We consider the system zipWith.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    false : [] --> bool 
    gcd : [nat * nat] --> nat 
    gcdlists : [list * list] --> list 
    if : [bool * nat * nat] --> nat 
    le : [nat * nat] --> bool 
    minus : [nat * nat] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 
    true : [] --> bool 
    zipWith : [nat -> nat -> nat * list * list] --> list 

  Rules:

    le(0, x) => true 
    le(s(x), 0) => false 
    le(s(x), s(y)) => le(x, y) 
    minus(x, 0) => x 
    minus(s(x), s(y)) => minus(x, y) 
    gcd(0, x) => 0 
    gcd(s(x), 0) => 0 
    gcd(s(x), s(y)) => if(le(y, x), s(x), s(y)) 
    if(true, s(x), s(y)) => gcd(minus(x, y), s(y)) 
    if(false, s(x), s(y)) => gcd(minus(y, x), s(x)) 
    zipWith(f, x, nil) => nil 
    zipWith(f, nil, x) => nil 
    zipWith(f, cons(x, y), cons(z, u)) => cons(f x z, zipWith(f, y, u)) 
    gcdlists(x, y) => zipWith(/\z./\u.gcd(z, u), x, y) 

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:

  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  gcd(0, X) => 0 
  gcd(s(X), 0) => 0 
  gcd(s(X), s(Y)) => if(le(Y, X), s(X), s(Y)) 
  if(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) 
  if(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) 

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
 || gcd(0,%X) -> 0
 || gcd(s(%X),0) -> 0
 || gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 || if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 || if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 || le(0,%X) -> true
 || le(s(%X),0) -> false
 || le(s(%X),s(%Y)) -> le(%X,%Y)
 || minus(s(%X),s(%Y)) -> minus(%X,%Y)
 || minus(%X,0) -> %X
 || ~PAIR(%X,%Y) -> %X
 || ~PAIR(%X,%Y) -> %Y
 || )
 || 
 || Problem 1: 
 || 
 || Dependency Pairs Processor:
 || -> Pairs:
 ||  GCD(s(%X),s(%Y)) -> IF(le(%Y,%X),s(%X),s(%Y))
 ||  GCD(s(%X),s(%Y)) -> LE(%Y,%X)
 ||  IF(false,s(%X),s(%Y)) -> GCD(minus(%Y,%X),s(%X))
 ||  IF(false,s(%X),s(%Y)) -> MINUS(%Y,%X)
 ||  IF(true,s(%X),s(%Y)) -> GCD(minus(%X,%Y),s(%Y))
 ||  IF(true,s(%X),s(%Y)) -> MINUS(%X,%Y)
 ||  LE(s(%X),s(%Y)) -> LE(%X,%Y)
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || 
 || Problem 1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  GCD(s(%X),s(%Y)) -> IF(le(%Y,%X),s(%X),s(%Y))
 ||  GCD(s(%X),s(%Y)) -> LE(%Y,%X)
 ||  IF(false,s(%X),s(%Y)) -> GCD(minus(%Y,%X),s(%X))
 ||  IF(false,s(%X),s(%Y)) -> MINUS(%Y,%X)
 ||  IF(true,s(%X),s(%Y)) -> GCD(minus(%X,%Y),s(%Y))
 ||  IF(true,s(%X),s(%Y)) -> MINUS(%X,%Y)
 ||  LE(s(%X),s(%Y)) -> LE(%X,%Y)
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Strongly Connected Components:
 || ->->Cycle:
 || ->->-> Pairs:
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || ->->-> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->->Cycle:
 || ->->-> Pairs:
 ||  LE(s(%X),s(%Y)) -> LE(%X,%Y)
 || ->->-> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->->Cycle:
 || ->->-> Pairs:
 ||  GCD(s(%X),s(%Y)) -> IF(le(%Y,%X),s(%X),s(%Y))
 ||  IF(false,s(%X),s(%Y)) -> GCD(minus(%Y,%X),s(%X))
 ||  IF(true,s(%X),s(%Y)) -> GCD(minus(%X,%Y),s(%Y))
 || ->->-> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || 
 || 
 || The problem is decomposed in 3 subproblems.
 || 
 || Problem 1.1: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Projection:
 ||  pi(MINUS) = 1
 || 
 || Problem 1.1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~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:
 ||  LE(s(%X),s(%Y)) -> LE(%X,%Y)
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || ->Projection:
 ||  pi(LE) = 1
 || 
 || Problem 1.2: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~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)) -> IF(le(%Y,%X),s(%X),s(%Y))
 ||  IF(false,s(%X),s(%Y)) -> GCD(minus(%Y,%X),s(%X))
 ||  IF(true,s(%X),s(%Y)) -> GCD(minus(%X,%Y),s(%Y))
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~PAIR(%X,%Y) -> %X
 ||  ~PAIR(%X,%Y) -> %Y
 || -> Usable rules:
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->Interpretation type:
 ||  Linear
 || ->Coefficients:
 ||  Natural Numbers
 || ->Dimension:
 ||  1
 || ->Bound:
 ||  2
 || ->Interpretation:
 ||  
 || [le](X1,X2) = X1 + 2.X2 + 2
 || [minus](X1,X2) = 2.X1
 || [0] = 2
 || [false] = 1
 || [s](X) = 2.X + 2
 || [true] = 2
 || [GCD](X1,X2) = 2.X1 + 2.X2 + 2
 || [IF](X1,X2,X3) = 2.X2 + 2.X3 + 1
 || 
 || Problem 1.3: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  IF(false,s(%X),s(%Y)) -> GCD(minus(%Y,%X),s(%X))
 ||  IF(true,s(%X),s(%Y)) -> GCD(minus(%X,%Y),s(%Y))
 || -> Rules:
 ||  gcd(0,%X) -> 0
 ||  gcd(s(%X),0) -> 0
 ||  gcd(s(%X),s(%Y)) -> if(le(%Y,%X),s(%X),s(%Y))
 ||  if(false,s(%X),s(%Y)) -> gcd(minus(%Y,%X),s(%X))
 ||  if(true,s(%X),s(%Y)) -> gcd(minus(%X,%Y),s(%Y))
 ||  le(0,%X) -> true
 ||  le(s(%X),0) -> false
 ||  le(s(%X),s(%Y)) -> le(%X,%Y)
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 ||  ~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] zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U)   
    1] gcdlists#(X, Y) =#> zipWith#(/\x./\y.gcd(x, y), X, Y)   
    2] gcdlists#(X, Y) =#> gcd#(Z, U)   

  Rules R_0:

    le(0, X) => true 
    le(s(X), 0) => false 
    le(s(X), s(Y)) => le(X, Y) 
    minus(X, 0) => X 
    minus(s(X), s(Y)) => minus(X, Y) 
    gcd(0, X) => 0 
    gcd(s(X), 0) => 0 
    gcd(s(X), s(Y)) => if(le(Y, X), s(X), s(Y)) 
    if(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) 
    if(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) 
    zipWith(F, X, nil) => nil 
    zipWith(F, nil, X) => nil 
    zipWith(F, cons(X, Y), cons(Z, U)) => cons(F X Z, zipWith(F, Y, U)) 
    gcdlists(X, Y) => zipWith(/\x./\y.gcd(x, y), X, Y) 

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 :  

This graph has the following strongly connected components:

  P_1:

    zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U)   

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).

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(zipWith#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(zipWith#(F, cons(X, Y), cons(Z, U))) = cons(X, Y) |> Y = nu(zipWith#(F, Y, U)) 

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.