3.93/1.99 YES 4.04/2.02 We consider the system theBenchmark. 4.04/2.02 4.04/2.02 Alphabet: 4.04/2.02 4.04/2.02 0 : [] --> a 4.04/2.02 cons : [d * e] --> e 4.04/2.02 edge : [a * a * b] --> b 4.04/2.02 empty : [] --> b 4.04/2.02 eq : [a * a] --> c 4.04/2.02 false : [] --> c 4.04/2.02 filter : [d -> c * e] --> e 4.04/2.02 filter2 : [c * d -> c * d * e] --> e 4.04/2.02 if!fac6220reach!fac62201 : [c * a * a * b * b] --> c 4.04/2.02 if!fac6220reach!fac62202 : [c * a * a * b * b] --> c 4.04/2.02 map : [d -> d * e] --> e 4.04/2.02 nil : [] --> e 4.04/2.02 or : [c * c] --> c 4.04/2.02 reach : [a * a * b * b] --> c 4.04/2.02 s : [a] --> a 4.04/2.02 true : [] --> c 4.04/2.02 union : [b * b] --> b 4.04/2.02 4.04/2.02 Rules: 4.04/2.02 4.04/2.02 eq(0, 0) => true 4.04/2.02 eq(0, s(x)) => false 4.04/2.02 eq(s(x), 0) => false 4.04/2.02 eq(s(x), s(y)) => eq(x, y) 4.04/2.02 or(true, x) => true 4.04/2.02 or(false, x) => x 4.04/2.02 union(empty, x) => x 4.04/2.02 union(edge(x, y, z), u) => edge(x, y, union(z, u)) 4.04/2.02 reach(x, y, empty, z) => false 4.04/2.02 reach(x, y, edge(z, u, v), w) => if!fac6220reach!fac62201(eq(x, z), x, y, edge(z, u, v), w) 4.04/2.02 if!fac6220reach!fac62201(true, x, y, edge(z, u, v), w) => if!fac6220reach!fac62202(eq(y, u), x, y, edge(z, u, v), w) 4.04/2.02 if!fac6220reach!fac62201(false, x, y, edge(z, u, v), w) => reach(x, y, v, edge(z, u, w)) 4.04/2.02 if!fac6220reach!fac62202(true, x, y, edge(z, u, v), w) => true 4.04/2.02 if!fac6220reach!fac62202(false, x, y, edge(z, u, v), w) => or(reach(x, y, v, w), reach(u, y, union(v, w), empty)) 4.04/2.02 map(f, nil) => nil 4.04/2.02 map(f, cons(x, y)) => cons(f x, map(f, y)) 4.04/2.02 filter(f, nil) => nil 4.04/2.02 filter(f, cons(x, y)) => filter2(f x, f, x, y) 4.04/2.02 filter2(true, f, x, y) => cons(x, filter(f, y)) 4.04/2.02 filter2(false, f, x, y) => filter(f, y) 4.04/2.02 4.04/2.02 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 4.04/2.02 4.04/2.02 We observe that the rules contain a first-order subset: 4.04/2.02 4.04/2.02 eq(0, 0) => true 4.04/2.02 eq(0, s(X)) => false 4.04/2.02 eq(s(X), 0) => false 4.04/2.02 eq(s(X), s(Y)) => eq(X, Y) 4.04/2.02 or(true, X) => true 4.04/2.02 or(false, X) => X 4.04/2.02 union(empty, X) => X 4.04/2.02 union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) 4.04/2.02 reach(X, Y, empty, Z) => false 4.04/2.02 reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 4.04/2.02 if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 4.04/2.02 if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 4.04/2.02 if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true 4.04/2.02 if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 4.04/2.02 4.04/2.02 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. 4.04/2.02 4.04/2.02 According to the external first-order termination prover, this system is indeed terminating: 4.04/2.02 4.04/2.02 || proof of resources/system.trs 4.04/2.02 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || Termination w.r.t. Q of the given QTRS could be proven: 4.04/2.02 || 4.04/2.02 || (0) QTRS 4.04/2.02 || (1) Overlay + Local Confluence [EQUIVALENT] 4.04/2.02 || (2) QTRS 4.04/2.02 || (3) DependencyPairsProof [EQUIVALENT] 4.04/2.02 || (4) QDP 4.04/2.02 || (5) DependencyGraphProof [EQUIVALENT] 4.04/2.02 || (6) AND 4.04/2.02 || (7) QDP 4.04/2.02 || (8) UsableRulesProof [EQUIVALENT] 4.04/2.02 || (9) QDP 4.04/2.02 || (10) QReductionProof [EQUIVALENT] 4.04/2.02 || (11) QDP 4.04/2.02 || (12) QDPSizeChangeProof [EQUIVALENT] 4.04/2.02 || (13) YES 4.04/2.02 || (14) QDP 4.04/2.02 || (15) UsableRulesProof [EQUIVALENT] 4.04/2.02 || (16) QDP 4.04/2.02 || (17) QReductionProof [EQUIVALENT] 4.04/2.02 || (18) QDP 4.04/2.02 || (19) QDPSizeChangeProof [EQUIVALENT] 4.04/2.02 || (20) YES 4.04/2.02 || (21) QDP 4.04/2.02 || (22) UsableRulesProof [EQUIVALENT] 4.04/2.02 || (23) QDP 4.04/2.02 || (24) QReductionProof [EQUIVALENT] 4.04/2.02 || (25) QDP 4.04/2.02 || (26) QDPOrderProof [EQUIVALENT] 4.04/2.02 || (27) QDP 4.04/2.02 || (28) DependencyGraphProof [EQUIVALENT] 4.04/2.02 || (29) QDP 4.04/2.02 || (30) UsableRulesProof [EQUIVALENT] 4.04/2.02 || (31) QDP 4.04/2.02 || (32) QReductionProof [EQUIVALENT] 4.04/2.02 || (33) QDP 4.04/2.02 || (34) QDPSizeChangeProof [EQUIVALENT] 4.04/2.02 || (35) YES 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (0) 4.04/2.02 || Obligation: 4.04/2.02 || Q restricted rewrite system: 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || Q is empty. 