15.30/6.32 WORST_CASE(Omega(n^1), O(n^1)) 15.47/6.33 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 15.47/6.33 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 15.47/6.33 15.47/6.33 15.47/6.33 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.47/6.33 15.47/6.33 (0) CpxTRS 15.47/6.33 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 15.47/6.33 (2) CpxWeightedTrs 15.47/6.33 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 15.47/6.33 (4) CpxTypedWeightedTrs 15.47/6.33 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 15.47/6.33 (6) CpxTypedWeightedCompleteTrs 15.47/6.33 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms] 15.47/6.33 (8) CpxRNTS 15.47/6.33 (9) CompleteCoflocoProof [FINISHED, 188 ms] 15.47/6.33 (10) BOUNDS(1, n^1) 15.47/6.33 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 15.47/6.33 (12) TRS for Loop Detection 15.47/6.33 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 15.47/6.33 (14) BEST 15.47/6.33 (15) proven lower bound 15.47/6.33 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 15.47/6.33 (17) BOUNDS(n^1, INF) 15.47/6.33 (18) TRS for Loop Detection 15.47/6.33 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (0) 15.47/6.33 Obligation: 15.47/6.33 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.47/6.33 15.47/6.33 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true 15.47/6.33 is_empty(cons(x, l)) -> false 15.47/6.33 hd(cons(x, l)) -> x 15.47/6.33 tl(cons(x, l)) -> l 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) 15.47/6.33 ifappend(l1, l2, nil) -> l2 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) 15.47/6.33 15.47/6.33 S is empty. 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 15.47/6.33 Transformed relative TRS to weighted TRS 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (2) 15.47/6.33 Obligation: 15.47/6.33 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 15.47/6.33 15.47/6.33 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true [1] 15.47/6.33 is_empty(cons(x, l)) -> false [1] 15.47/6.33 hd(cons(x, l)) -> x [1] 15.47/6.33 tl(cons(x, l)) -> l [1] 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) [1] 15.47/6.33 ifappend(l1, l2, nil) -> l2 [1] 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1] 15.47/6.33 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 15.47/6.33 Infered types. 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (4) 15.47/6.33 Obligation: 15.47/6.33 Runtime Complexity Weighted TRS with Types. 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true [1] 15.47/6.33 is_empty(cons(x, l)) -> false [1] 15.47/6.33 hd(cons(x, l)) -> x [1] 15.47/6.33 tl(cons(x, l)) -> l [1] 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) [1] 15.47/6.33 ifappend(l1, l2, nil) -> l2 [1] 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1] 15.47/6.33 15.47/6.33 The TRS has the following type information: 15.47/6.33 is_empty :: nil:cons -> true:false 15.47/6.33 nil :: nil:cons 15.47/6.33 true :: true:false 15.47/6.33 cons :: hd -> nil:cons -> nil:cons 15.47/6.33 false :: true:false 15.47/6.33 hd :: nil:cons -> hd 15.47/6.33 tl :: nil:cons -> nil:cons 15.47/6.33 append :: nil:cons -> nil:cons -> nil:cons 15.47/6.33 ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons 15.47/6.33 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (5) CompletionProof (UPPER BOUND(ID)) 15.47/6.33 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 15.47/6.33 15.47/6.33 hd(v0) -> null_hd [0] 15.47/6.33 tl(v0) -> null_tl [0] 15.47/6.33 is_empty(v0) -> null_is_empty [0] 15.47/6.33 ifappend(v0, v1, v2) -> null_ifappend [0] 15.47/6.33 15.47/6.33 And the following fresh constants: null_hd, null_tl, null_is_empty, null_ifappend 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (6) 15.47/6.33 Obligation: 15.47/6.33 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 15.47/6.33 15.47/6.33 Runtime Complexity Weighted TRS with Types. 