15.71/4.95 WORST_CASE(Omega(n^1), O(n^1)) 15.71/4.96 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 15.71/4.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 15.71/4.96 15.71/4.96 15.71/4.96 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.71/4.96 15.71/4.96 (0) CpxTRS 15.71/4.96 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 15.71/4.96 (2) CpxWeightedTrs 15.71/4.96 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 15.71/4.96 (4) CpxTypedWeightedTrs 15.71/4.96 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 15.71/4.96 (6) CpxTypedWeightedCompleteTrs 15.71/4.96 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 15.71/4.96 (8) CpxRNTS 15.71/4.96 (9) CompleteCoflocoProof [FINISHED, 255 ms] 15.71/4.96 (10) BOUNDS(1, n^1) 15.71/4.96 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 15.71/4.96 (12) TRS for Loop Detection 15.71/4.96 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 15.71/4.96 (14) BEST 15.71/4.96 (15) proven lower bound 15.71/4.96 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 15.71/4.96 (17) BOUNDS(n^1, INF) 15.71/4.96 (18) TRS for Loop Detection 15.71/4.96 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (0) 15.71/4.96 Obligation: 15.71/4.96 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.71/4.96 15.71/4.96 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one 15.71/4.96 dx(a) -> zero 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 15.71/4.96 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 15.71/4.96 15.71/4.96 S is empty. 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 15.71/4.96 Transformed relative TRS to weighted TRS 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (2) 15.71/4.96 Obligation: 15.71/4.96 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 15.71/4.96 15.71/4.96 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one [1] 15.71/4.96 dx(a) -> zero [1] 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 15.71/4.96 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] 15.71/4.96 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 15.71/4.96 Infered types. 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (4) 15.71/4.96 Obligation: 15.71/4.96 Runtime Complexity Weighted TRS with Types. 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one [1] 15.71/4.96 dx(a) -> zero [1] 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 15.71/4.96 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] 15.71/4.96 15.71/4.96 The TRS has the following type information: 15.71/4.96 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 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 15.71/4.96 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 15.71/4.96 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 15.71/4.96 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 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 15.71/4.96 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 15.71/4.96 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (5) CompletionProof (UPPER BOUND(ID)) 15.71/4.96 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 15.71/4.96 none 15.71/4.96 15.71/4.96 And the following fresh constants: none 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (6) 15.71/4.96 Obligation: 15.71/4.96 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 15.71/4.96 15.71/4.96 Runtime Complexity Weighted TRS with Types. 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one [1] 15.71/4.96 dx(a) -> zero [1] 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 15.71/4.96 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] 15.71/4.96 15.71/4.96 The TRS has the following type information: 15.71/4.96 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 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 15.71/4.96 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 15.71/4.96 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 15.71/4.96 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 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 15.71/4.96 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 