/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) Overlay + Local Confluence [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 9 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) QReductionProof [EQUIVALENT, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) QDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) QDP (17) QReductionProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) QDP (24) QReductionProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPOrderProof [EQUIVALENT, 58 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 0 ms] (29) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) 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: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) ---------------------------------------- (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: HALF(s(s(x))) -> HALF(x) INC(s(x)) -> INC(x) BITS(x) -> BITITER(x, 0) BITITER(x, y) -> IF(zero(x), x, inc(y)) BITITER(x, y) -> ZERO(x) BITITER(x, y) -> INC(y) IF(true, x, y) -> P(y) IF(false, x, y) -> BITITER(half(x), y) IF(false, x, y) -> HALF(x) The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: INC(s(x)) -> INC(x) The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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: INC(s(x)) -> INC(x) R is empty. The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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]. half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: INC(s(x)) -> INC(x) 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: *INC(s(x)) -> INC(x) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: HALF(s(s(x))) -> HALF(x) The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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: HALF(s(s(x))) -> HALF(x) R is empty. The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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]. half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: HALF(s(s(x))) -> HALF(x) 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: *HALF(s(s(x))) -> HALF(x) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: IF(false, x, y) -> BITITER(half(x), y) BITITER(x, y) -> IF(zero(x), x, inc(y)) The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) inc(0) -> 0 inc(s(x)) -> s(inc(x)) zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(x)) -> x bits(x) -> bitIter(x, 0) bitIter(x, y) -> if(zero(x), x, inc(y)) if(true, x, y) -> p(y) if(false, x, y) -> bitIter(half(x), y) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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: IF(false, x, y) -> BITITER(half(x), y) BITITER(x, y) -> IF(zero(x), x, inc(y)) The TRS R consists of the following rules: zero(0) -> true zero(s(x)) -> false inc(0) -> 0 inc(s(x)) -> s(inc(x)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) 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]. p(0) p(s(x0)) bits(x0) bitIter(x0, x1) if(true, x0, x1) if(false, x0, x1) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: IF(false, x, y) -> BITITER(half(x), y) BITITER(x, y) -> IF(zero(x), x, inc(y)) The TRS R consists of the following rules: zero(0) -> true zero(s(x)) -> false inc(0) -> 0 inc(s(x)) -> s(inc(x)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) 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(false, x, y) -> BITITER(half(x), y) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO,RATPOLO]: POL(0) = 0 POL(BITITER(x_1, x_2)) = [4]x_1 POL(IF(x_1, x_2, x_3)) = [4]x_1 + [2]x_2 POL(false) = [1/4] POL(half(x_1)) = [1/2]x_1 POL(inc(x_1)) = 0 POL(s(x_1)) = [1/2] + x_1 POL(true) = 0 POL(zero(x_1)) = [1/2]x_1 The value of delta used in the strict ordering is 1. The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) zero(0) -> true zero(s(x)) -> false ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: BITITER(x, y) -> IF(zero(x), x, inc(y)) The TRS R consists of the following rules: zero(0) -> true zero(s(x)) -> false inc(0) -> 0 inc(s(x)) -> s(inc(x)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) The set Q consists of the following terms: half(0) half(s(0)) half(s(s(x0))) inc(0) inc(s(x0)) zero(0) zero(s(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (29) TRUE