28.75/9.44 WORST_CASE(Omega(n^1), O(n^1)) 28.75/9.45 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.75/9.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.75/9.45 28.75/9.45 28.75/9.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.75/9.45 28.75/9.45 (0) CpxTRS 28.75/9.45 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 28.75/9.45 (2) CpxTRS 28.75/9.45 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 28.75/9.45 (4) CpxWeightedTrs 28.75/9.45 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 28.75/9.45 (6) CpxTypedWeightedTrs 28.75/9.45 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 28.75/9.45 (8) CpxTypedWeightedCompleteTrs 28.75/9.45 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 28.75/9.45 (10) CpxRNTS 28.75/9.45 (11) CompleteCoflocoProof [FINISHED, 474 ms] 28.75/9.45 (12) BOUNDS(1, n^1) 28.75/9.45 (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 28.75/9.45 (14) TRS for Loop Detection 28.75/9.45 (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 28.75/9.45 (16) BEST 28.75/9.45 (17) proven lower bound 28.75/9.45 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 28.75/9.45 (19) BOUNDS(n^1, INF) 28.75/9.45 (20) TRS for Loop Detection 28.75/9.45 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (0) 28.75/9.45 Obligation: 28.75/9.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.75/9.45 28.75/9.45 28.75/9.45 The TRS R consists of the following rules: 28.75/9.45 28.75/9.45 pred(s(x)) -> x 28.75/9.45 minus(x, 0) -> x 28.75/9.45 minus(x, s(y)) -> pred(minus(x, y)) 28.75/9.45 quot(0, s(y)) -> 0 28.75/9.45 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 28.75/9.45 log(s(0)) -> 0 28.75/9.45 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) 28.75/9.45 28.75/9.45 S is empty. 28.75/9.45 Rewrite Strategy: FULL 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 28.75/9.45 Converted rc-obligation to irc-obligation. 28.75/9.45 28.75/9.45 The duplicating contexts are: 28.75/9.45 quot(s(x), s([])) 28.75/9.45 28.75/9.45 28.75/9.45 The defined contexts are: 28.75/9.45 log(s([])) 28.75/9.45 quot([], s(x1)) 28.75/9.45 pred([]) 28.75/9.45 minus([], x1) 28.75/9.45 quot([], s(s(0))) 28.75/9.45 28.75/9.45 28.75/9.45 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (2) 28.75/9.45 Obligation: 28.75/9.45 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.75/9.45 28.75/9.45 28.75/9.45 The TRS R consists of the following rules: 28.75/9.45 28.75/9.45 pred(s(x)) -> x 28.75/9.45 minus(x, 0) -> x 28.75/9.45 minus(x, s(y)) -> pred(minus(x, y)) 28.75/9.45 quot(0, s(y)) -> 0 28.75/9.45 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 28.75/9.45 log(s(0)) -> 0 28.75/9.45 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) 28.75/9.45 28.75/9.45 S is empty. 