20.50/6.11 WORST_CASE(Omega(n^1), O(n^1)) 20.50/6.12 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 20.50/6.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 20.50/6.12 20.50/6.12 20.50/6.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.50/6.12 20.50/6.12 (0) CpxTRS 20.50/6.12 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 20.50/6.12 (2) CpxWeightedTrs 20.50/6.12 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 20.50/6.12 (4) CpxTypedWeightedTrs 20.50/6.12 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 20.50/6.12 (6) CpxTypedWeightedCompleteTrs 20.50/6.12 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 20.50/6.12 (8) CpxRNTS 20.50/6.12 (9) CompleteCoflocoProof [FINISHED, 200 ms] 20.50/6.12 (10) BOUNDS(1, n^1) 20.50/6.12 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 20.50/6.12 (12) TRS for Loop Detection 20.50/6.12 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 20.50/6.12 (14) BEST 20.50/6.12 (15) proven lower bound 20.50/6.12 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 20.50/6.12 (17) BOUNDS(n^1, INF) 20.50/6.12 (18) TRS for Loop Detection 20.50/6.12 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (0) 20.50/6.12 Obligation: 20.50/6.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.50/6.12 20.50/6.12 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false 20.50/6.12 nonZero(s(x)) -> true 20.50/6.12 p(0) -> 0 20.50/6.12 p(s(x)) -> x 20.50/6.12 id_inc(x) -> x 20.50/6.12 id_inc(x) -> s(x) 20.50/6.12 random(x) -> rand(x, 0) 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) 20.50/6.12 if(false, x, y) -> y 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) 20.50/6.12 20.50/6.12 S is empty. 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 20.50/6.12 Transformed relative TRS to weighted TRS 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (2) 20.50/6.12 Obligation: 20.50/6.12 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 20.50/6.12 20.50/6.12 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false [1] 20.50/6.12 nonZero(s(x)) -> true [1] 20.50/6.12 p(0) -> 0 [1] 20.50/6.12 p(s(x)) -> x [1] 20.50/6.12 id_inc(x) -> x [1] 20.50/6.12 id_inc(x) -> s(x) [1] 20.50/6.12 random(x) -> rand(x, 0) [1] 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) [1] 20.50/6.12 if(false, x, y) -> y [1] 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) [1] 20.50/6.12 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 20.50/6.12 Infered types. 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (4) 20.50/6.12 Obligation: 20.50/6.12 Runtime Complexity Weighted TRS with Types. 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false [1] 20.50/6.12 nonZero(s(x)) -> true [1] 20.50/6.12 p(0) -> 0 [1] 20.50/6.12 p(s(x)) -> x [1] 20.50/6.12 id_inc(x) -> x [1] 20.50/6.12 id_inc(x) -> s(x) [1] 20.50/6.12 random(x) -> rand(x, 0) [1] 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) [1] 20.50/6.12 if(false, x, y) -> y [1] 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) [1] 20.50/6.12 20.50/6.12 The TRS has the following type information: 20.50/6.12 nonZero :: 0:s -> false:true 20.50/6.12 0 :: 0:s 20.50/6.12 false :: false:true 20.50/6.12 s :: 0:s -> 0:s 20.50/6.12 true :: false:true 20.50/6.12 p :: 0:s -> 0:s 20.50/6.12 id_inc :: 0:s -> 0:s 20.50/6.12 random :: 0:s -> 0:s 20.50/6.12 