20.26/6.04 WORST_CASE(Omega(n^1), O(n^1)) 20.76/6.11 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 20.76/6.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 20.76/6.11 20.76/6.11 20.76/6.11 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.76/6.11 20.76/6.11 (0) CpxTRS 20.76/6.11 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 20.76/6.11 (2) CpxWeightedTrs 20.76/6.11 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 20.76/6.11 (4) CpxTypedWeightedTrs 20.76/6.11 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 20.76/6.11 (6) CpxTypedWeightedCompleteTrs 20.76/6.11 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 20.76/6.11 (8) CpxRNTS 20.76/6.11 (9) CompleteCoflocoProof [FINISHED, 411 ms] 20.76/6.11 (10) BOUNDS(1, n^1) 20.76/6.11 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 20.76/6.11 (12) CpxTRS 20.76/6.11 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 20.76/6.11 (14) typed CpxTrs 20.76/6.11 (15) OrderProof [LOWER BOUND(ID), 0 ms] 20.76/6.11 (16) typed CpxTrs 20.76/6.11 (17) RewriteLemmaProof [LOWER BOUND(ID), 272 ms] 20.76/6.11 (18) BEST 20.76/6.11 (19) proven lower bound 20.76/6.11 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 20.76/6.11 (21) BOUNDS(n^1, INF) 20.76/6.11 (22) typed CpxTrs 20.76/6.11 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (0) 20.76/6.11 Obligation: 20.76/6.11 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.76/6.11 20.76/6.11 20.76/6.11 The TRS R consists of the following rules: 20.76/6.11 20.76/6.11 min(X, 0) -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0, s(Y)) -> 0 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0)) -> 0 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) 20.76/6.11 20.76/6.11 S is empty. 20.76/6.11 Rewrite Strategy: INNERMOST 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 20.76/6.11 Transformed relative TRS to weighted TRS 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (2) 20.76/6.11 Obligation: 20.76/6.11 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 20.76/6.11 20.76/6.11 20.76/6.11 The TRS R consists of the following rules: 20.76/6.11 20.76/6.11 min(X, 0) -> X [1] 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) [1] 20.76/6.11 quot(0, s(Y)) -> 0 [1] 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1] 20.76/6.11 log(s(0)) -> 0 [1] 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1] 20.76/6.11 20.76/6.11 Rewrite Strategy: INNERMOST 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 20.76/6.11 Infered types. 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (4) 20.76/6.11 Obligation: 20.76/6.11 Runtime Complexity Weighted TRS with Types. 20.76/6.11 The TRS R consists of the following rules: 20.76/6.11 20.76/6.11 min(X, 0) -> X [1] 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) [1] 20.76/6.11 quot(0, s(Y)) -> 0 [1] 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1] 20.76/6.11 log(s(0)) -> 0 [1] 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1] 20.76/6.11 20.76/6.11 The TRS has the following type information: 20.76/6.11 min :: 0:s -> 0:s -> 0:s 20.76/6.11 0 :: 0:s 20.76/6.11 s :: 0:s -> 0:s 20.76/6.11 quot :: 0:s -> 0:s -> 0:s 20.76/6.11 log :: 0:s -> 0:s 20.76/6.11 20.76/6.11 Rewrite Strategy: INNERMOST 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (5) CompletionProof (UPPER BOUND(ID)) 20.76/6.11 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 20.76/6.11 20.76/6.11 min(v0, v1) -> null_min [0] 20.76/6.11 quot(v0, v1) -> null_quot [0] 20.76/6.11 log(v0) -> null_log [0] 20.76/6.11 20.76/6.11 And the following fresh constants: null_min, null_quot, null_log 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (6) 20.76/6.11 Obligation: 20.76/6.11 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 20.76/6.11 20.76/6.11 Runtime Complexity Weighted TRS with Types. 