306.94/291.54 WORST_CASE(Omega(n^2), ?) 306.95/291.55 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 306.95/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.95/291.55 306.95/291.55 306.95/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 306.95/291.55 306.95/291.55 (0) CpxTRS 306.95/291.55 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 306.95/291.55 (2) CpxTRS 306.95/291.55 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 306.95/291.55 (4) typed CpxTrs 306.95/291.55 (5) OrderProof [LOWER BOUND(ID), 0 ms] 306.95/291.55 (6) typed CpxTrs 306.95/291.55 (7) RewriteLemmaProof [LOWER BOUND(ID), 219 ms] 306.95/291.55 (8) BEST 306.95/291.55 (9) proven lower bound 306.95/291.55 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 306.95/291.55 (11) BOUNDS(n^1, INF) 306.95/291.55 (12) typed CpxTrs 306.95/291.55 (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms] 306.95/291.55 (14) proven lower bound 306.95/291.55 (15) LowerBoundPropagationProof [FINISHED, 0 ms] 306.95/291.55 (16) BOUNDS(n^2, INF) 306.95/291.55 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (0) 306.95/291.55 Obligation: 306.95/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 306.95/291.55 306.95/291.55 306.95/291.55 The TRS R consists of the following rules: 306.95/291.55 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0) -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0) -> 0 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 S is empty. 306.95/291.55 Rewrite Strategy: FULL 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 306.95/291.55 Renamed function symbols to avoid clashes with predefined symbol. 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (2) 306.95/291.55 Obligation: 306.95/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 306.95/291.55 306.95/291.55 306.95/291.55 The TRS R consists of the following rules: 306.95/291.55 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 S is empty. 306.95/291.55 Rewrite Strategy: FULL 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 306.95/291.55 Infered types. 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (4) 306.95/291.55 Obligation: 306.95/291.55 TRS: 306.95/291.55 Rules: 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 Types: 306.95/291.55 U11 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 tt :: tt 306.95/291.55 U12 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 activate :: s:0' -> s:0' 306.95/291.55 s :: s:0' -> s:0' 306.95/291.55 plus :: s:0' -> s:0' -> s:0' 306.95/291.55 U21 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 U22 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 x :: s:0' -> s:0' -> s:0' 306.95/291.55 0' :: s:0' 306.95/291.55 hole_s:0'1_0 :: s:0' 306.95/291.55 hole_tt2_0 :: tt 306.95/291.55 gen_s:0'3_0 :: Nat -> s:0' 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (5) OrderProof (LOWER BOUND(ID)) 306.95/291.55 Heuristically decided to analyse the following defined symbols: 306.95/291.55 plus, x 306.95/291.55 306.95/291.55 They will be analysed ascendingly in the following order: 306.95/291.55 plus < x 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (6) 306.95/291.55 Obligation: 306.95/291.55 TRS: 306.95/291.55 Rules: 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 Types: 306.95/291.55 U11 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 tt :: tt 306.95/291.55 U12 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 activate :: s:0' -> s:0' 306.95/291.55 s :: s:0' -> s:0' 306.95/291.55 plus :: s:0' -> s:0' -> s:0' 306.95/291.55 U21 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 U22 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 x :: s:0' -> s:0' -> s:0' 306.95/291.55 0' :: s:0' 306.95/291.55 hole_s:0'1_0 :: s:0' 306.95/291.55 hole_tt2_0 :: tt 306.95/291.55 gen_s:0'3_0 :: Nat -> s:0' 306.95/291.55 306.95/291.55 306.95/291.55 Generator Equations: 306.95/291.55 gen_s:0'3_0(0) <=> 0' 306.95/291.55 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 306.95/291.55 306.95/291.55 306.95/291.55 The following defined symbols remain to be analysed: 306.95/291.55 plus, x 306.95/291.55 306.95/291.55 They will be analysed ascendingly in the following order: 306.95/291.55 plus < x 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (7) RewriteLemmaProof (LOWER BOUND(ID)) 306.95/291.55 Proved the following rewrite lemma: 306.95/291.55 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 306.95/291.55 306.95/291.55 Induction Base: 306.95/291.55 plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1) 306.95/291.55 gen_s:0'3_0(a) 306.95/291.55 306.95/291.55 Induction Step: 306.95/291.55 plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1) 306.95/291.55 U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1) 306.95/291.55 U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1) 306.95/291.55 s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1) 306.95/291.55 s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1) 306.95/291.55 s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) ->_IH 306.95/291.55 s(gen_s:0'3_0(+(a, c6_0))) 306.95/291.55 306.95/291.55 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (8) 306.95/291.55 Complex Obligation (BEST) 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (9) 306.95/291.55 Obligation: 306.95/291.55 Proved the lower bound n^1 for the following obligation: 306.95/291.55 306.95/291.55 TRS: 306.95/291.55 Rules: 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 Types: 