22.11/7.49 WORST_CASE(Omega(n^1), O(n^1)) 22.11/7.49 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 22.11/7.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.11/7.49 22.11/7.49 22.11/7.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.11/7.49 22.11/7.49 (0) CpxTRS 22.11/7.49 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 22.11/7.49 (2) CpxTRS 22.11/7.49 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 22.11/7.49 (4) CpxWeightedTrs 22.11/7.49 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 22.11/7.49 (6) CpxTypedWeightedTrs 22.11/7.49 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 22.11/7.49 (8) CpxTypedWeightedCompleteTrs 22.11/7.49 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 22.11/7.49 (10) CpxRNTS 22.11/7.49 (11) CompleteCoflocoProof [FINISHED, 283 ms] 22.11/7.49 (12) BOUNDS(1, n^1) 22.11/7.49 (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 22.11/7.49 (14) TRS for Loop Detection 22.11/7.49 (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 22.11/7.49 (16) BEST 22.11/7.49 (17) proven lower bound 22.11/7.49 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 22.11/7.49 (19) BOUNDS(n^1, INF) 22.11/7.49 (20) TRS for Loop Detection 22.11/7.49 22.11/7.49 22.11/7.49 ---------------------------------------- 22.11/7.49 22.11/7.49 (0) 22.11/7.49 Obligation: 22.11/7.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.11/7.49 22.11/7.49 22.11/7.49 The TRS R consists of the following rules: 22.11/7.49 22.11/7.49 pred(s(x)) -> x 22.11/7.49 minus(x, 0) -> x 22.11/7.49 minus(x, s(y)) -> pred(minus(x, y)) 22.11/7.49 quot(0, s(y)) -> 0 22.11/7.49 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 22.11/7.49 22.11/7.49 S is empty. 22.11/7.49 Rewrite Strategy: FULL 22.11/7.49 ---------------------------------------- 22.11/7.49 22.11/7.49 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 22.11/7.49 Converted rc-obligation to irc-obligation. 22.11/7.49 22.11/7.49 The duplicating contexts are: 22.11/7.49 quot(s(x), s([])) 22.11/7.49 22.11/7.49 22.11/7.49 The defined contexts are: 22.11/7.49 quot([], s(x1)) 22.11/7.49 pred([]) 22.11/7.49 minus([], x1) 22.11/7.49 22.11/7.49 22.11/7.49 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 22.11/7.49 ---------------------------------------- 22.11/7.49 22.11/7.49 (2) 22.11/7.49 Obligation: 22.11/7.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 22.11/7.49 22.11/7.49 22.11/7.49 The TRS R consists of the following rules: 22.11/7.49 22.11/7.49 pred(s(x)) -> x 22.11/7.49 minus(x, 0) -> x 22.11/7.49 minus(x, s(y)) -> pred(minus(x, y)) 22.11/7.50 quot(0, s(y)) -> 0 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 22.11/7.50 22.11/7.50 S is empty. 22.11/7.50 Rewrite Strategy: INNERMOST 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 22.11/7.50 Transformed relative TRS to weighted TRS 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (4) 22.11/7.50 Obligation: 22.11/7.50 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 22.11/7.50 22.11/7.50 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x [1] 22.11/7.50 minus(x, 0) -> x [1] 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) [1] 22.11/7.50 quot(0, s(y)) -> 0 [1] 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 22.11/7.50 22.11/7.50 Rewrite Strategy: INNERMOST 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 22.11/7.50 Infered types. 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (6) 22.11/7.50 Obligation: 22.11/7.50 Runtime Complexity Weighted TRS with Types. 