We consider the system 429.

  Alphabet:

    asort : [natlist] --> natlist 
    cons : [nat * natlist] --> natlist 
    dsort : [natlist] --> natlist 
    insert : [nat * natlist * nat -> nat -> nat * nat -> nat -> nat] --> natlist 
    max : [nat * nat] --> nat 
    min : [nat * nat] --> nat 
    nil : [] --> natlist 
    sort : [natlist * nat -> nat -> nat * nat -> nat -> nat] --> natlist 

  Rules:

    insert(X, nil, /\x./\y.Y(x, y), /\z./\u.Z(z, u)) => cons(X, nil) 
    insert(X, cons(Y, Z), /\x./\y.U(x, y), /\z./\u.V(z, u)) => cons(U(X, Y), insert(V(X, Y), Z, /\v./\w.U(v, w), /\x'./\y'.V(x', y'))) 
    sort(nil, /\x./\y.X(x, y), /\z./\u.Y(z, u)) => nil 
    sort(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u)) => insert(X, sort(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', y')), /\z'./\u'.Z(z', u'), /\v'./\w'.U(v', w')) 
    asort(X) => sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) 
    dsort(X) => sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) 

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 [FuhKop19]).

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

  Dependency Pairs P_0:

    0] insert#(X, cons(Y, Z), /\x./\y.U(x, y), /\z./\u.V(z, u)) =#> insert#(V(X, Y), Z, /\v./\w.U(v, w), /\x'./\y'.V(x', y'))   
    1] sort#(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u)) =#> insert#(X, sort(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', y')), /\z'./\u'.Z(z', u'), /\v'./\w'.U(v', w'))   
    2] sort#(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u)) =#> sort#(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', y'))   
    3] asort#(X) =#> sort#(X, /\x./\y.min(x, y), /\z./\u.max(z, u))   
    4] dsort#(X) =#> sort#(X, /\x./\y.max(x, y), /\z./\u.min(z, u))   

  Rules R_0:

    insert(X, nil, /\x./\y.Y(x, y), /\z./\u.Z(z, u)) => cons(X, nil) 
    insert(X, cons(Y, Z), /\x./\y.U(x, y), /\z./\u.V(z, u)) => cons(U(X, Y), insert(V(X, Y), Z, /\v./\w.U(v, w), /\x'./\y'.V(x', y'))) 
    sort(nil, /\x./\y.X(x, y), /\z./\u.Y(z, u)) => nil 
    sort(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u)) => insert(X, sort(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', y')), /\z'./\u'.Z(z', u'), /\v'./\w'.U(v', w')) 
    asort(X) => sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) 
    dsort(X) => sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) 

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

We consider the dependency pair problem (P_0, R_0, computable, 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 : 0 
    * 2 : 1, 2 
    * 3 : 1, 2 
    * 4 : 1, 2 

This graph has the following strongly connected components:

  P_1:

    insert#(X, cons(Y, Z), /\x./\y.U(x, y), /\z./\u.V(z, u)) =#> insert#(V(X, Y), Z, /\v./\w.U(v, w), /\x'./\y'.V(x', y'))   

  P_2:

    sort#(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u)) =#> sort#(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', 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) and (P_2, R_0, m, f).

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

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(sort#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(sort#(cons(X, Y), /\x./\y.Z(x, y), /\z./\u.U(z, u))) = cons(X, Y) |> Y = nu(sort#(Y, /\v./\w.Z(v, w), /\x'./\y'.U(x', y'))) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, f).  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, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(insert#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(insert#(X, cons(Y, Z), /\x./\y.U(x, y), /\z./\u.V(z, u))) = cons(Y, Z) |> Z = nu(insert#(V(X, Y), Z, /\v./\w.U(v, w), /\x'./\y'.V(x', y'))) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, 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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[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.