880.14/291.50 WORST_CASE(Omega(n^1), O(n^2)) 880.14/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 880.14/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 880.14/291.52 880.14/291.52 880.14/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 880.14/291.52 880.14/291.52 (0) CpxTRS 880.14/291.52 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 880.14/291.52 (2) CpxWeightedTrs 880.14/291.52 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 880.14/291.52 (4) CpxTypedWeightedTrs 880.14/291.52 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 880.14/291.52 (6) CpxTypedWeightedCompleteTrs 880.14/291.52 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms] 880.14/291.52 (8) CpxRNTS 880.14/291.52 (9) CompleteCoflocoProof [FINISHED, 219 ms] 880.14/291.52 (10) BOUNDS(1, n^2) 880.14/291.52 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 880.14/291.52 (12) TRS for Loop Detection 880.14/291.52 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 880.14/291.52 (14) BEST 880.14/291.52 (15) proven lower bound 880.14/291.52 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 880.14/291.52 (17) BOUNDS(n^1, INF) 880.14/291.52 (18) TRS for Loop Detection 880.14/291.52 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (0) 880.14/291.52 Obligation: 880.14/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 880.14/291.52 880.14/291.52 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) 880.14/291.52 cond(false, x, y) -> 0 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) 880.14/291.52 ge(u, 0) -> true 880.14/291.52 ge(0, s(v)) -> false 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) 880.14/291.52 880.14/291.52 S is empty. 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 880.14/291.52 Transformed relative TRS to weighted TRS 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (2) 880.14/291.52 Obligation: 880.14/291.52 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 880.14/291.52 880.14/291.52 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 880.14/291.52 cond(false, x, y) -> 0 [1] 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) [1] 880.14/291.52 ge(u, 0) -> true [1] 880.14/291.52 ge(0, s(v)) -> false [1] 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) [1] 880.14/291.52 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 880.14/291.52 Infered types. 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (4) 880.14/291.52 Obligation: 880.14/291.52 Runtime Complexity Weighted TRS with Types. 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 880.14/291.52 cond(false, x, y) -> 0 [1] 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) [1] 880.14/291.52 ge(u, 0) -> true [1] 880.14/291.52 ge(0, s(v)) -> false [1] 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) [1] 880.14/291.52 880.14/291.52 The TRS has the following type information: 880.14/291.52 minus :: s:0 -> s:0 -> s:0 880.14/291.52 cond :: false:true -> s:0 -> s:0 -> s:0 880.14/291.52 ge :: s:0 -> s:0 -> false:true 880.14/291.52 s :: s:0 -> s:0 880.14/291.52 false :: false:true 880.14/291.52 0 :: s:0 880.14/291.52 true :: false:true 880.14/291.52 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (5) CompletionProof (UPPER BOUND(ID)) 880.14/291.52 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 880.14/291.52 none 880.14/291.52 880.14/291.52 And the following fresh constants: none 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (6) 880.14/291.52 Obligation: 880.14/291.52 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 880.14/291.52 880.14/291.52 Runtime Complexity Weighted TRS with Types. 