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (1) Overlay + Local Confluence (EQUIVALENT) 4.04/2.02 || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (2) 4.04/2.02 || Obligation: 4.04/2.02 || Q restricted rewrite system: 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (3) DependencyPairsProof (EQUIVALENT) 4.04/2.02 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (4) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) 4.04/2.02 || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> EQ(%X, %Z) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> EQ(%Y, %U) 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> OR(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> UNION(%V, %W) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (5) DependencyGraphProof (EQUIVALENT) 4.04/2.02 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (6) 4.04/2.02 || Complex Obligation (AND) 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (7) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (8) UsableRulesProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (9) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) 4.04/2.02 || 4.04/2.02 || R is empty. 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (10) QReductionProof (EQUIVALENT) 4.04/2.02 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (11) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) 4.04/2.02 || 4.04/2.02 || R is empty. 4.04/2.02 || Q is empty. 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (12) QDPSizeChangeProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || 4.04/2.02 || From the DPs we obtained the following set of size-change graphs: 4.04/2.02 || *UNION(edge(%X, %Y, %Z), %U) -> UNION(%Z, %U) 4.04/2.02 || The graph contains the following edges 1 > 1, 2 >= 2 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (13) 4.04/2.02 || YES 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (14) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (15) UsableRulesProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (16) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) 4.04/2.02 || 4.04/2.02 || R is empty. 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (17) QReductionProof (EQUIVALENT) 4.04/2.02 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (18) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || EQ(s(%X), s(%Y)) -> EQ(%X, %Y) 4.04/2.02 || 4.04/2.02 || R is empty. 4.04/2.02 || Q is empty. 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (19) QDPSizeChangeProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || 4.04/2.02 || From the DPs we obtained the following set of size-change graphs: 4.04/2.02 || *EQ(s(%X), s(%Y)) -> EQ(%X, %Y) 4.04/2.02 || The graph contains the following edges 1 > 1, 2 > 2 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (20) 4.04/2.02 || YES 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (21) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || or(true, %X) -> true 4.04/2.02 || or(false, %X) -> %X 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || reach(%X, %Y, empty, %Z) -> false 4.04/2.02 || reach(%X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> if!fac6220reach!fac62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || if!fac6220reach!fac62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> reach(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || if!fac6220reach!fac62202(true, %X, %Y, edge(%Z, %U, %V), %W) -> true 4.04/2.02 || if!fac6220reach!fac62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> or(reach(%X, %Y, %V, %W), reach(%U, %Y, union(%V, %W), empty)) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (22) UsableRulesProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (23) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (24) QReductionProof (EQUIVALENT) 4.04/2.02 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.04/2.02 || 4.04/2.02 || or(true, x0) 4.04/2.02 || or(false, x0) 4.04/2.02 || reach(x0, x1, empty, x2) 4.04/2.02 || reach(x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62201(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(true, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || if!fac6220reach!fac62202(false, x0, x1, edge(x2, x3, x4), x5) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (25) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (26) QDPOrderProof (EQUIVALENT) 4.04/2.02 || We use the reduction pair processor [LPAR04,JAR06]. 