1000.18/291.48 WORST_CASE(Omega(n^1), O(n^2)) 1000.25/291.49 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1000.25/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1000.25/291.49 1000.25/291.49 1000.25/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1000.25/291.49 1000.25/291.49 (0) CpxTRS 1000.25/291.49 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1000.25/291.49 (2) CpxWeightedTrs 1000.25/291.49 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1000.25/291.49 (4) CpxTypedWeightedTrs 1000.25/291.49 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 1000.25/291.49 (6) CpxTypedWeightedCompleteTrs 1000.25/291.49 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] 1000.25/291.49 (8) CpxRNTS 1000.25/291.49 (9) CompleteCoflocoProof [FINISHED, 580 ms] 1000.25/291.49 (10) BOUNDS(1, n^2) 1000.25/291.49 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1000.25/291.49 (12) TRS for Loop Detection 1000.25/291.49 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1000.25/291.49 (14) BEST 1000.25/291.49 (15) proven lower bound 1000.25/291.49 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1000.25/291.49 (17) BOUNDS(n^1, INF) 1000.25/291.49 (18) TRS for Loop Detection 1000.25/291.49 1000.25/291.49 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (0) 1000.25/291.49 Obligation: 1000.25/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1000.25/291.49 1000.25/291.49 1000.25/291.49 The TRS R consists of the following rules: 1000.25/291.49 1000.25/291.49 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 1000.25/291.49 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 1000.25/291.49 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 1000.25/291.49 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 1000.25/291.49 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 1000.25/291.49 gr(0, x) -> false 1000.25/291.49 gr(s(x), 0) -> true 1000.25/291.49 gr(s(x), s(y)) -> gr(x, y) 1000.25/291.49 add(0, x) -> x 1000.25/291.49 add(s(x), y) -> s(add(x, y)) 1000.25/291.49 p(0) -> 0 1000.25/291.49 p(s(x)) -> x 1000.25/291.49 1000.25/291.49 S is empty. 1000.25/291.49 Rewrite Strategy: INNERMOST 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1000.25/291.49 Transformed relative TRS to weighted TRS 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (2) 1000.25/291.49 Obligation: 1000.25/291.49 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 1000.25/291.49 1000.25/291.49 1000.25/291.49 The TRS R consists of the following rules: 1000.25/291.49 1000.25/291.49 cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] 1000.25/291.49 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1] 1000.25/291.49 cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] 1000.25/291.49 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1] 1000.25/291.49 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) [1] 1000.25/291.49 gr(0, x) -> false [1] 1000.25/291.49 gr(s(x), 0) -> true [1] 1000.25/291.49 gr(s(x), s(y)) -> gr(x, y) [1] 1000.25/291.49 add(0, x) -> x [1] 1000.25/291.49 add(s(x), y) -> s(add(x, y)) [1] 1000.25/291.49 p(0) -> 0 [1] 1000.25/291.49 p(s(x)) -> x [1] 1000.25/291.49 1000.25/291.49 Rewrite Strategy: INNERMOST 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1000.25/291.49 Infered types. 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (4) 1000.25/291.49 Obligation: 1000.25/291.49 Runtime Complexity Weighted TRS with Types. 