We consider the system onearg. Alphabet: 0 : [] --> nat add : [nat] --> nat -> nat eq : [nat] --> nat -> bool err : [] --> nat false : [] --> bool id : [] --> nat -> nat nul : [] --> nat -> bool pred : [nat] --> nat s : [nat] --> nat true : [] --> bool Rules: nul 0 => true nul s(x) => false nul err => false pred(0) => err pred(s(x)) => x id x => x eq(0) => nul eq(s(x)) => /\y.eq(x) pred(y) add(0) => id add(s(x)) => /\y.add(x) 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: nul(0) => true nul(s(X)) => false nul(err) => false pred(0) => err pred(s(X)) => X 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: || || nul(0) -> true || nul(s(%X)) -> false || nul(err) -> false || pred(0) -> err || pred(s(%X)) -> %X || 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: || || 0'(nul(x)) -> true'(x) || s(nul(x)) -> false'(x) || err'(nul(x)) -> false'(x) || 0'(pred(x)) -> err'(x) || s(pred(x)) -> x || 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: || || 0'(nul(x)) -> true'(x) || s(nul(x)) -> false'(x) || err'(nul(x)) -> false'(x) || 0'(pred(x)) -> err'(x) || s(pred(x)) -> x || id(x) -> x || || The certificate found is represented by the following graph. || The certificate consists of the following enumerated nodes: || 2, 4 || || Node 2 is start node and node 4 is final node. || || Those nodes are connected through the following edges: || || * 2 to 4 labelled true'_1(0), false'_1(0), err'_1(0), 0'_1(0), nul_1(0), s_1(0), pred_1(0), id_1(0), false'_1(1), true'_1(1), err'_1(1), 0'_1(1), nul_1(1), s_1(1), pred_1(1), 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 static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]). In order to do so, we start by eta-expanding the system, which gives: nul(0) => true nul(s(X)) => false nul(err) => false pred(0) => err pred(s(X)) => X id(X) => X eq(0, X) => nul(X) eq(s(X), Y) => (/\x.eq(X, pred(x))) Y add(0, X) => id(X) add(s(X), Y) => (/\x.add(X, s(x))) Y We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: 0] eq#(0, X) =#> nul#(X) 1] eq#(s(X), Y) =#> eq#(X, pred(Y)) 2] eq#(s(X), Y) =#> pred#(Y) 3] add#(0, X) =#> id#(X) 4] add#(s(X), Y) =#> add#(X, s(Y)) Rules R_0: nul(0) => true nul(s(X)) => false nul(err) => false pred(0) => err pred(s(X)) => X id(X) => X eq(0, X) => nul(X) eq(s(X), Y) => (/\x.eq(X, pred(x))) Y add(0, X) => id(X) add(s(X), Y) => (/\x.add(X, s(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 : * 1 : 0, 1, 2 * 2 : * 3 : * 4 : 3, 4 This graph has the following strongly connected components: P_1: eq#(s(X), Y) =#> eq#(X, pred(Y)) P_2: add#(s(X), Y) =#> add#(X, s(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(add#) = 1 Thus, we can orient the dependency pairs as follows: nu(add#(s(X), Y)) = s(X) |> X = nu(add#(X, s(Y))) 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(eq#) = 1 Thus, we can orient the dependency pairs as follows: nu(eq#(s(X), Y)) = s(X) |> X = nu(eq#(X, pred(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.