YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 44 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 2 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 4 ms] (9) QDP (10) QDPOrderProof [EQUIVALENT, 86 ms] (11) QDP (12) QDPOrderProof [EQUIVALENT, 52 ms] (13) QDP (14) QDPOrderProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 111 ms] (17) QDP (18) DependencyGraphProof [EQUIVALENT, 0 ms] (19) QDP (20) QDPOrderProof [EQUIVALENT, 73 ms] (21) QDP (22) PisEmptyProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) UsableRulesProof [EQUIVALENT, 3 ms] (26) QDP (27) QDPOrderProof [EQUIVALENT, 8 ms] (28) QDP (29) PisEmptyProof [EQUIVALENT, 0 ms] (30) YES (31) QDP (32) QDPOrderProof [EQUIVALENT, 75 ms] (33) QDP (34) PisEmptyProof [EQUIVALENT, 0 ms] (35) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) c(b(x1)) -> b(a(d(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a(x_1)) = x_1 POL(b(x_1)) = x_1 POL(c(x_1)) = 1 + x_1 POL(d(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: c(b(x1)) -> b(a(d(x1))) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. ---------------------------------------- (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: A(a(x1)) -> A(b(a(b(a(x1))))) A(a(x1)) -> B(a(b(a(x1)))) A(a(x1)) -> A(b(a(x1))) A(a(x1)) -> B(a(x1)) C(a(x1)) -> A(b(a(a(c(x1))))) C(a(x1)) -> B(a(a(c(x1)))) C(a(x1)) -> A(a(c(x1))) C(a(x1)) -> A(c(x1)) C(a(x1)) -> C(x1) B(b(b(x1))) -> A(b(x1)) C(b(x1)) -> A(a(c(x1))) C(b(x1)) -> A(c(x1)) C(b(x1)) -> C(x1) D(d(x1)) -> D(b(d(b(d(x1))))) D(d(x1)) -> B(d(b(d(x1)))) D(d(x1)) -> D(b(d(x1))) D(d(x1)) -> B(d(x1)) C(c(x1)) -> C(d(c(x1))) C(c(x1)) -> D(c(x1)) A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) A(a(a(x1))) -> B(x1) The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. 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 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(x1)) -> A(b(a(x1))) A(a(x1)) -> A(b(a(b(a(x1))))) A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) B(b(b(x1))) -> A(b(x1)) A(a(a(x1))) -> B(x1) The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) 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. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(x1)) -> A(b(a(x1))) A(a(x1)) -> A(b(a(b(a(x1))))) A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) B(b(b(x1))) -> A(b(x1)) A(a(a(x1))) -> B(x1) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A(a(a(x1))) -> B(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(a(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [1A, 1A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [1A, 0A, 0A]] * x_1 >>> <<< POL(B(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(x1)) -> A(b(a(x1))) A(a(x1)) -> A(b(a(b(a(x1))))) A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) B(b(b(x1))) -> A(b(x1)) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A(a(x1)) -> A(b(a(b(a(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(b(x_1)) = [[1A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(x1)) -> A(b(a(x1))) A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) B(b(b(x1))) -> A(b(x1)) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A(a(x1)) -> A(b(a(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(a(x_1)) = [[1A], [0A], [-I]] + [[0A, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(B(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) b(b(b(x1))) -> a(b(x1)) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(a(x1))) -> A(b(b(x1))) A(a(a(x1))) -> B(b(x1)) B(b(b(x1))) -> A(b(x1)) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A(a(a(x1))) -> B(b(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [0A, 0A, 1A], [0A, 0A, 1A]] * x_1 >>> <<< POL(b(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [1A, 0A, 0A], [1A, -I, -I]] * x_1 >>> <<< POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(a(x1))) -> A(b(b(x1))) B(b(b(x1))) -> A(b(x1)) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: A(a(a(x1))) -> A(b(b(x1))) The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A(a(a(x1))) -> A(b(b(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [1A, 0A, 0A], [0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) ---------------------------------------- (21) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: D(d(x1)) -> D(b(d(x1))) D(d(x1)) -> D(b(d(b(d(x1))))) The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: D(d(x1)) -> D(b(d(x1))) D(d(x1)) -> D(b(d(b(d(x1))))) The TRS R consists of the following rules: d(d(x1)) -> d(b(d(b(d(x1))))) b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. D(d(x1)) -> D(b(d(x1))) D(d(x1)) -> D(b(d(b(d(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(D(x_1)) = x_1 POL(a(x_1)) = x_1 POL(b(x_1)) = 0 POL(d(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(b(x1))) -> a(b(x1)) a(a(x1)) -> a(b(a(b(a(x1))))) a(a(a(x1))) -> a(b(b(x1))) ---------------------------------------- (28) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: d(d(x1)) -> d(b(d(b(d(x1))))) b(b(b(x1))) -> a(b(x1)) a(a(a(x1))) -> a(b(b(x1))) a(a(x1)) -> a(b(a(b(a(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (30) YES ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: C(b(x1)) -> C(x1) C(a(x1)) -> C(x1) C(c(x1)) -> C(d(c(x1))) The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C(b(x1)) -> C(x1) C(a(x1)) -> C(x1) C(c(x1)) -> C(d(c(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( C_1(x_1) ) = x_1 POL( d_1(x_1) ) = 0 POL( c_1(x_1) ) = 1 POL( a_1(x_1) ) = 2x_1 + 2 POL( b_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: d(d(x1)) -> d(b(d(b(d(x1))))) ---------------------------------------- (33) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a(a(x1)) -> a(b(a(b(a(x1))))) c(a(x1)) -> a(b(a(a(c(x1))))) b(b(b(x1))) -> a(b(x1)) c(b(x1)) -> a(a(c(x1))) d(d(x1)) -> d(b(d(b(d(x1))))) c(c(x1)) -> c(d(c(x1))) a(a(a(x1))) -> a(b(b(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (35) YES