1000.25/291.49 The TRS R consists of the following rules: 1000.25/291.49 1000.25/291.49 cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] 1000.25/291.49 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1] 1000.25/291.49 cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] 1000.25/291.49 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1] 1000.25/291.49 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) [1] 1000.25/291.49 gr(0, x) -> false [1] 1000.25/291.49 gr(s(x), 0) -> true [1] 1000.25/291.49 gr(s(x), s(y)) -> gr(x, y) [1] 1000.25/291.49 add(0, x) -> x [1] 1000.25/291.49 add(s(x), y) -> s(add(x, y)) [1] 1000.25/291.49 p(0) -> 0 [1] 1000.25/291.49 p(s(x)) -> x [1] 1000.25/291.49 1000.25/291.49 The TRS has the following type information: 1000.25/291.49 cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3 1000.25/291.49 true :: true:false 1000.25/291.49 cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3 1000.25/291.49 gr :: 0:s -> 0:s -> true:false 1000.25/291.49 0 :: 0:s 1000.25/291.49 add :: 0:s -> 0:s -> 0:s 1000.25/291.49 p :: 0:s -> 0:s 1000.25/291.49 false :: true:false 1000.25/291.49 cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3 1000.25/291.49 s :: 0:s -> 0:s 1000.25/291.49 1000.25/291.49 Rewrite Strategy: INNERMOST 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (5) CompletionProof (UPPER BOUND(ID)) 1000.25/291.49 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 1000.25/291.49 1000.25/291.49 cond1(v0, v1, v2) -> null_cond1 [0] 1000.25/291.49 1000.25/291.49 And the following fresh constants: null_cond1 1000.25/291.49 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (6) 1000.25/291.49 Obligation: 1000.25/291.49 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 1000.25/291.49 1000.25/291.49 Runtime Complexity Weighted TRS with Types. 1000.25/291.49 The TRS R consists of the following rules: 1000.25/291.49 1000.25/291.49 cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] 1000.25/291.49 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1] 1000.25/291.49 cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] 1000.25/291.49 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1] 1000.25/291.49 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) [1] 1000.25/291.49 gr(0, x) -> false [1] 1000.25/291.49 gr(s(x), 0) -> true [1] 1000.25/291.49 gr(s(x), s(y)) -> gr(x, y) [1] 1000.25/291.49 add(0, x) -> x [1] 1000.25/291.49 add(s(x), y) -> s(add(x, y)) [1] 1000.25/291.49 p(0) -> 0 [1] 1000.25/291.49 p(s(x)) -> x [1] 1000.25/291.49 cond1(v0, v1, v2) -> null_cond1 [0] 1000.25/291.49 1000.25/291.49 The TRS has the following type information: 1000.25/291.49 cond1 :: true:false -> 0:s -> 0:s -> null_cond1 1000.25/291.49 true :: true:false 1000.25/291.49 cond2 :: true:false -> 0:s -> 0:s -> null_cond1 1000.25/291.49 gr :: 0:s -> 0:s -> true:false 1000.25/291.49 0 :: 0:s 1000.25/291.49 add :: 0:s -> 0:s -> 0:s 1000.25/291.49 p :: 0:s -> 0:s 1000.25/291.49 false :: true:false 1000.25/291.49 cond3 :: true:false -> 0:s -> 0:s -> null_cond1 1000.25/291.49 s :: 0:s -> 0:s 1000.25/291.49 null_cond1 :: null_cond1 1000.25/291.49 1000.25/291.49 Rewrite Strategy: INNERMOST 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1000.25/291.49 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1000.25/291.49 The constant constructors are abstracted as follows: 1000.25/291.49 1000.25/291.49 true => 1 1000.25/291.49 0 => 0 1000.25/291.49 false => 0 1000.25/291.49 null_cond1 => 0 1000.25/291.49 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (8) 1000.25/291.49 Obligation: 1000.25/291.49 Complexity RNTS consisting of the following rules: 1000.25/291.49 1000.25/291.49 add(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 1000.25/291.49 add(z, z') -{ 1 }-> 1 + add(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 