306.95/291.55 U11 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 tt :: tt 306.95/291.55 U12 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 activate :: s:0' -> s:0' 306.95/291.55 s :: s:0' -> s:0' 306.95/291.55 plus :: s:0' -> s:0' -> s:0' 306.95/291.55 U21 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 U22 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 x :: s:0' -> s:0' -> s:0' 306.95/291.55 0' :: s:0' 306.95/291.55 hole_s:0'1_0 :: s:0' 306.95/291.55 hole_tt2_0 :: tt 306.95/291.55 gen_s:0'3_0 :: Nat -> s:0' 306.95/291.55 306.95/291.55 306.95/291.55 Generator Equations: 306.95/291.55 gen_s:0'3_0(0) <=> 0' 306.95/291.55 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 306.95/291.55 306.95/291.55 306.95/291.55 The following defined symbols remain to be analysed: 306.95/291.55 plus, x 306.95/291.55 306.95/291.55 They will be analysed ascendingly in the following order: 306.95/291.55 plus < x 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (10) LowerBoundPropagationProof (FINISHED) 306.95/291.55 Propagated lower bound. 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (11) 306.95/291.55 BOUNDS(n^1, INF) 306.95/291.55 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (12) 306.95/291.55 Obligation: 306.95/291.55 TRS: 306.95/291.55 Rules: 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 Types: 306.95/291.55 U11 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 tt :: tt 306.95/291.55 U12 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 activate :: s:0' -> s:0' 306.95/291.55 s :: s:0' -> s:0' 306.95/291.55 plus :: s:0' -> s:0' -> s:0' 306.95/291.55 U21 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 U22 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 x :: s:0' -> s:0' -> s:0' 306.95/291.55 0' :: s:0' 306.95/291.55 hole_s:0'1_0 :: s:0' 306.95/291.55 hole_tt2_0 :: tt 306.95/291.55 gen_s:0'3_0 :: Nat -> s:0' 306.95/291.55 306.95/291.55 306.95/291.55 Lemmas: 306.95/291.55 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 306.95/291.55 306.95/291.55 306.95/291.55 Generator Equations: 306.95/291.55 gen_s:0'3_0(0) <=> 0' 306.95/291.55 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 306.95/291.55 306.95/291.55 306.95/291.55 The following defined symbols remain to be analysed: 306.95/291.55 x 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (13) RewriteLemmaProof (LOWER BOUND(ID)) 306.95/291.55 Proved the following rewrite lemma: 306.95/291.55 x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) -> gen_s:0'3_0(*(n580_0, a)), rt in Omega(1 + a*n580_0 + n580_0) 306.95/291.55 306.95/291.55 Induction Base: 306.95/291.55 x(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1) 306.95/291.55 0' 306.95/291.55 306.95/291.55 Induction Step: 306.95/291.55 x(gen_s:0'3_0(a), gen_s:0'3_0(+(n580_0, 1))) ->_R^Omega(1) 306.95/291.55 U21(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) ->_R^Omega(1) 306.95/291.55 U22(tt, activate(gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 U22(tt, gen_s:0'3_0(n580_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 U22(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) ->_R^Omega(1) 306.95/291.55 plus(x(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 plus(x(gen_s:0'3_0(a), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 plus(x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) ->_IH 306.95/291.55 plus(gen_s:0'3_0(*(c581_0, a)), activate(gen_s:0'3_0(a))) ->_R^Omega(1) 306.95/291.55 plus(gen_s:0'3_0(*(n580_0, a)), gen_s:0'3_0(a)) ->_L^Omega(1 + a) 306.95/291.55 gen_s:0'3_0(+(a, *(n580_0, a))) 306.95/291.55 306.95/291.55 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (14) 306.95/291.55 Obligation: 306.95/291.55 Proved the lower bound n^2 for the following obligation: 306.95/291.55 306.95/291.55 TRS: 306.95/291.55 Rules: 306.95/291.55 U11(tt, M, N) -> U12(tt, activate(M), activate(N)) 306.95/291.55 U12(tt, M, N) -> s(plus(activate(N), activate(M))) 306.95/291.55 U21(tt, M, N) -> U22(tt, activate(M), activate(N)) 306.95/291.55 U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) 306.95/291.55 plus(N, 0') -> N 306.95/291.55 plus(N, s(M)) -> U11(tt, M, N) 306.95/291.55 x(N, 0') -> 0' 306.95/291.55 x(N, s(M)) -> U21(tt, M, N) 306.95/291.55 activate(X) -> X 306.95/291.55 306.95/291.55 Types: 306.95/291.55 U11 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 tt :: tt 306.95/291.55 U12 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 activate :: s:0' -> s:0' 306.95/291.55 s :: s:0' -> s:0' 306.95/291.55 plus :: s:0' -> s:0' -> s:0' 306.95/291.55 U21 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 U22 :: tt -> s:0' -> s:0' -> s:0' 306.95/291.55 x :: s:0' -> s:0' -> s:0' 306.95/291.55 0' :: s:0' 306.95/291.55 hole_s:0'1_0 :: s:0' 306.95/291.55 hole_tt2_0 :: tt 306.95/291.55 gen_s:0'3_0 :: Nat -> s:0' 306.95/291.55 306.95/291.55 306.95/291.55 Lemmas: 306.95/291.55 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 306.95/291.55 306.95/291.55 306.95/291.55 Generator Equations: 306.95/291.55 gen_s:0'3_0(0) <=> 0' 306.95/291.55 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 306.95/291.55 306.95/291.55 306.95/291.55 The following defined symbols remain to be analysed: 306.95/291.55 x 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (15) LowerBoundPropagationProof (FINISHED) 306.95/291.55 Propagated lower bound. 306.95/291.55 ---------------------------------------- 306.95/291.55 306.95/291.55 (16) 306.95/291.55 BOUNDS(n^2, INF) 306.95/291.58 EOF