3.75/1.75 YES 3.75/1.76 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.75/1.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.75/1.76 3.75/1.76 3.75/1.76 Quasi decreasingness of the given CTRS could be proven: 3.75/1.76 3.75/1.76 (0) CTRS 3.75/1.76 (1) CTRSToQTRSProof [SOUND, 0 ms] 3.75/1.76 (2) QTRS 3.75/1.76 (3) QTRSRRRProof [EQUIVALENT, 59 ms] 3.75/1.76 (4) QTRS 3.75/1.76 (5) QTRSRRRProof [EQUIVALENT, 0 ms] 3.75/1.76 (6) QTRS 3.75/1.76 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.75/1.76 (8) YES 3.75/1.76 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (0) 3.75/1.76 Obligation: 3.75/1.76 Conditional term rewrite system: 3.75/1.76 The TRS R consists of the following rules: 3.75/1.76 3.75/1.76 g(s(x)) -> x 3.75/1.76 h(s(x)) -> x 3.75/1.76 3.75/1.76 The conditional TRS C consists of the following conditional rules: 3.75/1.76 3.75/1.76 f(x, y) -> g(s(x)) <= c(g(x)) -> c(a) 3.75/1.76 f(x, y) -> h(s(x)) <= c(h(x)) -> c(a) 3.75/1.76 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (1) CTRSToQTRSProof (SOUND) 3.75/1.76 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (2) 3.75/1.76 Obligation: 3.75/1.76 Q restricted rewrite system: 3.75/1.76 The TRS R consists of the following rules: 3.75/1.76 3.75/1.76 f(x, y) -> U1(c(g(x)), x) 3.75/1.76 U1(c(a), x) -> g(s(x)) 3.75/1.76 f(x, y) -> U2(c(h(x)), x) 3.75/1.76 U2(c(a), x) -> h(s(x)) 3.75/1.76 g(s(x)) -> x 3.75/1.76 h(s(x)) -> x 3.75/1.76 3.75/1.76 Q is empty. 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (3) QTRSRRRProof (EQUIVALENT) 3.75/1.76 Used ordering: 3.75/1.76 Polynomial interpretation [POLO]: 3.75/1.76 3.75/1.76 POL(U1(x_1, x_2)) = x_1 + x_2 3.75/1.76 POL(U2(x_1, x_2)) = x_1 + x_2 3.75/1.76 POL(a) = 0 3.75/1.76 POL(c(x_1)) = x_1 3.75/1.76 POL(f(x_1, x_2)) = 1 + 2*x_1 + x_2 3.75/1.76 POL(g(x_1)) = x_1 3.75/1.76 POL(h(x_1)) = x_1 3.75/1.76 POL(s(x_1)) = x_1 3.75/1.76 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.75/1.76 3.75/1.76 f(x, y) -> U1(c(g(x)), x) 3.75/1.76 f(x, y) -> U2(c(h(x)), x) 3.75/1.76 3.75/1.76 3.75/1.76 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (4) 3.75/1.76 Obligation: 3.75/1.76 Q restricted rewrite system: 3.75/1.76 The TRS R consists of the following rules: 3.75/1.76 3.75/1.76 U1(c(a), x) -> g(s(x)) 3.75/1.76 U2(c(a), x) -> h(s(x)) 3.75/1.76 g(s(x)) -> x 3.75/1.76 h(s(x)) -> x 3.75/1.76 3.75/1.76 Q is empty. 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (5) QTRSRRRProof (EQUIVALENT) 3.75/1.76 Used ordering: 3.75/1.76 Knuth-Bendix order [KBO] with precedence:h_1 > s_1 > U2_2 > g_1 > a > U1_2 > c_1 3.75/1.76 3.75/1.76 and weight map: 3.75/1.76 3.75/1.76 a=1 3.75/1.76 c_1=1 3.75/1.76 g_1=1 3.75/1.76 s_1=1 3.75/1.76 h_1=1 3.75/1.76 U1_2=1 3.75/1.76 U2_2=1 3.75/1.76 3.75/1.76 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.75/1.76 3.75/1.76 U1(c(a), x) -> g(s(x)) 3.75/1.76 U2(c(a), x) -> h(s(x)) 3.75/1.76 g(s(x)) -> x 3.75/1.76 h(s(x)) -> x 3.75/1.76 3.75/1.76 3.75/1.76 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (6) 3.75/1.76 Obligation: 3.75/1.76 Q restricted rewrite system: 3.75/1.76 R is empty. 3.75/1.76 Q is empty. 3.75/1.76 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (7) RisEmptyProof (EQUIVALENT) 3.75/1.76 The TRS R is empty. Hence, termination is trivially proven. 3.75/1.76 ---------------------------------------- 3.75/1.76 3.75/1.76 (8) 3.75/1.76 YES 3.75/1.79 EOF