1000.25/291.49 cond1(z, z', z'') -{ 1 }-> cond2(gr(x, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 1000.25/291.49 cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1000.25/291.49 cond2(z, z', z'') -{ 1 }-> cond3(gr(y, 0), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 1000.25/291.49 cond2(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 1000.25/291.49 cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 1000.25/291.49 cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), x, p(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 1000.25/291.49 gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 1000.25/291.49 gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 1000.25/291.49 gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 1000.25/291.49 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 1000.25/291.49 p(z) -{ 1 }-> 0 :|: z = 0 1000.25/291.49 1000.25/291.49 Only complete derivations are relevant for the runtime complexity. 1000.25/291.49 1000.25/291.49 ---------------------------------------- 1000.25/291.49 1000.25/291.49 (9) CompleteCoflocoProof (FINISHED) 1000.25/291.49 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 1000.25/291.49 1000.25/291.49 eq(start(V1, V, V2),0,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 1000.25/291.49 eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 1000.25/291.49 eq(start(V1, V, V2),0,[cond3(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 1000.25/291.49 eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]). 1000.25/291.49 eq(start(V1, V, V2),0,[add(V1, V, Out)],[V1 >= 0,V >= 0]). 1000.25/291.49 eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]). 1000.25/291.49 eq(cond1(V1, V, V2, Out),1,[gr(V4, 0, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]). 1000.25/291.49 eq(cond2(V1, V, V2, Out),1,[add(V5, V6, Ret00),gr(Ret00, 0, Ret01),p(V5, Ret1),cond1(Ret01, Ret1, V6, Ret2)],[Out = Ret2,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]). 1000.25/291.49 eq(cond2(V1, V, V2, Out),1,[gr(V7, 0, Ret02),cond3(Ret02, V8, V7, Ret3)],[Out = Ret3,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]). 1000.25/291.49 eq(cond3(V1, V, V2, Out),1,[add(V9, V10, Ret001),gr(Ret001, 0, Ret03),p(V10, Ret21),cond1(Ret03, V9, Ret21, Ret4)],[Out = Ret4,V = V9,V2 = V10,V1 = 1,V9 >= 0,V10 >= 0]). 1000.25/291.49 eq(cond3(V1, V, V2, Out),1,[add(V12, V11, Ret002),gr(Ret002, 0, Ret04),cond1(Ret04, V12, V11, Ret5)],[Out = Ret5,V = V12,V2 = V11,V12 >= 0,V11 >= 0,V1 = 0]). 1000.25/291.49 eq(gr(V1, V, Out),1,[],[Out = 0,V = V13,V13 >= 0,V1 = 0]). 1000.25/291.49 eq(gr(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]). 1000.25/291.49 eq(gr(V1, V, Out),1,[gr(V16, V15, Ret6)],[Out = Ret6,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]). 1000.25/291.49 eq(add(V1, V, Out),1,[],[Out = V17,V = V17,V17 >= 0,V1 = 0]). 1000.25/291.49 eq(add(V1, V, Out),1,[add(V19, V18, Ret11)],[Out = 1 + Ret11,V19 >= 0,V18 >= 0,V1 = 1 + V19,V = V18]). 1000.25/291.49 eq(p(V1, Out),1,[],[Out = 0,V1 = 0]). 1000.25/291.49 eq(p(V1, Out),1,[],[Out = V20,V20 >= 0,V1 = 1 + V20]). 1000.25/291.50 eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V22 >= 0,V2 = V23,V21 >= 0,V1 = V22,V = V21,V23 >= 0]). 1000.25/291.50 input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]). 1000.25/291.50 input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]). 1000.25/291.50 input_output_vars(cond3(V1,V,V2,Out),[V1,V,V2],[Out]). 1000.25/291.50 input_output_vars(gr(V1,V,Out),[V1,V],[Out]). 1000.25/291.50 input_output_vars(add(V1,V,Out),[V1,V],[Out]). 1000.25/291.50 input_output_vars(p(V1,Out),[V1],[Out]). 