/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination of the given CSR could be proven: (0) CSR (1) CSDependencyPairsProof [EQUIVALENT, 13 ms] (2) QCSDP (3) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (4) QCSDP (5) QCSDPReductionPairProof [EQUIVALENT, 33 ms] (6) QCSDP (7) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (8) QCSDP (9) QCSDPSubtermProof [EQUIVALENT, 0 ms] (10) QCSDP (11) PIsEmptyProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) The replacement map contains the following entries: cons_1: empty set big_0: empty set cons_0: {1, 2} *top*_0: {1} inf_1: empty set s_0: {1} ---------------------------------------- (1) CSDependencyPairsProof (EQUIVALENT) Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. ---------------------------------------- (2) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {cons_0_2, *top*_0_1, s_0_1, *TOP*_0_1, CONS_0_2, S_0_1} are replacing on all positions. The symbols in {cons_1_2, inf_1_1, CONS_1_2, U_1} are not replacing on any position. The ordinary context-sensitive dependency pairs DP_o are: *TOP*_0(inf_1(x)) -> *TOP*_0(cons_0(x, inf_1(s_0(x)))) *TOP*_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x))) S_0(inf_1(x)) -> S_0(cons_0(x, inf_1(s_0(x)))) S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x))) CONS_0(inf_1(x), x0) -> CONS_0(cons_0(x, inf_1(s_0(x))), x0) CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x))) CONS_0(x0, inf_1(x)) -> CONS_1(x0, cons_0(x, inf_1(s_0(x)))) The collapsing dependency pairs are DP_c: *TOP*_0(inf_1(x)) -> x S_0(inf_1(x)) -> x CONS_0(inf_1(x), x0) -> x The hidden terms of R are: s_0(x0) cons_0(x0, inf_1(s_0(x0))) Every hiding context is built from: aprove.DPFramework.CSDPProblem.QCSDPProblem$1@7024b7c8 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@347a70dc Hence, the new unhiding pairs DP_u are : *TOP*_0(inf_1(x)) -> U(x) S_0(inf_1(x)) -> U(x) CONS_0(inf_1(x), x0) -> U(x) U(s_0(x_0)) -> U(x_0) U(cons_0(x_0, x_1)) -> U(x_0) U(s_0(x0)) -> S_0(x0) U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0))) The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) Q is empty. ---------------------------------------- (3) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 5 less nodes. The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> *TOP*_0(cons_0(z0, inf_1(s_0(z0)))) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules *TOP*_0(inf_1(x0)) -> *TOP*_0(cons_0(x0, inf_1(s_0(x0)))) and *TOP*_0(inf_1(z0)) -> U(z0) form no chain, because ECap^mu_R'(*TOP*_0(inf_1(z0))) = *TOP*_0(inf_1(z0)) does not unify with *TOP*_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> S_0(cons_0(z0, inf_1(s_0(z0)))) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules S_0(inf_1(x0)) -> S_0(cons_0(x0, inf_1(s_0(x0)))) and S_0(inf_1(z0)) -> U(z0) form no chain, because ECap^mu_R'(S_0(inf_1(z0))) = S_0(inf_1(z0)) does not unify with S_0(cons_0(x0, inf_1(s_0(x0)))). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> CONS_0(cons_0(z0, inf_1(s_0(z0))), z1) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> CONS_0(z0, inf_1(s_0(z0))) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) The rules CONS_0(inf_1(x0), x1) -> CONS_0(cons_0(x0, inf_1(s_0(x0))), x1) and CONS_0(inf_1(z0), z1) -> U(z0) form no chain, because ECap^mu_R'(CONS_0(inf_1(z0), z1)) = CONS_0(inf_1(z0), x_1) does not unify with