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || The following pairs can be oriented strictly and are deleted. 4.04/2.02 || 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, %W) 4.04/2.02 || IF!FAC6220REACH!FAC62202(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%U, %Y, union(%V, %W), empty) 4.04/2.02 || The remaining pairs can at least be oriented weakly. 4.04/2.02 || Used ordering: Polynomial interpretation [POLO]: 4.04/2.02 || 4.04/2.02 || POL(0) = 0 4.04/2.02 || POL(IF!FAC6220REACH!FAC62201(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5 4.04/2.02 || POL(IF!FAC6220REACH!FAC62202(x_1, x_2, x_3, x_4, x_5)) = x_4 + x_5 4.04/2.02 || POL(REACH(x_1, x_2, x_3, x_4)) = x_3 + x_4 4.04/2.02 || POL(edge(x_1, x_2, x_3)) = 1 + x_3 4.04/2.02 || POL(empty) = 0 4.04/2.02 || POL(eq(x_1, x_2)) = 0 4.04/2.02 || POL(false) = 0 4.04/2.02 || POL(s(x_1)) = 0 4.04/2.02 || POL(true) = 0 4.04/2.02 || POL(union(x_1, x_2)) = x_1 + x_2 4.04/2.02 || 4.04/2.02 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4.04/2.02 || 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (27) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(true, %X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62202(eq(%Y, %U), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (28) DependencyGraphProof (EQUIVALENT) 4.04/2.02 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (29) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || union(empty, %X) -> %X 4.04/2.02 || union(edge(%X, %Y, %Z), %U) -> edge(%X, %Y, union(%Z, %U)) 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (30) UsableRulesProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (31) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (32) QReductionProof (EQUIVALENT) 4.04/2.02 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.04/2.02 || 4.04/2.02 || union(empty, x0) 4.04/2.02 || union(edge(x0, x1, x2), x3) 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (33) 4.04/2.02 || Obligation: 4.04/2.02 || Q DP problem: 4.04/2.02 || The TRS P consists of the following rules: 4.04/2.02 || 4.04/2.02 || IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || 4.04/2.02 || The TRS R consists of the following rules: 4.04/2.02 || 4.04/2.02 || eq(0, 0) -> true 4.04/2.02 || eq(0, s(%X)) -> false 4.04/2.02 || eq(s(%X), 0) -> false 4.04/2.02 || eq(s(%X), s(%Y)) -> eq(%X, %Y) 4.04/2.02 || 4.04/2.02 || The set Q consists of the following terms: 4.04/2.02 || 4.04/2.02 || eq(0, 0) 4.04/2.02 || eq(0, s(x0)) 4.04/2.02 || eq(s(x0), 0) 4.04/2.02 || eq(s(x0), s(x1)) 4.04/2.02 || 4.04/2.02 || We have to consider all minimal (P,Q,R)-chains. 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (34) QDPSizeChangeProof (EQUIVALENT) 4.04/2.02 || 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. 4.04/2.02 || 4.04/2.02 || From the DPs we obtained the following set of size-change graphs: 4.04/2.02 || *REACH(%X, %Y, edge(%Z, %U, %V), %W) -> IF!FAC6220REACH!FAC62201(eq(%X, %Z), %X, %Y, edge(%Z, %U, %V), %W) 4.04/2.02 || The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || *IF!FAC6220REACH!FAC62201(false, %X, %Y, edge(%Z, %U, %V), %W) -> REACH(%X, %Y, %V, edge(%Z, %U, %W)) 4.04/2.02 || The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3 4.04/2.02 || 4.04/2.02 || 4.04/2.02 || ---------------------------------------- 4.04/2.02 || 4.04/2.02 || (35) 4.04/2.02 || YES 4.04/2.02 || 4.04/2.02 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 [Kop13]). 