We consider the system DicosmoKesner93. Alphabet: app : [] --> arrAB -> A -> B case : [] --> SAB -> (A -> C) -> (B -> C) -> C inl : [] --> A -> SAB inr : [] --> B -> SAB lam : [] --> (A -> B) -> arrAB pair : [] --> A -> B -> PAB piA : [] --> PAB -> A piB : [] --> PAB -> B Rules: app (lam (/\x.f x)) y => f y lam (/\x.app y x) => y piA (pair x y) => x piB (pair x y) => y pair (piA x) (piB x) => x case (inl x) (/\y.f y) (/\z.g z) => f x case (inr x) (/\y.f y) (/\z.g z) => g x Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: Alphabet: app : [arrAB * A] --> B case : [SAB * A -> C * B -> C] --> C inl : [A] --> SAB inr : [B] --> SAB lam : [A -> B] --> arrAB pair : [A * B] --> PAB piA : [PAB] --> A piB : [PAB] --> B ~AP1 : [A -> B * A] --> B ~AP2 : [A -> C * A] --> C ~AP3 : [B -> C * B] --> C Rules: app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) lam(/\x.app(X, x)) => X piA(pair(X, Y)) => X piB(pair(X, Y)) => Y pair(piA(X), piB(X)) => X case(inl(X), /\x.~AP2(F, x), /\y.~AP3(G, y)) => ~AP2(F, X) case(inr(X), /\x.~AP2(F, x), /\y.~AP3(G, y)) => ~AP3(G, X) app(lam(/\x.app(X, x)), Y) => app(X, Y) ~AP1(F, X) => F X ~AP2(F, X) => F X ~AP3(F, X) => F X Symbols ~AP1, ~AP2, and ~AP3 are encodings for application that are only used in innocuous ways. We can simplify the program (without losing non-termination) by removing them. Additionally, we can remove some (now-)redundant rules. This gives: Alphabet: app : [arrAB * A] --> B case : [SAB * A -> C * B -> C] --> C inl : [A] --> SAB inr : [B] --> SAB lam : [A -> B] --> arrAB pair : [A * B] --> PAB piA : [PAB] --> A piB : [PAB] --> B Rules: app(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X piA(pair(X, Y)) => X piB(pair(X, Y)) => Y pair(piA(X), piB(X)) => X case(inl(X), /\x.Y(x), /\y.Z(y)) => Y(X) case(inr(X), /\x.Y(x), /\y.Z(y)) => Z(X) We observe that the rules contain a first-order subset: piA(pair(X, Y)) => X piB(pair(X, Y)) => Y pair(piA(X), piB(X)) => X 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 the external first-order termination prover, this system is indeed Ce-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) QTRSRRRProof [EQUIVALENT] || (2) QTRS || (3) RisEmptyProof [EQUIVALENT] || (4) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || piA(pair(%X, %Y)) -> %X || piB(pair(%X, %Y)) -> %Y || pair(piA(%X), piB(%X)) -> %X || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || || ---------------------------------------- || || (1) QTRSRRRProof (EQUIVALENT) || Used ordering: || Polynomial interpretation [POLO]: || || POL(pair(x_1, x_2)) = 1 + x_1 + x_2 || POL(piA(x_1)) = 1 + 2*x_1 || POL(piB(x_1)) = 1 + x_1 || POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: || || piA(pair(%X, %Y)) -> %X || piB(pair(%X, %Y)) -> %Y || pair(piA(%X), piB(%X)) -> %X || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || || || || ---------------------------------------- || || (2) || Obligation: || Q restricted rewrite system: || R is empty. || Q is empty. || || ---------------------------------------- || || (3) RisEmptyProof (EQUIVALENT) || The TRS R is empty. Hence, termination is trivially proven. || ---------------------------------------- || || (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]). We thus obtain the following dependency pair problem (P_0, R_0, computable, all): Dependency Pairs P_0: Rules R_0: app(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X piA(pair(X, Y)) => X piB(pair(X, Y)) => Y pair(piA(X), piB(X)) => X case(inl(X), /\x.Y(x), /\y.Z(y)) => Y(X) case(inr(X), /\x.Y(x), /\y.Z(y)) => Z(X) Thus, the original system is terminating if (P_0, R_0, computable, all) is finite. We consider the dependency pair problem (P_0, R_0, computable, all). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 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. [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. [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.