3.59/1.80 YES 3.59/1.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.59/1.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.59/1.81 3.59/1.81 3.59/1.81 Termination w.r.t. Q of the given QTRS could be proven: 3.59/1.81 3.59/1.81 (0) QTRS 3.59/1.81 (1) QTRSToCSRProof [SOUND, 0 ms] 3.59/1.81 (2) CSR 3.59/1.81 (3) CSRRRRProof [EQUIVALENT, 39 ms] 3.59/1.81 (4) CSR 3.59/1.81 (5) RisEmptyProof [EQUIVALENT, 0 ms] 3.59/1.81 (6) YES 3.59/1.81 3.59/1.81 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (0) 3.59/1.81 Obligation: 3.59/1.81 Q restricted rewrite system: 3.59/1.81 The TRS R consists of the following rules: 3.59/1.81 3.59/1.81 active(fst(0, Z)) -> mark(nil) 3.59/1.81 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 3.59/1.81 active(from(X)) -> mark(cons(X, from(s(X)))) 3.59/1.81 active(add(0, X)) -> mark(X) 3.59/1.81 active(add(s(X), Y)) -> mark(s(add(X, Y))) 3.59/1.81 active(len(nil)) -> mark(0) 3.59/1.81 active(len(cons(X, Z))) -> mark(s(len(Z))) 3.59/1.81 active(cons(X1, X2)) -> cons(active(X1), X2) 3.59/1.81 active(fst(X1, X2)) -> fst(active(X1), X2) 3.59/1.81 active(fst(X1, X2)) -> fst(X1, active(X2)) 3.59/1.81 active(from(X)) -> from(active(X)) 3.59/1.81 active(add(X1, X2)) -> add(active(X1), X2) 3.59/1.81 active(add(X1, X2)) -> add(X1, active(X2)) 3.59/1.81 active(len(X)) -> len(active(X)) 3.59/1.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.59/1.81 fst(mark(X1), X2) -> mark(fst(X1, X2)) 3.59/1.81 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 3.59/1.81 from(mark(X)) -> mark(from(X)) 3.59/1.81 add(mark(X1), X2) -> mark(add(X1, X2)) 3.59/1.81 add(X1, mark(X2)) -> mark(add(X1, X2)) 3.59/1.81 len(mark(X)) -> mark(len(X)) 3.59/1.81 proper(0) -> ok(0) 3.59/1.81 proper(s(X)) -> s(proper(X)) 3.59/1.81 proper(nil) -> ok(nil) 3.59/1.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.59/1.81 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 3.59/1.81 proper(from(X)) -> from(proper(X)) 3.59/1.81 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 3.59/1.81 proper(len(X)) -> len(proper(X)) 3.59/1.81 s(ok(X)) -> ok(s(X)) 3.59/1.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.59/1.81 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 3.59/1.81 from(ok(X)) -> ok(from(X)) 3.59/1.81 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 3.59/1.81 len(ok(X)) -> ok(len(X)) 3.59/1.81 top(mark(X)) -> top(proper(X)) 3.59/1.81 top(ok(X)) -> top(active(X)) 3.59/1.81 3.59/1.81 The set Q consists of the following terms: 3.59/1.81 3.59/1.81 active(from(x0)) 3.59/1.81 active(cons(x0, x1)) 3.59/1.81 active(fst(x0, x1)) 3.59/1.81 active(add(x0, x1)) 3.59/1.81 active(len(x0)) 3.59/1.81 cons(mark(x0), x1) 3.59/1.81 fst(mark(x0), x1) 3.59/1.81 fst(x0, mark(x1)) 3.59/1.81 from(mark(x0)) 3.59/1.81 add(mark(x0), x1) 3.59/1.81 add(x0, mark(x1)) 3.59/1.81 len(mark(x0)) 3.59/1.81 proper(0) 3.59/1.81 proper(s(x0)) 3.59/1.81 proper(nil) 3.59/1.81 proper(cons(x0, x1)) 3.59/1.81 proper(fst(x0, x1)) 3.59/1.81 proper(from(x0)) 3.59/1.81 proper(add(x0, x1)) 3.59/1.81 proper(len(x0)) 