28.75/9.45 Rewrite Strategy: INNERMOST 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 28.75/9.45 Transformed relative TRS to weighted TRS 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (4) 28.75/9.45 Obligation: 28.75/9.45 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 28.75/9.45 28.75/9.45 28.75/9.45 The TRS R consists of the following rules: 28.75/9.45 28.75/9.45 pred(s(x)) -> x [1] 28.75/9.45 minus(x, 0) -> x [1] 28.75/9.45 minus(x, s(y)) -> pred(minus(x, y)) [1] 28.75/9.45 quot(0, s(y)) -> 0 [1] 28.75/9.45 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 28.75/9.45 log(s(0)) -> 0 [1] 28.75/9.45 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1] 28.75/9.45 28.75/9.45 Rewrite Strategy: INNERMOST 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 28.75/9.45 Infered types. 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (6) 28.75/9.45 Obligation: 28.75/9.45 Runtime Complexity Weighted TRS with Types. 28.75/9.45 The TRS R consists of the following rules: 28.75/9.45 28.75/9.45 pred(s(x)) -> x [1] 28.75/9.45 minus(x, 0) -> x [1] 28.75/9.45 minus(x, s(y)) -> pred(minus(x, y)) [1] 28.75/9.45 quot(0, s(y)) -> 0 [1] 28.75/9.45 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 28.75/9.45 log(s(0)) -> 0 [1] 28.75/9.45 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1] 28.75/9.45 28.75/9.45 The TRS has the following type information: 28.75/9.45 pred :: s:0 -> s:0 28.75/9.45 s :: s:0 -> s:0 28.75/9.45 minus :: s:0 -> s:0 -> s:0 28.75/9.45 0 :: s:0 28.75/9.45 quot :: s:0 -> s:0 -> s:0 28.75/9.45 log :: s:0 -> s:0 28.75/9.45 28.75/9.45 Rewrite Strategy: INNERMOST 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (7) CompletionProof (UPPER BOUND(ID)) 28.75/9.45 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 28.75/9.45 28.75/9.45 pred(v0) -> null_pred [0] 28.75/9.45 quot(v0, v1) -> null_quot [0] 28.75/9.45 log(v0) -> null_log [0] 28.75/9.45 minus(v0, v1) -> null_minus [0] 28.75/9.45 28.75/9.45 And the following fresh constants: null_pred, null_quot, null_log, null_minus 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (8) 28.75/9.45 Obligation: 28.75/9.45 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 28.75/9.45 28.75/9.45 Runtime Complexity Weighted TRS with Types. 28.75/9.45 The TRS R consists of the following rules: 28.75/9.45 28.75/9.45 pred(s(x)) -> x [1] 28.75/9.45 minus(x, 0) -> x [1] 28.75/9.45 minus(x, s(y)) -> pred(minus(x, y)) [1] 28.75/9.45 quot(0, s(y)) -> 0 [1] 28.75/9.45 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 28.75/9.45 log(s(0)) -> 0 [1] 28.75/9.45 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1] 28.75/9.45 pred(v0) -> null_pred [0] 28.75/9.45 quot(v0, v1) -> null_quot [0] 28.75/9.45 log(v0) -> null_log [0] 28.75/9.45 minus(v0, v1) -> null_minus [0] 28.75/9.45 28.75/9.45 The TRS has the following type information: 28.75/9.45 pred :: s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 s :: s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 minus :: s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 0 :: s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 quot :: s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 log :: s:0:null_pred:null_quot:null_log:null_minus -> s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 null_pred :: s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 