3.53/1.70 YES 3.53/1.70 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.53/1.70 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.53/1.70 3.53/1.70 3.53/1.70 Quasi decreasingness of the given CTRS could be proven: 3.53/1.70 3.53/1.70 (0) CTRS 3.53/1.70 (1) CTRSToQTRSProof [SOUND, 0 ms] 3.53/1.70 (2) QTRS 3.53/1.70 (3) QTRSRRRProof [EQUIVALENT, 27 ms] 3.53/1.70 (4) QTRS 3.53/1.70 (5) RisEmptyProof [EQUIVALENT, 0 ms] 3.53/1.70 (6) YES 3.53/1.70 3.53/1.70 3.53/1.70 ---------------------------------------- 3.53/1.70 3.53/1.70 (0) 3.53/1.70 Obligation: 3.53/1.70 Conditional term rewrite system: 3.53/1.70 The TRS R consists of the following rules: 3.53/1.70 3.53/1.70 pos(s(0)) -> true 3.53/1.70 pos(0) -> false 3.53/1.70 3.53/1.70 The conditional TRS C consists of the following conditional rules: 3.53/1.70 3.53/1.70 pos(s(x)) -> true <= pos(x) -> true 3.53/1.70 pos(p(x)) -> false <= pos(x) -> false 3.53/1.70 3.53/1.70 3.53/1.70 ---------------------------------------- 3.53/1.70 3.53/1.70 (1) CTRSToQTRSProof (SOUND) 3.53/1.70 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 3.53/1.70 ---------------------------------------- 3.53/1.70 3.53/1.70 (2) 3.53/1.70 Obligation: 3.53/1.70 Q restricted rewrite system: 3.53/1.70 The TRS R consists of the following rules: 3.53/1.70 3.53/1.70 pos(s(x)) -> U1(pos(x)) 3.53/1.70 U1(true) -> true 3.53/1.71 pos(p(x)) -> U2(pos(x)) 3.53/1.71 U2(false) -> false 3.53/1.71 pos(s(0)) -> true 3.53/1.71 pos(0) -> false 3.53/1.71 3.53/1.71 Q is empty. 3.53/1.71 3.53/1.71 ---------------------------------------- 3.53/1.71 3.53/1.71 (3) QTRSRRRProof (EQUIVALENT) 3.53/1.71 Used ordering: 3.53/1.71 Knuth-Bendix order [KBO] with precedence:U2_1 > U1_1 > 0 > pos_1 > false > p_1 > true > s_1 3.53/1.71 3.53/1.71 and weight map: 3.53/1.71 3.53/1.71 true=6 3.53/1.71 false=4 3.53/1.71 0=1 3.53/1.71 pos_1=3 3.53/1.71 s_1=2 3.53/1.71 U1_1=1 3.53/1.71 p_1=1 3.53/1.71 U2_1=0 3.53/1.71 3.53/1.71 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.53/1.71 3.53/1.71 pos(s(x)) -> U1(pos(x)) 3.53/1.71 U1(true) -> true 3.53/1.71 pos(p(x)) -> U2(pos(x)) 3.53/1.71 U2(false) -> false 3.53/1.71 pos(s(0)) -> true 3.53/1.71 pos(0) -> false 3.53/1.71 3.53/1.71 3.53/1.71 3.53/1.71 3.53/1.71 ---------------------------------------- 3.53/1.71 3.53/1.71 (4) 3.53/1.71 Obligation: 3.53/1.71 Q restricted rewrite system: 3.53/1.71 R is empty. 3.53/1.71 Q is empty. 3.53/1.71 3.53/1.71 ---------------------------------------- 3.53/1.71 3.53/1.71 (5) RisEmptyProof (EQUIVALENT) 3.53/1.71 The TRS R is empty. Hence, termination is trivially proven. 3.53/1.71 ---------------------------------------- 3.53/1.71 3.53/1.71 (6) 3.53/1.71 YES 3.59/1.82 EOF