23.61/8.00 WORST_CASE(Omega(n^1), O(n^1)) 23.82/8.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.82/8.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.82/8.01 23.82/8.01 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.82/8.01 23.82/8.01 (0) CpxTRS 23.82/8.01 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 23.82/8.01 (2) CpxTRS 23.82/8.01 (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 23.82/8.01 (4) CpxTRS 23.82/8.01 (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 23.82/8.01 (6) CpxWeightedTrs 23.82/8.01 (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms] 23.82/8.01 (8) CpxTypedWeightedTrs 23.82/8.01 (9) CompletionProof [UPPER BOUND(ID), 0 ms] 23.82/8.01 (10) CpxTypedWeightedCompleteTrs 23.82/8.01 (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 23.82/8.01 (12) CpxRNTS 23.82/8.01 (13) CompleteCoflocoProof [FINISHED, 290 ms] 23.82/8.01 (14) BOUNDS(1, n^1) 23.82/8.01 (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 23.82/8.01 (16) TRS for Loop Detection 23.82/8.01 (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 23.82/8.01 (18) BEST 23.82/8.01 (19) proven lower bound 23.82/8.01 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 23.82/8.01 (21) BOUNDS(n^1, INF) 23.82/8.01 (22) TRS for Loop Detection 23.82/8.01 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (0) 23.82/8.01 Obligation: 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 23.82/8.01 plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0))) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: FULL 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.82/8.01 The following defined symbols can occur below the 0th argument of quot: minus 23.82/8.01 The following defined symbols can occur below the 0th argument of minus: minus 23.82/8.01 23.82/8.01 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.82/8.01 plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 23.82/8.01 plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0))) 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (2) 23.82/8.01 Obligation: 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: FULL 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (3) RcToIrcProof (BOTH BOUNDS(ID, ID)) 23.82/8.01 Converted rc-obligation to irc-obligation. 23.82/8.01 23.82/8.01 The duplicating contexts are: 23.82/8.01 quot(s(x), s([])) 23.82/8.01 23.82/8.01 23.82/8.01 The defined contexts are: 23.82/8.01 quot([], s(x1)) 23.82/8.01 minus([], x1) 23.82/8.01 23.82/8.01 23.82/8.01 [] just represents basic- or constructor-terms in the following defined contexts: 23.82/8.01 quot([], s(x1)) 23.82/8.01 23.82/8.01 23.82/8.01 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (4) 23.82/8.01 Obligation: 23.82/8.01 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: INNERMOST 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 23.82/8.01 Transformed relative TRS to weighted TRS 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (6) 23.82/8.01 Obligation: 23.82/8.01 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x [1] 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) [1] 23.82/8.01 quot(0, s(y)) -> 0 [1] 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 23.82/8.01 plus(0, y) -> y [1] 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) [1] 23.82/8.01 23.82/8.01 Rewrite Strategy: INNERMOST 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 23.82/8.01 Infered types. 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (8) 23.82/8.01 Obligation: 23.82/8.01 Runtime Complexity Weighted TRS with Types. 