882.75/291.49 WORST_CASE(Omega(n^1), O(n^2)) 882.82/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 882.82/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 882.82/291.51 882.82/291.51 882.82/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.82/291.51 882.82/291.51 (0) CpxTRS 882.82/291.51 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 882.82/291.51 (2) CpxTRS 882.82/291.51 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 882.82/291.51 (4) CpxWeightedTrs 882.82/291.51 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 882.82/291.51 (6) CpxTypedWeightedTrs 882.82/291.51 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 882.82/291.51 (8) CpxTypedWeightedCompleteTrs 882.82/291.51 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] 882.82/291.51 (10) CpxRNTS 882.82/291.51 (11) CompleteCoflocoProof [FINISHED, 278 ms] 882.82/291.51 (12) BOUNDS(1, n^2) 882.82/291.51 (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 882.82/291.51 (14) TRS for Loop Detection 882.82/291.51 (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 882.82/291.51 (16) BEST 882.82/291.51 (17) proven lower bound 882.82/291.51 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 882.82/291.51 (19) BOUNDS(n^1, INF) 882.82/291.51 (20) TRS for Loop Detection 882.82/291.51 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (0) 882.82/291.51 Obligation: 882.82/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 882.82/291.51 gt(0, v) -> false 882.82/291.51 gt(s(u), 0) -> true 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) 882.82/291.51 882.82/291.51 S is empty. 882.82/291.51 Rewrite Strategy: FULL 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 882.82/291.51 Converted rc-obligation to irc-obligation. 882.82/291.51 882.82/291.51 The duplicating contexts are: 882.82/291.51 f(true, [], y) 882.82/291.51 f(true, x, []) 882.82/291.51 882.82/291.51 882.82/291.51 The defined contexts are: 882.82/291.51 f([], s(x1), s(s(x2))) 882.82/291.51 882.82/291.51 882.82/291.51 [] just represents basic- or constructor-terms in the following defined contexts: 882.82/291.51 f([], s(x1), s(s(x2))) 882.82/291.51 882.82/291.51 882.82/291.51 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (2) 882.82/291.51 Obligation: 882.82/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 882.82/291.51 gt(0, v) -> false 882.82/291.51 gt(s(u), 0) -> true 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) 882.82/291.51 882.82/291.51 S is empty. 882.82/291.51 Rewrite Strategy: INNERMOST 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 882.82/291.51 Transformed relative TRS to weighted TRS 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (4) 882.82/291.51 Obligation: 882.82/291.51 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 882.82/291.51 gt(0, v) -> false [1] 882.82/291.51 gt(s(u), 0) -> true [1] 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 882.82/291.51 882.82/291.51 Rewrite Strategy: INNERMOST 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 882.82/291.51 Infered types. 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (6) 882.82/291.51 Obligation: 882.82/291.51 Runtime Complexity Weighted TRS with Types. 