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) [1] 880.14/291.52 cond(false, x, y) -> 0 [1] 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) [1] 880.14/291.52 ge(u, 0) -> true [1] 880.14/291.52 ge(0, s(v)) -> false [1] 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) [1] 880.14/291.52 880.14/291.52 The TRS has the following type information: 880.14/291.52 minus :: s:0 -> s:0 -> s:0 880.14/291.52 cond :: false:true -> s:0 -> s:0 -> s:0 880.14/291.52 ge :: s:0 -> s:0 -> false:true 880.14/291.52 s :: s:0 -> s:0 880.14/291.52 false :: false:true 880.14/291.52 0 :: s:0 880.14/291.52 true :: false:true 880.14/291.52 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 880.14/291.52 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 880.14/291.52 The constant constructors are abstracted as follows: 880.14/291.52 880.14/291.52 false => 0 880.14/291.52 0 => 0 880.14/291.52 true => 1 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (8) 880.14/291.52 Obligation: 880.14/291.52 Complexity RNTS consisting of the following rules: 880.14/291.52 880.14/291.52 cond(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 880.14/291.52 cond(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 880.14/291.52 ge(z, z') -{ 1 }-> ge(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 880.14/291.52 ge(z, z') -{ 1 }-> 1 :|: z = u, z' = 0, u >= 0 880.14/291.52 ge(z, z') -{ 1 }-> 0 :|: v >= 0, z' = 1 + v, z = 0 880.14/291.52 minus(z, z') -{ 1 }-> cond(ge(x, 1 + y), x, y) :|: x >= 0, y >= 0, z = x, z' = y 880.14/291.52 880.14/291.52 Only complete derivations are relevant for the runtime complexity. 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (9) CompleteCoflocoProof (FINISHED) 880.14/291.52 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 880.14/291.52 880.14/291.52 eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 880.14/291.52 eq(start(V1, V, V5),0,[cond(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]). 880.14/291.52 eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]). 880.14/291.52 eq(minus(V1, V, Out),1,[ge(V3, 1 + V2, Ret0),cond(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]). 880.14/291.52 eq(cond(V1, V, V5, Out),1,[],[Out = 0,V = V4,V5 = V6,V4 >= 0,V6 >= 0,V1 = 0]). 880.14/291.52 eq(cond(V1, V, V5, Out),1,[minus(V8, 1 + V7, Ret1)],[Out = 1 + Ret1,V = V8,V5 = V7,V1 = 1,V8 >= 0,V7 >= 0]). 880.14/291.52 eq(ge(V1, V, Out),1,[],[Out = 1,V1 = V9,V = 0,V9 >= 0]). 880.14/291.52 eq(ge(V1, V, Out),1,[],[Out = 0,V10 >= 0,V = 1 + V10,V1 = 0]). 880.14/291.52 eq(ge(V1, V, Out),1,[ge(V12, V11, Ret2)],[Out = Ret2,V11 >= 0,V = 1 + V11,V1 = 1 + V12,V12 >= 0]). 880.14/291.52 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 880.14/291.52 input_output_vars(cond(V1,V,V5,Out),[V1,V,V5],[Out]). 880.14/291.52 input_output_vars(ge(V1,V,Out),[V1,V],[Out]). 880.14/291.52 880.14/291.52 880.14/291.52 CoFloCo proof output: 880.14/291.52 Preprocessing Cost Relations 880.14/291.52 ===================================== 880.14/291.52 880.14/291.52 #### Computed strongly connected components 880.14/291.52 0. recursive : [ge/3] 880.14/291.52 1. recursive : [cond/4,minus/3] 880.14/291.52 2. non_recursive : [start/3] 880.14/291.52 880.14/291.52 #### Obtained direct recursion through partial evaluation 880.14/291.52 0. SCC is partially evaluated into ge/3 880.14/291.52 1. SCC is partially evaluated into minus/3 880.14/291.52 2. SCC is partially evaluated into start/3 880.14/291.52 880.14/291.52 Control-Flow Refinement of Cost Relations 880.14/291.52 ===================================== 880.14/291.52 880.14/291.52 ### Specialization of cost equations ge/3 880.14/291.52 * CE 9 is refined into CE [10] 880.14/291.52 * CE 7 is refined into CE [11] 880.14/291.52 * CE 8 is refined into CE [12] 880.14/291.52 880.14/291.52 880.14/291.52 ### Cost equations --> "Loop" of ge/3 880.14/291.52 * CEs [11] --> Loop 8 880.14/291.52 * CEs [12] --> Loop 9 880.14/291.52 * CEs [10] --> Loop 10 880.14/291.52 880.14/291.52 ### Ranking functions of CR ge(V1,V,Out) 880.14/291.52 * RF of phase [10]: [V,V1] 880.14/291.52 880.14/291.52 #### Partial ranking functions of CR ge(V1,V,Out) 880.14/291.52 * Partial RF of phase [10]: 880.14/291.52 - RF of loop [10:1]: 880.14/291.52 V 880.14/291.52 V1 880.14/291.52 880.14/291.52 