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true [1] 15.47/6.33 is_empty(cons(x, l)) -> false [1] 15.47/6.33 hd(cons(x, l)) -> x [1] 15.47/6.33 tl(cons(x, l)) -> l [1] 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) [1] 15.47/6.33 ifappend(l1, l2, nil) -> l2 [1] 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1] 15.47/6.33 hd(v0) -> null_hd [0] 15.47/6.33 tl(v0) -> null_tl [0] 15.47/6.33 is_empty(v0) -> null_is_empty [0] 15.47/6.33 ifappend(v0, v1, v2) -> null_ifappend [0] 15.47/6.33 15.47/6.33 The TRS has the following type information: 15.47/6.33 is_empty :: nil:cons:null_tl:null_ifappend -> true:false:null_is_empty 15.47/6.33 nil :: nil:cons:null_tl:null_ifappend 15.47/6.33 true :: true:false:null_is_empty 15.47/6.33 cons :: null_hd -> nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend 15.47/6.33 false :: true:false:null_is_empty 15.47/6.33 hd :: nil:cons:null_tl:null_ifappend -> null_hd 15.47/6.33 tl :: nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend 15.47/6.33 append :: nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend 15.47/6.33 ifappend :: nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend -> nil:cons:null_tl:null_ifappend 15.47/6.33 null_hd :: null_hd 15.47/6.33 null_tl :: nil:cons:null_tl:null_ifappend 15.47/6.33 null_is_empty :: true:false:null_is_empty 15.47/6.33 null_ifappend :: nil:cons:null_tl:null_ifappend 15.47/6.33 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 15.47/6.33 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 15.47/6.33 The constant constructors are abstracted as follows: 15.47/6.33 15.47/6.33 nil => 0 15.47/6.33 true => 2 15.47/6.33 false => 1 15.47/6.33 null_hd => 0 15.47/6.33 null_tl => 0 15.47/6.33 null_is_empty => 0 15.47/6.33 null_ifappend => 0 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (8) 15.47/6.33 Obligation: 15.47/6.33 Complexity RNTS consisting of the following rules: 15.47/6.33 15.47/6.33 append(z, z') -{ 1 }-> ifappend(l1, l2, l1) :|: z = l1, z' = l2, l1 >= 0, l2 >= 0 15.47/6.33 hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l 15.47/6.33 hd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 15.47/6.33 ifappend(z, z', z'') -{ 1 }-> l2 :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0 15.47/6.33 ifappend(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 15.47/6.33 ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, l2) :|: z'' = 1 + x + l, z = l1, x >= 0, l >= 0, z' = l2, l1 >= 0, l2 >= 0 15.47/6.33 is_empty(z) -{ 1 }-> 2 :|: z = 0 15.47/6.33 is_empty(z) -{ 1 }-> 1 :|: x >= 0, l >= 0, z = 1 + x + l 15.47/6.33 is_empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 15.47/6.33 tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l 15.47/6.33 tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 15.47/6.33 15.47/6.33 Only complete derivations are relevant for the runtime complexity. 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (9) CompleteCoflocoProof (FINISHED) 15.47/6.33 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 15.47/6.33 15.47/6.33 eq(start(V, V7, V10),0,[fun(V, Out)],[V >= 0]). 