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 882.82/291.51 gt(0, v) -> false [1] 882.82/291.51 gt(s(u), 0) -> true [1] 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 882.82/291.51 882.82/291.51 The TRS has the following type information: 882.82/291.51 f :: true:false -> s:0 -> s:0 -> f 882.82/291.51 true :: true:false 882.82/291.51 gt :: s:0 -> s:0 -> true:false 882.82/291.51 s :: s:0 -> s:0 882.82/291.51 0 :: s:0 882.82/291.51 false :: true:false 882.82/291.51 882.82/291.51 Rewrite Strategy: INNERMOST 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (7) CompletionProof (UPPER BOUND(ID)) 882.82/291.51 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 882.82/291.51 882.82/291.51 f(v0, v1, v2) -> null_f [0] 882.82/291.51 882.82/291.51 And the following fresh constants: null_f 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (8) 882.82/291.51 Obligation: 882.82/291.51 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 882.82/291.51 882.82/291.51 Runtime Complexity Weighted TRS with Types. 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 882.82/291.51 gt(0, v) -> false [1] 882.82/291.51 gt(s(u), 0) -> true [1] 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 882.82/291.51 f(v0, v1, v2) -> null_f [0] 882.82/291.51 882.82/291.51 The TRS has the following type information: 882.82/291.51 f :: true:false -> s:0 -> s:0 -> null_f 882.82/291.51 true :: true:false 882.82/291.51 gt :: s:0 -> s:0 -> true:false 882.82/291.51 s :: s:0 -> s:0 882.82/291.51 0 :: s:0 882.82/291.51 false :: true:false 882.82/291.51 null_f :: null_f 882.82/291.51 882.82/291.51 Rewrite Strategy: INNERMOST 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 882.82/291.51 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 882.82/291.51 The constant constructors are abstracted as follows: 882.82/291.51 882.82/291.51 true => 1 882.82/291.51 0 => 0 882.82/291.51 false => 0 882.82/291.51 null_f => 0 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (10) 882.82/291.51 Obligation: 882.82/291.51 Complexity RNTS consisting of the following rules: 882.82/291.51 882.82/291.51 f(z, z', z'') -{ 1 }-> f(gt(x, y), 1 + x, 1 + (1 + y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 882.82/291.51 f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 882.82/291.51 gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 882.82/291.51 gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 882.82/291.51 gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0 882.82/291.51 882.82/291.51 Only complete derivations are relevant for the runtime complexity. 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (11) CompleteCoflocoProof (FINISHED) 882.82/291.51 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 882.82/291.51 882.82/291.51 eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 882.82/291.51 eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]). 882.82/291.51 eq(f(V1, V, V2, Out),1,[gt(V4, V3, Ret0),f(Ret0, 1 + V4, 1 + (1 + V3), Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]). 882.82/291.51 eq(gt(V1, V, Out),1,[],[Out = 0,V5 >= 0,V = V5,V1 = 0]). 882.82/291.51 eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V6,V = 0,V6 >= 0]). 882.82/291.51 eq(gt(V1, V, Out),1,[gt(V7, V8, Ret1)],[Out = Ret1,V8 >= 0,V = 1 + V8,V1 = 1 + V7,V7 >= 0]). 882.82/291.51 eq(f(V1, V, V2, Out),0,[],[Out = 0,V10 >= 0,V2 = V11,V9 >= 0,V1 = V10,V = V9,V11 >= 0]). 882.82/291.51 input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]). 882.82/291.51 input_output_vars(gt(V1,V,Out),[V1,V],[Out]). 