4.04/2.02 4.04/2.02 We thus obtain the following dependency pair problem (P_0, R_0, static, formative): 4.04/2.02 4.04/2.02 Dependency Pairs P_0: 4.04/2.02 4.04/2.02 0] map#(F, cons(X, Y)) =#> map#(F, Y) 4.04/2.02 1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 4.04/2.02 2] filter2#(true, F, X, Y) =#> filter#(F, Y) 4.04/2.02 3] filter2#(false, F, X, Y) =#> filter#(F, Y) 4.04/2.02 4.04/2.02 Rules R_0: 4.04/2.02 4.04/2.02 eq(0, 0) => true 4.04/2.02 eq(0, s(X)) => false 4.04/2.02 eq(s(X), 0) => false 4.04/2.02 eq(s(X), s(Y)) => eq(X, Y) 4.04/2.02 or(true, X) => true 4.04/2.02 or(false, X) => X 4.04/2.02 union(empty, X) => X 4.04/2.02 union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) 4.04/2.02 reach(X, Y, empty, Z) => false 4.04/2.02 reach(X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) 4.04/2.02 if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) 4.04/2.02 if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 4.04/2.02 if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true 4.04/2.02 if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 4.04/2.02 map(F, nil) => nil 4.04/2.02 map(F, cons(X, Y)) => cons(F X, map(F, Y)) 4.04/2.02 filter(F, nil) => nil 4.04/2.02 filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 4.04/2.02 filter2(true, F, X, Y) => cons(X, filter(F, Y)) 4.04/2.02 filter2(false, F, X, Y) => filter(F, Y) 4.04/2.02 4.04/2.02 Thus, the original system is terminating if (P_0, R_0, static, formative) is finite. 4.04/2.02 4.04/2.02 We consider the dependency pair problem (P_0, R_0, static, formative). 4.04/2.02 4.04/2.02 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 4.04/2.02 4.04/2.02 * 0 : 0 4.04/2.02 * 1 : 2, 3 4.04/2.02 * 2 : 1 4.04/2.02 * 3 : 1 4.04/2.02 4.04/2.02 This graph has the following strongly connected components: 4.04/2.02 4.04/2.02 P_1: 4.04/2.02 4.04/2.02 map#(F, cons(X, Y)) =#> map#(F, Y) 4.04/2.02 4.04/2.02 P_2: 4.04/2.02 4.04/2.02 filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 4.04/2.02 filter2#(true, F, X, Y) =#> filter#(F, Y) 4.04/2.02 filter2#(false, F, X, Y) =#> filter#(F, Y) 4.04/2.02 4.04/2.02 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). 4.04/2.02 4.04/2.02 Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite. 4.04/2.02 4.04/2.02 We consider the dependency pair problem (P_2, R_0, static, formative). 4.04/2.02 4.04/2.02 We apply the subterm criterion with the following projection function: 4.04/2.02 4.04/2.02 nu(filter2#) = 4 4.04/2.02 nu(filter#) = 2 4.04/2.02 4.04/2.02 Thus, we can orient the dependency pairs as follows: 4.04/2.02 4.04/2.02 nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 4.04/2.02 nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 4.04/2.02 nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 4.04/2.02 4.04/2.02 By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by (P_3, R_0, static, f), where P_3 contains: 4.04/2.02 4.04/2.02 filter2#(true, F, X, Y) =#> filter#(F, Y) 4.04/2.02 filter2#(false, F, X, Y) =#> filter#(F, Y) 4.04/2.02 4.04/2.02 Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite. 4.04/2.02 4.04/2.02 We consider the dependency pair problem (P_3, R_0, static, formative). 4.04/2.02 4.04/2.02 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 4.04/2.02 4.04/2.02 * 0 : 4.04/2.02 * 1 : 4.04/2.02 4.04/2.02 This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 4.04/2.02 4.04/2.02 Thus, the original system is terminating if (P_1, R_0, static, formative) is finite. 4.04/2.02 4.04/2.02 We consider the dependency pair problem (P_1, R_0, static, formative). 4.04/2.02 4.04/2.02 We apply the subterm criterion with the following projection function: 4.04/2.02 4.04/2.02 nu(map#) = 2 4.04/2.02 4.04/2.02 Thus, we can orient the dependency pairs as follows: 4.04/2.02 4.04/2.02 nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 4.04/2.02 4.04/2.02 By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. 4.04/2.02 4.04/2.02 As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. 4.04/2.02 4.04/2.02 4.04/2.02 +++ Citations +++ 4.04/2.02 4.04/2.02 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 4.04/2.02 [Kop13] C. Kop. Static Dependency Pairs with Accessibility. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013. 4.04/2.02 [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. 4.04/2.02 EOF