22.48/7.28 WORST_CASE(Omega(n^1), O(n^1)) 22.48/7.29 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.48/7.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.48/7.29 22.48/7.29 22.48/7.29 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.48/7.29 22.48/7.29 (0) CpxTRS 22.48/7.29 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 22.48/7.29 (2) CpxWeightedTrs 22.48/7.29 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 22.48/7.29 (4) CpxTypedWeightedTrs 22.48/7.29 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 22.48/7.29 (6) CpxTypedWeightedCompleteTrs 22.48/7.29 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 22.48/7.29 (8) CpxRNTS 22.48/7.29 (9) CompleteCoflocoProof [FINISHED, 248 ms] 22.48/7.29 (10) BOUNDS(1, n^1) 22.48/7.29 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 22.48/7.29 (12) TRS for Loop Detection 22.48/7.29 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 22.48/7.29 (14) BEST 22.48/7.29 (15) proven lower bound 22.48/7.29 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 22.48/7.29 (17) BOUNDS(n^1, INF) 22.48/7.29 (18) TRS for Loop Detection 22.48/7.29 22.48/7.29 22.48/7.29 ---------------------------------------- 22.48/7.29 22.48/7.29 (0) 22.48/7.29 Obligation: 22.48/7.29 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.48/7.29 22.48/7.29 22.48/7.29 The TRS R consists of the following rules: 22.48/7.29 22.48/7.29 D(t) -> 1 22.48/7.29 D(constant) -> 0 22.48/7.29 D(+(x, y)) -> +(D(x), D(y)) 22.48/7.29 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 22.48/7.29 D(-(x, y)) -> -(D(x), D(y)) 22.48/7.29 D(minus(x)) -> minus(D(x)) 22.48/7.29 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) 22.48/7.29 D(ln(x)) -> div(D(x), x) 22.48/7.29 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) 22.48/7.29 22.48/7.29 S is empty. 22.48/7.29 Rewrite Strategy: INNERMOST 22.48/7.29 ---------------------------------------- 22.48/7.29 22.48/7.29 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 22.48/7.29 Transformed relative TRS to weighted TRS 22.48/7.29 ---------------------------------------- 22.48/7.29 22.48/7.29 (2) 22.48/7.29 Obligation: 22.48/7.29 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 22.48/7.29 22.48/7.29 22.48/7.29 The TRS R consists of the following rules: 22.48/7.29 22.48/7.29 D(t) -> 1 [1] 22.48/7.29 D(constant) -> 0 [1] 22.48/7.29 D(+(x, y)) -> +(D(x), D(y)) [1] 22.48/7.29 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] 22.48/7.29 D(-(x, y)) -> -(D(x), D(y)) [1] 22.48/7.29 D(minus(x)) -> minus(D(x)) [1] 22.48/7.29 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1] 22.48/7.29 D(ln(x)) -> div(D(x), x) [1] 22.48/7.29 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1] 22.48/7.29 22.48/7.29 Rewrite Strategy: INNERMOST 22.48/7.29 ---------------------------------------- 22.48/7.29 22.48/7.29 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 22.48/7.29 Infered types. 22.48/7.29 ---------------------------------------- 22.48/7.29 22.48/7.29 (4) 22.48/7.29 Obligation: 22.48/7.29 Runtime Complexity Weighted TRS with Types. 22.48/7.29 The TRS R consists of the following rules: 22.48/7.29 22.48/7.29 D(t) -> 1 [1] 22.48/7.29 D(constant) -> 0 [1] 22.48/7.29 D(+(x, y)) -> +(D(x), D(y)) [1] 22.48/7.29 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] 22.48/7.29 D(-(x, y)) -> -(D(x), D(y)) [1] 22.48/7.29 D(minus(x)) -> minus(D(x)) [1] 22.48/7.29 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1] 22.48/7.29 D(ln(x)) -> div(D(x), x) [1] 22.48/7.29 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1] 22.48/7.29 22.48/7.29 The TRS has the following type information: 22.48/7.29 D :: t:1:constant:0:+:*:-:minus:div:2:pow:ln -> t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.29 t :: t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 1 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 constant :: t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 0 