3.61/1.66 YES 3.61/1.67 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.61/1.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.61/1.67 3.61/1.67 3.61/1.67 Termination w.r.t. Q of the given QTRS could be proven: 3.61/1.67 3.61/1.67 (0) QTRS 3.61/1.67 (1) QTRSToCSRProof [SOUND, 0 ms] 3.61/1.67 (2) CSR 3.61/1.67 (3) CSRRRRProof [EQUIVALENT, 29 ms] 3.61/1.67 (4) CSR 3.61/1.67 (5) CSRRRRProof [EQUIVALENT, 0 ms] 3.61/1.67 (6) CSR 3.61/1.67 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.61/1.67 (8) YES 3.61/1.67 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (0) 3.61/1.67 Obligation: 3.61/1.67 Q restricted rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 active(f(0)) -> mark(cons(0, f(s(0)))) 3.61/1.67 active(f(s(0))) -> mark(f(p(s(0)))) 3.61/1.67 active(p(s(0))) -> mark(0) 3.61/1.67 active(f(X)) -> f(active(X)) 3.61/1.67 active(cons(X1, X2)) -> cons(active(X1), X2) 3.61/1.67 active(s(X)) -> s(active(X)) 3.61/1.67 active(p(X)) -> p(active(X)) 3.61/1.67 f(mark(X)) -> mark(f(X)) 3.61/1.67 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.61/1.67 s(mark(X)) -> mark(s(X)) 3.61/1.67 p(mark(X)) -> mark(p(X)) 3.61/1.67 proper(f(X)) -> f(proper(X)) 3.61/1.67 proper(0) -> ok(0) 3.61/1.67 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.61/1.67 proper(s(X)) -> s(proper(X)) 3.61/1.67 proper(p(X)) -> p(proper(X)) 3.61/1.67 f(ok(X)) -> ok(f(X)) 3.61/1.67 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.61/1.67 s(ok(X)) -> ok(s(X)) 3.61/1.67 p(ok(X)) -> ok(p(X)) 3.61/1.67 top(mark(X)) -> top(proper(X)) 3.61/1.67 top(ok(X)) -> top(active(X)) 3.61/1.67 3.61/1.67 The set Q consists of the following terms: 3.61/1.67 3.61/1.67 active(f(x0)) 3.61/1.67 active(cons(x0, x1)) 3.61/1.67 active(s(x0)) 3.61/1.67 active(p(x0)) 3.61/1.67 f(mark(x0)) 3.61/1.67 cons(mark(x0), x1) 3.61/1.67 s(mark(x0)) 3.61/1.67 p(mark(x0)) 3.61/1.67 proper(f(x0)) 3.61/1.67 proper(0) 3.61/1.67 proper(cons(x0, x1)) 3.61/1.67 proper(s(x0)) 3.61/1.67 proper(p(x0)) 3.61/1.67 f(ok(x0)) 3.61/1.67 cons(ok(x0), ok(x1)) 3.61/1.67 s(ok(x0)) 3.61/1.67 p(ok(x0)) 3.61/1.67 top(mark(x0)) 3.61/1.67 top(ok(x0)) 3.61/1.67 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (1) QTRSToCSRProof (SOUND) 3.61/1.67 The following Q TRS is given: Q restricted rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 active(f(0)) -> mark(cons(0, f(s(0)))) 3.61/1.67 active(f(s(0))) -> mark(f(p(s(0)))) 3.61/1.67 active(p(s(0))) -> mark(0) 3.61/1.67 active(f(X)) -> f(active(X)) 3.61/1.67 active(cons(X1, X2)) -> cons(active(X1), X2) 3.61/1.67 active(s(X)) -> s(active(X)) 3.61/1.67 active(p(X)) -> p(active(X)) 3.61/1.67 f(mark(X)) -> mark(f(X)) 3.61/1.67 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.61/1.67 s(mark(X)) -> mark(s(X)) 3.61/1.67 p(mark(X)) -> mark(p(X)) 3.61/1.67 proper(f(X)) -> f(proper(X)) 3.61/1.67 proper(0) -> ok(0) 3.61/1.67 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.61/1.67 proper(s(X)) -> s(proper(X)) 3.61/1.67 proper(p(X)) -> p(proper(X)) 3.61/1.67 f(ok(X)) -> ok(f(X)) 3.61/1.67 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.61/1.67 s(ok(X)) -> ok(s(X)) 3.61/1.67 p(ok(X)) -> ok(p(X)) 3.61/1.67 top(mark(X)) -> top(proper(X)) 3.61/1.67 top(ok(X)) -> top(active(X)) 3.61/1.67 3.61/1.67 The set Q consists of the following terms: 3.61/1.67 3.61/1.67 active(f(x0)) 3.61/1.67 active(cons(x0, x1)) 3.61/1.67 active(s(x0)) 3.61/1.67 active(p(x0)) 3.61/1.67 f(mark(x0)) 3.61/1.67 cons(mark(x0), x1) 3.61/1.67 s(mark(x0)) 3.61/1.67 p(mark(x0)) 3.61/1.67 proper(f(x0)) 3.61/1.67 proper(0) 3.61/1.67 proper(cons(x0, x1)) 3.61/1.67 