We consider the system Applicative_first_order_05__#3.22. Alphabet: 0 : [] --> a 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 plus : [a * a] --> a s : [a] --> a times : [a * a] --> a true : [] --> b Rules: times(x, plus(y, s(z))) => plus(times(x, plus(y, times(s(z), 0))), times(x, s(z))) times(x, 0) => 0 times(x, s(y)) => plus(times(x, y), x) plus(x, 0) => x plus(x, s(y)) => s(plus(x, y)) 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: times(X, plus(Y, s(Z))) => plus(times(X, plus(Y, times(s(Z), 0))), times(X, s(Z))) times(X, 0) => 0 times(X, s(Y)) => plus(times(X, Y), X) plus(X, 0) => X plus(X, s(Y)) => s(plus(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 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) DependencyPairsProof [EQUIVALENT] || (2) QDP || (3) DependencyGraphProof [EQUIVALENT] || (4) AND || (5) QDP || (6) UsableRulesProof [EQUIVALENT] || (7) QDP || (8) QDPSizeChangeProof [EQUIVALENT] || (9) YES || (10) QDP || (11) UsableRulesProof [EQUIVALENT] || (12) QDP || (13) MRRProof [EQUIVALENT] || (14) QDP || (15) MRRProof [EQUIVALENT] || (16) QDP || (17) MRRProof [EQUIVALENT] || (18) QDP || (19) DependencyGraphProof [EQUIVALENT] || (20) TRUE || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || times(%X, plus(%Y, s(%Z))) -> plus(times(%X, plus(%Y, times(s(%Z), 0))), times(%X, s(%Z))) || times(%X, 0) -> 0 || times(%X, s(%Y)) -> plus(times(%X, %Y), %X) || plus(%X, 0) -> %X || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || || ---------------------------------------- || || (1) DependencyPairsProof (EQUIVALENT) || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. || ---------------------------------------- || || (2) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, plus(%Y, s(%Z))) -> PLUS(times(%X, plus(%Y, times(s(%Z), 0))), times(%X, s(%Z))) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || TIMES(%X, plus(%Y, s(%Z))) -> PLUS(%Y, times(s(%Z), 0)) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(s(%Z), 0) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, s(%Z)) || TIMES(%X, s(%Y)) -> PLUS(times(%X, %Y), %X) || TIMES(%X, s(%Y)) -> TIMES(%X, %Y) || PLUS(%X, s(%Y)) -> PLUS(%X, %Y) || || The TRS R consists of the following rules: || || times(%X, plus(%Y, s(%Z))) -> plus(times(%X, plus(%Y, times(s(%Z), 0))), times(%X, s(%Z))) || times(%X, 0) -> 0 || times(%X, s(%Y)) -> plus(times(%X, %Y), %X) || plus(%X, 0) -> %X || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (3) DependencyGraphProof (EQUIVALENT) || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. || ---------------------------------------- || || (4) || Complex Obligation (AND) || || ---------------------------------------- || || (5) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || PLUS(%X, s(%Y)) -> PLUS(%X, %Y) || || The TRS R consists of the following rules: || || times(%X, plus(%Y, s(%Z))) -> plus(times(%X, plus(%Y, times(s(%Z), 0))), times(%X, s(%Z))) || times(%X, 0) -> 0 || times(%X, s(%Y)) -> plus(times(%X, %Y), %X) || plus(%X, 0) -> %X || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (6) UsableRulesProof (EQUIVALENT) || We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (7) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || PLUS(%X, s(%Y)) -> PLUS(%X, %Y) || || R is empty. || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (8) QDPSizeChangeProof (EQUIVALENT) || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. || || From the DPs we obtained the following set of size-change graphs: || *PLUS(%X, s(%Y)) -> PLUS(%X, %Y) || The graph contains the following edges 1 >= 1, 2 > 2 || || || ---------------------------------------- || || (9) || YES || || ---------------------------------------- || || (10) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, s(%Z)) || TIMES(%X, s(%Y)) -> TIMES(%X, %Y) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || || The TRS R consists of the following rules: || || times(%X, plus(%Y, s(%Z))) -> plus(times(%X, plus(%Y, times(s(%Z), 0))), times(%X, s(%Z))) || times(%X, 0) -> 0 || times(%X, s(%Y)) -> plus(times(%X, %Y), %X) || plus(%X, 0) -> %X || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || ~PAIR(%X, %Y) -> %X || ~PAIR(%X, %Y) -> %Y || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (11) UsableRulesProof (EQUIVALENT) || We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (12) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, s(%Z)) || TIMES(%X, s(%Y)) -> TIMES(%X, %Y) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || || The TRS R consists of the following rules: || || times(%X, 0) -> 0 || plus(%X, 0) -> %X || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (13) MRRProof (EQUIVALENT) || By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. || || Strictly oriented dependency pairs: || || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, s(%Z)) || || Strictly oriented rules of the TRS R: || || plus(%X, 0) -> %X || || Used ordering: Polynomial interpretation [POLO]: || || POL(0) = 0 || POL(TIMES(x_1, x_2)) = x_1 + 2*x_2 || POL(plus(x_1, x_2)) = 1 + x_1 + 2*x_2 || POL(s(x_1)) = x_1 || POL(times(x_1, x_2)) = x_1 + 2*x_2 || || || ---------------------------------------- || || (14) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, s(%Y)) -> TIMES(%X, %Y) || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || || The TRS R consists of the following rules: || || times(%X, 