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 + :: 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 22.48/7.30 * :: 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 22.48/7.30 - :: 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 22.48/7.30 minus :: t:1:constant:0:+:*:-:minus:div:2:pow:ln -> t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 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 22.48/7.30 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 22.48/7.30 2 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 ln :: t:1:constant:0:+:*:-:minus:div:2:pow:ln -> t:1:constant:0:+:*:-:minus:div:2:pow:ln 22.48/7.30 22.48/7.30 Rewrite Strategy: INNERMOST 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (5) CompletionProof (UPPER BOUND(ID)) 22.48/7.30 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 22.48/7.30 22.48/7.30 D(v0) -> null_D [0] 22.48/7.30 22.48/7.30 And the following fresh constants: null_D 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (6) 22.48/7.30 Obligation: 22.48/7.30 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 22.48/7.30 22.48/7.30 Runtime Complexity Weighted TRS with Types. 22.48/7.30 The TRS R consists of the following rules: 22.48/7.30 22.48/7.30 D(t) -> 1 [1] 22.48/7.30 D(constant) -> 0 [1] 22.48/7.30 D(+(x, y)) -> +(D(x), D(y)) [1] 22.48/7.30 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] 22.48/7.30 D(-(x, y)) -> -(D(x), D(y)) [1] 22.48/7.30 D(minus(x)) -> minus(D(x)) [1] 22.48/7.30 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1] 22.48/7.30 D(ln(x)) -> div(D(x), x) [1] 22.48/7.30 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1] 22.48/7.30 D(v0) -> null_D [0] 22.48/7.30 22.48/7.30 The TRS has the following type information: 22.48/7.30 D :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 t :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 1 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 constant :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 0 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 + :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 * :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 - :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 minus :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 div :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 pow :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 2 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 ln :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D -> t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 null_D :: t:1:constant:0:+:*:-:minus:div:2:pow:ln:null_D 22.48/7.30 22.48/7.30 Rewrite Strategy: INNERMOST 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 22.48/7.30 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 22.48/7.30 The constant constructors are abstracted as follows: 22.48/7.30 22.48/7.30 t => 4 22.48/7.30 1 => 1 22.48/7.30 constant => 3 22.48/7.30 0 => 0 22.48/7.30 2 => 2 22.48/7.30 null_D => 0 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (8) 22.48/7.30 Obligation: 22.48/7.30 Complexity RNTS consisting of the following rules: 22.48/7.30 22.48/7.30 D(z) -{ 1 }-> 1 :|: z = 4 22.48/7.30 D(z) -{ 1 }-> 0 :|: z = 3 22.48/7.30 D(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 22.48/7.30 D(z) -{ 1 }-> 1 + D(x) :|: x >= 0, z = 1 + x 22.48/7.30 D(z) -{ 1 }-> 1 + D(x) + x :|: x >= 0, z = 1 + x 22.48/7.30 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 22.48/7.30 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 22.48/7.30 D(z) -{ 1 }-> 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0 22.48/7.30 D(z) -{ 1 }-> 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (1 + x)) + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 22.48/7.30 22.48/7.30 Only complete derivations are relevant for the runtime complexity. 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (9) CompleteCoflocoProof (FINISHED) 22.48/7.30 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 22.48/7.30 22.48/7.30 eq(start(V),0,[fun(V, Out)],[V >= 0]). 