20.76/6.11 The TRS R consists of the following rules: 20.76/6.11 20.76/6.11 min(X, 0) -> X [1] 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) [1] 20.76/6.11 quot(0, s(Y)) -> 0 [1] 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1] 20.76/6.11 log(s(0)) -> 0 [1] 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1] 20.76/6.11 min(v0, v1) -> null_min [0] 20.76/6.11 quot(v0, v1) -> null_quot [0] 20.76/6.11 log(v0) -> null_log [0] 20.76/6.11 20.76/6.11 The TRS has the following type information: 20.76/6.11 min :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log 20.76/6.11 0 :: 0:s:null_min:null_quot:null_log 20.76/6.11 s :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log 20.76/6.11 quot :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log 20.76/6.11 log :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log 20.76/6.11 null_min :: 0:s:null_min:null_quot:null_log 20.76/6.11 null_quot :: 0:s:null_min:null_quot:null_log 20.76/6.11 null_log :: 0:s:null_min:null_quot:null_log 20.76/6.11 20.76/6.11 Rewrite Strategy: INNERMOST 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 20.76/6.11 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 20.76/6.11 The constant constructors are abstracted as follows: 20.76/6.11 20.76/6.11 0 => 0 20.76/6.11 null_min => 0 20.76/6.11 null_quot => 0 20.76/6.11 null_log => 0 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (8) 20.76/6.11 Obligation: 20.76/6.11 Complexity RNTS consisting of the following rules: 20.76/6.11 20.76/6.11 log(z) -{ 1 }-> 0 :|: z = 1 + 0 20.76/6.11 log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 20.76/6.11 log(z) -{ 1 }-> 1 + log(1 + quot(X, 1 + (1 + 0))) :|: z = 1 + (1 + X), X >= 0 20.76/6.11 min(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 20.76/6.11 min(z, z') -{ 1 }-> min(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 20.76/6.11 min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 20.76/6.11 quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 20.76/6.11 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 20.76/6.11 quot(z, z') -{ 1 }-> 1 + quot(min(X, Y), 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 20.76/6.11 20.76/6.11 Only complete derivations are relevant for the runtime complexity. 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (9) CompleteCoflocoProof (FINISHED) 20.76/6.11 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 20.76/6.11 20.76/6.11 eq(start(V1, V),0,[min(V1, V, Out)],[V1 >= 0,V >= 0]). 20.76/6.11 eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]). 20.76/6.11 eq(start(V1, V),0,[log(V1, Out)],[V1 >= 0]). 20.76/6.11 eq(min(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]). 20.76/6.11 eq(min(V1, V, Out),1,[min(X2, Y1, Ret)],[Out = Ret,V1 = 1 + X2,Y1 >= 0,V = 1 + Y1,X2 >= 0]). 20.76/6.11 eq(quot(V1, V, Out),1,[],[Out = 0,Y2 >= 0,V = 1 + Y2,V1 = 0]). 