We consider the system AotoYamada_05__011.

  Alphabet:

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

  Rules:

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

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 dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] plus s(X) Y =#> plus X Y   
    1] plus s(X) Y =#> plus X   
    2] plus s(X) Y =#> plus#   
    3] map(F) cons(X, Y) =#> F(X)   
    4] map(F) cons(X, Y) =#> map(F) Y   
    5] map(F) cons(X, Y) =#> map#(F)   
    6] curry(F, X) Y =#> F(X, Y)   
    7] inc X =#> map(curry(plus, s(0))) X   
    8] inc# =#> map#(curry(plus, s(0)))   
    9] inc# =#> curry#(plus, s(0))   
    10] inc# =#> plus#   

  Rules R_0:

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

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 : 0, 1, 2 
    * 1 :  
    * 2 :  
    * 3 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 4 : 3, 4, 5 
    * 5 :  
    * 6 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 7 : 3, 4, 5 
    * 8 :  
    * 9 :  
    * 10 :  

This graph has the following strongly connected components:

  P_1:

    plus s(X) Y =#> plus X Y   

  P_2:

    map(F) cons(X, Y) =#> F(X)   
    map(F) cons(X, Y) =#> map(F) Y   
    curry(F, X) Y =#> F(X, Y)   
    inc X =#> map(curry(plus, s(0))) 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) and (P_2, R_0, m, f).

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).

The formative rules of (P_2, R_0) are R_1 ::=

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

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_2, R_0, minimal, formative) by (P_2, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

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

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( curry(F, X) ) = #argfun-curry#(/\x.F X x) 
  pi( inc(X) ) = #argfun-inc#(map(#argfun-curry#(/\x.plus s(0) x), X), map(#argfun-curry#(/\y.plus s(0) y), X)) 

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

The following interpretation satisfies the requirements:

  #argfun-curry# = \G0y1.3 + G0(y1) 
  #argfun-inc# = \y0y1.3 + max(y0, y1) 
  0 = 0 
  cons = \y0y1.1 + y0 + y1 
  curry = \G0y1y2.0 
  inc = \y0.0 
  map = \G0y1.y1 + y1G0(y1) 
  plus = \y0y1.0 
  s = \y0.0 

Using this interpretation, the requirements translate to:

  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + x2 + F0(1 + x1 + x2) + x1F0(1 + x1 + x2) + x2F0(1 + x1 + x2) > F0(x1) = [[_F0(_x1)]] 
  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + x2 + F0(1 + x1 + x2) + x1F0(1 + x1 + x2) + x2F0(1 + x1 + x2) > x2 + x2F0(x2) = [[map(_F0, _x2)]] 
  [[#argfun-curry#(/\x._F0 _x1 x) _x2]] = max(x2, 3 + max(x1, x2, F0(x1,x2))) >= F0(x1,x2) = [[_F0(_x1, _x2)]] 
  [[#argfun-inc#(map(#argfun-curry#(/\x.plus s(0) x), _x0), map(#argfun-curry#(/\y.plus s(0) y), _x0))]] = 3 + 4x0 + x0x0 > 4x0 + x0x0 = [[map(#argfun-curry#(/\x.plus s(0) x), _x0)]] 
  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + x2 + F0(1 + x1 + x2) + x1F0(1 + x1 + x2) + x2F0(1 + x1 + x2) >= 1 + x2 + x2F0(x2) + max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[#argfun-inc#(map(#argfun-curry#(/\x.plus s(0) x), _x0), map(#argfun-curry#(/\y.plus s(0) y), _x0))]] = 3 + 4x0 + x0x0 >= 4x0 + x0x0 = [[map(#argfun-curry#(/\x.plus s(0) x), _x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 consists of:

  curry(F, X) Y =#> F(X, Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, minimal, formative).

This combination (P_3, R_1) has no formative rules!  We will name the empty set of rules:R_2.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_1, minimal, formative) by (P_3, R_2, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_3, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_2, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  curry(F, X, Y) >? F(X, Y) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( curry(F, X, Y) ) = #argfun-curry#(F X Y) 

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

The following interpretation satisfies the requirements:

  #argfun-curry# = \y0.1 + y0 
  curry = \G0y1y2.0 

Using this interpretation, the requirements translate to:

  [[#argfun-curry#(_F0 _x1 _x2)]] = 1 + max(x1, x2, F0(x1,x2)) > F0(x1,x2) = [[_F0(_x1, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, 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 (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(plus) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(plus s(X) Y) = s(X) |> X = nu(plus X Y) 

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.