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x [1] 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) [1] 23.82/8.01 quot(0, s(y)) -> 0 [1] 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 23.82/8.01 plus(0, y) -> y [1] 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) [1] 23.82/8.01 23.82/8.01 The TRS has the following type information: 23.82/8.01 minus :: 0:s -> 0:s -> 0:s 23.82/8.01 0 :: 0:s 23.82/8.01 s :: 0:s -> 0:s 23.82/8.01 quot :: 0:s -> 0:s -> 0:s 23.82/8.01 plus :: 0:s -> 0:s -> 0:s 23.82/8.01 23.82/8.01 Rewrite Strategy: INNERMOST 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (9) CompletionProof (UPPER BOUND(ID)) 23.82/8.01 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 23.82/8.01 23.82/8.01 minus(v0, v1) -> null_minus [0] 23.82/8.01 quot(v0, v1) -> null_quot [0] 23.82/8.01 plus(v0, v1) -> null_plus [0] 23.82/8.01 23.82/8.01 And the following fresh constants: null_minus, null_quot, null_plus 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (10) 23.82/8.01 Obligation: 23.82/8.01 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 23.82/8.01 23.82/8.01 Runtime Complexity Weighted TRS with Types. 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x [1] 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) [1] 23.82/8.01 quot(0, s(y)) -> 0 [1] 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 23.82/8.01 plus(0, y) -> y [1] 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) [1] 23.82/8.01 minus(v0, v1) -> null_minus [0] 23.82/8.01 quot(v0, v1) -> null_quot [0] 23.82/8.01 plus(v0, v1) -> null_plus [0] 23.82/8.01 23.82/8.01 The TRS has the following type information: 23.82/8.01 minus :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus 23.82/8.01 0 :: 0:s:null_minus:null_quot:null_plus 23.82/8.01 s :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus 23.82/8.01 quot :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus 23.82/8.01 plus :: 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus -> 0:s:null_minus:null_quot:null_plus 23.82/8.01 null_minus :: 0:s:null_minus:null_quot:null_plus 23.82/8.01 null_quot :: 0:s:null_minus:null_quot:null_plus 23.82/8.01 null_plus :: 0:s:null_minus:null_quot:null_plus 23.82/8.01 23.82/8.01 Rewrite Strategy: INNERMOST 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 23.82/8.01 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 23.82/8.01 The constant constructors are abstracted as follows: 23.82/8.01 23.82/8.01 0 => 0 23.82/8.01 null_minus => 0 23.82/8.01 null_quot => 0 23.82/8.01 null_plus => 0 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (12) 23.82/8.01 Obligation: 23.82/8.01 Complexity RNTS consisting of the following rules: 23.82/8.01 23.82/8.01 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 23.82/8.01 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 23.82/8.01 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 23.82/8.01 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y 23.82/8.01 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 23.82/8.01 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 23.82/8.01 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 23.82/8.01 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 23.82/8.01 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 23.82/8.01 23.82/8.01 Only complete derivations are relevant for the runtime complexity. 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (13) CompleteCoflocoProof (FINISHED) 23.82/8.01 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 23.82/8.01 23.82/8.01 eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 23.82/8.01 eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]). 