3.31/1.72 YES 3.48/1.73 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.48/1.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.48/1.73 3.48/1.73 3.48/1.73 Termination w.r.t. Q of the given QTRS could be proven: 3.48/1.73 3.48/1.73 (0) QTRS 3.48/1.73 (1) QTRSToCSRProof [SOUND, 0 ms] 3.48/1.73 (2) CSR 3.48/1.73 (3) CSRRRRProof [EQUIVALENT, 63 ms] 3.48/1.73 (4) CSR 3.48/1.73 (5) CSRRRRProof [EQUIVALENT, 0 ms] 3.48/1.73 (6) CSR 3.48/1.73 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.48/1.73 (8) YES 3.48/1.73 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (0) 3.48/1.73 Obligation: 3.48/1.73 Q restricted rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 active(zeros) -> mark(cons(0, zeros)) 3.48/1.73 active(tail(cons(X, XS))) -> mark(XS) 3.48/1.73 active(cons(X1, X2)) -> cons(active(X1), X2) 3.48/1.73 active(tail(X)) -> tail(active(X)) 3.48/1.73 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.48/1.73 tail(mark(X)) -> mark(tail(X)) 3.48/1.73 proper(zeros) -> ok(zeros) 3.48/1.73 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.48/1.73 proper(0) -> ok(0) 3.48/1.73 proper(tail(X)) -> tail(proper(X)) 3.48/1.73 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.48/1.73 tail(ok(X)) -> ok(tail(X)) 3.48/1.73 top(mark(X)) -> top(proper(X)) 3.48/1.73 top(ok(X)) -> top(active(X)) 3.48/1.73 3.48/1.73 The set Q consists of the following terms: 3.48/1.73 3.48/1.73 active(zeros) 3.48/1.73 active(cons(x0, x1)) 3.48/1.73 active(tail(x0)) 3.48/1.73 cons(mark(x0), x1) 3.48/1.73 tail(mark(x0)) 3.48/1.73 proper(zeros) 3.48/1.73 proper(cons(x0, x1)) 3.48/1.73 proper(0) 3.48/1.73 proper(tail(x0)) 3.48/1.73 cons(ok(x0), ok(x1)) 3.48/1.73 tail(ok(x0)) 3.48/1.73 top(mark(x0)) 3.48/1.73 top(ok(x0)) 3.48/1.73 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (1) QTRSToCSRProof (SOUND) 3.48/1.73 The following Q TRS is given: Q restricted rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 active(zeros) -> mark(cons(0, zeros)) 3.48/1.73 active(tail(cons(X, XS))) -> mark(XS) 3.48/1.73 active(cons(X1, X2)) -> cons(active(X1), X2) 3.48/1.73 active(tail(X)) -> tail(active(X)) 3.48/1.73 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.48/1.73 tail(mark(X)) -> mark(tail(X)) 3.48/1.73 proper(zeros) -> ok(zeros) 3.48/1.73 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.48/1.73 proper(0) -> ok(0) 3.48/1.73 proper(tail(X)) -> tail(proper(X)) 3.48/1.73 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.48/1.73 tail(ok(X)) -> ok(tail(X)) 3.48/1.73 top(mark(X)) -> top(proper(X)) 3.48/1.73 top(ok(X)) -> top(active(X)) 3.48/1.73 3.48/1.73 The set Q consists of the following terms: 3.48/1.73 3.48/1.73 active(zeros) 3.48/1.73 active(cons(x0, x1)) 3.48/1.73 active(tail(x0)) 3.48/1.73 cons(mark(x0), x1) 3.48/1.73 tail(mark(x0)) 3.48/1.73 proper(zeros) 3.48/1.73 proper(cons(x0, x1)) 3.48/1.73 proper(0) 3.48/1.73 proper(tail(x0)) 3.48/1.73 cons(ok(x0), ok(x1)) 3.48/1.73 tail(ok(x0)) 3.48/1.73 top(mark(x0)) 3.48/1.73 top(ok(x0)) 3.48/1.73 3.48/1.73 Special symbols used for the transformation (see [GM04]): 3.48/1.73 top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 3.48/1.73 The replacement map contains the following entries: 3.48/1.73 3.48/1.73 zeros: empty set 3.48/1.73 cons: {1} 3.48/1.73 0: empty set 3.48/1.73 tail: {1} 3.48/1.73 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. 