rand :: 0:s -> 0:s -> 0:s 20.50/6.12 if :: false:true -> 0:s -> 0:s -> 0:s 20.50/6.12 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (5) CompletionProof (UPPER BOUND(ID)) 20.50/6.12 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 20.50/6.12 none 20.50/6.12 20.50/6.12 And the following fresh constants: none 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (6) 20.50/6.12 Obligation: 20.50/6.12 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 20.50/6.12 20.50/6.12 Runtime Complexity Weighted TRS with Types. 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false [1] 20.50/6.12 nonZero(s(x)) -> true [1] 20.50/6.12 p(0) -> 0 [1] 20.50/6.12 p(s(x)) -> x [1] 20.50/6.12 id_inc(x) -> x [1] 20.50/6.12 id_inc(x) -> s(x) [1] 20.50/6.12 random(x) -> rand(x, 0) [1] 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) [1] 20.50/6.12 if(false, x, y) -> y [1] 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) [1] 20.50/6.12 20.50/6.12 The TRS has the following type information: 20.50/6.12 nonZero :: 0:s -> false:true 20.50/6.12 0 :: 0:s 20.50/6.12 false :: false:true 20.50/6.12 s :: 0:s -> 0:s 20.50/6.12 true :: false:true 20.50/6.12 p :: 0:s -> 0:s 20.50/6.12 id_inc :: 0:s -> 0:s 20.50/6.12 random :: 0:s -> 0:s 20.50/6.12 rand :: 0:s -> 0:s -> 0:s 20.50/6.12 if :: false:true -> 0:s -> 0:s -> 0:s 20.50/6.12 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 20.50/6.12 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 20.50/6.12 The constant constructors are abstracted as follows: 20.50/6.12 20.50/6.12 0 => 0 20.50/6.12 false => 0 20.50/6.12 true => 1 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (8) 20.50/6.12 Obligation: 20.50/6.12 Complexity RNTS consisting of the following rules: 20.50/6.12 20.50/6.12 id_inc(z) -{ 1 }-> x :|: x >= 0, z = x 20.50/6.12 id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x 20.50/6.12 if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 20.50/6.12 if(z, z', z'') -{ 1 }-> rand(p(x), id_inc(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 20.50/6.12 nonZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x 20.50/6.12 nonZero(z) -{ 1 }-> 0 :|: z = 0 20.50/6.12 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 20.50/6.12 p(z) -{ 1 }-> 0 :|: z = 0 20.50/6.12 rand(z, z') -{ 1 }-> if(nonZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y 20.50/6.12 random(z) -{ 1 }-> rand(x, 0) :|: x >= 0, z = x 20.50/6.12 20.50/6.12 Only complete derivations are relevant for the runtime complexity. 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (9) CompleteCoflocoProof (FINISHED) 20.50/6.12 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 20.50/6.12 20.50/6.12 eq(start(V, V6, V9),0,[nonZero(V, Out)],[V >= 0]). 20.50/6.12 eq(start(V, V6, V9),0,[p(V, Out)],[V >= 0]). 20.50/6.12 eq(start(V, V6, V9),0,[fun(V, Out)],[V >= 0]). 20.50/6.12 eq(start(V, V6, V9),0,[random(V, Out)],[V >= 0]). 20.50/6.12 eq(start(V, V6, V9),0,[rand(V, V6, Out)],[V >= 0,V6 >= 0]). 20.50/6.12 eq(start(V, V6, V9),0,[if(V, V6, V9, Out)],[V >= 0,V6 >= 0,V9 >= 0]). 20.50/6.12 eq(nonZero(V, Out),1,[],[Out = 0,V = 0]). 20.50/6.12 eq(nonZero(V, Out),1,[],[Out = 1,V1 >= 0,V = 1 + V1]). 20.50/6.12 eq(p(V, Out),1,[],[Out = 0,V = 0]). 20.50/6.12 eq(p(V, Out),1,[],[Out = V2,V2 >= 0,V = 1 + V2]). 20.50/6.12 eq(fun(V, Out),1,[],[Out = V3,V3 >= 0,V = V3]). 