23.82/8.01 eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]). 23.82/8.01 eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]). 23.82/8.01 eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]). 23.82/8.01 eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V5,V5 >= 0,V1 = 0]). 23.82/8.01 eq(quot(V1, V, Out),1,[minus(V7, V6, Ret10),quot(Ret10, 1 + V6, Ret1)],[Out = 1 + Ret1,V = 1 + V6,V7 >= 0,V6 >= 0,V1 = 1 + V7]). 23.82/8.01 eq(plus(V1, V, Out),1,[],[Out = V8,V8 >= 0,V1 = 0,V = V8]). 23.82/8.01 eq(plus(V1, V, Out),1,[plus(V9, V10, Ret11)],[Out = 1 + Ret11,V9 >= 0,V10 >= 0,V1 = 1 + V9,V = V10]). 23.82/8.01 eq(minus(V1, V, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V1 = V12,V = V11]). 23.82/8.01 eq(quot(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]). 23.82/8.01 eq(plus(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]). 23.82/8.01 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 23.82/8.01 input_output_vars(quot(V1,V,Out),[V1,V],[Out]). 23.82/8.01 input_output_vars(plus(V1,V,Out),[V1,V],[Out]). 23.82/8.01 23.82/8.01 23.82/8.01 CoFloCo proof output: 23.82/8.01 Preprocessing Cost Relations 23.82/8.01 ===================================== 23.82/8.01 23.82/8.01 #### Computed strongly connected components 23.82/8.01 0. recursive : [minus/3] 23.82/8.01 1. recursive : [plus/3] 23.82/8.01 2. recursive : [quot/3] 23.82/8.01 3. non_recursive : [start/2] 23.82/8.01 23.82/8.01 #### Obtained direct recursion through partial evaluation 23.82/8.01 0. SCC is partially evaluated into minus/3 23.82/8.01 1. SCC is partially evaluated into plus/3 23.82/8.01 2. SCC is partially evaluated into quot/3 23.82/8.01 3. SCC is partially evaluated into start/2 23.82/8.01 23.82/8.01 Control-Flow Refinement of Cost Relations 23.82/8.01 ===================================== 23.82/8.01 23.82/8.01 ### Specialization of cost equations minus/3 23.82/8.01 * CE 6 is refined into CE [13] 23.82/8.01 * CE 4 is refined into CE [14] 23.82/8.01 * CE 5 is refined into CE [15] 23.82/8.01 23.82/8.01 23.82/8.01 ### Cost equations --> "Loop" of minus/3 23.82/8.01 * CEs [15] --> Loop 10 23.82/8.01 * CEs [13] --> Loop 11 23.82/8.01 * CEs [14] --> Loop 12 23.82/8.01 23.82/8.01 ### Ranking functions of CR minus(V1,V,Out) 23.82/8.01 * RF of phase [10]: [V,V1] 23.82/8.01 23.82/8.01 #### Partial ranking functions of CR minus(V1,V,Out) 23.82/8.01 * Partial RF of phase [10]: 23.82/8.01 - RF of loop [10:1]: 23.82/8.01 V 23.82/8.01 V1 23.82/8.01 23.82/8.01 23.82/8.01 ### Specialization of cost equations plus/3 23.82/8.01 * CE 12 is refined into CE [16] 23.82/8.01 * CE 10 is refined into CE [17] 23.82/8.01 * CE 11 is refined into CE [18] 23.82/8.01 23.82/8.01 23.82/8.01 ### Cost equations --> "Loop" of plus/3 23.82/8.01 * CEs [18] --> Loop 13 23.82/8.01 * CEs [16] --> Loop 14 23.82/8.01 * CEs [17] --> Loop 15 23.82/8.01 23.82/8.01 ### Ranking functions of CR plus(V1,V,Out) 23.82/8.01 * RF of phase [13]: [V1] 23.82/8.01 23.82/8.01 #### Partial ranking functions of CR plus(V1,V,Out) 23.82/8.01 * Partial RF of phase [13]: 23.82/8.01 - RF of loop [13:1]: 23.82/8.01 V1 23.82/8.01 23.82/8.01 23.82/8.01 ### Specialization of cost equations quot/3 23.82/8.01 * CE 7 is refined into CE [19] 23.82/8.01 * CE 9 is refined into CE [20] 23.82/8.01 * CE 8 is refined into CE [21,22,23] 23.82/8.01 23.82/8.01 23.82/8.01 ### Cost equations --> "Loop" of quot/3 23.82/8.01 * CEs [23] --> Loop 16 23.82/8.01 * CEs [22] --> Loop 17 23.82/8.01 * CEs [21] --> Loop 