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 nattprover, this system is indeed Ce-terminating:

 || Input TRS:
 ||     1: min(PeRCenTX,0()) -> 0()
 ||     2: min(0(),PeRCenTX) -> 0()
 ||     3: min(s(PeRCenTX),s(PeRCenTY)) -> s(min(PeRCenTX,PeRCenTY))
 ||     4: diff(PeRCenTX,0()) -> PeRCenTX
 ||     5: diff(0(),PeRCenTX) -> PeRCenTX
 ||     6: diff(s(PeRCenTX),s(PeRCenTY)) -> diff(PeRCenTX,PeRCenTY)
 ||     7: gcd(s(PeRCenTX),0()) -> s(PeRCenTX)
 ||     8: gcd(0(),s(PeRCenTX)) -> s(PeRCenTX)
 ||     9: gcd(s(PeRCenTX),s(PeRCenTY)) -> gcd(diff(PeRCenTX,PeRCenTY),s(min(PeRCenTX,PeRCenTY)))
 ||     10: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX
 ||     11: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY
 || Number of strict rules: 11
 || Direct POLO(bPol) ... failed.
 || Uncurrying ... failed.
 || Dependency Pairs:
 ||    #1: #diff(s(PeRCenTX),s(PeRCenTY)) -> #diff(PeRCenTX,PeRCenTY)
 ||    #2: #gcd(s(PeRCenTX),s(PeRCenTY)) -> #gcd(diff(PeRCenTX,PeRCenTY),s(min(PeRCenTX,PeRCenTY)))
 ||    #3: #gcd(s(PeRCenTX),s(PeRCenTY)) -> #diff(PeRCenTX,PeRCenTY)
 ||    #4: #gcd(s(PeRCenTX),s(PeRCenTY)) -> #min(PeRCenTX,PeRCenTY)
 ||    #5: #min(s(PeRCenTX),s(PeRCenTY)) -> #min(PeRCenTX,PeRCenTY)
 || Number of SCCs: 3, DPs: 3
 ||   SCC { #1 }
 || POLO(Sum)... succeeded.
 ||       TIlDePAIR	w: 0
 ||       diff	w: 0
 ||       s 	w: x1 + 1
 ||       gcd	w: 0
 ||       #min	w: 0
 ||       0 	w: 0
 ||       #TIlDePAIR	w: 0
 ||       #diff	w: x1 + x2
 ||       min	w: 0
 ||       #gcd	w: 0
 ||     USABLE RULES: { }
 ||     Removed DPs: #1
 || Number of SCCs: 2, DPs: 2
 ||   SCC { #5 }
 || POLO(Sum)... succeeded.
 ||       TIlDePAIR	w: 0
 ||       diff	w: 0
 ||       s 	w: x1 + 1
 ||       gcd	w: 0
 ||       #min	w: x1
 ||       0 	w: 0
 ||       #TIlDePAIR	w: 0
 ||       #diff	w: 0
 ||       min	w: 0
 ||       #gcd	w: 0
 ||     USABLE RULES: { }
 ||     Removed DPs: #5
 || Number of SCCs: 1, DPs: 1
 ||   SCC { #2 }
 || POLO(Sum)... POLO(max)... succeeded.
 ||       TIlDePAIR	w: 0
 ||       diff	w: max(x1 + 1, x2 + 1)
 ||       s 	w: x1 + 4
 ||       gcd	w: 0
 ||       #min	w: 0
 ||       0 	w: 1
 ||       #TIlDePAIR	w: 0
 ||       #diff	w: 0
 ||       min	w: max(x1 + 1)
 ||       #gcd	w: max(x1 + 2, x2)
 ||     USABLE RULES: { 1..6 }
 ||     Removed DPs: #2
 || Number of SCCs: 0, DPs: 0
 || 
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.