4.96/2.16 MAYBE 4.96/2.17 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.96/2.17 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.96/2.17 4.96/2.17 4.96/2.17 Quasi decreasingness of the given CTRS could not be shown: 4.96/2.17 4.96/2.17 (0) CTRS 4.96/2.17 (1) CTRSToQTRSProof [SOUND, 0 ms] 4.96/2.17 (2) QTRS 4.96/2.17 (3) QTRSRRRProof [EQUIVALENT, 64 ms] 4.96/2.17 (4) QTRS 4.96/2.17 (5) QTRSRRRProof [EQUIVALENT, 0 ms] 4.96/2.17 (6) QTRS 4.96/2.17 (7) AAECC Innermost [EQUIVALENT, 0 ms] 4.96/2.17 (8) QTRS 4.96/2.17 (9) DependencyPairsProof [EQUIVALENT, 0 ms] 4.96/2.17 (10) QDP 4.96/2.17 (11) UsableRulesProof [EQUIVALENT, 0 ms] 4.96/2.17 (12) QDP 4.96/2.17 (13) QReductionProof [EQUIVALENT, 0 ms] 4.96/2.17 (14) QDP 4.96/2.17 (15) NonTerminationLoopProof [COMPLETE, 0 ms] 4.96/2.17 (16) NO 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (0) 4.96/2.17 Obligation: 4.96/2.17 Conditional term rewrite system: 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 g(s(x)) -> x 4.96/2.17 h(s(x)) -> x 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 The conditional TRS C consists of the following conditional rules: 4.96/2.17 4.96/2.17 f(x, y) -> g(x) <= c(g(x)) -> c(a) 4.96/2.17 f(x, y) -> h(x) <= c(h(x)) -> c(a) 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (1) CTRSToQTRSProof (SOUND) 4.96/2.17 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (2) 4.96/2.17 Obligation: 4.96/2.17 Q restricted rewrite system: 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 f(x, y) -> U1(c(g(x)), x) 4.96/2.17 U1(c(a), x) -> g(x) 4.96/2.17 f(x, y) -> U2(c(h(x)), x) 4.96/2.17 U2(c(a), x) -> h(x) 4.96/2.17 g(s(x)) -> x 4.96/2.17 h(s(x)) -> x 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 Q is empty. 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (3) QTRSRRRProof (EQUIVALENT) 4.96/2.17 Used ordering: 4.96/2.17 Polynomial interpretation [POLO]: 4.96/2.17 4.96/2.17 POL(U1(x_1, x_2)) = x_1 + x_2 4.96/2.17 POL(U2(x_1, x_2)) = x_1 + x_2 4.96/2.17 POL(a) = 0 4.96/2.17 POL(b) = 0 4.96/2.17 POL(c(x_1)) = x_1 4.96/2.17 POL(f(x_1, x_2)) = 1 + 2*x_1 + x_2 4.96/2.17 POL(g(x_1)) = x_1 4.96/2.17 POL(h(x_1)) = x_1 4.96/2.17 POL(s(x_1)) = 2 + x_1 4.96/2.17 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.96/2.17 4.96/2.17 f(x, y) -> U1(c(g(x)), x) 4.96/2.17 f(x, y) -> U2(c(h(x)), x) 4.96/2.17 g(s(x)) -> x 4.96/2.17 h(s(x)) -> x 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (4) 4.96/2.17 Obligation: 4.96/2.17 Q restricted rewrite system: 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 U1(c(a), x) -> g(x) 4.96/2.17 U2(c(a), x) -> h(x) 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 Q is empty. 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (5) QTRSRRRProof (EQUIVALENT) 4.96/2.17 Used ordering: 4.96/2.17 Knuth-Bendix order [KBO] with precedence:b > U2_2 > h_1 > U1_2 > g_1 > a > c_1 4.96/2.17 4.96/2.17 and weight map: 4.96/2.17 4.96/2.17 a=1 4.96/2.17 b=1 4.96/2.17 c_1=1 4.96/2.17 g_1=2 4.96/2.17 h_1=2 4.96/2.17 U1_2=0 4.96/2.17 U2_2=0 4.96/2.17 4.96/2.17 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.96/2.17 4.96/2.17 U1(c(a), x) -> g(x) 4.96/2.17 U2(c(a), x) -> h(x) 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (6) 4.96/2.17 Obligation: 4.96/2.17 Q restricted rewrite system: 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 Q is empty. 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (7) AAECC Innermost (EQUIVALENT) 4.96/2.17 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none 4.96/2.17 4.96/2.17 The TRS R 2 is 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 The signature Sigma is {b} 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (8) 4.96/2.17 Obligation: 4.96/2.17 Q restricted rewrite system: 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 The set Q consists of the following terms: 4.96/2.17 4.96/2.17 b 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (9) DependencyPairsProof (EQUIVALENT) 4.96/2.17 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (10) 4.96/2.17 Obligation: 4.96/2.17 Q DP problem: 4.96/2.17 The TRS P consists of the following rules: 4.96/2.17 4.96/2.17 B -> B 4.96/2.17 4.96/2.17 The TRS R consists of the following rules: 4.96/2.17 4.96/2.17 b -> b 4.96/2.17 4.96/2.17 The set Q consists of the following terms: 4.96/2.17 4.96/2.17 b 4.96/2.17 4.96/2.17 We have to consider all minimal (P,Q,R)-chains. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (11) UsableRulesProof (EQUIVALENT) 4.96/2.17 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. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (12) 4.96/2.17 Obligation: 4.96/2.17 Q DP problem: 4.96/2.17 The TRS P consists of the following rules: 4.96/2.17 4.96/2.17 B -> B 4.96/2.17 4.96/2.17 R is empty. 4.96/2.17 The set Q consists of the following terms: 4.96/2.17 4.96/2.17 b 4.96/2.17 4.96/2.17 We have to consider all minimal (P,Q,R)-chains. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (13) QReductionProof (EQUIVALENT) 4.96/2.17 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.96/2.17 4.96/2.17 b 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (14) 4.96/2.17 Obligation: 4.96/2.17 Q DP problem: 4.96/2.17 The TRS P consists of the following rules: 4.96/2.17 4.96/2.17 B -> B 4.96/2.17 4.96/2.17 R is empty. 4.96/2.17 Q is empty. 4.96/2.17 We have to consider all minimal (P,Q,R)-chains. 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (15) NonTerminationLoopProof (COMPLETE) 4.96/2.17 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 4.96/2.17 Found a loop by semiunifying a rule from P directly. 4.96/2.17 4.96/2.17 s = B evaluates to t =B 4.96/2.17 4.96/2.17 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 4.96/2.17 * Matcher: [ ] 4.96/2.17 * Semiunifier: [ ] 4.96/2.17 4.96/2.17 -------------------------------------------------------------------------------- 4.96/2.17 Rewriting sequence 4.96/2.17 4.96/2.17 The DP semiunifies directly so there is only one rewrite step from B to B. 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 4.96/2.17 ---------------------------------------- 4.96/2.17 4.96/2.17 (16) 4.96/2.17 NO 5.11/2.22 EOF