20.50/6.12 eq(fun(V, Out),1,[],[Out = 1 + V4,V4 >= 0,V = V4]). 20.50/6.12 eq(random(V, Out),1,[rand(V5, 0, Ret)],[Out = Ret,V5 >= 0,V = V5]). 20.50/6.12 eq(rand(V, V6, Out),1,[nonZero(V7, Ret0),if(Ret0, V7, V8, Ret1)],[Out = Ret1,V7 >= 0,V8 >= 0,V = V7,V6 = V8]). 20.50/6.12 eq(if(V, V6, V9, Out),1,[],[Out = V11,V6 = V10,V9 = V11,V10 >= 0,V11 >= 0,V = 0]). 20.50/6.12 eq(if(V, V6, V9, Out),1,[p(V13, Ret01),fun(V12, Ret11),rand(Ret01, Ret11, Ret2)],[Out = Ret2,V6 = V13,V9 = V12,V = 1,V13 >= 0,V12 >= 0]). 20.50/6.12 input_output_vars(nonZero(V,Out),[V],[Out]). 20.50/6.12 input_output_vars(p(V,Out),[V],[Out]). 20.50/6.12 input_output_vars(fun(V,Out),[V],[Out]). 20.50/6.12 input_output_vars(random(V,Out),[V],[Out]). 20.50/6.12 input_output_vars(rand(V,V6,Out),[V,V6],[Out]). 20.50/6.12 input_output_vars(if(V,V6,V9,Out),[V,V6,V9],[Out]). 20.50/6.12 20.50/6.12 20.50/6.12 CoFloCo proof output: 20.50/6.12 Preprocessing Cost Relations 20.50/6.12 ===================================== 20.50/6.12 20.50/6.12 #### Computed strongly connected components 20.50/6.12 0. non_recursive : [fun/2] 20.50/6.12 1. non_recursive : [p/2] 20.50/6.12 2. non_recursive : [nonZero/2] 20.50/6.12 3. recursive : [if/4,rand/3] 20.50/6.12 4. non_recursive : [random/2] 20.50/6.12 5. non_recursive : [start/3] 20.50/6.12 20.50/6.12 #### Obtained direct recursion through partial evaluation 20.50/6.12 0. SCC is partially evaluated into fun/2 20.50/6.12 1. SCC is partially evaluated into p/2 20.50/6.12 2. SCC is partially evaluated into nonZero/2 20.50/6.12 3. SCC is partially evaluated into rand/3 20.50/6.12 4. SCC is completely evaluated into other SCCs 20.50/6.12 5. SCC is partially evaluated into start/3 20.50/6.12 20.50/6.12 Control-Flow Refinement of Cost Relations 20.50/6.12 ===================================== 20.50/6.12 20.50/6.12 ### Specialization of cost equations fun/2 20.50/6.12 * CE 10 is refined into CE [16] 20.50/6.12 * CE 11 is refined into CE [17] 20.50/6.12 20.50/6.12 20.50/6.12 ### Cost equations --> "Loop" of fun/2 20.50/6.12 * CEs [16] --> Loop 11 20.50/6.12 * CEs [17] --> Loop 12 20.50/6.12 20.50/6.12 ### Ranking functions of CR fun(V,Out) 20.50/6.12 20.50/6.12 #### Partial ranking functions of CR fun(V,Out) 20.50/6.12 20.50/6.12 20.50/6.12 ### Specialization of cost equations p/2 20.50/6.12 * CE 9 is refined into CE [18] 20.50/6.12 * CE 8 is refined into CE [19] 20.50/6.12 20.50/6.12 20.50/6.12 ### Cost equations --> "Loop" of p/2 20.50/6.12 * CEs [18] --> Loop 13 20.50/6.12 * CEs [19] --> Loop 14 20.50/6.12 20.50/6.12 ### Ranking functions of CR p(V,Out) 20.50/6.12 20.50/6.12 #### Partial ranking functions of CR p(V,Out) 20.50/6.12 20.50/6.12 20.50/6.12 ### Specialization of cost equations nonZero/2 20.50/6.12 * CE 15 is refined into CE [20] 20.50/6.12 * CE 14 is refined into CE [21] 20.50/6.12 20.50/6.12 20.50/6.12 ### Cost equations --> "Loop" of nonZero/2 20.50/6.12 * CEs [20] --> Loop 15 20.50/6.12 * CEs [21] --> Loop 16 20.50/6.12 20.50/6.12 ### Ranking functions of CR nonZero(V,Out) 20.50/6.12 20.50/6.12 #### Partial ranking functions of CR nonZero(V,Out) 20.50/6.12 20.50/6.12 20.50/6.12 ### Specialization of cost equations rand/3 20.50/6.12 * CE 13 is refined into CE [22] 20.50/6.12 * CE 12 is refined into CE [23,24] 20.50/6.12 20.50/6.12 20.50/6.12 ### Cost equations --> "Loop" of rand/3 20.50/6.12 * CEs [24] --> Loop 17 20.50/6.12 * CEs [23] --> Loop 18 20.50/6.12 * CEs [22] --> Loop 19 20.50/6.12 20.50/6.12 ### Ranking functions of CR rand(V,V6,Out) 20.50/6.12 * RF of phase [17,18]: [V] 20.50/6.12 20.50/6.12 #### Partial ranking functions of CR rand(V,V6,Out) 20.50/6.12 * Partial RF of phase [17,18]: 20.50/6.12 - RF of loop [17:1,18:1]: 20.50/6.12 V 20.50/6.12 20.50/6.12 20.50/6.12 ### Specialization of cost equations start/3 20.50/6.12 * CE 1 is refined into CE [25,26,27,28,29,30] 20.50/6.12 * CE 2 is refined into CE [31] 20.50/6.12 * CE 3 is refined into CE [32,33] 20.50/6.12 * CE 4 is refined into CE [34,35] 20.50/6.12 * CE 5 is refined into CE [36,37] 20.50/6.12 * CE 6 is refined into CE [38,39] 20.50/6.12 * CE 7 is refined into CE [40,41] 20.50/6.12 20.50/6.12 20.50/6.12 ### Cost equations --> "Loop" of start/3 20.50/6.12 * CEs [28,30] --> Loop 20 20.50/6.12 * CEs [27,29] --> Loop 21 20.50/6.12 * CEs [25,26] --> Loop 22 20.50/6.12 * CEs [31,32,33,34,35,36,37,38,39,40,41] --> Loop 23 20.50/6.12 20.50/6.12 ### Ranking functions of CR start(V,V6,V9) 20.50/6.12 20.50/6.12 #### Partial ranking functions of CR start(V,V6,V9) 20.50/6.12 20.50/6.12 20.50/6.12 Computing Bounds 20.50/6.12 ===================================== 20.50/6.12 20.50/6.12 #### Cost of chains of fun(V,Out): 20.50/6.12 * Chain [12]: 1 20.50/6.12 with precondition: [V+1=Out,V>=0] 20.50/6.12 20.50/6.12 * Chain [11]: 1 20.50/6.12 with precondition: [V=Out,V>=0] 20.50/6.12 20.50/6.12 20.50/6.12 #### Cost of chains of p(V,Out): 20.50/6.12 * Chain [14]: 1 20.50/6.12 with precondition: [V=0,Out=0] 20.50/6.12 20.50/6.12 * Chain [13]: 1 20.50/6.12 with precondition: [V=Out+1,V>=1] 20.50/6.12 20.50/6.12 20.50/6.12 #### Cost of chains of nonZero(V,Out): 20.50/6.12 * Chain [16]: 1 20.50/6.12 with precondition: [V=0,Out=0] 20.50/6.12 20.50/6.12 * Chain [15]: 1 20.50/6.12 with precondition: [Out=1,V>=1] 20.50/6.12 20.50/6.12 20.50/6.12 #### Cost of chains of rand(V,V6,Out): 20.50/6.12 * Chain [[17,18],19]: 10*it(17)+3 20.50/6.12 Such that:aux(3) =< V 20.50/6.12 it(17) =< aux(3) 20.50/6.12 20.50/6.12 with precondition: [V>=1,V6>=0,Out>=V6,V+V6>=Out] 20.50/6.12 20.50/6.12 * Chain [19]: 3 20.50/6.12 with precondition: [V=0,V6=Out,V6>=0] 20.50/6.12 20.50/6.12 20.50/6.12 #### Cost of chains of start(V,V6,V9): 20.50/6.12 * Chain [23]: 20*s(2)+4 20.50/6.12 Such that:aux(4) =< V 20.50/6.12 s(2) =< aux(4) 20.50/6.12 20.50/6.12 with precondition: [V>=0] 20.50/6.12 20.50/6.12 * Chain [22]: 6 20.50/6.12 with precondition: [V=1,V6=0,V9>=0] 20.50/6.12 20.50/6.12 * Chain [21]: 6 20.50/6.12 with precondition: [V=1,V6=1,V9>=0] 20.50/6.12 20.50/6.12 * Chain [20]: 20*s(6)+6 20.50/6.12 Such that:aux(5) =< V6 20.50/6.12 s(6) =< aux(5) 20.50/6.12 20.50/6.12 with precondition: [V=1,V6>=2,V9>=0] 20.50/6.12 20.50/6.12 20.50/6.12 Closed-form bounds of start(V,V6,V9): 20.50/6.12 ------------------------------------- 20.50/6.12 * Chain [23] with precondition: [V>=0] 20.50/6.12 - Upper bound: 20*V+4 20.50/6.12 - Complexity: n 20.50/6.12 * Chain [22] with precondition: [V=1,V6=0,V9>=0] 20.50/6.12 - Upper bound: 6 20.50/6.12 - Complexity: constant 20.50/6.12 * Chain [21] with precondition: [V=1,V6=1,V9>=0] 20.50/6.12 - Upper bound: 6 20.50/6.12 - Complexity: constant 20.50/6.12 * Chain [20] with precondition: [V=1,V6>=2,V9>=0] 20.50/6.12 - Upper bound: 20*V6+6 20.50/6.12 - Complexity: n 20.50/6.12 20.50/6.12 ### Maximum cost of start(V,V6,V9): max([20*V,2,nat(V6)*20+2])+4 20.50/6.12 Asymptotic class: n 20.50/6.12 * Total analysis performed in 134 ms. 20.50/6.12 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (10) 20.50/6.12 BOUNDS(1, n^1) 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 20.50/6.12 Transformed a relative TRS into a decreasing-loop problem. 