null_quot :: s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 null_log :: s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 null_minus :: s:0:null_pred:null_quot:null_log:null_minus 28.75/9.45 28.75/9.45 Rewrite Strategy: INNERMOST 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 28.75/9.45 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 28.75/9.45 The constant constructors are abstracted as follows: 28.75/9.45 28.75/9.45 0 => 0 28.75/9.45 null_pred => 0 28.75/9.45 null_quot => 0 28.75/9.45 null_log => 0 28.75/9.45 null_minus => 0 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (10) 28.75/9.45 Obligation: 28.75/9.45 Complexity RNTS consisting of the following rules: 28.75/9.45 28.75/9.45 log(z) -{ 1 }-> 0 :|: z = 1 + 0 28.75/9.45 log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 28.75/9.45 log(z) -{ 1 }-> 1 + log(1 + quot(x, 1 + (1 + 0))) :|: x >= 0, z = 1 + (1 + x) 28.75/9.45 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 28.75/9.45 minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x 28.75/9.45 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 28.75/9.45 pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 28.75/9.45 pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 28.75/9.45 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 28.75/9.45 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 28.75/9.45 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 28.75/9.45 28.75/9.45 Only complete derivations are relevant for the runtime complexity. 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (11) CompleteCoflocoProof (FINISHED) 28.75/9.45 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 28.75/9.45 28.75/9.45 eq(start(V, V2),0,[pred(V, Out)],[V >= 0]). 28.75/9.45 eq(start(V, V2),0,[minus(V, V2, Out)],[V >= 0,V2 >= 0]). 28.75/9.45 eq(start(V, V2),0,[quot(V, V2, Out)],[V >= 0,V2 >= 0]). 28.75/9.45 eq(start(V, V2),0,[log(V, Out)],[V >= 0]). 28.75/9.45 eq(pred(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]). 28.75/9.45 eq(minus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]). 28.75/9.45 eq(minus(V, V2, Out),1,[minus(V4, V5, Ret0),pred(Ret0, Ret)],[Out = Ret,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]). 28.75/9.45 eq(quot(V, V2, Out),1,[],[Out = 0,V2 = 1 + V6,V6 >= 0,V = 0]). 28.75/9.45 eq(quot(V, V2, Out),1,[minus(V7, V8, Ret10),quot(Ret10, 1 + V8, Ret1)],[Out = 1 + Ret1,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). 28.75/9.45 eq(log(V, Out),1,[],[Out = 0,V = 1]). 28.75/9.45 eq(log(V, Out),1,[quot(V9, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V9 >= 0,V = 2 + V9]). 28.75/9.45 eq(pred(V, Out),0,[],[Out = 0,V10 >= 0,V = V10]). 28.75/9.45 eq(quot(V, V2, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V = V12,V2 = V11]). 28.75/9.45 eq(log(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]). 28.75/9.45 eq(minus(V, V2, Out),0,[],[Out = 0,V14 >= 0,V15 >= 0,V = V14,V2 = V15]). 28.75/9.45 input_output_vars(pred(V,Out),[V],[Out]). 28.75/9.45 input_output_vars(minus(V,V2,Out),[V,V2],[Out]). 