15.71/4.96 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 15.71/4.96 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 15.71/4.96 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 15.71/4.96 The constant constructors are abstracted as follows: 15.71/4.96 15.71/4.96 one => 1 15.71/4.96 a => 0 15.71/4.96 zero => 3 15.71/4.96 two => 2 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (8) 15.71/4.96 Obligation: 15.71/4.96 Complexity RNTS consisting of the following rules: 15.71/4.96 15.71/4.96 dx(z) -{ 1 }-> 3 :|: z = 0 15.71/4.96 dx(z) -{ 1 }-> 1 :|: X >= 0, z = X 15.71/4.96 dx(z) -{ 1 }-> 1 + dx(ALPHA) :|: ALPHA >= 0, z = 1 + ALPHA 15.71/4.96 dx(z) -{ 1 }-> 1 + dx(ALPHA) + ALPHA :|: ALPHA >= 0, z = 1 + ALPHA 15.71/4.96 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 15.71/4.96 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 15.71/4.96 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 15.71/4.96 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 15.71/4.96 15.71/4.96 Only complete derivations are relevant for the runtime complexity. 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (9) CompleteCoflocoProof (FINISHED) 15.71/4.96 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 15.71/4.96 15.71/4.96 eq(start(V),0,[dx(V, Out)],[V >= 0]). 15.71/4.96 eq(dx(V, Out),1,[],[Out = 1,X1 >= 0,V = X1]). 15.71/4.96 eq(dx(V, Out),1,[],[Out = 3,V = 0]). 15.71/4.96 eq(dx(V, Out),1,[dx(ALPHA1, Ret01),dx(BETA1, Ret1)],[Out = 1 + Ret01 + Ret1,ALPHA1 >= 0,BETA1 >= 0,V = 1 + ALPHA1 + BETA1]). 15.71/4.96 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]). 15.71/4.96 eq(dx(V, Out),1,[dx(ALPHA3, Ret12)],[Out = 1 + Ret12,ALPHA3 >= 0,V = 1 + ALPHA3]). 15.71/4.96 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]). 15.71/4.96 eq(dx(V, Out),1,[dx(ALPHA5, Ret012)],[Out = 1 + ALPHA5 + Ret012,ALPHA5 >= 0,V = 1 + ALPHA5]). 15.71/4.96 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]). 15.71/4.96 input_output_vars(dx(V,Out),[V],[Out]). 15.71/4.96 15.71/4.96 15.71/4.96 CoFloCo proof output: 15.71/4.96 Preprocessing Cost Relations 15.71/4.96 ===================================== 15.71/4.96 15.71/4.96 #### Computed strongly connected components 15.71/4.96 0. recursive [multiple] : [dx/2] 15.71/4.96 1. non_recursive : [start/1] 15.71/4.96 15.71/4.96 #### Obtained direct recursion through partial evaluation 15.71/4.96 0. SCC is partially evaluated into dx/2 15.71/4.96 1. SCC is partially evaluated into start/1 15.71/4.96 15.71/4.96 Control-Flow Refinement of Cost Relations 15.71/4.96 ===================================== 15.71/4.96 15.71/4.96 ### Specialization of cost equations dx/2 15.71/4.96 * CE 9 is refined into CE [10] 15.71/4.96 * CE 7 is refined into CE [11] 15.71/4.96 * CE 5 is refined into CE [12] 15.71/4.96 * CE 4 is refined into CE [13] 15.71/4.96 * CE 8 is refined into CE [14] 15.71/4.96 * CE 6 is refined into CE [15] 15.71/4.96 * CE 2 is refined into CE [16] 15.71/4.96 * CE 3 is refined into CE [17] 15.71/4.96 15.71/4.96 15.71/4.96 ### Cost equations --> "Loop" of dx/2 15.71/4.96 * CEs [16] --> Loop 10 15.71/4.96 * CEs [17] --> Loop 11 15.71/4.96 * CEs [14] --> Loop 12 15.71/4.96 * CEs [15] --> Loop 13 15.71/4.96 * CEs [10] --> Loop 14 15.71/4.96 * CEs [11] --> Loop 15 15.71/4.96 * CEs [12] --> Loop 16 15.71/4.96 * CEs [13] --> Loop 17 15.71/4.96 15.71/4.96 ### Ranking functions of CR dx(V,Out) 15.71/4.96 * RF of phase [12,13,14,15,16,17]: [V] 15.71/4.96 15.71/4.96 #### Partial ranking functions of CR dx(V,Out) 15.71/4.96 * Partial RF of phase [12,13,14,15,16,17]: 15.71/4.96 - RF of loop [12:1,13:1,14:1,14:2,15:1,15:2,16:1,16:2,17:1,17:2]: 15.71/4.96 V 15.71/4.96 15.71/4.96 15.71/4.96 ### Specialization of cost equations start/1 15.71/4.96 * CE 1 is refined into CE [18,19,20] 15.71/4.96 15.71/4.96 15.71/4.96 ### Cost equations --> "Loop" of start/1 15.71/4.96 * CEs [18,19,20] --> Loop 18 15.71/4.96 15.71/4.96 ### Ranking functions of CR start(V) 15.71/4.96 15.71/4.96 #### Partial ranking functions of CR start(V) 15.71/4.96 15.71/4.96 15.71/4.96 Computing Bounds 15.71/4.96 ===================================== 15.71/4.96 15.71/4.96 #### Cost of chains of dx(V,Out): 15.71/4.96 * Chain [11]: 1 15.71/4.96 with precondition: [V=0,Out=3] 15.71/4.96 15.71/4.96 * Chain [10]: 1 15.71/4.96 with precondition: [Out=1,V>=0] 15.71/4.96 15.71/4.96 * Chain [multiple([12,13,14,15,16,17],[[11],[10]])]: 6*it(12)+2*it([10])+0 15.71/4.96 Such that:aux(5) =< 1 15.71/4.96 aux(6) =< V 15.71/4.96 it(12) =< aux(6) 15.71/4.96 it([10]) =< it(12)+it(12)+it(12)+it(12)+aux(5) 15.71/4.96 15.71/4.96 with precondition: [V>=1,Out>=2] 15.71/4.96 15.71/4.96 15.71/4.96 #### Cost of chains of start(V): 15.71/4.96 * Chain [18]: 6*s(3)+2*s(4)+1 15.71/4.96 Such that:s(1) =< 1 15.71/4.96 s(2) =< V 15.71/4.96 s(3) =< s(2) 15.71/4.96 s(4) =< s(3)+s(3)+s(3)+s(3)+s(1) 15.71/4.96 15.71/4.96 with precondition: [V>=0] 15.71/4.96 15.71/4.96 15.71/4.96 Closed-form bounds of start(V): 15.71/4.96 ------------------------------------- 15.71/4.96 * Chain [18] with precondition: [V>=0] 15.71/4.96 - Upper bound: 14*V+3 15.71/4.96 - Complexity: n 15.71/4.96 15.71/4.96 ### Maximum cost of start(V): 14*V+3 15.71/4.96 Asymptotic class: n 15.71/4.96 * Total analysis performed in 161 ms. 15.71/4.96 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (10) 15.71/4.96 BOUNDS(1, n^1) 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 15.71/4.96 Transformed a relative TRS into a decreasing-loop problem. 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (12) 15.71/4.96 Obligation: 15.71/4.96 Analyzing the following TRS for decreasing loops: 15.71/4.96 15.71/4.96 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.71/4.96 15.71/4.96 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one 15.71/4.96 dx(a) -> zero 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 15.71/4.96 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 15.71/4.96 15.71/4.96 S is empty. 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (13) DecreasingLoopProof (LOWER BOUND(ID)) 15.71/4.96 The following loop(s) give(s) rise to the lower bound Omega(n^1): 15.71/4.96 15.71/4.96 The rewrite sequence 15.71/4.96 15.71/4.96 dx(exp(ALPHA, BETA)) ->^+ plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 15.71/4.96 15.71/4.96 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,1]. 15.71/4.96 15.71/4.96 The pumping substitution is [ALPHA / exp(ALPHA, BETA)]. 15.71/4.96 15.71/4.96 The result substitution is [ ]. 15.71/4.96 15.71/4.96 15.71/4.96 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (14) 15.71/4.96 Complex Obligation (BEST) 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (15) 15.71/4.96 Obligation: 15.71/4.96 Proved the lower bound n^1 for the following obligation: 15.71/4.96 15.71/4.96 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.71/4.96 15.71/4.96 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one 15.71/4.96 dx(a) -> zero 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 15.71/4.96 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 15.71/4.96 15.71/4.96 S is empty. 15.71/4.96 Rewrite Strategy: INNERMOST 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (16) LowerBoundPropagationProof (FINISHED) 15.71/4.96 Propagated lower bound. 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (17) 15.71/4.96 BOUNDS(n^1, INF) 15.71/4.96 15.71/4.96 ---------------------------------------- 15.71/4.96 15.71/4.96 (18) 15.71/4.96 Obligation: 15.71/4.96 Analyzing the following TRS for decreasing loops: 15.71/4.96 15.71/4.96 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.71/4.96 15.71/4.96 15.71/4.96 The TRS R consists of the following rules: 15.71/4.96 15.71/4.96 dx(X) -> one 15.71/4.96 dx(a) -> zero 15.71/4.96 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 15.71/4.96 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 15.71/4.96 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 15.71/4.96 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 15.71/4.96 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 15.71/4.96 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 15.71/4.96 15.71/4.96 S is empty. 15.71/4.96 Rewrite Strategy: INNERMOST 16.07/5.05 EOF