3.36/1.77 YES 3.69/1.78 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.69/1.78 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.69/1.78 3.69/1.78 3.69/1.78 Quasi decreasingness of the given CTRS could be proven: 3.69/1.78 3.69/1.78 (0) CTRS 3.69/1.78 (1) CTRSToQTRSProof [SOUND, 0 ms] 3.69/1.78 (2) QTRS 3.69/1.78 (3) QTRSRRRProof [EQUIVALENT, 55 ms] 3.69/1.78 (4) QTRS 3.69/1.78 (5) QTRSRRRProof [EQUIVALENT, 0 ms] 3.69/1.78 (6) QTRS 3.69/1.78 (7) QTRSRRRProof [EQUIVALENT, 0 ms] 3.69/1.78 (8) QTRS 3.69/1.78 (9) RisEmptyProof [EQUIVALENT, 0 ms] 3.69/1.78 (10) YES 3.69/1.78 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (0) 3.69/1.78 Obligation: 3.69/1.78 Conditional term rewrite system: 3.69/1.78 The TRS R consists of the following rules: 3.69/1.78 3.69/1.78 a -> c 3.69/1.78 a -> d 3.69/1.78 b -> c 3.69/1.78 b -> d 3.69/1.78 g(x, x) -> h(x, x) 3.69/1.78 3.69/1.78 The conditional TRS C consists of the following conditional rules: 3.69/1.78 3.69/1.78 f(x) -> x <= x -> c 3.69/1.78 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (1) CTRSToQTRSProof (SOUND) 3.69/1.78 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (2) 3.69/1.78 Obligation: 3.69/1.78 Q restricted rewrite system: 3.69/1.78 The TRS R consists of the following rules: 3.69/1.78 3.69/1.78 f(x) -> U1(x, x) 3.69/1.78 U1(c, x) -> x 3.69/1.78 a -> c 3.69/1.78 a -> d 3.69/1.78 b -> c 3.69/1.78 b -> d 3.69/1.78 g(x, x) -> h(x, x) 3.69/1.78 3.69/1.78 Q is empty. 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (3) QTRSRRRProof (EQUIVALENT) 3.69/1.78 Used ordering: 3.69/1.78 Polynomial interpretation [POLO]: 3.69/1.78 3.69/1.78 POL(U1(x_1, x_2)) = 2 + x_1 + x_2 3.69/1.78 POL(a) = 1 3.69/1.78 POL(b) = 1 3.69/1.78 POL(c) = 0 3.69/1.78 POL(d) = 0 3.69/1.78 POL(f(x_1)) = 2 + 2*x_1 3.69/1.78 POL(g(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 3.69/1.78 POL(h(x_1, x_2)) = 1 + x_1 + x_2 3.69/1.78 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.69/1.78 3.69/1.78 U1(c, x) -> x 3.69/1.78 a -> c 3.69/1.78 a -> d 3.69/1.78 b -> c 3.69/1.78 b -> d 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (4) 3.69/1.78 Obligation: 3.69/1.78 Q restricted rewrite system: 3.69/1.78 The TRS R consists of the following rules: 3.69/1.78 3.69/1.78 f(x) -> U1(x, x) 3.69/1.78 g(x, x) -> h(x, x) 3.69/1.78 3.69/1.78 Q is empty. 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (5) QTRSRRRProof (EQUIVALENT) 3.69/1.78 Used ordering: 3.69/1.78 Polynomial interpretation [POLO]: 3.69/1.78 3.69/1.78 POL(U1(x_1, x_2)) = 1 + x_1 + x_2 3.69/1.78 POL(f(x_1)) = 1 + 2*x_1 3.69/1.78 POL(g(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 3.69/1.78 POL(h(x_1, x_2)) = x_1 + x_2 3.69/1.78 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.69/1.78 3.69/1.78 g(x, x) -> h(x, x) 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (6) 3.69/1.78 Obligation: 3.69/1.78 Q restricted rewrite system: 3.69/1.78 The TRS R consists of the following rules: 3.69/1.78 3.69/1.78 f(x) -> U1(x, x) 3.69/1.78 3.69/1.78 Q is empty. 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (7) QTRSRRRProof (EQUIVALENT) 3.69/1.78 Used ordering: 3.69/1.78 Quasi precedence: 3.69/1.78 f_1 > U1_2 3.69/1.78 3.69/1.78 3.69/1.78 Status: 3.69/1.78 f_1: multiset status 3.69/1.78 U1_2: multiset status 3.69/1.78 3.69/1.78 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.69/1.78 3.69/1.78 f(x) -> U1(x, x) 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (8) 3.69/1.78 Obligation: 3.69/1.78 Q restricted rewrite system: 3.69/1.78 R is empty. 3.69/1.78 Q is empty. 3.69/1.78 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (9) RisEmptyProof (EQUIVALENT) 3.69/1.78 The TRS R is empty. Hence, termination is trivially proven. 3.69/1.78 ---------------------------------------- 3.69/1.78 3.69/1.78 (10) 3.69/1.78 YES 3.72/1.81 EOF