/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (8) CpxRNTS (9) CompleteCoflocoProof [FINISHED, 221 ms] (10) BOUNDS(1, n^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), 1198 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: dx(X) -> one dx(a) -> zero dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: dx(X) -> one [1] dx(a) -> zero [1] dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: dx(X) -> one [1] dx(a) -> zero [1] dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] The TRS has the following type information: dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln Rewrite Strategy: INNERMOST ---------------------------------------- (5) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: dx(X) -> one [1] dx(a) -> zero [1] dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] The TRS has the following type information: dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: one => 1 a => 0 zero => 3 two => 2 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: dx(z) -{ 1 }-> 3 :|: z = 0 dx(z) -{ 1 }-> 1 :|: X >= 0, z = X dx(z) -{ 1 }-> 1 + dx(ALPHA) :|: ALPHA >= 0, z = 1 + ALPHA dx(z) -{ 1 }-> 1 + dx(ALPHA) + ALPHA :|: ALPHA >= 0, z = 1 + ALPHA dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (9) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V),0,[dx(V, Out)],[V >= 0]). eq(dx(V, Out),1,[],[Out = 1,X1 >= 0,V = X1]). eq(dx(V, Out),1,[],[Out = 3,V = 0]). eq(dx(V, Out),1,[dx(ALPHA1, Ret01),dx(BETA1, Ret1)],[Out = 1 + Ret01 + Ret1,ALPHA1 >= 0,BETA1 >= 0,V = 1 + ALPHA1 + BETA1]). eq(dx(V, Out),1,[dx(ALPHA2, Ret011),dx(BETA2, Ret11)],[Out = 3 + ALPHA2 + BETA2 + Ret011 + Ret11,ALPHA2 >= 0,BETA2 >= 0,V = 1 + ALPHA2 + BETA2]). eq(dx(V, Out),1,[dx(ALPHA3, Ret12)],[Out = 1 + Ret12,ALPHA3 >= 0,V = 1 + ALPHA3]). eq(dx(V, Out),1,[dx(ALPHA4, Ret0101),dx(BETA3, Ret1101)],[Out = 7 + ALPHA4 + 2*BETA3 + Ret0101 + Ret1101,ALPHA4 >= 0,BETA3 >= 0,V = 1 + ALPHA4 + BETA3]). eq(dx(V, Out),1,[dx(ALPHA5, Ret012)],[Out = 1 + ALPHA5 + Ret012,ALPHA5 >= 0,V = 1 + ALPHA5]). eq(dx(V, Out),1,[dx(ALPHA6, Ret0111),dx(BETA4, Ret111)],[Out = 10 + 3*ALPHA6 + 3*BETA4 + Ret0111 + Ret111,ALPHA6 >= 0,BETA4 >= 0,V = 1 + ALPHA6 + BETA4]). input_output_vars(dx(V,Out),[V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive [multiple] : [dx/2] 1. non_recursive : [start/1] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into dx/2 1. SCC is partially evaluated into start/1 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations dx/2 * CE 9 is refined into CE [10] * CE 7 is refined into CE [11] * CE 5 is refined into CE [12] * CE 4 is refined into CE [13] * CE 8 is refined into CE [14] * CE 6 is refined into CE [15] * CE 2 is refined into CE [16] * CE 3 is refined into CE [17] ### Cost equations --> "Loop" of dx/2 * CEs [16] --> Loop 10 * CEs [17] --> Loop 11 * CEs [14] --> Loop 12 * CEs [15] --> Loop 13 * CEs [10] --> Loop 14 * CEs [11] --> Loop 15 * CEs [12] --> Loop 16 * CEs [13] --> Loop 17 ### Ranking functions of CR dx(V,Out) * RF of phase [12,13,14,15,16,17]: [V] #### Partial ranking functions of CR dx(V,Out) * Partial RF of phase [12,13,14,15,16,17]: - RF of loop [12:1,13:1,14:1,14:2,15:1,15:2,16:1,16:2,17:1,17:2]: V ### Specialization of cost equations start/1 * CE 1 is refined into CE [18,19,20] ### Cost equations --> "Loop" of start/1 * CEs [18,19,20] --> Loop 18 ### Ranking functions of CR start(V) #### Partial ranking functions of CR start(V) Computing Bounds ===================================== #### Cost of chains of dx(V,Out): * Chain [11]: 1 with precondition: [V=0,Out=3] * Chain [10]: 1 with precondition: [Out=1,V>=0] * Chain [multiple([12,13,14,15,16,17],[[11],[10]])]: 6*it(12)+2*it([10])+0 Such that:aux(5) =< 1 aux(6) =< V it(12) =< aux(6) it([10]) =< it(12)+it(12)+it(12)+it(12)+aux(5) with precondition: [V>=1,Out>=2] #### Cost of chains of start(V): * Chain [18]: 6*s(3)+2*s(4)+1 Such that:s(1) =< 1 s(2) =< V s(3) =< s(2) s(4) =< s(3)+s(3)+s(3)+s(3)+s(1) with precondition: [V>=0] Closed-form bounds of start(V): ------------------------------------- * Chain [18] with precondition: [V>=0] - Upper bound: 14*V+3 - Complexity: n ### Maximum cost of start(V): 14*V+3 Asymptotic class: n * Total analysis performed in 161 ms. ---------------------------------------- (10) BOUNDS(1, n^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: dx(X) -> one dx(a) -> zero dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: dx(X) -> one dx(a) -> zero dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) Types: dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln ---------------------------------------- (15) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: dx ---------------------------------------- (16) Obligation: Innermost TRS: Rules: dx(X) -> one dx(a) -> zero dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) Types: dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln Generator Equations: gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0) <=> a gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(x, 1)) <=> plus(a, gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(x)) The following defined symbols remain to be analysed: dx ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0)) -> *3_0, rt in Omega(n4_0) Induction Base: dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0)) Induction Step: dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(n4_0, 1))) ->_R^Omega(1) plus(dx(a), dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) ->_R^Omega(1) plus(one, dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) ->_IH plus(one, *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: dx(X) -> one dx(a) -> zero dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) Types: dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln Generator Equations: gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0) <=> a gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(x, 1)) <=> plus(a, gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(x)) The following defined symbols remain to be analysed: dx ---------------------------------------- (19) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (20) BOUNDS(n^1, INF)