20.76/6.11 eq(quot(V1, V, Out),1,[min(X3, Y3, Ret10),quot(Ret10, 1 + Y3, Ret1)],[Out = 1 + Ret1,V1 = 1 + X3,Y3 >= 0,V = 1 + Y3,X3 >= 0]). 20.76/6.11 eq(log(V1, Out),1,[],[Out = 0,V1 = 1]). 20.76/6.11 eq(log(V1, Out),1,[quot(X4, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V1 = 2 + X4,X4 >= 0]). 20.76/6.11 eq(min(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]). 20.76/6.11 eq(quot(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]). 20.76/6.11 eq(log(V1, Out),0,[],[Out = 0,V6 >= 0,V1 = V6]). 20.76/6.11 input_output_vars(min(V1,V,Out),[V1,V],[Out]). 20.76/6.11 input_output_vars(quot(V1,V,Out),[V1,V],[Out]). 20.76/6.11 input_output_vars(log(V1,Out),[V1],[Out]). 20.76/6.11 20.76/6.11 20.76/6.11 CoFloCo proof output: 20.76/6.11 Preprocessing Cost Relations 20.76/6.11 ===================================== 20.76/6.11 20.76/6.11 #### Computed strongly connected components 20.76/6.11 0. recursive : [min/3] 20.76/6.11 1. recursive : [quot/3] 20.76/6.11 2. recursive : [log/2] 20.76/6.11 3. non_recursive : [start/2] 20.76/6.11 20.76/6.11 #### Obtained direct recursion through partial evaluation 20.76/6.11 0. SCC is partially evaluated into min/3 20.76/6.11 1. SCC is partially evaluated into quot/3 20.76/6.11 2. SCC is partially evaluated into log/2 20.76/6.11 3. SCC is partially evaluated into start/2 20.76/6.11 20.76/6.11 Control-Flow Refinement of Cost Relations 20.76/6.11 ===================================== 20.76/6.11 20.76/6.11 ### Specialization of cost equations min/3 20.76/6.11 * CE 6 is refined into CE [13] 20.76/6.11 * CE 4 is refined into CE [14] 20.76/6.11 * CE 5 is refined into CE [15] 20.76/6.11 20.76/6.11 20.76/6.11 ### Cost equations --> "Loop" of min/3 20.76/6.11 * CEs [15] --> Loop 9 20.76/6.11 * CEs [13] --> Loop 10 20.76/6.11 * CEs [14] --> Loop 11 20.76/6.11 20.76/6.11 ### Ranking functions of CR min(V1,V,Out) 20.76/6.11 * RF of phase [9]: [V,V1] 20.76/6.11 20.76/6.11 #### Partial ranking functions of CR min(V1,V,Out) 20.76/6.11 * Partial RF of phase [9]: 20.76/6.11 - RF of loop [9:1]: 20.76/6.11 V 20.76/6.11 V1 20.76/6.11 20.76/6.11 20.76/6.11 ### Specialization of cost equations quot/3 20.76/6.11 * CE 7 is refined into CE [16] 20.76/6.11 * CE 9 is refined into CE [17] 20.76/6.11 * CE 8 is refined into CE [18,19,20] 20.76/6.11 20.76/6.11 20.76/6.11 ### Cost equations --> "Loop" of quot/3 20.76/6.11 * CEs [20] --> Loop 12 20.76/6.11 * CEs [19] --> Loop 13 20.76/6.11 * CEs [18] --> Loop 14 20.76/6.11 * CEs [16,17] --> Loop 15 20.76/6.11 20.76/6.11 ### Ranking functions of CR quot(V1,V,Out) 20.76/6.11 * RF of phase [12]: [V1-1,V1-V+1] 20.76/6.11 * RF of phase [14]: [V1] 20.76/6.11 20.76/6.11 #### Partial ranking functions of CR quot(V1,V,Out) 20.76/6.11 * Partial RF of phase [12]: 20.76/6.11 - RF of loop [12:1]: 20.76/6.11 V1-1 20.76/6.11 V1-V+1 20.76/6.11 * Partial RF of phase [14]: 20.76/6.11 - RF of loop [14:1]: 20.76/6.11 V1 20.76/6.11 20.76/6.11 20.76/6.11 ### Specialization of cost equations log/2 20.76/6.11 * CE 10 is refined into CE [21] 20.76/6.11 * CE 12 is refined into CE [22] 20.76/6.11 * CE 11 is refined into CE [23,24,25,26] 20.76/6.11 20.76/6.11 20.76/6.11 ### Cost equations --> "Loop" of log/2 20.76/6.11 * CEs [26] --> Loop 16 20.76/6.11 * CEs [25] --> Loop 17 20.76/6.11 * CEs [24] --> Loop 18 20.76/6.11 * CEs [23] --> Loop 19 20.76/6.11 * CEs [21,22] --> Loop 20 20.76/6.11 20.76/6.11 ### Ranking functions of CR log(V1,Out) 20.76/6.11 * RF of phase [16,17]: [V1-3,V1/2-3/2] 20.76/6.11 20.76/6.11 #### Partial ranking functions of CR log(V1,Out) 20.76/6.11 * Partial RF of phase [16,17]: 20.76/6.11 - RF of loop [16:1]: 20.76/6.11 V1/2-2 20.76/6.11 - RF of loop [17:1]: 20.76/6.11 V1-3 20.76/6.11 20.76/6.11 20.76/6.11 ### Specialization of cost equations start/2 20.76/6.11 * CE 1 is refined into CE [27,28,29] 20.76/6.11 * CE 2 is refined into CE [30,31,32,33,34] 20.76/6.11 * CE 3 is refined into CE [35,36,37,38,39,40] 20.76/6.11 20.76/6.11 20.76/6.11 ### Cost equations --> "Loop" of start/2 20.76/6.11 * CEs [30] --> Loop 21 20.76/6.11 * CEs [27,28,29,31,32,33,34,35,36,37,38,39,40] --> Loop 22 20.76/6.11 20.76/6.11 ### Ranking functions of CR start(V1,V) 20.76/6.11 20.76/6.11 #### Partial ranking functions of CR start(V1,V) 20.76/6.11 20.76/6.11 20.76/6.11 Computing Bounds 20.76/6.11 ===================================== 20.76/6.11 20.76/6.11 #### Cost of chains of min(V1,V,Out): 20.76/6.11 * Chain [[9],11]: 1*it(9)+1 20.76/6.11 Such that:it(9) =< V 20.76/6.11 20.76/6.11 with precondition: [V1=Out+V,V>=1,V1>=V] 20.76/6.11 20.76/6.11 * Chain [[9],10]: 1*it(9)+0 20.76/6.11 Such that:it(9) =< V 20.76/6.11 20.76/6.11 with precondition: [Out=0,V1>=1,V>=1] 20.76/6.11 20.76/6.11 * Chain [11]: 1 20.76/6.11 with precondition: [V=0,V1=Out,V1>=0] 20.76/6.11 20.76/6.11 * Chain [10]: 0 20.76/6.11 with precondition: [Out=0,V1>=0,V>=0] 20.76/6.11 20.76/6.11 20.76/6.11 #### Cost of chains of quot(V1,V,Out): 20.76/6.11 * Chain [[14],15]: 2*it(14)+1 20.76/6.11 Such that:it(14) =< Out 20.76/6.11 20.76/6.11 with precondition: [V=1,Out>=1,V1>=Out] 20.76/6.11 20.76/6.11 * Chain [[14],13,15]: 2*it(14)+1*s(2)+2 20.76/6.11 Such that:s(2) =< 1 20.76/6.11 it(14) =< Out 20.76/6.11 20.76/6.11 with precondition: [V=1,Out>=2,V1>=Out] 20.76/6.11 20.76/6.11 * Chain [[12],15]: 2*it(12)+1*s(5)+1 20.76/6.11 Such that:it(12) =< V1-V+1 20.76/6.11 aux(3) =< V1 20.76/6.11 it(12) =< aux(3) 20.76/6.11 s(5) =< aux(3) 20.76/6.11 20.76/6.11 with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 20.76/6.11 20.76/6.11 * Chain [[12],13,15]: 2*it(12)+1*s(2)+1*s(5)+2 20.76/6.11 Such that:it(12) =< V1-V+1 20.76/6.11 s(2) =< V 20.76/6.11 aux(4) =< V1 20.76/6.11 it(12) =< aux(4) 20.76/6.11 s(5) =< aux(4) 20.76/6.11 20.76/6.11 with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 20.76/6.11 20.76/6.11 * Chain [15]: 1 20.76/6.11 with precondition: [Out=0,V1>=0,V>=0] 20.76/6.11 20.76/6.11 * Chain [13,15]: 1*s(2)+2 20.76/6.11 Such that:s(2) =< V 20.76/6.11 20.76/6.11 with precondition: [Out=1,V1>=1,V>=1] 20.76/6.11 20.76/6.11 20.76/6.11 #### Cost of chains of log(V1,Out): 20.76/6.11 * Chain [[16,17],20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1 20.76/6.11 Such that:s(25) =< 2*V1 20.76/6.11 aux(14) =< 5/2*V1 20.76/6.11 aux(13) =< 5/2*V1+27/2 20.76/6.11 aux(15) =< V1 20.76/6.11 aux(16) =< V1/2 20.76/6.11 aux(8) =< aux(15) 20.76/6.11 it(16) =< aux(15) 20.76/6.11 it(17) =< aux(15) 20.76/6.11 aux(8) =< aux(16) 20.76/6.11 it(16) =< aux(16) 20.76/6.11 it(17) =< aux(16) 20.76/6.11 it(17) =< aux(13) 20.76/6.11 s(23) =< aux(13) 20.76/6.11 it(17) =< aux(14) 20.76/6.11 s(23) =< aux(14) 20.76/6.11 s(22) =< aux(8)*2 20.76/6.11 s(24) =< s(25) 20.76/6.11 s(21) =< s(23) 20.76/6.11 20.76/6.11 with precondition: [Out>=1,V1>=3*Out+1] 20.76/6.11 