0) -> 0 || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (15) MRRProof (EQUIVALENT) || By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. || || Strictly oriented dependency pairs: || || TIMES(%X, s(%Y)) -> TIMES(%X, %Y) || || || Used ordering: Polynomial interpretation [POLO]: || || POL(0) = 0 || POL(TIMES(x_1, x_2)) = x_1 + 2*x_2 || POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2 || POL(s(x_1)) = 1 + x_1 || POL(times(x_1, x_2)) = x_1 + x_2 || || || ---------------------------------------- || || (16) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || || The TRS R consists of the following rules: || || times(%X, 0) -> 0 || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (17) MRRProof (EQUIVALENT) || By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. || || || Strictly oriented rules of the TRS R: || || plus(%X, s(%Y)) -> s(plus(%X, %Y)) || || Used ordering: Polynomial interpretation [POLO]: || || POL(0) = 0 || POL(TIMES(x_1, x_2)) = x_1 + 2*x_2 || POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2 || POL(s(x_1)) = 2 + x_1 || POL(times(x_1, x_2)) = x_1 + x_2 || || || ---------------------------------------- || || (18) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || TIMES(%X, plus(%Y, s(%Z))) -> TIMES(%X, plus(%Y, times(s(%Z), 0))) || || The TRS R consists of the following rules: || || times(%X, 0) -> 0 || || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (19) DependencyGraphProof (EQUIVALENT) || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. || ---------------------------------------- || || (20) || TRUE || We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs. After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] map#(F, cons(X, Y)) =#> F(X) 1] map#(F, cons(X, Y)) =#> map#(F, Y) 2] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 3] filter#(F, cons(X, Y)) =#> F(X) 4] filter2#(true, F, X, Y) =#> filter#(F, Y) 5] filter2#(false, F, X, Y) =#> filter#(F, Y) Rules R_0: times(X, plus(Y, s(Z))) => plus(times(X, plus(Y, times(s(Z), 0))), times(X, s(Z))) times(X, 0) => 0 times(X, s(Y)) => plus(times(X, Y), X) plus(X, 0) => X plus(X, s(Y)) => s(plus(X, Y)) 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, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_0, minimal, formative). The formative rules of (P_0, R_0) are R_1 ::= map(F, cons(X, Y)) => cons(F X, map(F, Y)) 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) By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative). Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_1, minimal, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70]. Thus, we must orient: map#(F, cons(X, Y)) >? F(X) map#(F, cons(X, Y)) >? map#(F, Y) filter#(F, cons(X, Y)) >? filter2#(F X, F, X, Y) filter#(F, cons(X, Y)) >? F(X) filter2#(true, F, X, Y) >? filter#(F, Y) filter2#(false, F, X, Y) >? filter#(F, Y) map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 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) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: cons = \y0y1.2 + y1 + 2y0 false = 3 filter = \G0y1.y1 + 2y1G0(y1) filter2 = \y0G1y2y3.2 + y3 + 2y2 + 2y3G1(y3) filter2# = \y0G1y2y3.y3 + 2y3G1(y3) + 2G1(y3) filter# = \G0y1.y1 + 2y1G0(y1) + 2G0(y1) map = \G0y1.2y1 + y1G0(y1) map# = \G0y1.3 + 2G0(y1) + y1G0(y1) true = 3 Using this interpretation, the requirements translate to: [[map#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) > F0(x1) = [[_F0(_x1)]] [[map#(_F0, cons(_x1, _x2))]] = 3 + 2x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) >= 3 + 2F0(x2) + x2F0(x2) = [[map#(_F0, _x2)]] [[filter#(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 6F0(2 + x2 + 2x1) > x2 + 2x2F0(x2) + 2F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] [[filter#(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 6F0(2 + x2 + 2x1) > F0(x1) = [[_F0(_x1)]] [[filter2#(true, _F0, _x1, _x2)]] = x2 + 2x2F0(x2) + 2F0(x2) >= x2 + 2x2F0(x2) + 2F0(x2) = [[filter#(_F0, _x2)]] [[filter2#(false, _F0, _x1, _x2)]] = x2 + 2x2F0(x2) + 2F0(x2) >= x2 + 2x2F0(x2) + 2F0(x2) = [[filter#(_F0, _x2)]] [[map(_F0, cons(_x1, _x2))]] = 4 + 2x2 + 4x1 + 2x1F0(2 + x2 + 2x1) + 2F0(2 + x2 + 2x1) + x2F0(2 + x2 + 2x1) >= 2 + 2x2 + x2F0(x2) + 2max(x1, F0(x1)) = [[cons(_F0 _x1, map(_F0, _x2))]] [[filter(_F0, cons(_x1, _x2))]] = 2 + x2 + 2x1 + 2x2F0(2 + x2 + 2x1) + 4x1F0(2 + x2 + 2x1) + 4F0(2 + x2 + 2x1) >= 2 + x2 + 2x1 + 2x2F0(x2) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] [[filter2(true, _F0, _x1, _x2)]] = 2 + x2 + 2x1 + 2x2F0(x2) >= 2 + x2 + 2x1 + 2x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] [[filter2(false, _F0, _x1, _x2)]] = 2 + x2 + 2x1 + 2x2F0(x2) >= x2 + 2x2F0(x2) = [[filter(_F0, _x2)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of: map#(F, cons(X, Y)) =#> map#(F, Y) filter2#(true, F, X, Y) =#> filter#(F, Y) filter2#(false, F, X, Y) =#> filter#(F, Y) Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_1, R_1, 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 : 0 * 1 : * 2 : This graph has the following strongly connected components: P_2: map#(F, cons(X, Y)) =#> map#(F, Y) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_1, R_1, m, f) by (P_2, R_1, m, f). Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_2, R_1, minimal, 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_2, R_1, minimal, f) by ({}, R_1, 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.