880.14/291.52 ### Specialization of cost equations minus/3 880.14/291.52 * CE 6 is refined into CE [13,14] 880.14/291.52 * CE 5 is refined into CE [15] 880.14/291.52 880.14/291.52 880.14/291.52 ### Cost equations --> "Loop" of minus/3 880.14/291.52 * CEs [15] --> Loop 11 880.14/291.52 * CEs [14] --> Loop 12 880.14/291.52 * CEs [13] --> Loop 13 880.14/291.52 880.14/291.52 ### Ranking functions of CR minus(V1,V,Out) 880.14/291.52 * RF of phase [11]: [V1-V] 880.14/291.52 880.14/291.52 #### Partial ranking functions of CR minus(V1,V,Out) 880.14/291.52 * Partial RF of phase [11]: 880.14/291.52 - RF of loop [11:1]: 880.14/291.52 V1-V 880.14/291.52 880.14/291.52 880.14/291.52 ### Specialization of cost equations start/3 880.14/291.52 * CE 1 is refined into CE [16,17,18] 880.14/291.52 * CE 2 is refined into CE [19] 880.14/291.52 * CE 3 is refined into CE [20,21,22] 880.14/291.52 * CE 4 is refined into CE [23,24,25,26] 880.14/291.52 880.14/291.52 880.14/291.52 ### Cost equations --> "Loop" of start/3 880.14/291.52 * CEs [26] --> Loop 14 880.14/291.52 * CEs [22] --> Loop 15 880.14/291.52 * CEs [18] --> Loop 16 880.14/291.52 * CEs [17,21,25] --> Loop 17 880.14/291.52 * CEs [16,24] --> Loop 18 880.14/291.52 * CEs [19,20,23] --> Loop 19 880.14/291.52 880.14/291.52 ### Ranking functions of CR start(V1,V,V5) 880.14/291.52 880.14/291.52 #### Partial ranking functions of CR start(V1,V,V5) 880.14/291.52 880.14/291.52 880.14/291.52 Computing Bounds 880.14/291.52 ===================================== 880.14/291.52 880.14/291.52 #### Cost of chains of ge(V1,V,Out): 880.14/291.52 * Chain [[10],9]: 1*it(10)+1 880.14/291.52 Such that:it(10) =< V1 880.14/291.52 880.14/291.52 with precondition: [Out=0,V1>=1,V>=V1+1] 880.14/291.52 880.14/291.52 * Chain [[10],8]: 1*it(10)+1 880.14/291.52 Such that:it(10) =< V 880.14/291.52 880.14/291.52 with precondition: [Out=1,V>=1,V1>=V] 880.14/291.52 880.14/291.52 * Chain [9]: 1 880.14/291.52 with precondition: [V1=0,Out=0,V>=1] 880.14/291.52 880.14/291.52 * Chain [8]: 1 880.14/291.52 with precondition: [V=0,Out=1,V1>=0] 880.14/291.52 880.14/291.52 880.14/291.52 #### Cost of chains of minus(V1,V,Out): 880.14/291.52 * Chain [[11],12]: 3*it(11)+1*s(1)+1*s(4)+3 880.14/291.52 Such that:it(11) =< Out 880.14/291.52 aux(2) =< V+Out 880.14/291.52 s(1) =< aux(2) 880.14/291.52 s(4) =< it(11)*aux(2) 880.14/291.52 880.14/291.52 with precondition: [V1=Out+V,V>=0,V1>=V+1] 880.14/291.52 880.14/291.52 * Chain [13]: 3 880.14/291.52 with precondition: [V1=0,Out=0,V>=0] 880.14/291.52 880.14/291.52 * Chain [12]: 1*s(1)+3 880.14/291.52 Such that:s(1) =< V1 880.14/291.52 880.14/291.52 with precondition: [Out=0,V1>=1,V>=V1] 880.14/291.52 880.14/291.52 880.14/291.52 #### Cost of chains of start(V1,V,V5): 880.14/291.52 * Chain [19]: 3 880.14/291.52 with precondition: [V1=0,V>=0] 880.14/291.52 880.14/291.52 * Chain [18]: 4 880.14/291.52 with precondition: [V=0,V1>=0] 880.14/291.52 880.14/291.52 * Chain [17]: 1*s(5)+2*s(6)+4 880.14/291.52 Such that:s(5) =< V 880.14/291.52 aux(3) =< V1 880.14/291.52 s(6) =< aux(3) 880.14/291.52 880.14/291.52 with precondition: [V1>=1,V>=V1] 880.14/291.52 880.14/291.52 * Chain [16]: 3*s(8)+1*s(10)+1*s(11)+4 880.14/291.52 Such that:s(9) =< V 880.14/291.52 s(8) =< V-V5 880.14/291.52 s(10) =< s(9) 880.14/291.52 s(11) =< s(8)*s(9) 880.14/291.52 880.14/291.52 with precondition: [V1=1,V5>=0,V>=V5+2] 880.14/291.52 880.14/291.52 * Chain [15]: 3*s(12)+1*s(14)+1*s(15)+3 880.14/291.52 Such that:s(13) =< V1 880.14/291.52 s(12) =< V1-V 880.14/291.52 s(14) =< s(13) 880.14/291.52 s(15) =< s(12)*s(13) 880.14/291.52 880.14/291.52 with precondition: [V>=0,V1>=V+1] 880.14/291.52 880.14/291.52 * Chain [14]: 1*s(16)+1 880.14/291.52 Such that:s(16) =< V 880.14/291.52 880.14/291.52 with precondition: [V>=1,V1>=V] 880.14/291.52 880.14/291.52 880.14/291.52 Closed-form bounds of start(V1,V,V5): 880.14/291.52 ------------------------------------- 880.14/291.52 * Chain [19] with