15.47/6.33 eq(start(V, V7, V10),0,[hd(V, Out)],[V >= 0]). 15.47/6.33 eq(start(V, V7, V10),0,[tl(V, Out)],[V >= 0]). 15.47/6.33 eq(start(V, V7, V10),0,[append(V, V7, Out)],[V >= 0,V7 >= 0]). 15.47/6.33 eq(start(V, V7, V10),0,[ifappend(V, V7, V10, Out)],[V >= 0,V7 >= 0,V10 >= 0]). 15.47/6.33 eq(fun(V, Out),1,[],[Out = 2,V = 0]). 15.47/6.33 eq(fun(V, Out),1,[],[Out = 1,V1 >= 0,V2 >= 0,V = 1 + V1 + V2]). 15.47/6.33 eq(hd(V, Out),1,[],[Out = V3,V3 >= 0,V4 >= 0,V = 1 + V3 + V4]). 15.47/6.33 eq(tl(V, Out),1,[],[Out = V6,V5 >= 0,V6 >= 0,V = 1 + V5 + V6]). 15.47/6.33 eq(append(V, V7, Out),1,[ifappend(V9, V8, V9, Ret)],[Out = Ret,V = V9,V7 = V8,V9 >= 0,V8 >= 0]). 15.47/6.33 eq(ifappend(V, V7, V10, Out),1,[],[Out = V11,V10 = 0,V = V12,V7 = V11,V12 >= 0,V11 >= 0]). 15.47/6.33 eq(ifappend(V, V7, V10, Out),1,[append(V15, V16, Ret1)],[Out = 1 + Ret1 + V13,V10 = 1 + V13 + V15,V = V14,V13 >= 0,V15 >= 0,V7 = V16,V14 >= 0,V16 >= 0]). 15.47/6.33 eq(hd(V, Out),0,[],[Out = 0,V17 >= 0,V = V17]). 15.47/6.33 eq(tl(V, Out),0,[],[Out = 0,V18 >= 0,V = V18]). 15.47/6.33 eq(fun(V, Out),0,[],[Out = 0,V19 >= 0,V = V19]). 15.47/6.33 eq(ifappend(V, V7, V10, Out),0,[],[Out = 0,V20 >= 0,V10 = V22,V21 >= 0,V = V20,V7 = V21,V22 >= 0]). 15.47/6.33 input_output_vars(fun(V,Out),[V],[Out]). 15.47/6.33 input_output_vars(hd(V,Out),[V],[Out]). 15.47/6.33 input_output_vars(tl(V,Out),[V],[Out]). 15.47/6.33 input_output_vars(append(V,V7,Out),[V,V7],[Out]). 15.47/6.33 input_output_vars(ifappend(V,V7,V10,Out),[V,V7,V10],[Out]). 15.47/6.33 15.47/6.33 15.47/6.33 CoFloCo proof output: 15.47/6.33 Preprocessing Cost Relations 15.47/6.33 ===================================== 15.47/6.33 15.47/6.33 #### Computed strongly connected components 15.47/6.33 0. recursive : [append/3,ifappend/4] 15.47/6.33 1. non_recursive : [fun/2] 15.47/6.33 2. non_recursive : [hd/2] 15.47/6.33 3. non_recursive : [tl/2] 15.47/6.33 4. non_recursive : [start/3] 15.47/6.33 15.47/6.33 #### Obtained direct recursion through partial evaluation 15.47/6.33 0. SCC is partially evaluated into append/3 15.47/6.33 1. SCC is partially evaluated into fun/2 15.47/6.33 2. SCC is partially evaluated into hd/2 15.47/6.33 3. SCC is partially evaluated into tl/2 15.47/6.33 4. SCC is partially evaluated into start/3 15.47/6.33 15.47/6.33 Control-Flow Refinement of Cost Relations 15.47/6.33 ===================================== 15.47/6.33 15.47/6.33 ### Specialization of cost equations append/3 15.47/6.33 * CE 8 is refined into CE [18] 15.47/6.33 * CE 10 is refined into CE [19] 15.47/6.33 * CE 9 is refined into CE [20] 15.47/6.33 15.47/6.33 15.47/6.33 ### Cost equations --> "Loop" of append/3 15.47/6.33 * CEs [20] --> Loop 12 15.47/6.33 * CEs [18] --> Loop 13 15.47/6.33 * CEs [19] --> Loop 14 15.47/6.33 15.47/6.33 ### Ranking functions of CR append(V,V7,Out) 15.47/6.33 * RF of phase [12]: [V] 15.47/6.33 15.47/6.33 #### Partial ranking functions of CR append(V,V7,Out) 15.47/6.33 * Partial RF of phase [12]: 15.47/6.33 - RF of loop [12:1]: 15.47/6.33 V 15.47/6.33 15.47/6.33 15.47/6.33 ### Specialization of cost equations fun/2 15.47/6.33 * CE 12 is refined into CE [21] 15.47/6.33 * CE 13 is refined into CE [22] 15.47/6.33 * CE 11 is refined into CE [23] 15.47/6.33 15.47/6.33 15.47/6.33 ### Cost equations --> "Loop" of fun/2 15.47/6.33 * CEs [21] --> Loop 15 15.47/6.33 * CEs [22] --> Loop 16 15.47/6.33 * CEs [23] --> Loop 17 15.47/6.33 15.47/6.33 ### Ranking functions of CR fun(V,Out) 15.47/6.33 15.47/6.33 #### Partial ranking functions of CR fun(V,Out) 15.47/6.33 15.47/6.33 15.47/6.33 ### Specialization of cost equations hd/2 15.47/6.33 * CE 14 is refined into CE [24] 15.47/6.33 * CE 15 is refined into CE [25] 15.47/6.33 15.47/6.33 15.47/6.33 ### Cost equations --> "Loop" of hd/2 15.47/6.33 * CEs [24] --> Loop 18 15.47/6.33 * CEs [25] --> Loop 19 15.47/6.33 15.47/6.33 ### Ranking functions of CR hd(V,Out) 15.47/6.33 15.47/6.33 #### Partial ranking functions of CR hd(V,Out) 15.47/6.33 15.47/6.33 15.47/6.33 ### Specialization of cost equations tl/2 15.47/6.33 * CE 16 is refined into CE [26] 15.47/6.33 * CE 17 is refined into CE [27] 15.47/6.33 15.47/6.33 15.47/6.33 ### Cost equations --> "Loop" of tl/2 15.47/6.33 * CEs [26] --> Loop 20 15.47/6.33 * CEs [27] --> Loop 21 15.47/6.33 15.47/6.33 ### Ranking functions of CR tl(V,Out) 15.47/6.33 15.47/6.33 #### Partial ranking functions of CR tl(V,Out) 15.47/6.33 15.47/6.33 15.47/6.33 ### Specialization of cost equations start/3 15.47/6.33 * CE 1 is refined into CE [28] 15.47/6.33 * CE 2 is refined into CE [29,30,31,32] 15.47/6.33 * CE 3 is refined into CE [33] 15.47/6.33 * CE 4 is refined into CE [34,35,36] 15.47/6.33 * CE 5 is refined into CE [37,38] 15.47/6.33 * CE 6 is refined into CE [39,40] 15.47/6.33 * CE 7 is refined into CE [41,42,43,44] 15.47/6.33 15.47/6.33 15.47/6.33 ### Cost equations --> "Loop" of start/3 15.47/6.33 * CEs [33] --> Loop 22 15.47/6.33 * CEs [28,29,30,31,32,34,35,36,37,38,39,40,41,42,43,44] --> Loop 23 15.47/6.33 15.47/6.33 ### Ranking functions of CR start(V,V7,V10) 15.47/6.33 15.47/6.33 #### Partial ranking functions of CR start(V,V7,V10) 15.47/6.33 15.47/6.33 15.47/6.33 Computing Bounds 15.47/6.33 ===================================== 15.47/6.33 15.47/6.33 #### Cost of chains of append(V,V7,Out): 15.47/6.33 * Chain [[12],14]: 2*it(12)+2 15.47/6.33 Such that:it(12) =< -V7+Out 15.47/6.33 15.47/6.33 with precondition: [V+V7=Out,V>=1,V7>=0] 15.47/6.33 15.47/6.33 * Chain [[12],13]: 2*it(12)+1 15.47/6.33 Such that:it(12) =< Out 15.47/6.33 15.47/6.33 with precondition: [V7>=0,Out>=1,V>=Out] 15.47/6.33 15.47/6.33 * Chain [14]: 2 15.47/6.33 with precondition: [V=0,V7=Out,V7>=0] 15.47/6.33 15.47/6.33 * Chain [13]: 1 15.47/6.33 with precondition: [Out=0,V>=0,V7>=0] 15.47/6.33 15.47/6.33 15.47/6.33 #### Cost of chains of fun(V,Out): 15.47/6.33 * Chain [17]: 1 15.47/6.33 with precondition: [V=0,Out=2] 15.47/6.33 15.47/6.33 * Chain [16]: 0 15.47/6.33 with precondition: [Out=0,V>=0] 15.47/6.33 15.47/6.33 * Chain [15]: 1 15.47/6.33 with precondition: [Out=1,V>=1] 15.47/6.33 15.47/6.33 15.47/6.33 #### Cost of chains of hd(V,Out): 15.47/6.33 * Chain [19]: 0 15.47/6.33 with precondition: [Out=0,V>=0] 15.47/6.33 15.47/6.33 * Chain [18]: 1 15.47/6.33 with precondition: [Out>=0,V>=Out+1] 15.47/6.33 15.47/6.33 15.47/6.33 #### Cost of chains of tl(V,Out): 15.47/6.33 * Chain [21]: 0 15.47/6.33 with precondition: [Out=0,V>=0] 15.47/6.33 15.47/6.33 * Chain [20]: 1 15.47/6.33 with precondition: [Out>=0,V>=Out+1] 15.47/6.33 15.47/6.33 15.47/6.33 #### Cost of chains of start(V,V7,V10): 15.47/6.33 * Chain [23]: 4*s(1)+4*s(3)+3 15.47/6.33 Such that:aux(1) =< V 15.47/6.33 aux(2) =< V10 15.47/6.33 s(3) =< aux(1) 15.47/6.33 s(1) =< aux(2) 15.47/6.33 15.47/6.33 with precondition: [V>=0] 15.47/6.33 15.47/6.33 * Chain [22]: 1 15.47/6.33 with precondition: [V10=0,V>=0,V7>=0] 15.47/6.33 15.47/6.33 15.47/6.33 Closed-form