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x [1] 22.11/7.50 minus(x, 0) -> x [1] 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) [1] 22.11/7.50 quot(0, s(y)) -> 0 [1] 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 22.11/7.50 22.11/7.50 The TRS has the following type information: 22.11/7.50 pred :: s:0 -> s:0 22.11/7.50 s :: s:0 -> s:0 22.11/7.50 minus :: s:0 -> s:0 -> s:0 22.11/7.50 0 :: s:0 22.11/7.50 quot :: s:0 -> s:0 -> s:0 22.11/7.50 22.11/7.50 Rewrite Strategy: INNERMOST 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (7) CompletionProof (UPPER BOUND(ID)) 22.11/7.50 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 22.11/7.50 22.11/7.50 pred(v0) -> null_pred [0] 22.11/7.50 quot(v0, v1) -> null_quot [0] 22.11/7.50 minus(v0, v1) -> null_minus [0] 22.11/7.50 22.11/7.50 And the following fresh constants: null_pred, null_quot, null_minus 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (8) 22.11/7.50 Obligation: 22.11/7.50 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 22.11/7.50 22.11/7.50 Runtime Complexity Weighted TRS with Types. 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x [1] 22.11/7.50 minus(x, 0) -> x [1] 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) [1] 22.11/7.50 quot(0, s(y)) -> 0 [1] 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 22.11/7.50 pred(v0) -> null_pred [0] 22.11/7.50 quot(v0, v1) -> null_quot [0] 22.11/7.50 minus(v0, v1) -> null_minus [0] 22.11/7.50 22.11/7.50 The TRS has the following type information: 22.11/7.50 pred :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus 22.11/7.50 s :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus 22.11/7.50 minus :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus 22.11/7.50 0 :: s:0:null_pred:null_quot:null_minus 22.11/7.50 quot :: s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus -> s:0:null_pred:null_quot:null_minus 22.11/7.50 null_pred :: s:0:null_pred:null_quot:null_minus 22.11/7.50 null_quot :: s:0:null_pred:null_quot:null_minus 22.11/7.50 null_minus :: s:0:null_pred:null_quot:null_minus 22.11/7.50 22.11/7.50 Rewrite Strategy: INNERMOST 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 22.11/7.50 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 22.11/7.50 The constant constructors are abstracted as follows: 22.11/7.50 22.11/7.50 0 => 0 22.11/7.50 null_pred => 0 22.11/7.50 null_quot => 0 22.11/7.50 null_minus => 0 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (10) 22.11/7.50 Obligation: 22.11/7.50 Complexity RNTS consisting of the following rules: 22.11/7.50 22.11/7.50 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 22.11/7.50 minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x 22.11/7.50 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 22.11/7.50 pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 22.11/7.50 pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 22.11/7.50 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 22.11/7.50 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 22.11/7.50 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 22.11/7.50 22.11/7.50 Only complete derivations are relevant for the runtime complexity. 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (11) CompleteCoflocoProof (FINISHED) 22.11/7.50 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 22.11/7.50 22.11/7.50 eq(start(V, V2),0,[pred(V, Out)],[V >= 0]). 22.11/7.50 eq(start(V, V2),0,[minus(V, V2, Out)],[V >= 0,V2 >= 0]). 