18 23.82/8.01 * CEs [19,20] --> Loop 19 23.82/8.01 23.82/8.01 ### Ranking functions of CR quot(V1,V,Out) 23.82/8.01 * RF of phase [16]: [V1-1,V1-V+1] 23.82/8.01 * RF of phase [18]: [V1] 23.82/8.01 23.82/8.01 #### Partial ranking functions of CR quot(V1,V,Out) 23.82/8.01 * Partial RF of phase [16]: 23.82/8.01 - RF of loop [16:1]: 23.82/8.01 V1-1 23.82/8.01 V1-V+1 23.82/8.01 * Partial RF of phase [18]: 23.82/8.01 - RF of loop [18:1]: 23.82/8.01 V1 23.82/8.01 23.82/8.01 23.82/8.01 ### Specialization of cost equations start/2 23.82/8.01 * CE 1 is refined into CE [24,25,26] 23.82/8.01 * CE 2 is refined into CE [27,28,29,30,31] 23.82/8.01 * CE 3 is refined into CE [32,33,34,35] 23.82/8.01 23.82/8.01 23.82/8.01 ### Cost equations --> "Loop" of start/2 23.82/8.01 * CEs [27] --> Loop 20 23.82/8.01 * CEs [24] --> Loop 21 23.82/8.01 * CEs [25,26,28,29,30,31,32,33,34,35] --> Loop 22 23.82/8.01 23.82/8.01 ### Ranking functions of CR start(V1,V) 23.82/8.01 23.82/8.01 #### Partial ranking functions of CR start(V1,V) 23.82/8.01 23.82/8.01 23.82/8.01 Computing Bounds 23.82/8.01 ===================================== 23.82/8.01 23.82/8.01 #### Cost of chains of minus(V1,V,Out): 23.82/8.01 * Chain [[10],12]: 1*it(10)+1 23.82/8.01 Such that:it(10) =< V 23.82/8.01 23.82/8.01 with precondition: [V1=Out+V,V>=1,V1>=V] 23.82/8.01 23.82/8.01 * Chain [[10],11]: 1*it(10)+0 23.82/8.01 Such that:it(10) =< V 23.82/8.01 23.82/8.01 with precondition: [Out=0,V1>=1,V>=1] 23.82/8.01 23.82/8.01 * Chain [12]: 1 23.82/8.01 with precondition: [V=0,V1=Out,V1>=0] 23.82/8.01 23.82/8.01 * Chain [11]: 0 23.82/8.01 with precondition: [Out=0,V1>=0,V>=0] 23.82/8.01 23.82/8.01 23.82/8.01 #### Cost of chains of plus(V1,V,Out): 23.82/8.01 * Chain [[13],15]: 1*it(13)+1 23.82/8.01 Such that:it(13) =< -V+Out 23.82/8.01 23.82/8.01 with precondition: [V+V1=Out,V1>=1,V>=0] 23.82/8.01 23.82/8.01 * Chain [[13],14]: 1*it(13)+0 23.82/8.01 Such that:it(13) =< Out 23.82/8.01 23.82/8.01 with precondition: [V>=0,Out>=1,V1>=Out] 23.82/8.01 23.82/8.01 * Chain [15]: 1 23.82/8.01 with precondition: [V1=0,V=Out,V>=0] 23.82/8.01 23.82/8.01 * Chain [14]: 0 23.82/8.01 with precondition: [Out=0,V1>=0,V>=0] 23.82/8.01 23.82/8.01 23.82/8.01 #### Cost of chains of quot(V1,V,Out): 23.82/8.01 * Chain [[18],19]: 2*it(18)+1 23.82/8.01 Such that:it(18) =< Out 23.82/8.01 23.82/8.01 with precondition: [V=1,Out>=1,V1>=Out] 23.82/8.01 23.82/8.01 * Chain [[18],17,19]: 2*it(18)+1*s(2)+2 23.82/8.01 Such that:s(2) =< 1 23.82/8.01 it(18) =< Out 23.82/8.01 23.82/8.01 with precondition: [V=1,Out>=2,V1>=Out] 23.82/8.01 23.82/8.01 * Chain [[16],19]: 2*it(16)+1*s(5)+1 23.82/8.01 Such that:it(16) =< V1-V+1 23.82/8.01 aux(3) =< V1 23.82/8.01 it(16) =< aux(3) 23.82/8.01 s(5) =< aux(3) 23.82/8.01 23.82/8.01 with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 23.82/8.01 23.82/8.01 * Chain [[16],17,19]: 2*it(16)+1*s(2)+1*s(5)+2 23.82/8.01 Such that:it(16) =< V1-V+1 23.82/8.01 s(2) =< V 23.82/8.01 aux(4) =< V1 23.82/8.01 it(16) =< aux(4) 23.82/8.01 s(5) =< aux(4) 23.82/8.01 23.82/8.01 with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 23.82/8.01 23.82/8.01 * Chain [19]: 1 23.82/8.01 with precondition: [Out=0,V1>=0,V>=0] 23.82/8.01 23.82/8.01 * Chain [17,19]: 1*s(2)+2 23.82/8.01 Such that:s(2) =< V 23.82/8.01 23.82/8.01 with precondition: [Out=1,V1>=1,V>=1] 23.82/8.01 23.82/8.01 23.82/8.01 #### Cost of chains of start(V1,V): 23.82/8.01 * Chain [22]: 4*s(9)+4*s(12)+4*s(14)+2 23.82/8.01 Such that:aux(6) =< V1 23.82/8.01 aux(7) =< V1-V+1 23.82/8.01 aux(8) =< V 23.82/8.01 s(14) =< aux(6) 23.82/8.01 