3.59/1.81 s(ok(x0)) 3.59/1.81 cons(ok(x0), ok(x1)) 3.59/1.81 fst(ok(x0), ok(x1)) 3.59/1.81 from(ok(x0)) 3.59/1.81 add(ok(x0), ok(x1)) 3.59/1.81 len(ok(x0)) 3.59/1.81 top(mark(x0)) 3.59/1.81 top(ok(x0)) 3.59/1.81 3.59/1.81 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (1) QTRSToCSRProof (SOUND) 3.59/1.81 The following Q TRS is given: Q restricted rewrite system: 3.59/1.81 The TRS R consists of the following rules: 3.59/1.81 3.59/1.81 active(fst(0, Z)) -> mark(nil) 3.59/1.81 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 3.59/1.81 active(from(X)) -> mark(cons(X, from(s(X)))) 3.59/1.81 active(add(0, X)) -> mark(X) 3.59/1.81 active(add(s(X), Y)) -> mark(s(add(X, Y))) 3.59/1.81 active(len(nil)) -> mark(0) 3.59/1.81 active(len(cons(X, Z))) -> mark(s(len(Z))) 3.59/1.81 active(cons(X1, X2)) -> cons(active(X1), X2) 3.59/1.81 active(fst(X1, X2)) -> fst(active(X1), X2) 3.59/1.81 active(fst(X1, X2)) -> fst(X1, active(X2)) 3.59/1.81 active(from(X)) -> from(active(X)) 3.59/1.81 active(add(X1, X2)) -> add(active(X1), X2) 3.59/1.81 active(add(X1, X2)) -> add(X1, active(X2)) 3.59/1.81 active(len(X)) -> len(active(X)) 3.59/1.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 3.59/1.81 fst(mark(X1), X2) -> mark(fst(X1, X2)) 3.59/1.81 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 3.59/1.81 from(mark(X)) -> mark(from(X)) 3.59/1.81 add(mark(X1), X2) -> mark(add(X1, X2)) 3.59/1.81 add(X1, mark(X2)) -> mark(add(X1, X2)) 3.59/1.81 len(mark(X)) -> mark(len(X)) 3.59/1.81 proper(0) -> ok(0) 3.59/1.81 proper(s(X)) -> s(proper(X)) 3.59/1.81 proper(nil) -> ok(nil) 3.59/1.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 3.59/1.81 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 3.59/1.81 proper(from(X)) -> from(proper(X)) 3.59/1.81 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 3.59/1.81 proper(len(X)) -> len(proper(X)) 3.59/1.81 s(ok(X)) -> ok(s(X)) 3.59/1.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 3.59/1.81 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 3.59/1.81 from(ok(X)) -> ok(from(X)) 3.59/1.81 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 3.59/1.81 len(ok(X)) -> ok(len(X)) 3.59/1.81 top(mark(X)) -> top(proper(X)) 3.59/1.81 top(ok(X)) -> top(active(X)) 3.59/1.81 3.59/1.81 The set Q consists of the following terms: 3.59/1.81 3.59/1.81 active(from(x0)) 3.59/1.81 active(cons(x0, x1)) 3.59/1.81 active(fst(x0, x1)) 3.59/1.81 active(add(x0, x1)) 3.59/1.81 active(len(x0)) 3.59/1.81 cons(mark(x0), x1) 3.59/1.81 fst(mark(x0), x1) 3.59/1.81 fst(x0, mark(x1)) 3.59/1.81 from(mark(x0)) 3.59/1.81 add(mark(x0), x1) 3.59/1.81 add(x0, mark(x1)) 3.59/1.81 len(mark(x0)) 3.59/1.81 proper(0) 3.59/1.81 proper(s(x0)) 3.59/1.81 proper(nil) 3.59/1.81 proper(cons(x0, x1)) 3.59/1.81 proper(fst(x0, x1)) 3.59/1.81 proper(from(x0)) 3.59/1.81 proper(add(x0, x1)) 3.59/1.81 proper(len(x0)) 3.59/1.81 s(ok(x0)) 3.59/1.81 cons(ok(x0), ok(x1)) 3.59/1.81 fst(ok(x0), ok(x1)) 3.59/1.81 from(ok(x0)) 3.59/1.81 add(ok(x0), ok(x1)) 3.59/1.81 len(ok(x0)) 3.59/1.81 top(mark(x0)) 3.59/1.81 