882.82/291.51 882.82/291.51 882.82/291.51 CoFloCo proof output: 882.82/291.51 Preprocessing Cost Relations 882.82/291.51 ===================================== 882.82/291.51 882.82/291.51 #### Computed strongly connected components 882.82/291.51 0. recursive : [gt/3] 882.82/291.51 1. recursive : [f/4] 882.82/291.51 2. non_recursive : [start/3] 882.82/291.51 882.82/291.51 #### Obtained direct recursion through partial evaluation 882.82/291.51 0. SCC is partially evaluated into gt/3 882.82/291.51 1. SCC is partially evaluated into f/4 882.82/291.51 2. SCC is partially evaluated into start/3 882.82/291.51 882.82/291.51 Control-Flow Refinement of Cost Relations 882.82/291.51 ===================================== 882.82/291.51 882.82/291.51 ### Specialization of cost equations gt/3 882.82/291.51 * CE 7 is refined into CE [8] 882.82/291.51 * CE 6 is refined into CE [9] 882.82/291.51 * CE 5 is refined into CE [10] 882.82/291.51 882.82/291.51 882.82/291.51 ### Cost equations --> "Loop" of gt/3 882.82/291.51 * CEs [9] --> Loop 7 882.82/291.51 * CEs [10] --> Loop 8 882.82/291.51 * CEs [8] --> Loop 9 882.82/291.51 882.82/291.51 ### Ranking functions of CR gt(V1,V,Out) 882.82/291.51 * RF of phase [9]: [V,V1] 882.82/291.51 882.82/291.51 #### Partial ranking functions of CR gt(V1,V,Out) 882.82/291.51 * Partial RF of phase [9]: 882.82/291.51 - RF of loop [9:1]: 882.82/291.51 V 882.82/291.51 V1 882.82/291.51 882.82/291.51 882.82/291.51 ### Specialization of cost equations f/4 882.82/291.51 * CE 4 is refined into CE [11] 882.82/291.51 * CE 3 is refined into CE [12,13,14,15] 882.82/291.51 882.82/291.51 882.82/291.51 ### Cost equations --> "Loop" of f/4 882.82/291.51 * CEs [15] --> Loop 10 882.82/291.51 * CEs [14] --> Loop 11 882.82/291.51 * CEs [13] --> Loop 12 882.82/291.51 * CEs [12] --> Loop 13 882.82/291.51 * CEs [11] --> Loop 14 882.82/291.51 882.82/291.51 ### Ranking functions of CR f(V1,V,V2,Out) 882.82/291.51 * RF of phase [10]: [V-V2] 882.82/291.51 882.82/291.51 #### Partial ranking functions of CR f(V1,V,V2,Out) 882.82/291.51 * Partial RF of phase [10]: 882.82/291.51 - RF of loop [10:1]: 882.82/291.51 V-V2 882.82/291.51 882.82/291.51 882.82/291.51 ### Specialization of cost equations start/3 882.82/291.51 * CE 1 is refined into CE [16,17,18,19] 882.82/291.51 * CE 2 is refined into CE [20,21,22,23] 882.82/291.51 882.82/291.51 882.82/291.51 ### Cost equations --> "Loop" of start/3 882.82/291.51 * CEs [23] --> Loop 15 882.82/291.51 * CEs [21] --> Loop 16 882.82/291.51 * CEs [19] --> Loop 17 882.82/291.51 * CEs [18,22] --> Loop 18 882.82/291.51 * CEs [16,17] --> Loop 19 882.82/291.51 * CEs [20] --> Loop 20 882.82/291.51 882.82/291.51 ### Ranking functions of CR start(V1,V,V2) 882.82/291.51 882.82/291.51 #### Partial ranking functions of CR start(V1,V,V2) 882.82/291.51 882.82/291.51 882.82/291.51 Computing Bounds 882.82/291.51 ===================================== 882.82/291.51 882.82/291.51 #### Cost of chains of gt(V1,V,Out): 882.82/291.51 * Chain [[9],8]: 1*it(9)+1 882.82/291.51 Such that:it(9) =< V1 882.82/291.51 882.82/291.51 with precondition: [Out=0,V1>=1,V>=V1] 882.82/291.51 882.82/291.51 * Chain [[9],7]: 1*it(9)+1 882.82/291.51 Such that:it(9) =< V 882.82/291.51 882.82/291.51 with precondition: [Out=1,V>=1,V1>=V+1] 882.82/291.51 882.82/291.51 * Chain [8]: 1 882.82/291.51 with precondition: [V1=0,Out=0,V>=0] 882.82/291.51 882.82/291.51 * Chain [7]: 1 882.82/291.51 