s(12) =< aux(7) 23.82/8.01 s(9) =< aux(8) 23.82/8.01 s(12) =< aux(6) 23.82/8.01 23.82/8.01 with precondition: [V1>=0,V>=0] 23.82/8.01 23.82/8.01 * Chain [21]: 1 23.82/8.01 with precondition: [V=0,V1>=0] 23.82/8.01 23.82/8.01 * Chain [20]: 1*s(21)+4*s(23)+2 23.82/8.01 Such that:s(21) =< 1 23.82/8.01 s(22) =< V1 23.82/8.01 s(23) =< s(22) 23.82/8.01 23.82/8.01 with precondition: [V=1,V1>=1] 23.82/8.01 23.82/8.01 23.82/8.01 Closed-form bounds of start(V1,V): 23.82/8.01 ------------------------------------- 23.82/8.01 * Chain [22] with precondition: [V1>=0,V>=0] 23.82/8.01 - Upper bound: 4*V1+4*V+2+nat(V1-V+1)*4 23.82/8.01 - Complexity: n 23.82/8.01 * Chain [21] with precondition: [V=0,V1>=0] 23.82/8.01 - Upper bound: 1 23.82/8.01 - Complexity: constant 23.82/8.01 * Chain [20] with precondition: [V=1,V1>=1] 23.82/8.01 - Upper bound: 4*V1+3 23.82/8.01 - Complexity: n 23.82/8.01 23.82/8.01 ### Maximum cost of start(V1,V): 4*V1+1+max([1,nat(V1-V+1)*4+4*V])+1 23.82/8.01 Asymptotic class: n 23.82/8.01 * Total analysis performed in 216 ms. 23.82/8.01 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (14) 23.82/8.01 BOUNDS(1, n^1) 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 23.82/8.01 Transformed a relative TRS into a decreasing-loop problem. 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (16) 23.82/8.01 Obligation: 23.82/8.01 Analyzing the following TRS for decreasing loops: 23.82/8.01 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 23.82/8.01 plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0))) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: FULL 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (17) DecreasingLoopProof (LOWER BOUND(ID)) 23.82/8.01 The following loop(s) give(s) rise to the lower bound Omega(n^1): 23.82/8.01 23.82/8.01 The rewrite sequence 23.82/8.01 23.82/8.01 minus(s(x), s(y)) ->^+ minus(x, y) 23.82/8.01 23.82/8.01 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 23.82/8.01 23.82/8.01 The pumping substitution is [x / s(x), y / s(y)]. 23.82/8.01 23.82/8.01 The result substitution is [ ]. 23.82/8.01 23.82/8.01 23.82/8.01 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (18) 23.82/8.01 Complex Obligation (BEST) 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (19) 23.82/8.01 Obligation: 23.82/8.01 Proved the lower bound n^1 for the following obligation: 23.82/8.01 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 23.82/8.01 plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0))) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: FULL 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (20) LowerBoundPropagationProof (FINISHED) 23.82/8.01 Propagated lower bound. 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (21) 23.82/8.01 BOUNDS(n^1, INF) 23.82/8.01 23.82/8.01 ---------------------------------------- 23.82/8.01 23.82/8.01 (22) 23.82/8.01 Obligation: 23.82/8.01 Analyzing the following TRS for decreasing loops: 23.82/8.01 23.82/8.01 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.82/8.01 23.82/8.01 23.82/8.01 The TRS R consists of the following rules: 23.82/8.01 23.82/8.01 minus(x, 0) -> x 23.82/8.01 minus(s(x), s(y)) -> minus(x, y) 23.82/8.01 quot(0, s(y)) -> 0 23.82/8.01 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 23.82/8.01 plus(0, y) -> y 23.82/8.01 plus(s(x), y) -> s(plus(x, y)) 23.82/8.01 plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 23.82/8.01 plus(plus(x, s(0)), plus(y, s(s(z)))) -> plus(plus(y, s(s(z))), plus(x, s(0))) 23.82/8.01 23.82/8.01 S is empty. 23.82/8.01 Rewrite Strategy: FULL 23.82/8.05 EOF