/export/starexec/sandbox/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: 0 : [] --> a cons : [d * e] --> e edge : [a * a * b] --> b empty : [] --> b eq : [a * a] --> c false : [] --> c filter : [d -> c * e] --> e filter2 : [c * d -> c * d * e] --> e if!fac6220reach!fac62201 : [c * a * a * b * b] --> c if!fac6220reach!fac62202 : [c * a * a * b * b] --> c map : [d -> d * e] --> e nil : [] --> e or : [c * c] --> c reach : [a * a * b * b] --> c s : [a] --> a true : [] --> c union : [b * b] --> b Rules: eq(0, 0) => true eq(0, s(x)) => false eq(s(x), 0) => false eq(s(x), s(y)) => eq(x, y) or(true, x) => true or(false, x) => x union(empty, x) => x union(edge(x, y, z), u) => edge(x, y, union(z, u)) reach(x, y, empty, z) => false reach(x, y, edge(z, u, v), w) => if!fac6220reach!fac62201(eq(x, z), x, y, edge(z, u, v), w) if!fac6220reach!fac62201(true, x, y, edge(z, u, v), w) => if!fac6220reach!fac62202(eq(y, u), x, y, edge(z, u, v), w) if!fac6220reach!fac62201(false, x, y, edge(z, u, v), w) => reach(x, y, v, edge(z, u, w)) if!fac6220reach!fac62202(true, x, y, edge(z, u, v), w) => true if!fac6220reach!fac62202(false, x, y, edge(z, u, v), w) => or(reach(x, y, v, w), reach(u, y, union(v, w), empty)) 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: eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) or(true, X) => true or(false, X) => X union(empty, X) => X union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) reach(X, Y, empty, Z) => false reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 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) Overlay + Local Confluence [EQUIVALENT] || (2) QTRS || (3) DependencyPairsProof [EQUIVALENT] || (4) QDP || (5) DependencyGraphProof [EQUIVALENT] || (6) AND || (7) QDP || (8) UsableRulesProof [EQUIVALENT] || (9) QDP || (10) QReductionProof [EQUIVALENT] || (11) QDP || (12) QDPSizeChangeProof [EQUIVALENT] || (13) YES || (14) QDP || (15) UsableRulesProof [EQUIVALENT] || (16) QDP || (17) QReductionProof [EQUIVALENT] || (18) QDP || (19) QDPSizeChangeProof [EQUIVALENT] || (20) YES || (21) QDP || (22) UsableRulesProof [EQUIVALENT] || (23) QDP || (24) QReductionProof [EQUIVALENT] || (25) QDP || (26) QDPOrderProof [EQUIVALENT] || (27) QDP || (28) DependencyGraphProof [EQUIVALENT] || (29) QDP || (30) UsableRulesProof [EQUIVALENT] || (31) QDP || (32) QReductionProof [EQUIVALENT] || (33) QDP || (34) QDPSizeChangeProof [EQUIVALENT] || (35) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || Q is empty. || || ---------------------------------------- || || (1) Overlay + Local Confluence (EQUIVALENT) || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. || ---------------------------------------- || || (2) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || || ---------------------------------------- || || (3) DependencyPairsProof (EQUIVALENT) || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. || ---------------------------------------- || || (4) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> EQ(%X, %Z) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> EQ(%Y, %U) || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> OR(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> UNION(%V, %W) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (5) DependencyGraphProof (EQUIVALENT) || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes. || ---------------------------------------- || || (6) || Complex Obligation (AND) || || ---------------------------------------- || || (7) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (8) UsableRulesProof (EQUIVALENT) || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (9) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) || || R is empty. || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (10) QReductionProof (EQUIVALENT) || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || || ---------------------------------------- || || (11) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) || || R is empty. || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (12) 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: || *UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) || The graph contains the following edges 1 > 1, 2 >= 2 || || || ---------------------------------------- || || (13) || YES || || ---------------------------------------- || || (14) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (15) UsableRulesProof (EQUIVALENT) || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (16) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) || || R is empty. || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (17) QReductionProof (EQUIVALENT) || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || || ---------------------------------------- || || (18) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) || || R is empty. || Q is empty. || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (19) 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: || *EQ(s(%X), s(%Y)) -> EQ(%X, %Y) || The graph contains the following edges 1 > 1, 2 > 2 || || || ---------------------------------------- || || (20) || YES || || ---------------------------------------- || || (21) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || or(true, %X) -> true || or(false, %X) -> %X || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || reach(%X, %Y, empty, %Z) -> false || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (22) UsableRulesProof (EQUIVALENT) || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (23) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || || The TRS R consists of the following rules: || || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || or(true, x0) || or(false, x0) || union(empty, x0) || union(edge(x0, x1, x2), x3) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (24) QReductionProof (EQUIVALENT) || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. || || or(true, x0) || or(false, x0) || reach(x0, x1, empty, x2) || reach(x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) || || || ---------------------------------------- || || (25) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || || The TRS R consists of the following rules: || || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || union(empty, x0) || union(edge(x0, x1, x2), x3) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (26) QDPOrderProof (EQUIVALENT) || We use the reduction pair processor [LPAR04,JAR06]. || || || The following pairs can be oriented strictly and are deleted. || || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) || The remaining pairs can at least be oriented weakly. || Used ordering: Polynomial interpretation [POLO]: || || POL(0) = 0 || POL(IF!FAC6220REACH!FAC62201(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5 || POL(IF!FAC6220REACH!FAC62202(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5 || POL(REACH(x_1, x_2, x_3, x_4)) = x_3 + x_4 || POL(edge(x_1, x_2, x_3)) = 1 + x_3 || POL(empty) = 0 || POL(eq(x_1, x_2)) = 0 || POL(false) = 0 || POL(s(x_1)) = 0 || POL(true) = 0 || POL(union(x_1, x_2)) = x_1 + x_2 || || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: || || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || || || ---------------------------------------- || || (27) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || || The TRS R consists of the following rules: || || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || union(empty, x0) || union(edge(x0, x1, x2), x3) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (28) DependencyGraphProof (EQUIVALENT) || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. || ---------------------------------------- || || (29) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || || The TRS R consists of the following rules: || || union(empty, %X) -> %X || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || union(empty, x0) || union(edge(x0, x1, x2), x3) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (30) UsableRulesProof (EQUIVALENT) || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. || ---------------------------------------- || || (31) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || union(empty, x0) || union(edge(x0, x1, x2), x3) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (32) QReductionProof (EQUIVALENT) || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. || || union(empty, x0) || union(edge(x0, x1, x2), x3) || || || ---------------------------------------- || || (33) || Obligation: || Q DP problem: || The TRS P consists of the following rules: || || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || || The TRS R consists of the following rules: || || eq(0, 0) -> true || eq(0, s(%X)) -> false || eq(s(%X), 0) -> false || eq(s(%X), s(%Y)) -> eq(%X, %Y) || || The set Q consists of the following terms: || || eq(0, 0) || eq(0, s(x0)) || eq(s(x0), 0) || eq(s(x0), s(x1)) || || We have to consider all minimal (P,Q,R)-chains. || ---------------------------------------- || || (34) 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: || *REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) || The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5 || || || *IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) || The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3 || || || ---------------------------------------- || || (35) || 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 in [Kop12, Ch. 7.8]). We thus obtain the following dependency pair problem (P_0, R_0, minimal, 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: eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) or(true, X) => true or(false, X) => X union(empty, X) => X union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) reach(X, Y, empty, Z) => false reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 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). 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, minimal, formative) and (P_2, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_2, R_0, minimal, 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 [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by (P_3, R_0, minimal, 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, minimal, formative) and (P_3, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_3, 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 : 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, 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(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 [Kop12, Thm. 7.35], 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 +++ [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.