20.76/6.11 * Chain [[16,17],19,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+3 20.76/6.11 Such that:s(25) =< 2*V1 20.76/6.11 aux(14) =< 5/2*V1 20.76/6.11 aux(13) =< 5/2*V1+27/2 20.76/6.11 aux(17) =< V1 20.76/6.11 aux(18) =< V1/2 20.76/6.11 aux(8) =< aux(17) 20.76/6.11 it(16) =< aux(17) 20.76/6.11 it(17) =< aux(17) 20.76/6.11 aux(8) =< aux(18) 20.76/6.11 it(16) =< aux(18) 20.76/6.11 it(17) =< aux(18) 20.76/6.11 it(17) =< aux(13) 20.76/6.11 s(23) =< aux(13) 20.76/6.11 it(17) =< aux(14) 20.76/6.11 s(23) =< aux(14) 20.76/6.11 s(22) =< aux(8)*2 20.76/6.11 s(24) =< s(25) 20.76/6.11 s(21) =< s(23) 20.76/6.11 20.76/6.11 with precondition: [Out>=2,V1+2>=3*Out] 20.76/6.11 20.76/6.11 * Chain [[16,17],18,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1*s(26)+4 20.76/6.11 Such that:s(26) =< 2 20.76/6.11 s(25) =< 2*V1 20.76/6.11 aux(14) =< 5/2*V1 20.76/6.11 aux(13) =< 5/2*V1+27/2 20.76/6.11 aux(19) =< V1 20.76/6.11 aux(20) =< V1/2 20.76/6.11 aux(8) =< aux(19) 20.76/6.11 it(16) =< aux(19) 20.76/6.11 it(17) =< aux(19) 20.76/6.11 aux(8) =< aux(20) 20.76/6.11 it(16) =< aux(20) 20.76/6.11 it(17) =< aux(20) 20.76/6.11 it(17) =< aux(13) 20.76/6.11 s(23) =< aux(13) 20.76/6.11 it(17) =< aux(14) 20.76/6.11 s(23) =< aux(14) 20.76/6.11 s(22) =< aux(8)*2 20.76/6.11 s(24) =< s(25) 20.76/6.11 s(21) =< s(23) 20.76/6.11 20.76/6.11 with precondition: [Out>=2,V1+3>=4*Out] 20.76/6.11 20.76/6.11 * Chain [[16,17],18,19,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1*s(26)+6 20.76/6.11 Such that:s(26) =< 2 20.76/6.11 s(25) =< 2*V1 20.76/6.11 aux(14) =< 5/2*V1 20.76/6.11 aux(13) =< 5/2*V1+27/2 20.76/6.11 aux(21) =< V1 20.76/6.11 aux(22) =< V1/2 20.76/6.11 aux(8) =< aux(21) 20.76/6.11 it(16) =< aux(21) 20.76/6.11 it(17) =< aux(21) 20.76/6.11 aux(8) =< aux(22) 20.76/6.11 it(16) =< aux(22) 20.76/6.11 it(17) =< aux(22) 20.76/6.11 it(17) =< aux(13) 20.76/6.11 s(23) =< aux(13) 20.76/6.11 it(17) =< aux(14) 20.76/6.11 s(23) =< aux(14) 20.76/6.11 s(22) =< aux(8)*2 20.76/6.11 s(24) =< s(25) 20.76/6.11 s(21) =< s(23) 20.76/6.11 20.76/6.11 with precondition: [Out>=3,V1+7>=4*Out] 20.76/6.11 20.76/6.11 * Chain [20]: 1 20.76/6.11 with precondition: [Out=0,V1>=0] 20.76/6.11 20.76/6.11 * Chain [19,20]: 3 20.76/6.11 with precondition: [Out=1,V1>=2] 20.76/6.11 20.76/6.11 * Chain [18,20]: 1*s(26)+4 20.76/6.11 Such that:s(26) =< 2 20.76/6.11 20.76/6.11 with precondition: [Out=1,V1>=3] 20.76/6.11 20.76/6.11 * Chain [18,19,20]: 1*s(26)+6 20.76/6.11 Such that:s(26) =< 2 20.76/6.11 20.76/6.11 with precondition: [Out=2,V1>=3] 20.76/6.11 20.76/6.11 20.76/6.11 #### Cost of chains of start(V1,V): 20.76/6.11 * Chain [22]: 4*s(53)+4*s(56)+2*s(58)+3*s(63)+12*s(70)+8*s(71)+4*s(73)+12*s(74)+12*s(75)+6 20.76/6.11 Such that:aux(28) =< 2 20.76/6.11 aux(29) =< V1 20.76/6.11 aux(30) =< V1-V+1 20.76/6.11 aux(31) =< 2*V1 20.76/6.11 aux(32) =< V1/2 20.76/6.11 aux(33) =< 5/2*V1 20.76/6.11 aux(34) =< 5/2*V1+27/2 20.76/6.11 aux(35) =< V 20.76/6.11 s(63) =< aux(28) 20.76/6.11 s(56) =< aux(30) 20.76/6.11 s(53) =< aux(35) 20.76/6.11 s(69) =< aux(29) 20.76/6.11 s(70) =< aux(29) 20.76/6.11 s(71) =< aux(29) 20.76/6.11 s(69) =< aux(32) 20.76/6.11 s(70) =< aux(32) 20.76/6.11 s(71) =< aux(32) 20.76/6.11 s(71) =< aux(34) 20.76/6.11 s(72) =< aux(34) 20.76/6.11 s(71) =< aux(33) 20.76/6.11 s(72) =< aux(33) 20.76/6.11 s(73) =< s(69)*2 20.76/6.11 s(74) =< aux(31) 20.76/6.11 