22.48/7.30 eq(fun(V, Out),1,[],[Out = 1,V = 4]). 22.48/7.30 eq(fun(V, Out),1,[],[Out = 0,V = 3]). 22.48/7.30 eq(fun(V, Out),1,[fun(V2, Ret01),fun(V1, Ret1)],[Out = 1 + Ret01 + Ret1,V = 1 + V1 + V2,V2 >= 0,V1 >= 0]). 22.48/7.30 eq(fun(V, Out),1,[fun(V3, Ret011),fun(V4, Ret11)],[Out = 3 + Ret011 + Ret11 + V3 + V4,V = 1 + V3 + V4,V3 >= 0,V4 >= 0]). 22.48/7.30 eq(fun(V, Out),1,[fun(V5, Ret12)],[Out = 1 + Ret12,V5 >= 0,V = 1 + V5]). 22.48/7.30 eq(fun(V, Out),1,[fun(V6, Ret0101),fun(V7, Ret1011)],[Out = 7 + Ret0101 + Ret1011 + V6 + 2*V7,V = 1 + V6 + V7,V6 >= 0,V7 >= 0]). 22.48/7.30 eq(fun(V, Out),1,[fun(V8, Ret012)],[Out = 1 + Ret012 + V8,V8 >= 0,V = 1 + V8]). 22.48/7.30 eq(fun(V, Out),1,[fun(V10, Ret0111),fun(V9, Ret111)],[Out = 10 + Ret0111 + Ret111 + 3*V10 + 3*V9,V = 1 + V10 + V9,V10 >= 0,V9 >= 0]). 22.48/7.30 eq(fun(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]). 22.48/7.30 input_output_vars(fun(V,Out),[V],[Out]). 22.48/7.30 22.48/7.30 22.48/7.30 CoFloCo proof output: 22.48/7.30 Preprocessing Cost Relations 22.48/7.30 ===================================== 22.48/7.30 22.48/7.30 #### Computed strongly connected components 22.48/7.30 0. recursive [multiple] : [fun/2] 22.48/7.30 1. non_recursive : [start/1] 22.48/7.30 22.48/7.30 #### Obtained direct recursion through partial evaluation 22.48/7.30 0. SCC is partially evaluated into fun/2 22.48/7.30 1. SCC is partially evaluated into start/1 22.48/7.30 22.48/7.30 Control-Flow Refinement of Cost Relations 22.48/7.30 ===================================== 22.48/7.30 22.48/7.30 ### Specialization of cost equations fun/2 22.48/7.30 * CE 2 is refined into CE [11] 22.48/7.30 * CE 3 is refined into CE [12] 22.48/7.30 * CE 10 is refined into CE [13] 22.48/7.30 * CE 9 is refined into CE [14] 22.48/7.30 * CE 7 is refined into CE [15] 22.48/7.30 * CE 5 is refined into CE [16] 22.48/7.30 * CE 4 is refined into CE [17] 22.48/7.30 * CE 8 is refined into CE [18] 22.48/7.30 * CE 6 is refined into CE [19] 22.48/7.30 22.48/7.30 22.48/7.30 ### Cost equations --> "Loop" of fun/2 22.48/7.30 * CEs [18] --> Loop 10 22.48/7.30 * CEs [19] --> Loop 11 22.48/7.30 * CEs [14] --> Loop 12 22.48/7.30 * CEs [15] --> Loop 13 22.48/7.30 * CEs [16] --> Loop 14 22.48/7.30 * CEs [17] --> Loop 15 22.48/7.30 * CEs [11] --> Loop 16 22.48/7.30 * CEs [12,13] --> Loop 17 22.48/7.30 22.48/7.30 ### Ranking functions of CR fun(V,Out) 22.48/7.30 * RF of phase [10,11,12,13,14,15]: [V] 22.48/7.30 22.48/7.30 #### Partial ranking functions of CR fun(V,Out) 22.48/7.30 * Partial RF of phase [10,11,12,13,14,15]: 22.48/7.30 - RF of loop [10:1,11:1,12:1,12:2,13:1,13:2,14:1,14:2,15:1,15:2]: 22.48/7.30 V 22.48/7.30 22.48/7.30 22.48/7.30 ### Specialization of cost equations start/1 22.48/7.30 * CE 1 is refined into CE [20,21] 22.48/7.30 22.48/7.30 22.48/7.30 ### Cost equations --> "Loop" of start/1 22.48/7.30 * CEs [20,21] --> Loop 18 22.48/7.30 22.48/7.30 ### Ranking functions of CR start(V) 22.48/7.30 22.48/7.30 #### Partial ranking functions of CR start(V) 22.48/7.30 22.48/7.30 22.48/7.30 Computing Bounds 22.48/7.30 ===================================== 22.48/7.30 22.48/7.30 #### Cost of chains of fun(V,Out): 22.48/7.30 * Chain [17]: 1 22.48/7.30 with precondition: [Out=0,V>=0] 22.48/7.30 22.48/7.30 * Chain [16]: 1 22.48/7.30 with precondition: [V=4,Out=1] 22.48/7.30 22.48/7.30 * Chain [multiple([10,11,12,13,14,15],[[17],[16]])]: 6*it(10)+1*it([16])+1*it([17])+0 22.48/7.30 Such that:aux(5) =< 1 22.48/7.30 aux(6) =< V 22.48/7.30 it(10) =< aux(6) 22.48/7.30 it([16]) =< aux(6) 22.48/7.30 it([16]) =< it(10)+it(10)+it(10)+it(10)+aux(5) 22.48/7.30 it([17]) =< it(10)+it(10)+it(10)+it(10)+aux(5) 22.48/7.30 22.48/7.30 with precondition: [V>=1,Out>=1] 22.48/7.30 22.48/7.30 22.48/7.30 #### Cost of chains of start(V): 22.48/7.30 * Chain [18]: 6*s(8)+1*s(9)+1*s(10)+1 22.48/7.30 Such that:s(6) =< 1 22.48/7.30 