CONS_0(cons_0(x0, inf_1(s_0(x0))), x1). R' = ( big_0, cons_1(x, cons_1(y, z))) ( big_0, cons_1(x, cons_0(y, z))) ( *top*_0(cons_0(x, inf_1(s_0(x)))), *top*_0(inf_1(x))) ( s_0(cons_0(x, inf_1(s_0(x)))), s_0(inf_1(x))) ( cons_0(cons_0(x, inf_1(s_0(x))), x0), cons_0(inf_1(x), x0)) ( cons_1(x0, cons_0(x, inf_1(s_0(x)))), cons_0(x0, inf_1(x))) ---------------------------------------- (4) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {cons_0_2, *top*_0_1, s_0_1, CONS_0_2, S_0_1} are replacing on all positions. The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position. The TRS P consists of the following rules: CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x))) CONS_0(inf_1(x), x0) -> U(x) U(s_0(x_0)) -> U(x_0) U(cons_0(x_0, x_1)) -> U(x_0) U(s_0(x0)) -> S_0(x0) S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x))) S_0(inf_1(x)) -> U(x) U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0))) The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) Q is empty. ---------------------------------------- (5) QCSDPReductionPairProof (EQUIVALENT) Using the order Polynomial interpretation with max and min functions [POLO,MAXPOLO]: POL(CONS_0(x_1, x_2)) = x_1 POL(S_0(x_1)) = x_1 POL(U(x_1)) = x_1 POL(big_0) = 0 POL(cons_0(x_1, x_2)) = x_1 POL(cons_1(x_1, x_2)) = x_1 POL(inf_1(x_1)) = 1 + x_1 POL(s_0(x_1)) = x_1 the following usable rules s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 could all be oriented weakly. Furthermore, the pairs CONS_0(inf_1(x), x0) -> CONS_0(x, inf_1(s_0(x))) CONS_0(inf_1(x), x0) -> U(x) S_0(inf_1(x)) -> CONS_0(x, inf_1(s_0(x))) S_0(inf_1(x)) -> U(x) could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES]. ---------------------------------------- (6) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {cons_0_2, *top*_0_1, s_0_1, S_0_1, CONS_0_2} are replacing on all positions. The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position. The TRS P consists of the following rules: U(s_0(x_0)) -> U(x_0) U(cons_0(x_0, x_1)) -> U(x_0) U(s_0(x0)) -> S_0(x0) U(cons_0(x0, inf_1(s_0(x0)))) -> CONS_0(x0, inf_1(s_0(x0))) The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) Q is empty. ---------------------------------------- (7) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 2 less nodes. ---------------------------------------- (8) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {cons_0_2, *top*_0_1, s_0_1} are replacing on all positions. The symbols in {cons_1_2, inf_1_1, U_1} are not replacing on any position. The TRS P consists of the following rules: U(s_0(x_0)) -> U(x_0) U(cons_0(x_0, x_1)) -> U(x_0) The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) Q is empty. ---------------------------------------- (9) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. U(s_0(x_0)) -> U(x_0) U(cons_0(x_0, x_1)) -> U(x_0) The remaining pairs can at least be oriented weakly. none Used ordering: Combined order from the following AFS and order. U(x1) = x1 Subterm Order ---------------------------------------- (10) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {cons_0_2, *top*_0_1, s_0_1} are replacing on all positions. The symbols in {cons_1_2, inf_1_1} are not replacing on any position. The TRS P consists of the following rules: none The TRS R consists of the following rules: cons_1(x, cons_1(y, z)) -> big_0 cons_1(x, cons_0(y, z)) -> big_0 *top*_0(inf_1(x)) -> *top*_0(cons_0(x, inf_1(s_0(x)))) s_0(inf_1(x)) -> s_0(cons_0(x, inf_1(s_0(x)))) cons_0(inf_1(x), x0) -> cons_0(cons_0(x, inf_1(s_0(x))), x0) cons_0(x0, inf_1(x)) -> cons_1(x0, cons_0(x, inf_1(s_0(x)))) Q is empty. ---------------------------------------- (11) PIsEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. ---------------------------------------- (12) YES