We consider the system Applicative_first_order_05__#3.38.

  Alphabet:

    0 : [] --> c 
    cons : [c * d] --> d 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    map : [c -> c * d] --> d 
    nil : [] --> d 
    rev : [d] --> d 
    rev1 : [c * d] --> c 
    rev2 : [c * d] --> d 
    s : [a] --> c 
    true : [] --> b 

  Rules:

    rev(nil) => nil 
    rev(cons(x, y)) => cons(rev1(x, y), rev2(x, y)) 
    rev1(0, nil) => 0 
    rev1(s(x), nil) => s(x) 
    rev1(x, cons(y, z)) => rev1(y, z) 
    rev2(x, nil) => nil 
    rev2(x, cons(y, z)) => rev(cons(x, rev2(y, z))) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(f, 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:

  rev(nil) => nil 
  rev(cons(X, Y)) => cons(rev1(X, Y), rev2(X, Y)) 
  rev1(0, nil) => 0 
  rev1(s(X), nil) => s(X) 
  rev1(X, cons(Y, Z)) => rev1(Y, Z) 
  rev2(X, nil) => nil 
  rev2(X, cons(Y, Z)) => rev(cons(X, rev2(Y, Z))) 

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 nattprover, this system is indeed terminating:

 || Input TRS:
 ||     1: rev(nil()) -> nil()
 ||     2: rev(cons(PeRCenTX,PeRCenTY)) -> cons(rev1(PeRCenTX,PeRCenTY),rev2(PeRCenTX,PeRCenTY))
 ||     3: rev1(0(),nil()) -> 0()
 ||     4: rev1(s(PeRCenTX),nil()) -> s(PeRCenTX)
 ||     5: rev1(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> rev1(PeRCenTY,PeRCenTZ)
 ||     6: rev2(PeRCenTX,nil()) -> nil()
 ||     7: rev2(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> rev(cons(PeRCenTX,rev2(PeRCenTY,PeRCenTZ)))
 || Number of strict rules: 7
 || Direct POLO(bPol) ... failed.
 || Uncurrying rev
 || 1: rev^1_nil() -> nil()
 || 2: rev^1_cons(PeRCenTX,PeRCenTY) -> cons(rev1(PeRCenTX,PeRCenTY),rev2(PeRCenTX,PeRCenTY))
 || 3: rev1(0(),nil()) -> 0()
 || 4: rev1(s(PeRCenTX),nil()) -> s(PeRCenTX)
 || 5: rev1(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> rev1(PeRCenTY,PeRCenTZ)
 || 6: rev2(PeRCenTX,nil()) -> nil()
 || 7: rev2(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> rev^1_cons(PeRCenTX,rev2(PeRCenTY,PeRCenTZ))
 || 8: rev(cons(_1,_2)) ->= rev^1_cons(_1,_2)
 || 9: rev(nil()) ->= rev^1_nil()
 || Number of strict rules: 7
 || Direct POLO(bPol) ... failed.
 || Dependency Pairs:
 ||    #1: #rev^1_cons(PeRCenTX,PeRCenTY) -> #rev1(PeRCenTX,PeRCenTY)
 ||    #2: #rev^1_cons(PeRCenTX,PeRCenTY) -> #rev2(PeRCenTX,PeRCenTY)
 ||    #3: #rev(nil()) ->? #rev^1_nil()
 ||    #4: #rev2(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> #rev^1_cons(PeRCenTX,rev2(PeRCenTY,PeRCenTZ))
 ||    #5: #rev2(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> #rev2(PeRCenTY,PeRCenTZ)
 ||    #6: #rev1(PeRCenTX,cons(PeRCenTY,PeRCenTZ)) -> #rev1(PeRCenTY,PeRCenTZ)
 ||    #7: #rev(cons(_1,_2)) ->? #rev^1_cons(_1,_2)
 || Number of SCCs: 2, DPs: 4
 ||   SCC { #6 }
 || POLO(Sum)... succeeded.
 ||       #rev	w: 0
 ||       s 	w: 0
 ||       rev1	w: 0
 ||       #rev1	w: x2
 ||       #rev2	w: 0
 ||       0 	w: 0
 ||       #rev^1_nil	w: 0
 ||       nil	w: 0
 ||       rev	w: 0
 ||       cons	w: x1 + x2 + 1
 ||       #rev^1_cons	w: 0
 ||       rev^1_cons	w: 0
 ||       rev2	w: 0
 ||       rev^1_nil	w: 0
 ||     USABLE RULES: { }
 ||     Removed DPs: #6
 || Number of SCCs: 1, DPs: 3
 ||   SCC { #2 #4 #5 }
 || POLO(Sum)... succeeded.
 ||       #rev	w: 0
 ||       s 	w: 1
 ||       rev1	w: x1 + x2 + 1
 ||       #rev1	w: 0
 ||       #rev2	w: x2
 ||       0 	w: 1
 ||       #rev^1_nil	w: 0
 ||       nil	w: 1
 ||       rev	w: 0
 ||       cons	w: x2 + 3
 ||       #rev^1_cons	w: x2 + 1
 ||       rev^1_cons	w: x2 + 3
 ||       rev2	w: x2
 ||       rev^1_nil	w: 0
 ||     USABLE RULES: { 2 6 7 }
 ||     Removed DPs: #2 #4 #5
 || 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] map#(F, cons(X, Y)) =#> map#(F, Y)   
    1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    rev(nil) => nil 
    rev(cons(X, Y)) => cons(rev1(X, Y), rev2(X, Y)) 
    rev1(0, nil) => 0 
    rev1(s(X), nil) => s(X) 
    rev1(X, cons(Y, Z)) => rev1(Y, Z) 
    rev2(X, nil) => nil 
    rev2(X, cons(Y, Z)) => rev(cons(X, rev2(Y, Z))) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, 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 : 2, 3 
    * 2 : 1 
    * 3 : 1 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, 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(filter2#) = 4 
  nu(filter#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by (P_3, R_0, computable, f), where P_3 contains:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_3, 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 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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(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_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.