3.71/1.75 YES 3.71/1.76 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.71/1.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.71/1.76 3.71/1.76 3.71/1.76 Termination w.r.t. Q of the given QTRS could be proven: 3.71/1.76 3.71/1.76 (0) QTRS 3.71/1.76 (1) QTRSRRRProof [EQUIVALENT, 59 ms] 3.71/1.76 (2) QTRS 3.71/1.76 (3) QTRSRRRProof [EQUIVALENT, 2 ms] 3.71/1.76 (4) QTRS 3.71/1.76 (5) RisEmptyProof [EQUIVALENT, 0 ms] 3.71/1.76 (6) YES 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (0) 3.71/1.76 Obligation: 3.71/1.76 Q restricted rewrite system: 3.71/1.76 The TRS R consists of the following rules: 3.71/1.76 3.71/1.76 active(f(g(X), Y)) -> mark(f(X, f(g(X), Y))) 3.71/1.76 active(f(X1, X2)) -> f(active(X1), X2) 3.71/1.76 active(g(X)) -> g(active(X)) 3.71/1.76 f(mark(X1), X2) -> mark(f(X1, X2)) 3.71/1.76 g(mark(X)) -> mark(g(X)) 3.71/1.76 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 3.71/1.76 proper(g(X)) -> g(proper(X)) 3.71/1.76 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 3.71/1.76 g(ok(X)) -> ok(g(X)) 3.71/1.76 top(mark(X)) -> top(proper(X)) 3.71/1.76 top(ok(X)) -> top(active(X)) 3.71/1.76 3.71/1.76 The set Q consists of the following terms: 3.71/1.76 3.71/1.76 active(f(x0, x1)) 3.71/1.76 active(g(x0)) 3.71/1.76 f(mark(x0), x1) 3.71/1.76 g(mark(x0)) 3.71/1.76 proper(f(x0, x1)) 3.71/1.76 proper(g(x0)) 3.71/1.76 f(ok(x0), ok(x1)) 3.71/1.76 g(ok(x0)) 3.71/1.76 top(mark(x0)) 3.71/1.76 top(ok(x0)) 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (1) QTRSRRRProof (EQUIVALENT) 3.71/1.76 Used ordering: 3.71/1.76 Polynomial interpretation [POLO]: 3.71/1.76 3.71/1.76 POL(active(x_1)) = 1 + 2*x_1 3.71/1.76 POL(f(x_1, x_2)) = x_1 + x_2 3.71/1.76 POL(g(x_1)) = x_1 3.71/1.76 POL(mark(x_1)) = x_1 3.71/1.76 POL(ok(x_1)) = 1 + 2*x_1 3.71/1.76 POL(proper(x_1)) = x_1 3.71/1.76 POL(top(x_1)) = x_1 3.71/1.76 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.71/1.76 3.71/1.76 active(f(g(X), Y)) -> mark(f(X, f(g(X), Y))) 3.71/1.76 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 3.71/1.76 3.71/1.76 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (2) 3.71/1.76 Obligation: 3.71/1.76 Q restricted rewrite system: 3.71/1.76 The TRS R consists of the following rules: 3.71/1.76 3.71/1.76 active(f(X1, X2)) -> f(active(X1), X2) 3.71/1.76 active(g(X)) -> g(active(X)) 3.71/1.76 f(mark(X1), X2) -> mark(f(X1, X2)) 3.71/1.76 g(mark(X)) -> mark(g(X)) 3.71/1.76 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 3.71/1.76 proper(g(X)) -> g(proper(X)) 3.71/1.76 g(ok(X)) -> ok(g(X)) 3.71/1.76 top(mark(X)) -> top(proper(X)) 3.71/1.76 top(ok(X)) -> top(active(X)) 3.71/1.76 3.71/1.76 The set Q consists of the following terms: 3.71/1.76 3.71/1.76 active(f(x0, x1)) 3.71/1.76 active(g(x0)) 3.71/1.76 f(mark(x0), x1) 3.71/1.76 g(mark(x0)) 3.71/1.76 proper(f(x0, x1)) 3.71/1.76 proper(g(x0)) 3.71/1.76 f(ok(x0), ok(x1)) 3.71/1.76 g(ok(x0)) 3.71/1.76 top(mark(x0)) 3.71/1.76 top(ok(x0)) 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (3) QTRSRRRProof (EQUIVALENT) 3.71/1.76 Used ordering: 3.71/1.76 Knuth-Bendix order [KBO] with precedence:proper_1 > top_1 > active_1 > g_1 > ok_1 > f_2 > mark_1 3.71/1.76 3.71/1.76 and weight map: 3.71/1.76 3.71/1.76 active_1=2 3.71/1.76 g_1=4 3.71/1.76 mark_1=1 3.71/1.76 proper_1=0 3.71/1.76 ok_1=3 3.71/1.76 top_1=1 3.71/1.76 f_2=0 3.71/1.76 3.71/1.76 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.71/1.76 3.71/1.76 active(f(X1, X2)) -> f(active(X1), X2) 3.71/1.76 active(g(X)) -> g(active(X)) 3.71/1.76 f(mark(X1), X2) -> mark(f(X1, X2)) 3.71/1.76 g(mark(X)) -> mark(g(X)) 3.71/1.76 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 3.71/1.76 proper(g(X)) -> g(proper(X)) 3.71/1.76 g(ok(X)) -> ok(g(X)) 3.71/1.76 top(mark(X)) -> top(proper(X)) 3.71/1.76 top(ok(X)) -> top(active(X)) 3.71/1.76 3.71/1.76 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (4) 3.71/1.76 Obligation: 3.71/1.76 Q restricted rewrite system: 3.71/1.76 R is empty. 3.71/1.76 The set Q consists of the following terms: 3.71/1.76 3.71/1.76 active(f(x0, x1)) 3.71/1.76 active(g(x0)) 3.71/1.76 f(mark(x0), x1) 3.71/1.76 g(mark(x0)) 3.71/1.76 proper(f(x0, x1)) 3.71/1.76 proper(g(x0)) 3.71/1.76 f(ok(x0), ok(x1)) 3.71/1.76 g(ok(x0)) 3.71/1.76 top(mark(x0)) 3.71/1.76 top(ok(x0)) 3.71/1.76 3.71/1.76 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (5) RisEmptyProof (EQUIVALENT) 3.71/1.76 The TRS R is empty. Hence, termination is trivially proven. 3.71/1.76 ---------------------------------------- 3.71/1.76 3.71/1.76 (6) 3.71/1.76 YES 3.71/1.77 EOF