3.67/1.81 YES 3.67/1.82 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.67/1.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.67/1.82 3.67/1.82 3.67/1.82 Termination w.r.t. Q of the given QTRS could be proven: 3.67/1.82 3.67/1.82 (0) QTRS 3.67/1.82 (1) QTRSRRRProof [EQUIVALENT, 60 ms] 3.67/1.82 (2) QTRS 3.67/1.82 (3) QTRSRRRProof [EQUIVALENT, 0 ms] 3.67/1.82 (4) QTRS 3.67/1.82 (5) QTRSRRRProof [EQUIVALENT, 6 ms] 3.67/1.82 (6) QTRS 3.67/1.82 (7) DependencyPairsProof [EQUIVALENT, 0 ms] 3.67/1.82 (8) QDP 3.67/1.82 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 3.67/1.82 (10) TRUE 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (0) 3.67/1.82 Obligation: 3.67/1.82 Q restricted rewrite system: 3.67/1.82 The TRS R consists of the following rules: 3.67/1.82 3.67/1.82 a__f(b, X, c) -> a__f(X, a__c, X) 3.67/1.82 a__c -> b 3.67/1.82 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 3.67/1.82 mark(c) -> a__c 3.67/1.82 mark(b) -> b 3.67/1.82 a__f(X1, X2, X3) -> f(X1, X2, X3) 3.67/1.82 a__c -> c 3.67/1.82 3.67/1.82 The set Q consists of the following terms: 3.67/1.82 3.67/1.82 a__c 3.67/1.82 mark(f(x0, x1, x2)) 3.67/1.82 mark(c) 3.67/1.82 mark(b) 3.67/1.82 a__f(x0, x1, x2) 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (1) QTRSRRRProof (EQUIVALENT) 3.67/1.82 Used ordering: 3.67/1.82 Polynomial interpretation [POLO]: 3.67/1.82 3.67/1.82 POL(a__c) = 0 3.67/1.82 POL(a__f(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 3.67/1.82 POL(b) = 0 3.67/1.82 POL(c) = 0 3.67/1.82 POL(f(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 3.67/1.82 POL(mark(x_1)) = 2*x_1 3.67/1.82 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.67/1.82 3.67/1.82 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (2) 3.67/1.82 Obligation: 3.67/1.82 Q restricted rewrite system: 3.67/1.82 The TRS R consists of the following rules: 3.67/1.82 3.67/1.82 a__f(b, X, c) -> a__f(X, a__c, X) 3.67/1.82 a__c -> b 3.67/1.82 mark(c) -> a__c 3.67/1.82 mark(b) -> b 3.67/1.82 a__f(X1, X2, X3) -> f(X1, X2, X3) 3.67/1.82 a__c -> c 3.67/1.82 3.67/1.82 The set Q consists of the following terms: 3.67/1.82 3.67/1.82 a__c 3.67/1.82 mark(f(x0, x1, x2)) 3.67/1.82 mark(c) 3.67/1.82 mark(b) 3.67/1.82 a__f(x0, x1, x2) 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (3) QTRSRRRProof (EQUIVALENT) 3.67/1.82 Used ordering: 3.67/1.82 Polynomial interpretation [POLO]: 3.67/1.82 3.67/1.82 POL(a__c) = 0 3.67/1.82 POL(a__f(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 3.67/1.82 POL(b) = 0 3.67/1.82 POL(c) = 0 3.67/1.82 POL(f(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 3.67/1.82 POL(mark(x_1)) = x_1 3.67/1.82 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.67/1.82 3.67/1.82 a__f(X1, X2, X3) -> f(X1, X2, X3) 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (4) 3.67/1.82 Obligation: 3.67/1.82 Q restricted rewrite system: 3.67/1.82 The TRS R consists of the following rules: 3.67/1.82 3.67/1.82 a__f(b, X, c) -> a__f(X, a__c, X) 3.67/1.82 a__c -> b 3.67/1.82 mark(c) -> a__c 3.67/1.82 mark(b) -> b 3.67/1.82 a__c -> c 3.67/1.82 3.67/1.82 The set Q consists of the following terms: 3.67/1.82 3.67/1.82 a__c 3.67/1.82 mark(f(x0, x1, x2)) 3.67/1.82 mark(c) 3.67/1.82 mark(b) 3.67/1.82 a__f(x0, x1, x2) 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (5) QTRSRRRProof (EQUIVALENT) 3.67/1.82 Used ordering: 3.67/1.82 Polynomial interpretation [POLO]: 3.67/1.82 3.67/1.82 POL(a__c) = 0 3.67/1.82 POL(a__f(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 3.67/1.82 POL(b) = 0 3.67/1.82 POL(c) = 0 3.67/1.82 POL(mark(x_1)) = 2 + 2*x_1 3.67/1.82 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.67/1.82 3.67/1.82 mark(c) -> a__c 3.67/1.82 mark(b) -> b 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (6) 3.67/1.82 Obligation: 3.67/1.82 Q restricted rewrite system: 3.67/1.82 The TRS R consists of the following rules: 3.67/1.82 3.67/1.82 a__f(b, X, c) -> a__f(X, a__c, X) 3.67/1.82 a__c -> b 3.67/1.82 a__c -> c 3.67/1.82 3.67/1.82 The set Q consists of the following terms: 3.67/1.82 3.67/1.82 a__c 3.67/1.82 mark(f(x0, x1, x2)) 3.67/1.82 mark(c) 3.67/1.82 mark(b) 3.67/1.82 a__f(x0, x1, x2) 3.67/1.82 3.67/1.82 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (7) DependencyPairsProof (EQUIVALENT) 3.67/1.82 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (8) 3.67/1.82 Obligation: 3.67/1.82 Q DP problem: 3.67/1.82 The TRS P consists of the following rules: 3.67/1.82 3.67/1.82 A__F(b, X, c) -> A__F(X, a__c, X) 3.67/1.82 A__F(b, X, c) -> A__C 3.67/1.82 3.67/1.82 The TRS R consists of the following rules: 3.67/1.82 3.67/1.82 a__f(b, X, c) -> a__f(X, a__c, X) 3.67/1.82 a__c -> b 3.67/1.82 a__c -> c 3.67/1.82 3.67/1.82 The set Q consists of the following terms: 3.67/1.82 3.67/1.82 a__c 3.67/1.82 mark(f(x0, x1, x2)) 3.67/1.82 mark(c) 3.67/1.82 mark(b) 3.67/1.82 a__f(x0, x1, x2) 3.67/1.82 3.67/1.82 We have to consider all minimal (P,Q,R)-chains. 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (9) DependencyGraphProof (EQUIVALENT) 3.67/1.82 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 3.67/1.82 ---------------------------------------- 3.67/1.82 3.67/1.82 (10) 3.67/1.82 TRUE 3.70/1.85 EOF