62202*(F, X, cons(Y, Z)) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [5], [12] and [13], by (Stat) 
  12] X >= X  by (Meta) 
  13] cons(Y, Z) > Z  because [14], by definition 
  14] cons*(Y, Z) >= Z  because [15], by (Select) 
  15] Z >= Z  by (Meta) 

We can thus remove the following rules:

  map!fac62202(F, X, cons(Y, Z)) => cons(F Y X, map!fac62202(F, X, Z)) 

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.