s(75) =< s(72) 20.76/6.11 s(56) =< aux(29) 20.76/6.11 s(58) =< aux(29) 20.76/6.11 20.76/6.11 with precondition: [V1>=0] 20.76/6.11 20.76/6.11 * Chain [21]: 1*s(102)+4*s(104)+2 20.76/6.11 Such that:s(102) =< 1 20.76/6.11 s(103) =< V1 20.76/6.11 s(104) =< s(103) 20.76/6.11 20.76/6.11 with precondition: [V=1,V1>=1] 20.76/6.11 20.76/6.11 20.76/6.11 Closed-form bounds of start(V1,V): 20.76/6.11 ------------------------------------- 20.76/6.11 * Chain [22] with precondition: [V1>=0] 20.76/6.11 - Upper bound: 30*V1+12+nat(V)*4+24*V1+(30*V1+162)+nat(V1-V+1)*4 20.76/6.11 - Complexity: n 20.76/6.11 * Chain [21] with precondition: [V=1,V1>=1] 20.76/6.11 - Upper bound: 4*V1+3 20.76/6.11 - Complexity: n 20.76/6.11 20.76/6.11 ### Maximum cost of start(V1,V): 26*V1+9+nat(V)*4+24*V1+(30*V1+162)+nat(V1-V+1)*4+(4*V1+3) 20.76/6.11 Asymptotic class: n 20.76/6.11 * Total analysis performed in 327 ms. 20.76/6.11 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (10) 20.76/6.11 BOUNDS(1, n^1) 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 20.76/6.11 Renamed function symbols to avoid clashes with predefined symbol. 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (12) 20.76/6.11 Obligation: 20.76/6.11 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 20.76/6.11 20.76/6.11 20.76/6.11 The TRS R consists of the following rules: 20.76/6.11 20.76/6.11 min(X, 0') -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0', s(Y)) -> 0' 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0')) -> 0' 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0')))))) 20.76/6.11 20.76/6.11 S is empty. 20.76/6.11 Rewrite Strategy: INNERMOST 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 20.76/6.11 Infered types. 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (14) 20.76/6.11 Obligation: 20.76/6.11 Innermost TRS: 20.76/6.11 Rules: 20.76/6.11 min(X, 0') -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0', s(Y)) -> 0' 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0')) -> 0' 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0')))))) 20.76/6.11 20.76/6.11 Types: 20.76/6.11 min :: 0':s -> 0':s -> 0':s 20.76/6.11 0' :: 0':s 20.76/6.11 s :: 0':s -> 0':s 20.76/6.11 quot :: 0':s -> 0':s -> 0':s 20.76/6.11 log :: 0':s -> 0':s 20.76/6.11 hole_0':s1_0 :: 0':s 20.76/6.11 gen_0':s2_0 :: Nat -> 0':s 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (15) OrderProof (LOWER BOUND(ID)) 20.76/6.11 Heuristically decided to analyse the following defined symbols: 20.76/6.11 min, quot, log 20.76/6.11 20.76/6.11 They will be analysed ascendingly in the following order: 20.76/6.11 min < quot 20.76/6.11 quot < log 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (16) 20.76/6.11 Obligation: 20.76/6.11 Innermost TRS: 20.76/6.11 Rules: 20.76/6.11 min(X, 0') -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0', s(Y)) -> 0' 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0')) -> 0' 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0')))))) 20.76/6.11 20.76/6.11 Types: 20.76/6.11 min :: 0':s -> 0':s -> 0':s 20.76/6.11 0' :: 0':s 20.76/6.11 s :: 0':s -> 0':s 20.76/6.11 quot :: 0':s -> 0':s -> 0':s 20.76/6.11 log :: 0':s -> 0':s 20.76/6.11 hole_0':s1_0 :: 0':s 20.76/6.11 gen_0':s2_0 :: Nat -> 0':s 20.76/6.11 20.76/6.11 20.76/6.11 Generator Equations: 20.76/6.11 gen_0':s2_0(0) <=> 0' 20.76/6.11 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 