We consider the system fuhkop12rta2.

  Alphabet:

    0 : [] --> nat 
    build : [nat] --> list 
    collapse : [list] --> nat 
    cons : [nat -> nat * list] --> list 
    diff : [nat * nat] --> nat 
    gcd : [nat * nat] --> nat 
    min : [nat * nat] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 

  Rules:

    min(x, 0) => 0 
    min(0, x) => 0 
    min(s(x), s(y)) => s(min(x, y)) 
    diff(x, 0) => x 
    diff(0, x) => x 
    diff(s(x), s(y)) => diff(x, y) 
    gcd(s(x), 0) => s(x) 
    gcd(0, s(x)) => s(x) 
    gcd(s(x), s(y)) => gcd(diff(x, y), s(min(x, y))) 
    collapse(nil) => 0 
    collapse(cons(f, x)) => f collapse(x) 
    build(0) => nil 
    build(s(x)) => cons(/\y.gcd(y, x), build(x)) 

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

We observe that the rules contain a first-order subset:

  min(X, 0) => 0 
  min(0, X) => 0 
  min(s(X), s(Y)) => s(min(X, Y)) 
  diff(X, 0) => X 
  diff(0, X) => X 
  diff(s(X), s(Y)) => diff(X, Y) 
  gcd(s(X), 0) => s(X) 
  gcd(0, s(X)) => s(X) 
  gcd(s(X), s(Y)) => gcd(diff(X, Y), s(min(X, Y))) 

Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.

According to mutermprover, this system is indeed Ce-terminating:

 || 
 || Problem 1: 
 || 
 || (VAR %X %Y)
 || (RULES
 || diff(0,%X) -> %X
 || diff(s(%X),s(%Y)) -> diff(%X,%Y)
 || diff(%X,0) -> %X
 || gcd(0,s(%X)) -> s(%X)
 || gcd(s(%X),0) -> s(%X)
 || gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 || min(0,%X) -> 0
 || min(s(%X),s(%Y)) -> s(min(%X,%Y))
 || min(%X,0) -> 0
 || ~PAIR(%X,%Y) -> %X
 || ~PAIR(%X,%Y) -> %Y
 || )
 || 
 || Problem 1: 
 || 
 || Dependency Pairs Processor:
 || -> Pairs:
 ||  DIFF(s(%X),s(%Y)) -> DIFF(%X,%Y)
 ||  GCD(s(%X),s(%Y)) -> DIFF(%X,%Y)
 ||  GCD(s(%X),s(%Y)) -> GCD(diff(%X,%Y),s(min(%X,%Y)))
 ||  GCD(s(%X),s(%Y)) -> MIN(%X,%Y)
 ||  MIN(s(%X),s(%Y)) -> MIN(%X,%Y)
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || 
 || Problem 1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  DIFF(s(%X),s(%Y)) -> DIFF(%X,%Y)
 ||  GCD(s(%X),s(%Y)) -> DIFF(%X,%Y)
 ||  GCD(s(%X),s(%Y)) -> GCD(diff(%X,%Y),s(min(%X,%Y)))
 ||  GCD(s(%X),s(%Y)) -> MIN(%X,%Y)
 ||  MIN(s(%X),s(%Y)) -> MIN(%X,%Y)
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Strongly Connected Components:
 || ->->Cycle:
 || ->->-> Pairs:
 ||  MIN(s(%X),s(%Y)) -> MIN(%X,%Y)
 || ->->-> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->->Cycle:
 || ->->-> Pairs:
 ||  DIFF(s(%X),s(%Y)) -> DIFF(%X,%Y)
 || ->->-> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->->Cycle:
 || ->->-> Pairs:
 ||  GCD(s(%X),s(%Y)) -> GCD(diff(%X,%Y),s(min(%X,%Y)))
 || ->->-> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || 
 || 
 || The problem is decomposed in 3 subproblems.
 || 
 || Problem 1.1: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  MIN(s(%X),s(%Y)) -> MIN(%X,%Y)
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Projection:
 ||  pi(MIN) = 1
 || 
 || Problem 1.1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
 || Problem 1.2: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  DIFF(s(%X),s(%Y)) -> DIFF(%X,%Y)
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Projection:
 ||  pi(DIFF) = 1
 || 
 || Problem 1.2: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
 || Problem 1.3: 
 || 
 || Reduction Pair Processor:
 || -> Pairs:
 ||  GCD(s(%X),s(%Y)) -> GCD(diff(%X,%Y),s(min(%X,%Y)))
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || -> Usable rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 || ->Interpretation type:
 ||  Linear
 || ->Coefficients:
 ||  Natural Numbers
 || ->Dimension:
 ||  1
 || ->Bound:
 ||  2
 || ->Interpretation:
 ||  
 || [diff](X1,X2) = X1 + X2 + 1
 || [min](X1,X2) = X1
 || [0] = 0
 || [s](X) = 2.X + 2
 || [GCD](X1,X2) = 2.X1 + X2
 || 
 || Problem 1.3: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  diff(0,%X) -> %X
 ||  diff(s(%X),s(%Y)) -> diff(%X,%Y)
 ||  diff(%X,0) -> %X
 ||  gcd(0,s(%X)) -> s(%X)
 ||  gcd(s(%X),0) -> s(%X)
 ||  gcd(s(%X),s(%Y)) -> gcd(diff(%X,%Y),s(min(%X,%Y)))
 ||  min(0,%X) -> 0
 ||  min(s(%X),s(%Y)) -> s(min(%X,%Y))
 ||  min(%X,0) -> 0
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
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] collapse#(cons(F, X)) =#> collapse#(X)   
    1] build#(s(X)) =#> gcd#(Y, X)   
    2] build#(s(X)) =#> build#(X)   

  Rules R_0:

    min(X, 0) => 0 
    min(0, X) => 0 
    min(s(X), s(Y)) => s(min(X, Y)) 
    diff(X, 0) => X 
    diff(0, X) => X 
    diff(s(X), s(Y)) => diff(X, Y) 
    gcd(s(X), 0) => s(X) 
    gcd(0, s(X)) => s(X) 
    gcd(s(X), s(Y)) => gcd(diff(X, Y), s(min(X, Y))) 
    collapse(nil) => 0 
    collapse(cons(F, X)) => F collapse(X) 
    build(0) => nil 
    build(s(X)) => cons(/\x.gcd(x, X), build(X)) 

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 :  
    * 2 : 1, 2 

This graph has the following strongly connected components:

  P_1:

    collapse#(cons(F, X)) =#> collapse#(X)   

  P_2:

    build#(s(X)) =#> build#(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, 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(build#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(build#(s(X))) = s(X) |> X = nu(build#(X)) 

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(collapse#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(collapse#(cons(F, X))) = cons(F, X) |> X = nu(collapse#(X)) 

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.