We consider the system Applicative_05__mapDivMinus.

  Alphabet:

    0 : [] --> c 
    cons : [a * b] --> b 
    div : [c * c] --> c 
    map : [a -> a * b] --> b 
    minus : [c * c] --> c 
    nil : [] --> b 
    s : [c] --> c 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    minus(x, 0) => x 
    minus(s(x), s(y)) => minus(x, y) 
    div(0, s(x)) => 0 
    div(s(x), s(y)) => s(div(minus(x, y), s(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:

  minus(X, 0) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  div(0, s(X)) => 0 
  div(s(X), s(Y)) => s(div(minus(X, Y), s(Y))) 

Moreover, the system is orthogonal.  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 terminating when seen as a many-sorted first-order TRS.

According to mutermprover, this system is indeed terminating:

 || 
 || Problem 1: 
 || 
 || (VAR %X %Y)
 || (RULES
 || div(0,s(%X)) -> 0
 || div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 || minus(s(%X),s(%Y)) -> minus(%X,%Y)
 || minus(%X,0) -> %X
 || )
 || 
 || Problem 1: 
 || 
 || Innermost Equivalent Processor:
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 ||  
 || 
 || Problem 1: 
 || 
 || Dependency Pairs Processor:
 || -> Pairs:
 ||  DIV(s(%X),s(%Y)) -> DIV(minus(%X,%Y),s(%Y))
 ||  DIV(s(%X),s(%Y)) -> MINUS(%X,%Y)
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || 
 || Problem 1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  DIV(s(%X),s(%Y)) -> DIV(minus(%X,%Y),s(%Y))
 ||  DIV(s(%X),s(%Y)) -> MINUS(%X,%Y)
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->Strongly Connected Components:
 || ->->Cycle:
 || ->->-> Pairs:
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || ->->-> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->->Cycle:
 || ->->-> Pairs:
 ||  DIV(s(%X),s(%Y)) -> DIV(minus(%X,%Y),s(%Y))
 || ->->-> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || 
 || 
 || The problem is decomposed in 2 subproblems.
 || 
 || Problem 1.1: 
 || 
 || Subterm Processor:
 || -> Pairs:
 ||  MINUS(s(%X),s(%Y)) -> MINUS(%X,%Y)
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->Projection:
 ||  pi(MINUS) = 1
 || 
 || Problem 1.1: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->Strongly Connected Components:
 ||  There is no strongly connected component
 || 
 || The problem is finite.
 || 
 || Problem 1.2: 
 || 
 || Reduction Pairs Processor:
 || -> Pairs:
 ||  DIV(s(%X),s(%Y)) -> DIV(minus(%X,%Y),s(%Y))
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || -> Usable rules:
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->Interpretation type:
 ||  Linear
 || ->Coefficients:
 ||  Natural Numbers
 || ->Dimension:
 ||  1
 || ->Bound:
 ||  2
 || ->Interpretation:
 ||  
 || [minus](X1,X2) = 2.X1 + 1
 || [0] = 0
 || [s](X) = 2.X + 2
 || [DIV](X1,X2) = 2.X1
 || 
 || Problem 1.2: 
 || 
 || SCC Processor:
 || -> Pairs:
 ||  Empty
 || -> Rules:
 ||  div(0,s(%X)) -> 0
 ||  div(s(%X),s(%Y)) -> s(div(minus(%X,%Y),s(%Y)))
 ||  minus(s(%X),s(%Y)) -> minus(%X,%Y)
 ||  minus(%X,0) -> %X
 || ->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] map#(F, cons(X, Y)) =#> map#(F, Y)   

  Rules R_0:

    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    minus(X, 0) => X 
    minus(s(X), s(Y)) => minus(X, Y) 
    div(0, s(X)) => 0 
    div(s(X), s(Y)) => s(div(minus(X, Y), s(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 apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_0, 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.