22.11/7.50 eq(start(V, V2),0,[quot(V, V2, Out)],[V >= 0,V2 >= 0]). 22.11/7.50 eq(pred(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]). 22.11/7.50 eq(minus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]). 22.11/7.50 eq(minus(V, V2, Out),1,[minus(V4, V5, Ret0),pred(Ret0, Ret)],[Out = Ret,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]). 22.11/7.50 eq(quot(V, V2, Out),1,[],[Out = 0,V2 = 1 + V6,V6 >= 0,V = 0]). 22.11/7.50 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]). 22.11/7.50 eq(pred(V, Out),0,[],[Out = 0,V9 >= 0,V = V9]). 22.11/7.50 eq(quot(V, V2, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V = V11,V2 = V10]). 22.11/7.50 eq(minus(V, V2, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V = V13,V2 = V12]). 22.11/7.50 input_output_vars(pred(V,Out),[V],[Out]). 22.11/7.50 input_output_vars(minus(V,V2,Out),[V,V2],[Out]). 22.11/7.50 input_output_vars(quot(V,V2,Out),[V,V2],[Out]). 22.11/7.50 22.11/7.50 22.11/7.50 CoFloCo proof output: 22.11/7.50 Preprocessing Cost Relations 22.11/7.50 ===================================== 22.11/7.50 22.11/7.50 #### Computed strongly connected components 22.11/7.50 0. non_recursive : [pred/2] 22.11/7.50 1. recursive [non_tail] : [minus/3] 22.11/7.50 2. recursive : [quot/3] 22.11/7.50 3. non_recursive : [start/2] 22.11/7.50 22.11/7.50 #### Obtained direct recursion through partial evaluation 22.11/7.50 0. SCC is partially evaluated into pred/2 22.11/7.50 1. SCC is partially evaluated into minus/3 22.11/7.50 2. SCC is partially evaluated into quot/3 22.11/7.50 3. SCC is partially evaluated into start/2 22.11/7.50 22.11/7.50 Control-Flow Refinement of Cost Relations 22.11/7.50 ===================================== 22.11/7.50 22.11/7.50 ### Specialization of cost equations pred/2 22.11/7.50 * CE 4 is refined into CE [12] 22.11/7.50 * CE 5 is refined into CE [13] 22.11/7.50 22.11/7.50 22.11/7.50 ### Cost equations --> "Loop" of pred/2 22.11/7.50 * CEs [12] --> Loop 9 22.11/7.50 * CEs [13] --> Loop 10 22.11/7.50 22.11/7.50 ### Ranking functions of CR pred(V,Out) 22.11/7.50 22.11/7.50 #### Partial ranking functions of CR pred(V,Out) 22.11/7.50 22.11/7.50 22.11/7.50 ### Specialization of cost equations minus/3 22.11/7.50 * CE 8 is refined into CE [14] 22.11/7.50 * CE 6 is refined into CE [15] 22.11/7.50 * CE 7 is refined into CE [16,17] 22.11/7.50 22.11/7.50 22.11/7.50 ### Cost equations --> "Loop" of minus/3 22.11/7.50 * CEs [17] --> Loop 11 22.11/7.50 * CEs [16] --> Loop 12 22.11/7.50 * CEs [14] --> Loop 13 22.11/7.50 * CEs [15] --> Loop 14 22.11/7.50 22.11/7.50 ### Ranking functions of CR minus(V,V2,Out) 22.11/7.50 * RF of phase [11]: [V2] 22.11/7.50 * RF of phase [12]: [V2] 22.11/7.50 22.11/7.50 #### Partial ranking functions of CR minus(V,V2,Out) 22.11/7.50 * Partial RF of phase [11]: 22.11/7.50 - RF of loop [11:1]: 22.11/7.50 V2 22.11/7.50 * Partial RF of phase [12]: 22.11/7.50 - RF of loop [12:1]: 22.11/7.50 V2 22.11/7.50 22.11/7.50 22.11/7.50 ### Specialization of cost equations quot/3 22.11/7.50 * CE 9 is refined into CE [18] 22.11/7.50 * CE 11 is refined into CE [19] 22.11/7.50 * CE 10 is refined into CE [20,21,22] 22.11/7.50 22.11/7.50 22.11/7.50 ### Cost equations --> "Loop" of quot/3 22.11/7.50 * CEs [22] --> Loop 15 22.11/7.50 * CEs [21] --> Loop 16 22.11/7.50 * CEs [20] --> Loop 17 22.11/7.50 * CEs [18,19] --> Loop 18 22.11/7.50 22.11/7.50 ### Ranking functions of CR quot(V,V2,Out) 22.11/7.50 * RF of phase [15]: [V-1,V-V2+1] 22.11/7.50 * RF of phase [17]: [V] 22.11/7.50 22.11/7.50 #### Partial ranking functions of CR quot(V,V2,Out) 22.11/7.50 * Partial RF of phase [15]: 22.11/7.50 - RF of loop [15:1]: 22.11/7.50 V-1 22.11/7.50 V-V2+1 22.11/7.50 * Partial RF of phase [17]: 22.11/7.50 - RF of loop [17:1]: 22.11/7.50 V 22.11/7.50 22.11/7.50 22.11/7.50 ### Specialization of cost equations start/2 22.11/7.50 * CE 1 is refined into CE [23,24] 22.11/7.50 * CE 2 is refined into CE [25,26,27] 22.11/7.50 * CE 3 is refined into CE [28,29,30,31,32] 22.11/7.50 22.11/7.50 22.11/7.50 ### Cost equations --> "Loop" of start/2 22.11/7.50 * CEs [28] --> Loop 19 22.11/7.50 * CEs [23,24,25,26,27,29,30,31,32] --> Loop 20 22.11/7.50 22.11/7.50 ### Ranking functions of CR start(V,V2) 22.11/7.50 22.11/7.50 #### Partial ranking functions of CR start(V,V2) 22.11/7.50 22.11/7.50 22.11/7.50 Computing Bounds 22.11/7.50 ===================================== 22.11/7.50 22.11/7.50 #### Cost of chains of pred(V,Out): 22.11/7.50 * Chain [10]: 0 22.11/7.50 with precondition: [Out=0,V>=0] 22.11/7.50 22.11/7.50 * Chain [9]: 1 22.11/7.50 with precondition: [V=Out+1,V>=1] 22.11/7.50 22.11/7.50 22.11/7.50 #### Cost of chains of minus(V,V2,Out): 22.11/7.50 * Chain [[12],[11],14]: 3*it(11)+1 22.11/7.50 Such that:aux(1) =< V2 22.11/7.50 it(11) =< aux(1) 22.11/7.50 22.11/7.50 with precondition: [Out=0,V>=1,V2>=2] 22.11/7.50 22.11/7.50 * Chain [[12],14]: 1*it(12)+1 22.11/7.50 Such that:it(12) =< V2 22.11/7.50 22.11/7.50 with precondition: [Out=0,V>=0,V2>=1] 22.11/7.50 22.11/7.50 * Chain [[12],13]: 1*it(12)+0 22.11/7.50 Such that:it(12) =< V2 22.11/7.50 22.11/7.50 with precondition: [Out=0,V>=0,V2>=1] 22.11/7.50 22.11/7.50 * Chain [[11],14]: 2*it(11)+1 22.11/7.50 Such that:it(11) =< V2 22.11/7.50 22.11/7.50 with precondition: [V=Out+V2,V2>=1,V>=V2] 22.11/7.50 22.11/7.50 * Chain [14]: 1 22.11/7.50 with precondition: [V2=0,V=Out,V>=0] 22.11/7.50 22.11/7.50 * Chain [13]: 0 22.11/7.50 with precondition: [Out=0,V>=0,V2>=0] 22.11/7.50 22.11/7.50 22.11/7.50 #### Cost of chains of quot(V,V2,Out): 22.11/7.50 * Chain [[17],18]: 2*it(17)+1 22.11/7.50 Such that:it(17) =< Out 22.11/7.50 22.11/7.50 with precondition: [V2=1,Out>=1,V>=Out] 22.11/7.50 22.11/7.50 * Chain [[17],16,18]: 2*it(17)+5*s(6)+3 22.11/7.50 Such that:s(5) =< 1 22.11/7.50 it(17) =< Out 22.11/7.50 s(6) =< s(5) 22.11/7.50 22.11/7.50 with precondition: [V2=1,Out>=2,V>=Out] 22.11/7.50 22.11/7.50 * Chain [[15],18]: 2*it(15)+2*s(9)+1 22.11/7.50 Such that:it(15) =< V-V2+1 22.11/7.50 aux(5) =< V 22.11/7.50 it(15) =< aux(5) 22.11/7.50 s(9) =< aux(5) 22.11/7.50 22.11/7.50 with precondition: [V2>=2,Out>=1,V+2>=2*Out+V2] 22.11/7.50 22.11/7.50 * Chain [[15],16,18]: 2*it(15)+5*s(6)+2*s(9)+3 22.11/7.50 Such that:it(15) =< V-V2+1 22.11/7.50 s(5) =< V2 22.11/7.50 aux(6) =< V 22.11/7.50 s(6) =< s(5) 22.11/7.50 it(15) =< aux(6) 22.11/7.50 s(9) =< aux(6) 22.11/7.50 22.11/7.50 with precondition: [V2>=2,Out>=2,V+3>=2*Out+V2] 22.11/7.50 22.11/7.50 * Chain [18]: 1 22.11/7.50 with precondition: [Out=0,V>=0,V2>=0] 22.11/7.50 22.11/7.50 * Chain [16,18]: 5*s(6)+3 22.11/7.50 Such that:s(5) =< V2 22.11/7.50 s(6) =< s(5) 22.11/7.50 22.11/7.50 with precondition: [Out=1,V>=1,V2>=1] 22.11/7.50 22.11/7.50 22.11/7.50 #### Cost of chains of start(V,V2): 22.11/7.50 * Chain [20]: 17*s(15)+4*s(19)+4*s(21)+3 22.11/7.50 Such that:aux(8) =< V 22.11/7.50 aux(9) =< V-V2+1 22.11/7.50 aux(10) =< V2 22.11/7.50 s(19) =< aux(9) 22.11/7.50 s(15) =< aux(10) 22.11/7.50 s(19) =< aux(8) 22.11/7.50 