precondition: [V1=0,V>=0] 880.14/291.52 - Upper bound: 3 880.14/291.52 - Complexity: constant 880.14/291.52 * Chain [18] with precondition: [V=0,V1>=0] 880.14/291.52 - Upper bound: 4 880.14/291.52 - Complexity: constant 880.14/291.52 * Chain [17] with precondition: [V1>=1,V>=V1] 880.14/291.52 - Upper bound: 2*V1+V+4 880.14/291.52 - Complexity: n 880.14/291.52 * Chain [16] with precondition: [V1=1,V5>=0,V>=V5+2] 880.14/291.52 - Upper bound: 3*V-3*V5+(V+4+(V-V5)*V) 880.14/291.52 - Complexity: n^2 880.14/291.52 * Chain [15] with precondition: [V>=0,V1>=V+1] 880.14/291.52 - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1) 880.14/291.52 - Complexity: n^2 880.14/291.52 * Chain [14] with precondition: [V>=1,V1>=V] 880.14/291.52 - Upper bound: V+1 880.14/291.52 - Complexity: n 880.14/291.52 880.14/291.52 ### Maximum cost of start(V1,V,V5): max([max([1,nat(V1-V)*V1+V1+nat(V1-V)*3])+2,max([2*V1+3,nat(V-V5)*V+3+nat(V-V5)*3])+V])+1 880.14/291.52 Asymptotic class: n^2 880.14/291.52 * Total analysis performed in 146 ms. 880.14/291.52 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (10) 880.14/291.52 BOUNDS(1, n^2) 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 880.14/291.52 Transformed a relative TRS into a decreasing-loop problem. 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (12) 880.14/291.52 Obligation: 880.14/291.52 Analyzing the following TRS for decreasing loops: 880.14/291.52 880.14/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 880.14/291.52 880.14/291.52 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) 880.14/291.52 cond(false, x, y) -> 0 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) 880.14/291.52 ge(u, 0) -> true 880.14/291.52 ge(0, s(v)) -> false 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) 880.14/291.52 880.14/291.52 S is empty. 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (13) DecreasingLoopProof (LOWER BOUND(ID)) 880.14/291.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 880.14/291.52 880.14/291.52 The rewrite sequence 880.14/291.52 880.14/291.52 ge(s(u), s(v)) ->^+ ge(u, v) 880.14/291.52 880.14/291.52 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 880.14/291.52 880.14/291.52 The pumping substitution is [u / s(u), v / s(v)]. 880.14/291.52 880.14/291.52 The result substitution is [ ]. 880.14/291.52 880.14/291.52 880.14/291.52 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (14) 880.14/291.52 Complex Obligation (BEST) 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (15) 880.14/291.52 Obligation: 880.14/291.52 Proved the lower bound n^1 for the following obligation: 880.14/291.52 880.14/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 880.14/291.52 880.14/291.52 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) 880.14/291.52 cond(false, x, y) -> 0 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) 880.14/291.52 ge(u, 0) -> true 880.14/291.52 ge(0, s(v)) -> false 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) 880.14/291.52 880.14/291.52 S is empty. 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (16) LowerBoundPropagationProof (FINISHED) 880.14/291.52 Propagated lower bound. 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (17) 880.14/291.52 BOUNDS(n^1, INF) 880.14/291.52 880.14/291.52 ---------------------------------------- 880.14/291.52 880.14/291.52 (18) 880.14/291.52 Obligation: 880.14/291.52 Analyzing the following TRS for decreasing loops: 880.14/291.52 880.14/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 880.14/291.52 880.14/291.52 880.14/291.52 The TRS R consists of the following rules: 880.14/291.52 880.14/291.52 minus(x, y) -> cond(ge(x, s(y)), x, y) 880.14/291.52 cond(false, x, y) -> 0 880.14/291.52 cond(true, x, y) -> s(minus(x, s(y))) 880.14/291.52 ge(u, 0) -> true 880.14/291.52 ge(0, s(v)) -> false 880.14/291.52 ge(s(u), s(v)) -> ge(u, v) 880.14/291.52 880.14/291.52 S is empty. 880.14/291.52 Rewrite Strategy: INNERMOST 880.14/291.56 EOF