28.75/9.45 input_output_vars(quot(V,V2,Out),[V,V2],[Out]). 28.75/9.45 input_output_vars(log(V,Out),[V],[Out]). 28.75/9.45 28.75/9.45 28.75/9.45 CoFloCo proof output: 28.75/9.45 Preprocessing Cost Relations 28.75/9.45 ===================================== 28.75/9.45 28.75/9.45 #### Computed strongly connected components 28.75/9.45 0. non_recursive : [pred/2] 28.75/9.45 1. recursive [non_tail] : [minus/3] 28.75/9.45 2. recursive : [quot/3] 28.75/9.45 3. recursive : [log/2] 28.75/9.45 4. non_recursive : [start/2] 28.75/9.45 28.75/9.45 #### Obtained direct recursion through partial evaluation 28.75/9.45 0. SCC is partially evaluated into pred/2 28.75/9.45 1. SCC is partially evaluated into minus/3 28.75/9.45 2. SCC is partially evaluated into quot/3 28.75/9.45 3. SCC is partially evaluated into log/2 28.75/9.45 4. SCC is partially evaluated into start/2 28.75/9.45 28.75/9.45 Control-Flow Refinement of Cost Relations 28.75/9.45 ===================================== 28.75/9.45 28.75/9.45 ### Specialization of cost equations pred/2 28.75/9.45 * CE 5 is refined into CE [16] 28.75/9.45 * CE 6 is refined into CE [17] 28.75/9.45 28.75/9.45 28.75/9.45 ### Cost equations --> "Loop" of pred/2 28.75/9.45 * CEs [16] --> Loop 11 28.75/9.45 * CEs [17] --> Loop 12 28.75/9.45 28.75/9.45 ### Ranking functions of CR pred(V,Out) 28.75/9.45 28.75/9.45 #### Partial ranking functions of CR pred(V,Out) 28.75/9.45 28.75/9.45 28.75/9.45 ### Specialization of cost equations minus/3 28.75/9.45 * CE 9 is refined into CE [18] 28.75/9.45 * CE 7 is refined into CE [19] 28.75/9.45 * CE 8 is refined into CE [20,21] 28.75/9.45 28.75/9.45 28.75/9.45 ### Cost equations --> "Loop" of minus/3 28.75/9.45 * CEs [21] --> Loop 13 28.75/9.45 * CEs [20] --> Loop 14 28.75/9.45 * CEs [18] --> Loop 15 28.75/9.45 * CEs [19] --> Loop 16 28.75/9.45 28.75/9.45 ### Ranking functions of CR minus(V,V2,Out) 28.75/9.45 * RF of phase [13]: [V2] 28.75/9.45 * RF of phase [14]: [V2] 28.75/9.45 28.75/9.45 #### Partial ranking functions of CR minus(V,V2,Out) 28.75/9.45 * Partial RF of phase [13]: 28.75/9.45 - RF of loop [13:1]: 28.75/9.45 V2 28.75/9.45 * Partial RF of phase [14]: 28.75/9.45 - RF of loop [14:1]: 28.75/9.45 V2 28.75/9.45 28.75/9.45 28.75/9.45 ### Specialization of cost equations quot/3 28.75/9.45 * CE 10 is refined into CE [22] 28.75/9.45 * CE 12 is refined into CE [23] 28.75/9.45 * CE 11 is refined into CE [24,25,26] 28.75/9.45 28.75/9.45 28.75/9.45 ### Cost equations --> "Loop" of quot/3 28.75/9.45 * CEs [26] --> Loop 17 28.75/9.45 * CEs [25] --> Loop 18 28.75/9.45 * CEs [24] --> Loop 19 28.75/9.45 * CEs [22,23] --> Loop 20 28.75/9.45 28.75/9.45 ### Ranking functions of CR quot(V,V2,Out) 28.75/9.45 * RF of phase [17]: [V-1,V-V2+1] 28.75/9.45 * RF of phase [19]: [V] 28.75/9.45 28.75/9.45 #### Partial ranking functions of CR quot(V,V2,Out) 28.75/9.45 * Partial RF of phase [17]: 28.75/9.45 - RF of loop [17:1]: 28.75/9.45 V-1 28.75/9.45 V-V2+1 28.75/9.45 * Partial RF of phase [19]: 28.75/9.45 - RF of loop [19:1]: 28.75/9.45 