1000.25/291.50 1000.25/291.50 1000.25/291.50 CoFloCo proof output: 1000.25/291.50 Preprocessing Cost Relations 1000.25/291.50 ===================================== 1000.25/291.50 1000.25/291.50 #### Computed strongly connected components 1000.25/291.50 0. recursive : [add/3] 1000.25/291.50 1. recursive : [gr/3] 1000.25/291.50 2. non_recursive : [p/2] 1000.25/291.50 3. recursive : [cond1/4,cond2/4,cond3/4] 1000.25/291.50 4. non_recursive : [start/3] 1000.25/291.50 1000.25/291.50 #### Obtained direct recursion through partial evaluation 1000.25/291.50 0. SCC is partially evaluated into add/3 1000.25/291.50 1. SCC is partially evaluated into gr/3 1000.25/291.50 2. SCC is partially evaluated into p/2 1000.25/291.50 3. SCC is partially evaluated into cond1/4 1000.25/291.50 4. SCC is partially evaluated into start/3 1000.25/291.50 1000.25/291.50 Control-Flow Refinement of Cost Relations 1000.25/291.50 ===================================== 1000.25/291.50 1000.25/291.50 ### Specialization of cost equations add/3 1000.25/291.50 * CE 11 is refined into CE [21] 1000.25/291.50 * CE 10 is refined into CE [22] 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Cost equations --> "Loop" of add/3 1000.25/291.50 * CEs [22] --> Loop 13 1000.25/291.50 * CEs [21] --> Loop 14 1000.25/291.50 1000.25/291.50 ### Ranking functions of CR add(V1,V,Out) 1000.25/291.50 * RF of phase [14]: [V1] 1000.25/291.50 1000.25/291.50 #### Partial ranking functions of CR add(V1,V,Out) 1000.25/291.50 * Partial RF of phase [14]: 1000.25/291.50 - RF of loop [14:1]: 1000.25/291.50 V1 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Specialization of cost equations gr/3 1000.25/291.50 * CE 14 is refined into CE [23] 1000.25/291.50 * CE 13 is refined into CE [24] 1000.25/291.50 * CE 12 is refined into CE [25] 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Cost equations --> "Loop" of gr/3 1000.25/291.50 * CEs [24] --> Loop 15 1000.25/291.50 * CEs [25] --> Loop 16 1000.25/291.50 * CEs [23] --> Loop 17 1000.25/291.50 1000.25/291.50 ### Ranking functions of CR gr(V1,V,Out) 1000.25/291.50 * RF of phase [17]: [V,V1] 1000.25/291.50 1000.25/291.50 #### Partial ranking functions of CR gr(V1,V,Out) 1000.25/291.50 * Partial RF of phase [17]: 1000.25/291.50 - RF of loop [17:1]: 1000.25/291.50 V 1000.25/291.50 V1 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Specialization of cost equations p/2 1000.25/291.50 * CE 16 is refined into CE [26] 1000.25/291.50 * CE 15 is refined into CE [27] 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Cost equations --> "Loop" of p/2 1000.25/291.50 * CEs [26] --> Loop 18 1000.25/291.50 * CEs [27] --> Loop 19 1000.25/291.50 1000.25/291.50 ### Ranking functions of CR p(V1,Out) 1000.25/291.50 1000.25/291.50 #### Partial ranking functions of CR p(V1,Out) 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Specialization of cost equations cond1/4 1000.25/291.50 * CE 20 is refined into CE [28] 1000.25/291.50 * CE 17 is refined into CE [29] 1000.25/291.50 * CE 18 is refined into CE [30] 1000.25/291.50 * CE 19 is refined into CE [31] 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Cost equations --> "Loop" of cond1/4 1000.25/291.50 * CEs [29] --> Loop 20 1000.25/291.50 * CEs [30] --> Loop 21 1000.25/291.50 * CEs [31] --> Loop 22 1000.25/291.50 * CEs [28] --> Loop 23 1000.25/291.50 1000.25/291.50 ### Ranking functions of CR cond1(V1,V,V2,Out) 1000.25/291.50 * RF of phase [20]: [V] 1000.25/291.50 * RF of phase [21]: [V2] 1000.25/291.50 1000.25/291.50 #### Partial ranking functions of CR cond1(V1,V,V2,Out) 1000.25/291.50 * Partial RF of phase [20]: 1000.25/291.50 - RF of loop [20:1]: 