WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (2) CdtProblem (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CdtProblem (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms] (8) CdtProblem (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (10) BOUNDS(1, 1) (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTRS (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (14) typed CpxTrs (15) OrderProof [LOWER BOUND(ID), 0 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 1585 ms] (18) proven lower bound (19) LowerBoundPropagationProof [FINISHED, 0 ms] (20) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (2) Obligation: Complexity Dependency Tuples Problem Rules: D(t) -> 1 D(constant) -> 0 D(+(z0, z1)) -> +(D(z0), D(z1)) D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) D(-(z0, z1)) -> -(D(z0), D(z1)) D(minus(z0)) -> minus(D(z0)) D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2))) D(ln(z0)) -> div(D(z0), z0) D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1))) Tuples: D'(t) -> c D'(constant) -> c1 D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) S tuples: D'(t) -> c D'(constant) -> c1 D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) K tuples:none Defined Rule Symbols: D_1 Defined Pair Symbols: D'_1 Compound Symbols: c, c1, c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2 ---------------------------------------- (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: D'(t) -> c D'(constant) -> c1 ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: D(t) -> 1 D(constant) -> 0 D(+(z0, z1)) -> +(D(z0), D(z1)) D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) D(-(z0, z1)) -> -(D(z0), D(z1)) D(minus(z0)) -> minus(D(z0)) D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2))) D(ln(z0)) -> div(D(z0), z0) D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1))) Tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) S tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) K tuples:none Defined Rule Symbols: D_1 Defined Pair Symbols: D'_1 Compound Symbols: c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2 ---------------------------------------- (5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: D(t) -> 1 D(constant) -> 0 D(+(z0, z1)) -> +(D(z0), D(z1)) D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) D(-(z0, z1)) -> -(D(z0), D(z1)) D(minus(z0)) -> minus(D(z0)) D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2))) D(ln(z0)) -> div(D(z0), z0) D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1))) ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) S tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: D'_1 Compound Symbols: c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2 ---------------------------------------- (7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) We considered the (Usable) Rules:none And the Tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 + x_2 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(-(x_1, x_2)) = [1] + x_1 + x_2 POL(D'(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 + x_2 POL(ln(x_1)) = [1] + x_1 POL(minus(x_1)) = [1] + x_1 POL(pow(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) S tuples:none K tuples: D'(+(z0, z1)) -> c2(D'(z0), D'(z1)) D'(*(z0, z1)) -> c3(D'(z0), D'(z1)) D'(-(z0, z1)) -> c4(D'(z0), D'(z1)) D'(minus(z0)) -> c5(D'(z0)) D'(div(z0, z1)) -> c6(D'(z0), D'(z1)) D'(ln(z0)) -> c7(D'(z0)) D'(pow(z0, z1)) -> c8(D'(z0), D'(z1)) Defined Rule Symbols:none Defined Pair Symbols: D'_1 Compound Symbols: c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2 ---------------------------------------- (9) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (10) BOUNDS(1, 1) ---------------------------------------- (11) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (12) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2'))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2'))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y))) Types: D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln +' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln *' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln - :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln 2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln ---------------------------------------- (15) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: D ---------------------------------------- (16) Obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2'))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y))) Types: D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln +' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln *' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln - :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln 2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln Generator Equations: gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) <=> t gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x)) The following defined symbols remain to be analysed: D ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) -> *3_0, rt in Omega(n4_0) Induction Base: D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0)) Induction Step: D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(n4_0, 1))) ->_R^Omega(1) +'(D(t), D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) ->_R^Omega(1) +'(1', D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) ->_IH +'(1', *3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2'))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y))) Types: D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln 0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln +' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln *' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln - :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln 2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln Generator Equations: gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) <=> t gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x)) The following defined symbols remain to be analysed: D ---------------------------------------- (19) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (20) BOUNDS(n^1, INF)