V 28.75/9.45 28.75/9.45 28.75/9.45 ### Specialization of cost equations log/2 28.75/9.45 * CE 13 is refined into CE [27] 28.75/9.45 * CE 15 is refined into CE [28] 28.75/9.45 * CE 14 is refined into CE [29,30,31,32] 28.75/9.45 28.75/9.45 28.75/9.45 ### Cost equations --> "Loop" of log/2 28.75/9.45 * CEs [32] --> Loop 21 28.75/9.45 * CEs [31] --> Loop 22 28.75/9.45 * CEs [30] --> Loop 23 28.75/9.45 * CEs [29] --> Loop 24 28.75/9.45 * CEs [27,28] --> Loop 25 28.75/9.45 28.75/9.45 ### Ranking functions of CR log(V,Out) 28.75/9.45 * RF of phase [21,22]: [V-3,V/2-3/2] 28.75/9.45 28.75/9.45 #### Partial ranking functions of CR log(V,Out) 28.75/9.45 * Partial RF of phase [21,22]: 28.75/9.45 - RF of loop [21:1]: 28.75/9.45 V/2-2 28.75/9.45 - RF of loop [22:1]: 28.75/9.45 V-3 28.75/9.45 28.75/9.45 28.75/9.45 ### Specialization of cost equations start/2 28.75/9.45 * CE 1 is refined into CE [33,34] 28.75/9.45 * CE 2 is refined into CE [35,36,37] 28.75/9.45 * CE 3 is refined into CE [38,39,40,41,42] 28.75/9.45 * CE 4 is refined into CE [43,44,45,46,47,48] 28.75/9.45 28.75/9.45 28.75/9.45 ### Cost equations --> "Loop" of start/2 28.75/9.45 * CEs [38] --> Loop 26 28.75/9.45 * CEs [33,34,35,36,37,39,40,41,42,43,44,45,46,47,48] --> Loop 27 28.75/9.45 28.75/9.45 ### Ranking functions of CR start(V,V2) 28.75/9.45 28.75/9.45 #### Partial ranking functions of CR start(V,V2) 28.75/9.45 28.75/9.45 28.75/9.45 Computing Bounds 28.75/9.45 ===================================== 28.75/9.45 28.75/9.45 #### Cost of chains of pred(V,Out): 28.75/9.45 * Chain [12]: 0 28.75/9.45 with precondition: [Out=0,V>=0] 28.75/9.45 28.75/9.45 * Chain [11]: 1 28.75/9.45 with precondition: [V=Out+1,V>=1] 28.75/9.45 28.75/9.45 28.75/9.45 #### Cost of chains of minus(V,V2,Out): 28.75/9.45 * Chain [[14],[13],16]: 3*it(13)+1 28.75/9.45 Such that:aux(1) =< V2 28.75/9.45 it(13) =< aux(1) 28.75/9.45 28.75/9.45 with precondition: [Out=0,V>=1,V2>=2] 28.75/9.45 28.75/9.45 * Chain [[14],16]: 1*it(14)+1 28.75/9.45 Such that:it(14) =< V2 28.75/9.45 28.75/9.45 with precondition: [Out=0,V>=0,V2>=1] 28.75/9.45 28.75/9.45 * Chain [[14],15]: 1*it(14)+0 28.75/9.45 Such that:it(14) =< V2 28.75/9.45 28.75/9.45 with precondition: [Out=0,V>=0,V2>=1] 28.75/9.45 28.75/9.45 * Chain [[13],16]: 2*it(13)+1 28.75/9.45 Such that:it(13) =< V2 28.75/9.45 28.75/9.45 with precondition: [V=Out+V2,V2>=1,V>=V2] 28.75/9.45 28.75/9.45 * Chain [16]: 1 28.75/9.45 with precondition: [V2=0,V=Out,V>=0] 28.75/9.45 28.75/9.45 * Chain [15]: 0 28.75/9.45 with precondition: [Out=0,V>=0,V2>=0] 28.75/9.45 28.75/9.45 28.75/9.45 #### Cost of chains of quot(V,V2,Out): 28.75/9.45 * Chain [[19],20]: 2*it(19)+1 28.75/9.45 Such that:it(19) =< Out 28.75/9.45 28.75/9.45 with precondition: [V2=1,Out>=1,V>=Out] 28.75/9.45 28.75/9.45 * Chain [[19],18,20]: 2*it(19)+5*s(6)+3 28.75/9.45 Such that:s(5) =< 1 28.75/9.45 it(19) =< Out 28.75/9.45 s(6) =< s(5) 28.75/9.45 28.75/9.45 with precondition: [V2=1,Out>=2,V>=Out] 28.75/9.45 28.75/9.45 * Chain [[17],20]: 2*it(17)+2*s(9)+1 28.75/9.45 Such that:it(17) =< V-V2+1 