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (2) 3.48/1.73 Obligation: 3.48/1.73 Context-sensitive rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 zeros -> cons(0, zeros) 3.48/1.73 tail(cons(X, XS)) -> XS 3.48/1.73 3.48/1.73 The replacement map contains the following entries: 3.48/1.73 3.48/1.73 zeros: empty set 3.48/1.73 cons: {1} 3.48/1.73 0: empty set 3.48/1.73 tail: {1} 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (3) CSRRRRProof (EQUIVALENT) 3.48/1.73 The following CSR is given: Context-sensitive rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 zeros -> cons(0, zeros) 3.48/1.73 tail(cons(X, XS)) -> XS 3.48/1.73 3.48/1.73 The replacement map contains the following entries: 3.48/1.73 3.48/1.73 zeros: empty set 3.48/1.73 cons: {1} 3.48/1.73 0: empty set 3.48/1.73 tail: {1} 3.48/1.73 Used ordering: 3.48/1.73 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 3.48/1.73 3.48/1.73 <<< 3.48/1.73 POL(zeros) = [[1], [1]] 3.48/1.73 >>> 3.48/1.73 3.48/1.73 <<< 3.48/1.73 POL(cons(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [0, 1]] * x_2 3.48/1.73 >>> 3.48/1.73 3.48/1.73 <<< 3.48/1.73 POL(0) = [[0], [0]] 3.48/1.73 >>> 3.48/1.73 3.48/1.73 <<< 3.48/1.73 POL(tail(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 3.48/1.73 >>> 3.48/1.73 3.48/1.73 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.48/1.73 3.48/1.73 tail(cons(X, XS)) -> XS 3.48/1.73 3.48/1.73 3.48/1.73 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (4) 3.48/1.73 Obligation: 3.48/1.73 Context-sensitive rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 zeros -> cons(0, zeros) 3.48/1.73 3.48/1.73 The replacement map contains the following entries: 3.48/1.73 3.48/1.73 zeros: empty set 3.48/1.73 cons: {1} 3.48/1.73 0: empty set 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (5) CSRRRRProof (EQUIVALENT) 3.48/1.73 The following CSR is given: Context-sensitive rewrite system: 3.48/1.73 The TRS R consists of the following rules: 3.48/1.73 3.48/1.73 zeros -> cons(0, zeros) 3.48/1.73 3.48/1.73 The replacement map contains the following entries: 3.48/1.73 3.48/1.73 zeros: empty set 3.48/1.73 cons: {1} 3.48/1.73 0: empty set 3.48/1.73 Used ordering: 3.48/1.73 Polynomial interpretation [POLO]: 3.48/1.73 3.48/1.73 POL(0) = 0 3.48/1.73 POL(cons(x_1, x_2)) = x_1 3.48/1.73 POL(zeros) = 1 3.48/1.73 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.48/1.73 3.48/1.73 zeros -> cons(0, zeros) 3.48/1.73 3.48/1.73 3.48/1.73 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (6) 3.48/1.73 Obligation: 3.48/1.73 Context-sensitive rewrite system: 3.48/1.73 R is empty. 3.48/1.73 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (7) RisEmptyProof (EQUIVALENT) 3.48/1.73 The CSR R is empty. Hence, termination is trivially proven. 3.48/1.73 ---------------------------------------- 3.48/1.73 3.48/1.73 (8) 3.48/1.73 YES 3.48/1.74 EOF