with precondition: [V=0,Out=1,V1>=1] 882.82/291.51 882.82/291.51 882.82/291.51 #### Cost of chains of f(V1,V,V2,Out): 882.82/291.51 * Chain [[10],14]: 2*it(10)+1*s(3)+0 882.82/291.51 Such that:it(10) =< V-V2 882.82/291.51 aux(1) =< 2*V-V2 882.82/291.51 s(3) =< it(10)*aux(1) 882.82/291.51 882.82/291.51 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 882.82/291.51 882.82/291.51 * Chain [[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+2 882.82/291.51 Such that:it(10) =< V-V2 882.82/291.51 aux(1) =< 2*V-V2 882.82/291.51 s(4) =< 2*V-V2+1 882.82/291.51 s(3) =< it(10)*aux(1) 882.82/291.51 882.82/291.51 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 882.82/291.51 882.82/291.51 * Chain [14]: 0 882.82/291.51 with precondition: [Out=0,V1>=0,V>=0,V2>=0] 882.82/291.51 882.82/291.51 * Chain [13,14]: 2 882.82/291.51 with precondition: [V1=1,V=0,Out=0,V2>=0] 882.82/291.51 882.82/291.51 * Chain [12,[10],14]: 2*it(10)+1*s(3)+2 882.82/291.51 Such that:it(10) =< V 882.82/291.51 aux(1) =< 2*V 882.82/291.51 s(3) =< it(10)*aux(1) 882.82/291.51 882.82/291.51 with precondition: [V1=1,V2=0,Out=0,V>=2] 882.82/291.51 882.82/291.51 * Chain [12,[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+4 882.82/291.51 Such that:it(10) =< V 882.82/291.51 aux(1) =< 2*V 882.82/291.51 s(4) =< 2*V+1 882.82/291.51 s(3) =< it(10)*aux(1) 882.82/291.51 882.82/291.51 with precondition: [V1=1,V2=0,Out=0,V>=2] 882.82/291.51 882.82/291.51 * Chain [12,14]: 2 882.82/291.51 with precondition: [V1=1,V2=0,Out=0,V>=1] 882.82/291.51 882.82/291.51 * Chain [12,11,14]: 1*s(4)+4 882.82/291.51 Such that:s(4) =< 3 882.82/291.51 882.82/291.51 with precondition: [V1=1,V=1,V2=0,Out=0] 882.82/291.51 882.82/291.51 * Chain [11,14]: 1*s(4)+2 882.82/291.51 Such that:s(4) =< V+1 882.82/291.51 882.82/291.51 with precondition: [V1=1,Out=0,V>=1,V2>=V] 882.82/291.51 882.82/291.51 882.82/291.51 #### Cost of chains of start(V1,V,V2): 882.82/291.51 * Chain [20]: 1 882.82/291.51 with precondition: [V1=0,V>=0] 882.82/291.51 882.82/291.51 * Chain [19]: 1*s(20)+1*s(21)+4*s(24)+2*s(25)+4 882.82/291.51 Such that:s(20) =< 3 882.82/291.51 s(22) =< V 882.82/291.51 s(23) =< 2*V 882.82/291.51 s(21) =< 2*V+1 882.82/291.51 s(24) =< s(22) 882.82/291.51 s(25) =< s(24)*s(23) 882.82/291.51 882.82/291.51 with precondition: [V1>=0,V>=0,V2>=0] 882.82/291.51 882.82/291.51 * Chain [18]: 1*s(26)+1*s(27)+2 882.82/291.51 Such that:s(27) =< V1 882.82/291.51 s(26) =< V+1 882.82/291.51 882.82/291.51 with precondition: [V1>=1,V>=V1] 882.82/291.51 882.82/291.51 * Chain [17]: 1*s(28)+4*s(31)+2*s(32)+2 882.82/291.51 Such that:s(29) =< V-V2 882.82/291.51 s(30) =< 2*V-V2 882.82/291.51 s(28) =< 2*V-V2+1 882.82/291.51 s(31) =< s(29) 882.82/291.51 s(32) =< s(31)*s(30) 882.82/291.51 882.82/291.51 with precondition: [V1=1,V2>=1,V>=V2+1] 882.82/291.51 882.82/291.51 * Chain [16]: 1 882.82/291.51 with precondition: [V=0,V1>=1] 882.82/291.51 882.82/291.51 * Chain [15]: 1*s(33)+1 882.82/291.51 Such that:s(33) =< V 882.82/291.51 882.82/291.51 with precondition: [V>=1,V1>=V+1] 882.82/291.51 882.82/291.51 882.82/291.51 Closed-form bounds of start(V1,V,V2): 882.82/291.51 ------------------------------------- 882.82/291.51 * Chain [20] with precondition: [V1=0,V>=0] 882.82/291.51 - Upper bound: 1 882.82/291.51 - Complexity: constant 882.82/291.51 * Chain [19] with precondition: [V1>=0,V>=0,V2>=0] 882.82/291.51 - Upper bound: 