top(ok(x0)) 3.59/1.81 3.59/1.81 Special symbols used for the transformation (see [GM04]): 3.59/1.81 top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 3.59/1.81 The replacement map contains the following entries: 3.59/1.81 3.59/1.81 fst: {1, 2} 3.59/1.81 0: empty set 3.59/1.81 nil: empty set 3.59/1.81 s: empty set 3.59/1.81 cons: {1} 3.59/1.81 from: {1} 3.59/1.81 add: {1, 2} 3.59/1.81 len: {1} 3.59/1.81 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. 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (2) 3.59/1.81 Obligation: 3.59/1.81 Context-sensitive rewrite system: 3.59/1.81 The TRS R consists of the following rules: 3.59/1.81 3.59/1.81 fst(0, Z) -> nil 3.59/1.81 fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) 3.59/1.81 from(X) -> cons(X, from(s(X))) 3.59/1.81 add(0, X) -> X 3.59/1.81 add(s(X), Y) -> s(add(X, Y)) 3.59/1.81 len(nil) -> 0 3.59/1.81 len(cons(X, Z)) -> s(len(Z)) 3.59/1.81 3.59/1.81 The replacement map contains the following entries: 3.59/1.81 3.59/1.81 fst: {1, 2} 3.59/1.81 0: empty set 3.59/1.81 nil: empty set 3.59/1.81 s: empty set 3.59/1.81 cons: {1} 3.59/1.81 from: {1} 3.59/1.81 add: {1, 2} 3.59/1.81 len: {1} 3.59/1.81 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (3) CSRRRRProof (EQUIVALENT) 3.59/1.81 The following CSR is given: Context-sensitive rewrite system: 3.59/1.81 The TRS R consists of the following rules: 3.59/1.81 3.59/1.81 fst(0, Z) -> nil 3.59/1.81 fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) 3.59/1.81 from(X) -> cons(X, from(s(X))) 3.59/1.81 add(0, X) -> X 3.59/1.81 add(s(X), Y) -> s(add(X, Y)) 3.59/1.81 len(nil) -> 0 3.59/1.81 len(cons(X, Z)) -> s(len(Z)) 3.59/1.81 3.59/1.81 The replacement map contains the following entries: 3.59/1.81 3.59/1.81 fst: {1, 2} 3.59/1.81 0: empty set 3.59/1.81 nil: empty set 3.59/1.81 s: empty set 3.59/1.81 cons: {1} 3.59/1.81 from: {1} 3.59/1.81 add: {1, 2} 3.59/1.81 len: {1} 3.59/1.81 Used ordering: 3.59/1.81 Polynomial interpretation [POLO]: 3.59/1.81 3.59/1.81 POL(0) = 2 3.59/1.81 POL(add(x_1, x_2)) = 1 + x_1 + 2*x_2 3.59/1.81 POL(cons(x_1, x_2)) = 1 + 2*x_1 3.59/1.81 POL(from(x_1)) = 2 + 2*x_1 3.59/1.81 POL(fst(x_1, x_2)) = 2*x_1 + 2*x_2 3.59/1.81 POL(len(x_1)) = 2 + x_1 3.59/1.81 POL(nil) = 2 3.59/1.81 POL(s(x_1)) = 0 3.59/1.81 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.59/1.81 3.59/1.81 fst(0, Z) -> nil 3.59/1.81 fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) 3.59/1.81 from(X) -> cons(X, from(s(X))) 3.59/1.81 add(0, X) -> X 3.59/1.81 add(s(X), Y) -> s(add(X, Y)) 3.59/1.81 len(nil) -> 0 3.59/1.81 len(cons(X, Z)) -> s(len(Z)) 3.59/1.81 3.59/1.81 3.59/1.81 3.59/1.81 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (4) 3.59/1.81 Obligation: 3.59/1.81 Context-sensitive rewrite system: 3.59/1.81 R is empty. 3.59/1.81 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (5) RisEmptyProof (EQUIVALENT) 3.59/1.81 The CSR R is empty. Hence, termination is trivially proven. 3.59/1.81 ---------------------------------------- 3.59/1.81 3.59/1.81 (6) 3.59/1.81 YES 3.70/1.85 EOF