bounds of start(V,V7,V10): 15.47/6.33 ------------------------------------- 15.47/6.33 * Chain [23] with precondition: [V>=0] 15.47/6.33 - Upper bound: 4*V+3+nat(V10)*4 15.47/6.33 - Complexity: n 15.47/6.33 * Chain [22] with precondition: [V10=0,V>=0,V7>=0] 15.47/6.33 - Upper bound: 1 15.47/6.33 - Complexity: constant 15.47/6.33 15.47/6.33 ### Maximum cost of start(V,V7,V10): 4*V+2+nat(V10)*4+1 15.47/6.33 Asymptotic class: n 15.47/6.33 * Total analysis performed in 119 ms. 15.47/6.33 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (10) 15.47/6.33 BOUNDS(1, n^1) 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 15.47/6.33 Transformed a relative TRS into a decreasing-loop problem. 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (12) 15.47/6.33 Obligation: 15.47/6.33 Analyzing the following TRS for decreasing loops: 15.47/6.33 15.47/6.33 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.47/6.33 15.47/6.33 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true 15.47/6.33 is_empty(cons(x, l)) -> false 15.47/6.33 hd(cons(x, l)) -> x 15.47/6.33 tl(cons(x, l)) -> l 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) 15.47/6.33 ifappend(l1, l2, nil) -> l2 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) 15.47/6.33 15.47/6.33 S is empty. 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (13) DecreasingLoopProof (LOWER BOUND(ID)) 15.47/6.33 The following loop(s) give(s) rise to the lower bound Omega(n^1): 15.47/6.33 15.47/6.33 The rewrite sequence 15.47/6.33 15.47/6.33 append(cons(x3_0, l4_0), l2) ->^+ cons(x3_0, append(l4_0, l2)) 15.47/6.33 15.47/6.33 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 15.47/6.33 15.47/6.33 The pumping substitution is [l4_0 / cons(x3_0, l4_0)]. 15.47/6.33 15.47/6.33 The result substitution is [ ]. 15.47/6.33 15.47/6.33 15.47/6.33 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (14) 15.47/6.33 Complex Obligation (BEST) 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (15) 15.47/6.33 Obligation: 15.47/6.33 Proved the lower bound n^1 for the following obligation: 15.47/6.33 15.47/6.33 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.47/6.33 15.47/6.33 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true 15.47/6.33 is_empty(cons(x, l)) -> false 15.47/6.33 hd(cons(x, l)) -> x 15.47/6.33 tl(cons(x, l)) -> l 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) 15.47/6.33 ifappend(l1, l2, nil) -> l2 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) 15.47/6.33 15.47/6.33 S is empty. 15.47/6.33 Rewrite Strategy: INNERMOST 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (16) LowerBoundPropagationProof (FINISHED) 15.47/6.33 Propagated lower bound. 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (17) 15.47/6.33 BOUNDS(n^1, INF) 15.47/6.33 15.47/6.33 ---------------------------------------- 15.47/6.33 15.47/6.33 (18) 15.47/6.33 Obligation: 15.47/6.33 Analyzing the following TRS for decreasing loops: 15.47/6.33 15.47/6.33 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 15.47/6.33 15.47/6.33 15.47/6.33 The TRS R consists of the following rules: 15.47/6.33 15.47/6.33 is_empty(nil) -> true 15.47/6.33 is_empty(cons(x, l)) -> false 15.47/6.33 hd(cons(x, l)) -> x 15.47/6.33 tl(cons(x, l)) -> l 15.47/6.33 append(l1, l2) -> ifappend(l1, l2, l1) 15.47/6.33 ifappend(l1, l2, nil) -> l2 15.47/6.33 ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) 15.47/6.33 15.47/6.33 S is empty. 15.47/6.33 Rewrite Strategy: INNERMOST 15.52/7.06 EOF