1000.25/291.50 V 1000.25/291.50 * Partial RF of phase [21]: 1000.25/291.50 - RF of loop [21:1]: 1000.25/291.50 V2 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Specialization of cost equations start/3 1000.25/291.50 * CE 1 is refined into CE [32,33,34,35,36,37] 1000.25/291.50 * CE 5 is refined into CE [38,39,40,41,42,43,44] 1000.25/291.50 * CE 2 is refined into CE [45,46,47,48] 1000.25/291.50 * CE 3 is refined into CE [49,50,51] 1000.25/291.50 * CE 4 is refined into CE [52,53,54,55,56] 1000.25/291.50 * CE 6 is refined into CE [57,58,59] 1000.25/291.50 * CE 7 is refined into CE [60,61,62,63] 1000.25/291.50 * CE 8 is refined into CE [64,65] 1000.25/291.50 * CE 9 is refined into CE [66,67] 1000.25/291.50 1000.25/291.50 1000.25/291.50 ### Cost equations --> "Loop" of start/3 1000.25/291.50 * CEs [41,42] --> Loop 24 1000.25/291.50 * CEs [35,36,37,43,44,59] --> Loop 25 1000.25/291.50 * CEs [33,34,39,40,57,58] --> Loop 26 1000.25/291.50 * CEs [32,38,61,62,63,65,67] --> Loop 27 1000.25/291.50 * CEs [45,46,47,48,49,50,51,52,53,54,55,56,60,64,66] --> Loop 28 1000.25/291.50 1000.25/291.50 ### Ranking functions of CR start(V1,V,V2) 1000.25/291.50 1000.25/291.50 #### Partial ranking functions of CR start(V1,V,V2) 1000.25/291.50 1000.25/291.50 1000.25/291.50 Computing Bounds 1000.25/291.50 ===================================== 1000.25/291.50 1000.25/291.50 #### Cost of chains of add(V1,V,Out): 1000.25/291.50 * Chain [[14],13]: 1*it(14)+1 1000.25/291.50 Such that:it(14) =< -V+Out 1000.25/291.50 1000.25/291.50 with precondition: [V+V1=Out,V1>=1,V>=0] 1000.25/291.50 1000.25/291.50 * Chain [13]: 1 1000.25/291.50 with precondition: [V1=0,V=Out,V>=0] 1000.25/291.50 1000.25/291.50 1000.25/291.50 #### Cost of chains of gr(V1,V,Out): 1000.25/291.50 * Chain [[17],16]: 1*it(17)+1 1000.25/291.50 Such that:it(17) =< V1 1000.25/291.50 1000.25/291.50 with precondition: [Out=0,V1>=1,V>=V1] 1000.25/291.50 1000.25/291.50 * Chain [[17],15]: 1*it(17)+1 1000.25/291.50 Such that:it(17) =< V 1000.25/291.50 1000.25/291.50 with precondition: [Out=1,V>=1,V1>=V+1] 1000.25/291.50 1000.25/291.50 * Chain [16]: 1 1000.25/291.50 with precondition: [V1=0,Out=0,V>=0] 1000.25/291.50 1000.25/291.50 * Chain [15]: 1 1000.25/291.50 with precondition: [V=0,Out=1,V1>=1] 1000.25/291.50 1000.25/291.50 1000.25/291.50 #### Cost of chains of p(V1,Out): 1000.25/291.50 * Chain [19]: 1 1000.25/291.50 with precondition: [V1=0,Out=0] 1000.25/291.50 1000.25/291.50 * Chain [18]: 1 1000.25/291.50 with precondition: [V1=Out+1,V1>=1] 1000.25/291.50 1000.25/291.50 1000.25/291.50 #### Cost of chains of cond1(V1,V,V2,Out): 1000.25/291.50 * Chain [[21],23]: 8*it(21)+0 1000.25/291.50 Such that:it(21) =< V2 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,V=0,Out=0,V2>=1] 1000.25/291.50 1000.25/291.50 * Chain [[21],22,23]: 8*it(21)+7 1000.25/291.50 Such that:it(21) =< V2 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,V=0,Out=0,V2>=1] 1000.25/291.50 1000.25/291.50 * Chain [[20],[21],23]: 6*it(20)+8*it(21)+1*s(3)+0 1000.25/291.50 Such that:it(21) =< V2 1000.25/291.50 aux(3) =< V 1000.25/291.50 it(20) =< aux(3) 1000.25/291.50 s(3) =< it(20)*aux(3) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,Out=0,V>=1,V2>=1] 1000.25/291.50 1000.25/291.50 * Chain [[20],[21],22,23]: 6*it(20)+8*it(21)+1*s(3)+7 1000.25/291.50 Such that:it(21) =< V2 1000.25/291.50 aux(4) =< V 1000.25/291.50 it(20) =< aux(4) 1000.25/291.50 s(3) =< it(20)*aux(4) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,Out=0,V>=1,V2>=1] 1000.25/291.50 1000.25/291.50 * Chain [[20],23]: 6*it(20)+1*s(3)+0 1000.25/291.50 Such that:aux(5) =< V 1000.25/291.50 