s(7) =< V 22.48/7.30 s(8) =< s(7) 22.48/7.30 s(9) =< s(7) 22.48/7.30 s(9) =< s(8)+s(8)+s(8)+s(8)+s(6) 22.48/7.30 s(10) =< s(8)+s(8)+s(8)+s(8)+s(6) 22.48/7.30 22.48/7.30 with precondition: [V>=0] 22.48/7.30 22.48/7.30 22.48/7.30 Closed-form bounds of start(V): 22.48/7.30 ------------------------------------- 22.48/7.30 * Chain [18] with precondition: [V>=0] 22.48/7.30 - Upper bound: 11*V+2 22.48/7.30 - Complexity: n 22.48/7.30 22.48/7.30 ### Maximum cost of start(V): 11*V+2 22.48/7.30 Asymptotic class: n 22.48/7.30 * Total analysis performed in 163 ms. 22.48/7.30 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (10) 22.48/7.30 BOUNDS(1, n^1) 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 22.48/7.30 Transformed a relative TRS into a decreasing-loop problem. 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (12) 22.48/7.30 Obligation: 22.48/7.30 Analyzing the following TRS for decreasing loops: 22.48/7.30 22.48/7.30 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.48/7.30 22.48/7.30 22.48/7.30 The TRS R consists of the following rules: 22.48/7.30 22.48/7.30 D(t) -> 1 22.48/7.30 D(constant) -> 0 22.48/7.30 D(+(x, y)) -> +(D(x), D(y)) 22.48/7.30 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 22.48/7.30 D(-(x, y)) -> -(D(x), D(y)) 22.48/7.30 D(minus(x)) -> minus(D(x)) 22.48/7.30 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) 22.48/7.30 D(ln(x)) -> div(D(x), x) 22.48/7.30 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) 22.48/7.30 22.48/7.30 S is empty. 22.48/7.30 Rewrite Strategy: INNERMOST 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (13) DecreasingLoopProof (LOWER BOUND(ID)) 22.48/7.30 The following loop(s) give(s) rise to the lower bound Omega(n^1): 22.48/7.30 22.48/7.30 The rewrite sequence 22.48/7.30 22.48/7.30 D(+(x, y)) ->^+ +(D(x), D(y)) 22.48/7.30 22.48/7.30 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 22.48/7.30 22.48/7.30 The pumping substitution is [x / +(x, y)]. 22.48/7.30 22.48/7.30 The result substitution is [ ]. 22.48/7.30 22.48/7.30 22.48/7.30 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (14) 22.48/7.30 Complex Obligation (BEST) 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (15) 22.48/7.30 Obligation: 22.48/7.30 Proved the lower bound n^1 for the following obligation: 22.48/7.30 22.48/7.30 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.48/7.30 22.48/7.30 22.48/7.30 The TRS R consists of the following rules: 22.48/7.30 22.48/7.30 D(t) -> 1 22.48/7.30 D(constant) -> 0 22.48/7.30 D(+(x, y)) -> +(D(x), D(y)) 22.48/7.30 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 22.48/7.30 D(-(x, y)) -> -(D(x), D(y)) 22.48/7.30 D(minus(x)) -> minus(D(x)) 22.48/7.30 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) 22.48/7.30 D(ln(x)) -> div(D(x), x) 22.48/7.30 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) 22.48/7.30 22.48/7.30 S is empty. 22.48/7.30 Rewrite Strategy: INNERMOST 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (16) LowerBoundPropagationProof (FINISHED) 22.48/7.30 Propagated lower bound. 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (17) 22.48/7.30 BOUNDS(n^1, INF) 22.48/7.30 22.48/7.30 ---------------------------------------- 22.48/7.30 22.48/7.30 (18) 22.48/7.30 Obligation: 22.48/7.30 Analyzing the following TRS for decreasing loops: 22.48/7.30 22.48/7.30 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.48/7.30 22.48/7.30 22.48/7.30 The TRS R consists of the following rules: 22.48/7.30 22.48/7.30 D(t) -> 1 22.48/7.30 D(constant) -> 0 22.48/7.30 D(+(x, y)) -> +(D(x), D(y)) 22.48/7.30 D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 22.48/7.30 D(-(x, y)) -> -(D(x), D(y)) 22.48/7.30 D(minus(x)) -> minus(D(x)) 22.48/7.30 D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) 22.48/7.30 D(ln(x)) -> div(D(x), x) 22.48/7.30 D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) 22.48/7.30 22.48/7.30 S is empty. 22.48/7.30 Rewrite Strategy: INNERMOST 22.72/8.54 EOF