3.85/1.81 YES 3.85/1.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.85/1.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.85/1.82 3.85/1.82 3.85/1.82 Quasi decreasingness of the given CTRS could be proven: 3.85/1.82 3.85/1.82 (0) CTRS 3.85/1.82 (1) CTRSToQTRSProof [SOUND, 0 ms] 3.85/1.82 (2) QTRS 3.85/1.82 (3) QTRSRRRProof [EQUIVALENT, 55 ms] 3.85/1.82 (4) QTRS 3.85/1.82 (5) QTRSRRRProof [EQUIVALENT, 11 ms] 3.85/1.82 (6) QTRS 3.85/1.82 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.85/1.82 (8) YES 3.85/1.82 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (0) 3.85/1.82 Obligation: 3.85/1.82 Conditional term rewrite system: 3.85/1.82 The TRS R consists of the following rules: 3.85/1.82 3.85/1.82 le(0, s(x)) -> true 3.85/1.82 le(x, 0) -> false 3.85/1.82 le(s(x), s(y)) -> le(x, y) 3.85/1.82 min(cons(x, nil)) -> x 3.85/1.82 3.85/1.82 The conditional TRS C consists of the following conditional rules: 3.85/1.82 3.85/1.82 min(cons(x, l)) -> x <= le(x, min(l)) -> true 3.85/1.82 min(cons(x, l)) -> min(l) <= le(x, min(l)) -> false 3.85/1.82 min(cons(x, l)) -> min(l) <= min(l) -> x 3.85/1.82 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (1) CTRSToQTRSProof (SOUND) 3.85/1.82 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (2) 3.85/1.82 Obligation: 3.85/1.82 Q restricted rewrite system: 3.85/1.82 The TRS R consists of the following rules: 3.85/1.82 3.85/1.82 min(cons(x, l)) -> U1(le(x, min(l)), x, l) 3.85/1.82 U1(true, x, l) -> x 3.85/1.82 U1(false, x, l) -> min(l) 3.85/1.82 min(cons(x, l)) -> U2(min(l), l) 3.85/1.82 U2(x, l) -> min(l) 3.85/1.82 le(0, s(x)) -> true 3.85/1.82 le(x, 0) -> false 3.85/1.82 le(s(x), s(y)) -> le(x, y) 3.85/1.82 min(cons(x, nil)) -> x 3.85/1.82 3.85/1.82 Q is empty. 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (3) QTRSRRRProof (EQUIVALENT) 3.85/1.82 Used ordering: 3.85/1.82 Polynomial interpretation [POLO]: 3.85/1.82 3.85/1.82 POL(0) = 0 3.85/1.82 POL(U1(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 3.85/1.82 POL(U2(x_1, x_2)) = x_1 + 2*x_2 3.85/1.82 POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2 3.85/1.82 POL(false) = 1 3.85/1.82 POL(le(x_1, x_2)) = 2 + x_1 + x_2 3.85/1.82 POL(min(x_1)) = 2*x_1 3.85/1.82 POL(nil) = 0 3.85/1.82 POL(s(x_1)) = 2*x_1 3.85/1.82 POL(true) = 0 3.85/1.82 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.85/1.82 3.85/1.82 U1(true, x, l) -> x 3.85/1.82 U1(false, x, l) -> min(l) 3.85/1.82 min(cons(x, l)) -> U2(min(l), l) 3.85/1.82 le(0, s(x)) -> true 3.85/1.82 le(x, 0) -> false 3.85/1.82 min(cons(x, nil)) -> x 3.85/1.82 3.85/1.82 3.85/1.82 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (4) 3.85/1.82 Obligation: 3.85/1.82 Q restricted rewrite system: 3.85/1.82 The TRS R consists of the following rules: 3.85/1.82 3.85/1.82 min(cons(x, l)) -> U1(le(x, min(l)), x, l) 3.85/1.82 U2(x, l) -> min(l) 3.85/1.82 le(s(x), s(y)) -> le(x, y) 3.85/1.82 3.85/1.82 Q is empty. 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (5) QTRSRRRProof (EQUIVALENT) 3.85/1.82 Used ordering: 3.85/1.82 Polynomial interpretation [POLO]: 3.85/1.82 3.85/1.82 POL(U1(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 3.85/1.82 POL(U2(x_1, x_2)) = 2 + x_1 + 2*x_2 3.85/1.82 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 3.85/1.82 POL(le(x_1, x_2)) = x_1 + x_2 3.85/1.82 POL(min(x_1)) = 2*x_1 3.85/1.82 POL(s(x_1)) = 1 + 2*x_1 3.85/1.82 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.85/1.82 3.85/1.82 min(cons(x, l)) -> U1(le(x, min(l)), x, l) 3.85/1.82 U2(x, l) -> min(l) 3.85/1.82 le(s(x), s(y)) -> le(x, y) 3.85/1.82 3.85/1.82 3.85/1.82 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (6) 3.85/1.82 Obligation: 3.85/1.82 Q restricted rewrite system: 3.85/1.82 R is empty. 3.85/1.82 Q is empty. 3.85/1.82 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (7) RisEmptyProof (EQUIVALENT) 3.85/1.82 The TRS R is empty. Hence, termination is trivially proven. 3.85/1.82 ---------------------------------------- 3.85/1.82 3.85/1.82 (8) 3.85/1.82 YES 3.85/1.86 EOF