4*V+7+2*V*(2*V)+(2*V+1) 882.82/291.51 - Complexity: n^2 882.82/291.51 * Chain [18] with precondition: [V1>=1,V>=V1] 882.82/291.51 - Upper bound: V1+V+3 882.82/291.51 - Complexity: n 882.82/291.51 * Chain [17] with precondition: [V1=1,V2>=1,V>=V2+1] 882.82/291.51 - Upper bound: 6*V-5*V2+3+(2*V-2*V2)*(2*V-V2) 882.82/291.51 - Complexity: n^2 882.82/291.51 * Chain [16] with precondition: [V=0,V1>=1] 882.82/291.51 - Upper bound: 1 882.82/291.51 - Complexity: constant 882.82/291.51 * Chain [15] with precondition: [V>=1,V1>=V+1] 882.82/291.51 - Upper bound: V+1 882.82/291.51 - Complexity: n 882.82/291.51 882.82/291.51 ### Maximum cost of start(V1,V,V2): max([max([V1+V+2,nat(2*V-V2+1)+1+nat(V-V2)*4+nat(V-V2)*2*nat(2*V-V2)]),3*V+6+2*V*(2*V)+(2*V+1)+V])+1 882.82/291.51 Asymptotic class: n^2 882.82/291.51 * Total analysis performed in 205 ms. 882.82/291.51 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (12) 882.82/291.51 BOUNDS(1, n^2) 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 882.82/291.51 Transformed a relative TRS into a decreasing-loop problem. 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (14) 882.82/291.51 Obligation: 882.82/291.51 Analyzing the following TRS for decreasing loops: 882.82/291.51 882.82/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 882.82/291.51 gt(0, v) -> false 882.82/291.51 gt(s(u), 0) -> true 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) 882.82/291.51 882.82/291.51 S is empty. 882.82/291.51 Rewrite Strategy: FULL 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (15) DecreasingLoopProof (LOWER BOUND(ID)) 882.82/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 882.82/291.51 882.82/291.51 The rewrite sequence 882.82/291.51 882.82/291.51 gt(s(u), s(v)) ->^+ gt(u, v) 882.82/291.51 882.82/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 882.82/291.51 882.82/291.51 The pumping substitution is [u / s(u), v / s(v)]. 882.82/291.51 882.82/291.51 The result substitution is [ ]. 882.82/291.51 882.82/291.51 882.82/291.51 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (16) 882.82/291.51 Complex Obligation (BEST) 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (17) 882.82/291.51 Obligation: 882.82/291.51 Proved the lower bound n^1 for the following obligation: 882.82/291.51 882.82/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 882.82/291.51 gt(0, v) -> false 882.82/291.51 gt(s(u), 0) -> true 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) 882.82/291.51 882.82/291.51 S is empty. 882.82/291.51 Rewrite Strategy: FULL 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (18) LowerBoundPropagationProof (FINISHED) 882.82/291.51 Propagated lower bound. 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (19) 882.82/291.51 BOUNDS(n^1, INF) 882.82/291.51 882.82/291.51 ---------------------------------------- 882.82/291.51 882.82/291.51 (20) 882.82/291.51 Obligation: 882.82/291.51 Analyzing the following TRS for decreasing loops: 882.82/291.51 882.82/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 882.82/291.51 882.82/291.51 882.82/291.51 The TRS R consists of the following rules: 882.82/291.51 882.82/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 882.82/291.51 gt(0, v) -> false 882.82/291.51 gt(s(u), 0) -> true 882.82/291.51 gt(s(u), s(v)) -> gt(u, v) 882.82/291.51 882.82/291.51 S is empty. 882.82/291.51 Rewrite Strategy: FULL 882.93/291.56 EOF