28.75/9.45 aux(5) =< V 28.75/9.45 it(17) =< aux(5) 28.75/9.45 s(9) =< aux(5) 28.75/9.45 28.75/9.45 with precondition: [V2>=2,Out>=1,V+2>=2*Out+V2] 28.75/9.45 28.75/9.45 * Chain [[17],18,20]: 2*it(17)+5*s(6)+2*s(9)+3 28.75/9.45 Such that:it(17) =< V-V2+1 28.75/9.45 s(5) =< V2 28.75/9.45 aux(6) =< V 28.75/9.45 s(6) =< s(5) 28.75/9.45 it(17) =< aux(6) 28.75/9.45 s(9) =< aux(6) 28.75/9.45 28.75/9.45 with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2] 28.75/9.45 28.75/9.45 * Chain [20]: 1 28.75/9.45 with precondition: [Out=0,V>=0,V2>=0] 28.75/9.45 28.75/9.45 * Chain [18,20]: 5*s(6)+3 28.75/9.45 Such that:s(5) =< V2 28.75/9.45 s(6) =< s(5) 28.75/9.45 28.75/9.45 with precondition: [Out=1,V>=1,V2>=1] 28.75/9.45 28.75/9.45 28.75/9.45 #### Cost of chains of log(V,Out): 28.75/9.45 * Chain [[21,22],25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+1 28.75/9.45 Such that:s(33) =< 2*V 28.75/9.45 aux(16) =< 5/2*V 28.75/9.45 aux(15) =< 5/2*V+27/2 28.75/9.45 aux(17) =< V 28.75/9.45 aux(18) =< V/2 28.75/9.45 aux(10) =< aux(17) 28.75/9.45 it(21) =< aux(17) 28.75/9.45 it(22) =< aux(17) 28.75/9.45 aux(10) =< aux(18) 28.75/9.45 it(21) =< aux(18) 28.75/9.45 it(22) =< aux(18) 28.75/9.45 it(22) =< aux(15) 28.75/9.45 s(31) =< aux(15) 28.75/9.45 it(22) =< aux(16) 28.75/9.45 s(31) =< aux(16) 28.75/9.45 s(30) =< aux(10)*2 28.75/9.45 s(32) =< s(33) 28.75/9.45 s(28) =< s(31) 28.75/9.45 s(29) =< s(30) 28.75/9.45 28.75/9.45 with precondition: [Out>=1,V>=3*Out+1] 28.75/9.45 28.75/9.45 * Chain [[21,22],24,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+3 28.75/9.45 Such that:s(33) =< 2*V 28.75/9.45 aux(16) =< 5/2*V 28.75/9.45 aux(15) =< 5/2*V+27/2 28.75/9.45 aux(19) =< V 28.75/9.45 aux(20) =< V/2 28.75/9.45 aux(10) =< aux(19) 28.75/9.45 it(21) =< aux(19) 28.75/9.45 it(22) =< aux(19) 28.75/9.45 aux(10) =< aux(20) 28.75/9.45 it(21) =< aux(20) 28.75/9.45 it(22) =< aux(20) 28.75/9.45 it(22) =< aux(15) 28.75/9.45 s(31) =< aux(15) 28.75/9.45 it(22) =< aux(16) 28.75/9.45 s(31) =< aux(16) 28.75/9.45 s(30) =< aux(10)*2 28.75/9.45 s(32) =< s(33) 28.75/9.45 s(28) =< s(31) 28.75/9.45 s(29) =< s(30) 28.75/9.45 28.75/9.45 with precondition: [Out>=2,V+2>=3*Out] 28.75/9.45 28.75/9.45 * Chain [[21,22],23,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+5 28.75/9.45 Such that:s(34) =< 2 28.75/9.45 s(33) =< 2*V 28.75/9.45 aux(16) =< 5/2*V 28.75/9.45 aux(15) =< 5/2*V+27/2 28.75/9.45 aux(21) =< V 28.75/9.45 aux(22) =< V/2 28.75/9.45 s(35) =< s(34) 28.75/9.45 aux(10) =< aux(21) 28.75/9.45 it(21) =< aux(21) 28.75/9.45 it(22) =< aux(21) 28.75/9.45 aux(10) =< aux(22) 28.75/9.45 it(21) =< aux(22) 28.75/9.45 it(22) =< aux(22) 28.75/9.45 it(22) =< aux(15) 28.75/9.45 s(31) =< aux(15) 28.75/9.45 it(22) =< aux(16) 28.75/9.45 s(31) =< aux(16) 28.75/9.45 s(30) =< aux(10)*2 28.75/9.45 s(32) =< s(33) 28.75/9.45 s(28) =< s(31) 28.75/9.45 s(29) =< s(30) 28.75/9.45 28.75/9.45 with precondition: [Out>=2,V+3>=4*Out] 28.75/9.45 28.75/9.45 * Chain [[21,22],23,24,25]: 4*it(21)+2*it(22)+4*s(28)+5*s(29)+4*s(32)+5*s(35)+7 28.75/9.45 Such