it(20) =< aux(5) 1000.25/291.50 s(3) =< it(20)*aux(5) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,Out=0,V>=1,V2>=0] 1000.25/291.50 1000.25/291.50 * Chain [[20],22,23]: 6*it(20)+1*s(3)+7 1000.25/291.50 Such that:aux(6) =< V 1000.25/291.50 it(20) =< aux(6) 1000.25/291.50 s(3) =< it(20)*aux(6) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,V2=0,Out=0,V>=1] 1000.25/291.50 1000.25/291.50 * Chain [23]: 0 1000.25/291.50 with precondition: [Out=0,V1>=0,V>=0,V2>=0] 1000.25/291.50 1000.25/291.50 * Chain [22,23]: 7 1000.25/291.50 with precondition: [V1=1,V=0,V2=0,Out=0] 1000.25/291.50 1000.25/291.50 1000.25/291.50 #### Cost of chains of start(V1,V,V2): 1000.25/291.50 * Chain [28]: 64*s(21)+78*s(22)+12*s(28)+13 1000.25/291.50 Such that:aux(13) =< V 1000.25/291.50 aux(14) =< V2 1000.25/291.50 s(22) =< aux(13) 1000.25/291.50 s(28) =< s(22)*aux(13) 1000.25/291.50 s(21) =< aux(14) 1000.25/291.50 1000.25/291.50 with precondition: [V1=0] 1000.25/291.50 1000.25/291.50 * Chain [27]: 2*s(45)+1*s(46)+11 1000.25/291.50 Such that:s(46) =< V 1000.25/291.50 aux(15) =< V1 1000.25/291.50 s(45) =< aux(15) 1000.25/291.50 1000.25/291.50 with precondition: [V1>=1] 1000.25/291.50 1000.25/291.50 * Chain [26]: 48*s(49)+11 1000.25/291.50 Such that:aux(16) =< V2 1000.25/291.50 s(49) =< aux(16) 1000.25/291.50 1000.25/291.50 with precondition: [V1>=0,V>=0,V2>=0] 1000.25/291.50 1000.25/291.50 * Chain [25]: 76*s(54)+1*s(55)+64*s(57)+12*s(63)+11 1000.25/291.50 Such that:s(55) =< 1 1000.25/291.50 aux(19) =< V 1000.25/291.50 aux(20) =< V2 1000.25/291.50 s(54) =< aux(19) 1000.25/291.50 s(57) =< aux(20) 1000.25/291.50 s(63) =< s(54)*aux(19) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,V>=1,V2>=0] 1000.25/291.50 1000.25/291.50 * Chain [24]: 26*s(76)+4*s(82)+11 1000.25/291.50 Such that:aux(22) =< V 1000.25/291.50 s(76) =< aux(22) 1000.25/291.50 s(82) =< s(76)*aux(22) 1000.25/291.50 1000.25/291.50 with precondition: [V1=1,V2=0,V>=1] 1000.25/291.50 1000.25/291.50 1000.25/291.50 Closed-form bounds of start(V1,V,V2): 1000.25/291.50 ------------------------------------- 1000.25/291.50 * Chain [28] with precondition: [V1=0] 1000.25/291.50 - Upper bound: nat(V)*78+13+nat(V)*12*nat(V)+nat(V2)*64 1000.25/291.50 - Complexity: n^2 1000.25/291.50 * Chain [27] with precondition: [V1>=1] 1000.25/291.50 - Upper bound: 2*V1+11+nat(V) 1000.25/291.50 - Complexity: n 1000.25/291.50 * Chain [26] with precondition: [V1>=0,V>=0,V2>=0] 1000.25/291.50 - Upper bound: 48*V2+11 1000.25/291.50 - Complexity: n 1000.25/291.50 * Chain [25] with precondition: [V1=1,V>=1,V2>=0] 1000.25/291.50 - Upper bound: 76*V+12+12*V*V+64*V2 1000.25/291.50 - Complexity: n^2 1000.25/291.50 * Chain [24] with precondition: [V1=1,V2=0,V>=1] 1000.25/291.50 - Upper bound: 26*V+11+4*V*V 1000.25/291.50 - Complexity: n^2 1000.25/291.50 1000.25/291.50 ### Maximum cost of start(V1,V,V2): max([nat(V2)*48,nat(V)+max([2*V1,nat(V)*50+1+nat(V)*8*nat(V)+nat(V2)*64+(nat(V)*2+1)+(nat(V)*4*nat(V)+nat(V)*25)])])+11 1000.25/291.50 Asymptotic class: n^2 1000.25/291.50 * Total analysis performed in 487 ms. 1000.25/291.50 1000.25/291.50 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (10) 1000.25/291.50 BOUNDS(1, n^2) 1000.25/291.50 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1000.25/291.50 Transformed a relative TRS into a decreasing-loop problem. 