20.76/6.11 20.76/6.11 20.76/6.11 The following defined symbols remain to be analysed: 20.76/6.11 min, quot, log 20.76/6.11 20.76/6.11 They will be analysed ascendingly in the following order: 20.76/6.11 min < quot 20.76/6.11 quot < log 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (17) RewriteLemmaProof (LOWER BOUND(ID)) 20.76/6.11 Proved the following rewrite lemma: 20.76/6.11 min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 20.76/6.11 20.76/6.11 Induction Base: 20.76/6.11 min(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1) 20.76/6.11 gen_0':s2_0(0) 20.76/6.11 20.76/6.11 Induction Step: 20.76/6.11 min(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1) 20.76/6.11 min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH 20.76/6.11 gen_0':s2_0(0) 20.76/6.11 20.76/6.11 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (18) 20.76/6.11 Complex Obligation (BEST) 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (19) 20.76/6.11 Obligation: 20.76/6.11 Proved the lower bound n^1 for the following obligation: 20.76/6.11 20.76/6.11 Innermost TRS: 20.76/6.11 Rules: 20.76/6.11 min(X, 0') -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0', s(Y)) -> 0' 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0')) -> 0' 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0')))))) 20.76/6.11 20.76/6.11 Types: 20.76/6.11 min :: 0':s -> 0':s -> 0':s 20.76/6.11 0' :: 0':s 20.76/6.11 s :: 0':s -> 0':s 20.76/6.11 quot :: 0':s -> 0':s -> 0':s 20.76/6.11 log :: 0':s -> 0':s 20.76/6.11 hole_0':s1_0 :: 0':s 20.76/6.11 gen_0':s2_0 :: Nat -> 0':s 20.76/6.11 20.76/6.11 20.76/6.11 Generator Equations: 20.76/6.11 gen_0':s2_0(0) <=> 0' 20.76/6.11 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 20.76/6.11 20.76/6.11 20.76/6.11 The following defined symbols remain to be analysed: 20.76/6.11 min, quot, log 20.76/6.11 20.76/6.11 They will be analysed ascendingly in the following order: 20.76/6.11 min < quot 20.76/6.11 quot < log 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (20) LowerBoundPropagationProof (FINISHED) 20.76/6.11 Propagated lower bound. 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (21) 20.76/6.11 BOUNDS(n^1, INF) 20.76/6.11 20.76/6.11 ---------------------------------------- 20.76/6.11 20.76/6.11 (22) 20.76/6.11 Obligation: 20.76/6.11 Innermost TRS: 20.76/6.11 Rules: 20.76/6.11 min(X, 0') -> X 20.76/6.11 min(s(X), s(Y)) -> min(X, Y) 20.76/6.11 quot(0', s(Y)) -> 0' 20.76/6.11 quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) 20.76/6.11 log(s(0')) -> 0' 20.76/6.11 log(s(s(X))) -> s(log(s(quot(X, s(s(0')))))) 20.76/6.11 20.76/6.11 Types: 20.76/6.11 min :: 0':s -> 0':s -> 0':s 20.76/6.11 0' :: 0':s 20.76/6.11 s :: 0':s -> 0':s 20.76/6.11 quot :: 0':s -> 0':s -> 0':s 20.76/6.11 log :: 0':s -> 0':s 20.76/6.11 hole_0':s1_0 :: 0':s 20.76/6.11 gen_0':s2_0 :: Nat -> 0':s 20.76/6.11 20.76/6.11 20.76/6.11 Lemmas: 20.76/6.11 min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 20.76/6.11 20.76/6.11 20.76/6.11 Generator Equations: 20.76/6.11 gen_0':s2_0(0) <=> 0' 20.76/6.11 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 20.76/6.11 20.76/6.11 20.76/6.11 The following defined symbols remain to be analysed: 20.76/6.11 quot, log 20.76/6.11 20.76/6.11 They will be analysed ascendingly in the following order: 20.76/6.11 quot < log 21.02/6.27 EOF