s(21) =< aux(8) 22.11/7.50 22.11/7.50 with precondition: [V>=0] 22.11/7.50 22.11/7.50 * Chain [19]: 4*s(29)+5*s(30)+3 22.11/7.50 Such that:s(27) =< 1 22.11/7.50 s(28) =< V 22.11/7.50 s(29) =< s(28) 22.11/7.50 s(30) =< s(27) 22.11/7.50 22.11/7.50 with precondition: [V2=1,V>=1] 22.11/7.50 22.11/7.50 22.11/7.50 Closed-form bounds of start(V,V2): 22.11/7.50 ------------------------------------- 22.11/7.50 * Chain [20] with precondition: [V>=0] 22.11/7.50 - Upper bound: 4*V+3+nat(V2)*17+nat(V-V2+1)*4 22.11/7.50 - Complexity: n 22.11/7.50 * Chain [19] with precondition: [V2=1,V>=1] 22.11/7.50 - Upper bound: 4*V+8 22.11/7.50 - Complexity: n 22.11/7.50 22.11/7.50 ### Maximum cost of start(V,V2): 4*V+3+max([5,nat(V-V2+1)*4+nat(V2)*17]) 22.11/7.50 Asymptotic class: n 22.11/7.50 * Total analysis performed in 201 ms. 22.11/7.50 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (12) 22.11/7.50 BOUNDS(1, n^1) 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 22.11/7.50 Transformed a relative TRS into a decreasing-loop problem. 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (14) 22.11/7.50 Obligation: 22.11/7.50 Analyzing the following TRS for decreasing loops: 22.11/7.50 22.11/7.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.11/7.50 22.11/7.50 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x 22.11/7.50 minus(x, 0) -> x 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) 22.11/7.50 quot(0, s(y)) -> 0 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 22.11/7.50 22.11/7.50 S is empty. 22.11/7.50 Rewrite Strategy: FULL 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (15) DecreasingLoopProof (LOWER BOUND(ID)) 22.11/7.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 22.11/7.50 22.11/7.50 The rewrite sequence 22.11/7.50 22.11/7.50 minus(x, s(y)) ->^+ pred(minus(x, y)) 22.11/7.50 22.11/7.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 22.11/7.50 22.11/7.50 The pumping substitution is [y / s(y)]. 22.11/7.50 22.11/7.50 The result substitution is [ ]. 22.11/7.50 22.11/7.50 22.11/7.50 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (16) 22.11/7.50 Complex Obligation (BEST) 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (17) 22.11/7.50 Obligation: 22.11/7.50 Proved the lower bound n^1 for the following obligation: 22.11/7.50 22.11/7.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.11/7.50 22.11/7.50 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x 22.11/7.50 minus(x, 0) -> x 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) 22.11/7.50 quot(0, s(y)) -> 0 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 22.11/7.50 22.11/7.50 S is empty. 22.11/7.50 Rewrite Strategy: FULL 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (18) LowerBoundPropagationProof (FINISHED) 22.11/7.50 Propagated lower bound. 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (19) 22.11/7.50 BOUNDS(n^1, INF) 22.11/7.50 22.11/7.50 ---------------------------------------- 22.11/7.50 22.11/7.50 (20) 22.11/7.50 Obligation: 22.11/7.50 Analyzing the following TRS for decreasing loops: 22.11/7.50 22.11/7.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.11/7.50 22.11/7.50 22.11/7.50 The TRS R consists of the following rules: 22.11/7.50 22.11/7.50 pred(s(x)) -> x 22.11/7.50 minus(x, 0) -> x 22.11/7.50 minus(x, s(y)) -> pred(minus(x, y)) 22.11/7.50 quot(0, s(y)) -> 0 22.11/7.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 22.11/7.50 22.11/7.50 S is empty. 22.11/7.50 Rewrite Strategy: FULL 22.15/11.25 EOF