We consider the system AotoYamada_05__023.

  Alphabet:

    0 : [] --> a 
    id : [] --> a -> a 
    plus : [a] --> a -> a 
    s : [a] --> a 

  Rules:

    id x => x 
    plus(0) => id 
    plus(s(x)) y => s(plus(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:

  id X => X 

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 the external first-order termination prover, this system is indeed terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRS Reverse [EQUIVALENT]
 || (2) QTRS
 || (3) RFCMatchBoundsTRSProof [EQUIVALENT]
 || (4) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    id(%X) -> %X
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRS Reverse (EQUIVALENT)
 || We applied the QTRS Reverse Processor [REVERSE].
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    id(x) -> x
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) RFCMatchBoundsTRSProof (EQUIVALENT)
 || Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
 || The following rules were used to construct the certificate:
 || 
 ||    id(x) -> x
 || 
 || The certificate found is represented by the following graph.
 || The certificate consists of the following enumerated nodes:
 || 3, 4
 || 
 || Node 3 is start node and node 4 is final node.
 || 
 || Those nodes are connected through the following edges:
 || 
 || * 3 to 4 labelled id_1(0), id_1(1)* 4 to 4 labelled #_1(0)
 || 
 || 
 || ----------------------------------------
 || 
 || (4)
 || YES
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] plus(0) X =#> id X   
    1] plus#(0) =#> id#   
    2] plus(s(X)) Y =#> plus(X) Y   
    3] plus(s(X)) Y =#> plus#(X)   

  Rules R_0:

    id X => X 
    plus(0) => id 
    plus(s(X)) Y => s(plus(X) Y) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, 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 :  
    * 2 : 0, 2, 3 
    * 3 : 1 

This graph has the following strongly connected components:

  P_1:

    plus(s(X)) Y =#> plus(X) 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).

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(plus) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(plus(s(X)) Y) = s(X) |> X = nu(plus(X) Y) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, 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.