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (12) 20.50/6.12 Obligation: 20.50/6.12 Analyzing the following TRS for decreasing loops: 20.50/6.12 20.50/6.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.50/6.12 20.50/6.12 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false 20.50/6.12 nonZero(s(x)) -> true 20.50/6.12 p(0) -> 0 20.50/6.12 p(s(x)) -> x 20.50/6.12 id_inc(x) -> x 20.50/6.12 id_inc(x) -> s(x) 20.50/6.12 random(x) -> rand(x, 0) 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) 20.50/6.12 if(false, x, y) -> y 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) 20.50/6.12 20.50/6.12 S is empty. 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (13) DecreasingLoopProof (LOWER BOUND(ID)) 20.50/6.12 The following loop(s) give(s) rise to the lower bound Omega(n^1): 20.50/6.12 20.50/6.12 The rewrite sequence 20.50/6.12 20.50/6.12 rand(s(x1_0), y) ->^+ rand(x1_0, id_inc(y)) 20.50/6.12 20.50/6.12 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 20.50/6.12 20.50/6.12 The pumping substitution is [x1_0 / s(x1_0)]. 20.50/6.12 20.50/6.12 The result substitution is [y / id_inc(y)]. 20.50/6.12 20.50/6.12 20.50/6.12 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (14) 20.50/6.12 Complex Obligation (BEST) 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (15) 20.50/6.12 Obligation: 20.50/6.12 Proved the lower bound n^1 for the following obligation: 20.50/6.12 20.50/6.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.50/6.12 20.50/6.12 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false 20.50/6.12 nonZero(s(x)) -> true 20.50/6.12 p(0) -> 0 20.50/6.12 p(s(x)) -> x 20.50/6.12 id_inc(x) -> x 20.50/6.12 id_inc(x) -> s(x) 20.50/6.12 random(x) -> rand(x, 0) 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) 20.50/6.12 if(false, x, y) -> y 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) 20.50/6.12 20.50/6.12 S is empty. 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (16) LowerBoundPropagationProof (FINISHED) 20.50/6.12 Propagated lower bound. 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (17) 20.50/6.12 BOUNDS(n^1, INF) 20.50/6.12 20.50/6.12 ---------------------------------------- 20.50/6.12 20.50/6.12 (18) 20.50/6.12 Obligation: 20.50/6.12 Analyzing the following TRS for decreasing loops: 20.50/6.12 20.50/6.12 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.50/6.12 20.50/6.12 20.50/6.12 The TRS R consists of the following rules: 20.50/6.12 20.50/6.12 nonZero(0) -> false 20.50/6.12 nonZero(s(x)) -> true 20.50/6.12 p(0) -> 0 20.50/6.12 p(s(x)) -> x 20.50/6.12 id_inc(x) -> x 20.50/6.12 id_inc(x) -> s(x) 20.50/6.12 random(x) -> rand(x, 0) 20.50/6.12 rand(x, y) -> if(nonZero(x), x, y) 20.50/6.12 if(false, x, y) -> y 20.50/6.12 if(true, x, y) -> rand(p(x), id_inc(y)) 20.50/6.12 20.50/6.12 S is empty. 20.50/6.12 Rewrite Strategy: INNERMOST 20.50/6.17 EOF