that:s(34) =< 2 28.75/9.45 s(33) =< 2*V 28.75/9.45 aux(16) =< 5/2*V 28.75/9.45 aux(15) =< 5/2*V+27/2 28.75/9.45 aux(23) =< V 28.75/9.45 aux(24) =< V/2 28.75/9.45 s(35) =< s(34) 28.75/9.45 aux(10) =< aux(23) 28.75/9.45 it(21) =< aux(23) 28.75/9.45 it(22) =< aux(23) 28.75/9.45 aux(10) =< aux(24) 28.75/9.45 it(21) =< aux(24) 28.75/9.45 it(22) =< aux(24) 28.75/9.45 it(22) =< aux(15) 28.75/9.45 s(31) =< aux(15) 28.75/9.45 it(22) =< aux(16) 28.75/9.45 s(31) =< aux(16) 28.75/9.45 s(30) =< aux(10)*2 28.75/9.45 s(32) =< s(33) 28.75/9.45 s(28) =< s(31) 28.75/9.45 s(29) =< s(30) 28.75/9.45 28.75/9.45 with precondition: [Out>=3,V+7>=4*Out] 28.75/9.45 28.75/9.45 * Chain [25]: 1 28.75/9.45 with precondition: [Out=0,V>=0] 28.75/9.45 28.75/9.45 * Chain [24,25]: 3 28.75/9.45 with precondition: [Out=1,V>=2] 28.75/9.45 28.75/9.45 * Chain [23,25]: 5*s(35)+5 28.75/9.45 Such that:s(34) =< 2 28.75/9.45 s(35) =< s(34) 28.75/9.45 28.75/9.45 with precondition: [Out=1,V>=3] 28.75/9.45 28.75/9.45 * Chain [23,24,25]: 5*s(35)+7 28.75/9.45 Such that:s(34) =< 2 28.75/9.45 s(35) =< s(34) 28.75/9.45 28.75/9.45 with precondition: [Out=2,V>=3] 28.75/9.45 28.75/9.45 28.75/9.45 #### Cost of chains of start(V,V2): 28.75/9.45 * Chain [27]: 17*s(67)+4*s(71)+4*s(73)+15*s(80)+16*s(87)+8*s(88)+16*s(91)+16*s(92)+20*s(93)+15 28.75/9.45 Such that:aux(30) =< 2 28.75/9.45 aux(31) =< V 28.75/9.45 aux(32) =< V-V2+1 28.75/9.45 aux(33) =< 2*V 28.75/9.45 aux(34) =< V/2 28.75/9.45 aux(35) =< 5/2*V 28.75/9.45 aux(36) =< 5/2*V+27/2 28.75/9.45 aux(37) =< V2 28.75/9.45 s(71) =< aux(32) 28.75/9.45 s(67) =< aux(37) 28.75/9.45 s(80) =< aux(30) 28.75/9.45 s(86) =< aux(31) 28.75/9.45 s(87) =< aux(31) 28.75/9.45 s(88) =< aux(31) 28.75/9.45 s(86) =< aux(34) 28.75/9.45 s(87) =< aux(34) 28.75/9.45 s(88) =< aux(34) 28.75/9.45 s(88) =< aux(36) 28.75/9.45 s(89) =< aux(36) 28.75/9.45 s(88) =< aux(35) 28.75/9.45 s(89) =< aux(35) 28.75/9.45 s(90) =< s(86)*2 28.75/9.45 s(91) =< aux(33) 28.75/9.45 s(92) =< s(89) 28.75/9.45 s(93) =< s(90) 28.75/9.45 s(71) =< aux(31) 28.75/9.45 s(73) =< aux(31) 28.75/9.45 28.75/9.45 with precondition: [V>=0] 28.75/9.45 28.75/9.45 * Chain [26]: 4*s(126)+5*s(127)+3 28.75/9.45 Such that:s(124) =< 1 28.75/9.45 s(125) =< V 28.75/9.45 s(126) =< s(125) 28.75/9.45 s(127) =< s(124) 28.75/9.45 28.75/9.45 with precondition: [V2=1,V>=1] 28.75/9.45 28.75/9.45 28.75/9.45 Closed-form bounds of start(V,V2): 28.75/9.45 ------------------------------------- 28.75/9.45 * Chain [27] with precondition: [V>=0] 28.75/9.45 - Upper bound: 68*V+45+nat(V2)*17+32*V+(40*V+216)+nat(V-V2+1)*4 28.75/9.45 - Complexity: n 28.75/9.45 * Chain [26] with precondition: [V2=1,V>=1] 28.75/9.45 - Upper bound: 4*V+8 28.75/9.45 - Complexity: n 28.75/9.45 28.75/9.45 ### Maximum cost of start(V,V2): 64*V+37+nat(V2)*17+32*V+(40*V+216)+nat(V-V2+1)*4+(4*V+8) 28.75/9.45 Asymptotic class: n 28.75/9.45 * Total analysis performed in 370 ms. 28.75/9.45 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (12) 28.75/9.45 BOUNDS(1, n^1) 28.75/9.45 28.75/9.45 ---------------------------------------- 28.75/9.45 28.75/9.45 (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 28.75/9.45 Transformed a relative TRS into a decreasing-loop problem. 