proper(s(x0)) 3.61/1.67 proper(p(x0)) 3.61/1.67 f(ok(x0)) 3.61/1.67 cons(ok(x0), ok(x1)) 3.61/1.67 s(ok(x0)) 3.61/1.67 p(ok(x0)) 3.61/1.67 top(mark(x0)) 3.61/1.67 top(ok(x0)) 3.61/1.67 3.61/1.67 Special symbols used for the transformation (see [GM04]): 3.61/1.67 top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 3.61/1.67 The replacement map contains the following entries: 3.61/1.67 3.61/1.67 f: {1} 3.61/1.67 0: empty set 3.61/1.67 cons: {1} 3.61/1.67 s: {1} 3.61/1.67 p: {1} 3.61/1.67 The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete. 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (2) 3.61/1.67 Obligation: 3.61/1.67 Context-sensitive rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 f(0) -> cons(0, f(s(0))) 3.61/1.67 f(s(0)) -> f(p(s(0))) 3.61/1.67 p(s(0)) -> 0 3.61/1.67 3.61/1.67 The replacement map contains the following entries: 3.61/1.67 3.61/1.67 f: {1} 3.61/1.67 0: empty set 3.61/1.67 cons: {1} 3.61/1.67 s: {1} 3.61/1.67 p: {1} 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (3) CSRRRRProof (EQUIVALENT) 3.61/1.67 The following CSR is given: Context-sensitive rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 f(0) -> cons(0, f(s(0))) 3.61/1.67 f(s(0)) -> f(p(s(0))) 3.61/1.67 p(s(0)) -> 0 3.61/1.67 3.61/1.67 The replacement map contains the following entries: 3.61/1.67 3.61/1.67 f: {1} 3.61/1.67 0: empty set 3.61/1.67 cons: {1} 3.61/1.67 s: {1} 3.61/1.67 p: {1} 3.61/1.67 Used ordering: 3.61/1.67 Polynomial interpretation [POLO]: 3.61/1.67 3.61/1.67 POL(0) = 2 3.61/1.67 POL(cons(x_1, x_2)) = x_1 3.61/1.67 POL(f(x_1)) = 1 + 2*x_1 3.61/1.67 POL(p(x_1)) = x_1 3.61/1.67 POL(s(x_1)) = 2*x_1 3.61/1.67 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.61/1.67 3.61/1.67 f(0) -> cons(0, f(s(0))) 3.61/1.67 p(s(0)) -> 0 3.61/1.67 3.61/1.67 3.61/1.67 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (4) 3.61/1.67 Obligation: 3.61/1.67 Context-sensitive rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 f(s(0)) -> f(p(s(0))) 3.61/1.67 3.61/1.67 The replacement map contains the following entries: 3.61/1.67 3.61/1.67 f: {1} 3.61/1.67 0: empty set 3.61/1.67 s: {1} 3.61/1.67 p: {1} 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (5) CSRRRRProof (EQUIVALENT) 3.61/1.67 The following CSR is given: Context-sensitive rewrite system: 3.61/1.67 The TRS R consists of the following rules: 3.61/1.67 3.61/1.67 f(s(0)) -> f(p(s(0))) 3.61/1.67 3.61/1.67 The replacement map contains the following entries: 3.61/1.67 3.61/1.67 f: {1} 3.61/1.67 0: empty set 3.61/1.67 s: {1} 3.61/1.67 p: {1} 3.61/1.67 Used ordering: 3.61/1.67 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 3.61/1.67 3.61/1.67 <<< 3.61/1.67 POL(f(x_1)) = [[0]] + [[1, 1]] * x_1 3.61/1.67 >>> 3.61/1.67 3.61/1.67 <<< 3.61/1.67 POL(s(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 3.61/1.67 >>> 3.61/1.67 3.61/1.67 <<< 3.61/1.67 POL(0) = [[1], [1]] 3.61/1.67 >>> 3.61/1.67 3.61/1.67 <<< 3.61/1.67 POL(p(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 3.61/1.67 >>> 3.61/1.67 3.61/1.67 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.61/1.67 3.61/1.67 f(s(0)) -> f(p(s(0))) 3.61/1.67 3.61/1.67 3.61/1.67 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (6) 3.61/1.67 Obligation: 3.61/1.67 Context-sensitive rewrite system: 3.61/1.67 R is empty. 3.61/1.67 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (7) RisEmptyProof (EQUIVALENT) 3.61/1.67 The CSR R is empty. Hence, termination is trivially proven. 3.61/1.67 ---------------------------------------- 3.61/1.67 3.61/1.67 (8) 3.61/1.67 YES 3.61/1.68 EOF