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (12) 1000.25/291.50 Obligation: 1000.25/291.50 Analyzing the following TRS for decreasing loops: 1000.25/291.50 1000.25/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1000.25/291.50 1000.25/291.50 1000.25/291.50 The TRS R consists of the following rules: 1000.25/291.50 1000.25/291.50 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 1000.25/291.50 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 1000.25/291.50 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 1000.25/291.50 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 1000.25/291.50 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 1000.25/291.50 gr(0, x) -> false 1000.25/291.50 gr(s(x), 0) -> true 1000.25/291.50 gr(s(x), s(y)) -> gr(x, y) 1000.25/291.50 add(0, x) -> x 1000.25/291.50 add(s(x), y) -> s(add(x, y)) 1000.25/291.50 p(0) -> 0 1000.25/291.50 p(s(x)) -> x 1000.25/291.50 1000.25/291.50 S is empty. 1000.25/291.50 Rewrite Strategy: INNERMOST 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (13) DecreasingLoopProof (LOWER BOUND(ID)) 1000.25/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1000.25/291.50 1000.25/291.50 The rewrite sequence 1000.25/291.50 1000.25/291.50 add(s(x), y) ->^+ s(add(x, y)) 1000.25/291.50 1000.25/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1000.25/291.50 1000.25/291.50 The pumping substitution is [x / s(x)]. 1000.25/291.50 1000.25/291.50 The result substitution is [ ]. 1000.25/291.50 1000.25/291.50 1000.25/291.50 1000.25/291.50 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (14) 1000.25/291.50 Complex Obligation (BEST) 1000.25/291.50 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (15) 1000.25/291.50 Obligation: 1000.25/291.50 Proved the lower bound n^1 for the following obligation: 1000.25/291.50 1000.25/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1000.25/291.50 1000.25/291.50 1000.25/291.50 The TRS R consists of the following rules: 1000.25/291.50 1000.25/291.50 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 1000.25/291.50 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 1000.25/291.50 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 1000.25/291.50 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 1000.25/291.50 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 1000.25/291.50 gr(0, x) -> false 1000.25/291.50 gr(s(x), 0) -> true 1000.25/291.50 gr(s(x), s(y)) -> gr(x, y) 1000.25/291.50 add(0, x) -> x 1000.25/291.50 add(s(x), y) -> s(add(x, y)) 1000.25/291.50 p(0) -> 0 1000.25/291.50 p(s(x)) -> x 1000.25/291.50 1000.25/291.50 S is empty. 1000.25/291.50 Rewrite Strategy: INNERMOST 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (16) LowerBoundPropagationProof (FINISHED) 1000.25/291.50 Propagated lower bound. 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (17) 1000.25/291.50 BOUNDS(n^1, INF) 1000.25/291.50 1000.25/291.50 ---------------------------------------- 1000.25/291.50 1000.25/291.50 (18) 1000.25/291.50 Obligation: 1000.25/291.50 Analyzing the following TRS for decreasing loops: 1000.25/291.50 1000.25/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1000.25/291.50 1000.25/291.50 1000.25/291.50 The TRS R consists of the following rules: 1000.25/291.50 1000.25/291.50 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 1000.25/291.50 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 1000.25/291.50 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 1000.25/291.50 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 1000.25/291.50 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 1000.25/291.50 gr(0, x) -> false 1000.25/291.50 gr(s(x), 0) -> true 1000.25/291.50 gr(s(x), s(y)) -> gr(x, y) 1000.25/291.50 add(0, x) -> x 1000.25/291.50 add(s(x), y) -> s(add(x, y)) 1000.25/291.50 p(0) -> 0 1000.25/291.50 p(s(x)) -> x 1000.25/291.50 1000.25/291.50 S is empty. 1000.25/291.50 Rewrite Strategy: INNERMOST 1000.46/291.60 EOF