28.75/9.45 ---------------------------------------- 28.75/9.46 28.75/9.46 (14) 28.75/9.46 Obligation: 28.75/9.46 Analyzing the following TRS for decreasing loops: 28.75/9.46 28.75/9.46 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.75/9.46 28.75/9.46 28.75/9.46 The TRS R consists of the following rules: 28.75/9.46 28.75/9.46 pred(s(x)) -> x 28.75/9.46 minus(x, 0) -> x 28.75/9.46 minus(x, s(y)) -> pred(minus(x, y)) 28.75/9.46 quot(0, s(y)) -> 0 28.75/9.46 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 28.75/9.46 log(s(0)) -> 0 28.75/9.46 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) 28.75/9.46 28.75/9.46 S is empty. 28.75/9.46 Rewrite Strategy: FULL 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (15) DecreasingLoopProof (LOWER BOUND(ID)) 28.75/9.46 The following loop(s) give(s) rise to the lower bound Omega(n^1): 28.75/9.46 28.75/9.46 The rewrite sequence 28.75/9.46 28.75/9.46 minus(x, s(y)) ->^+ pred(minus(x, y)) 28.75/9.46 28.75/9.46 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 28.75/9.46 28.75/9.46 The pumping substitution is [y / s(y)]. 28.75/9.46 28.75/9.46 The result substitution is [ ]. 28.75/9.46 28.75/9.46 28.75/9.46 28.75/9.46 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (16) 28.75/9.46 Complex Obligation (BEST) 28.75/9.46 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (17) 28.75/9.46 Obligation: 28.75/9.46 Proved the lower bound n^1 for the following obligation: 28.75/9.46 28.75/9.46 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.75/9.46 28.75/9.46 28.75/9.46 The TRS R consists of the following rules: 28.75/9.46 28.75/9.46 pred(s(x)) -> x 28.75/9.46 minus(x, 0) -> x 28.75/9.46 minus(x, s(y)) -> pred(minus(x, y)) 28.75/9.46 quot(0, s(y)) -> 0 28.75/9.46 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 28.75/9.46 log(s(0)) -> 0 28.75/9.46 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) 28.75/9.46 28.75/9.46 S is empty. 28.75/9.46 Rewrite Strategy: FULL 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (18) LowerBoundPropagationProof (FINISHED) 28.75/9.46 Propagated lower bound. 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (19) 28.75/9.46 BOUNDS(n^1, INF) 28.75/9.46 28.75/9.46 ---------------------------------------- 28.75/9.46 28.75/9.46 (20) 28.75/9.46 Obligation: 28.75/9.46 Analyzing the following TRS for decreasing loops: 28.75/9.46 28.75/9.46 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.75/9.46 28.75/9.46 28.75/9.46 The TRS R consists of the following rules: 28.75/9.46 28.75/9.46 pred(s(x)) -> x 28.75/9.46 minus(x, 0) -> x 28.75/9.46 minus(x, s(y)) -> pred(minus(x, y)) 28.75/9.46 quot(0, s(y)) -> 0 28.75/9.46 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 28.75/9.46 log(s(0)) -> 0 28.75/9.46 log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) 28.75/9.46 28.75/9.46 S is empty. 28.75/9.46 Rewrite Strategy: FULL 28.82/9.52 EOF