878.01/291.50 WORST_CASE(Omega(n^1), O(n^2)) 878.01/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 878.01/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 878.01/291.51 878.01/291.51 878.01/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 878.01/291.51 878.01/291.51 (0) CpxTRS 878.01/291.51 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 878.01/291.51 (2) CpxWeightedTrs 878.01/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 878.01/291.51 (4) CpxTypedWeightedTrs 878.01/291.51 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 878.01/291.51 (6) CpxTypedWeightedCompleteTrs 878.01/291.51 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 878.01/291.51 (8) CpxRNTS 878.01/291.51 (9) CompleteCoflocoProof [FINISHED, 277 ms] 878.01/291.51 (10) BOUNDS(1, n^2) 878.01/291.51 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 878.01/291.51 (12) CpxTRS 878.01/291.51 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 878.01/291.51 (14) typed CpxTrs 878.01/291.51 (15) OrderProof [LOWER BOUND(ID), 0 ms] 878.01/291.51 (16) typed CpxTrs 878.01/291.51 (17) RewriteLemmaProof [LOWER BOUND(ID), 308 ms] 878.01/291.51 (18) BEST 878.01/291.51 (19) proven lower bound 878.01/291.51 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 878.01/291.51 (21) BOUNDS(n^1, INF) 878.01/291.51 (22) typed CpxTrs 878.01/291.51 878.01/291.51 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (0) 878.01/291.51 Obligation: 878.01/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 878.01/291.51 878.01/291.51 878.01/291.51 The TRS R consists of the following rules: 878.01/291.51 878.01/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.51 gt(0, v) -> false 878.01/291.51 gt(s(u), 0) -> true 878.01/291.51 gt(s(u), s(v)) -> gt(u, v) 878.01/291.51 878.01/291.51 S is empty. 878.01/291.51 Rewrite Strategy: INNERMOST 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 878.01/291.51 Transformed relative TRS to weighted TRS 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (2) 878.01/291.51 Obligation: 878.01/291.51 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 878.01/291.51 878.01/291.51 878.01/291.51 The TRS R consists of the following rules: 878.01/291.51 878.01/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 878.01/291.51 gt(0, v) -> false [1] 878.01/291.51 gt(s(u), 0) -> true [1] 878.01/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 878.01/291.51 878.01/291.51 Rewrite Strategy: INNERMOST 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 878.01/291.51 Infered types. 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (4) 878.01/291.51 Obligation: 878.01/291.51 Runtime Complexity Weighted TRS with Types. 878.01/291.51 The TRS R consists of the following rules: 878.01/291.51 878.01/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 878.01/291.51 gt(0, v) -> false [1] 878.01/291.51 gt(s(u), 0) -> true [1] 878.01/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 878.01/291.51 878.01/291.51 The TRS has the following type information: 878.01/291.51 f :: true:false -> s:0 -> s:0 -> f 878.01/291.51 true :: true:false 878.01/291.51 gt :: s:0 -> s:0 -> true:false 878.01/291.51 s :: s:0 -> s:0 878.01/291.51 0 :: s:0 878.01/291.51 false :: true:false 878.01/291.51 878.01/291.51 Rewrite Strategy: INNERMOST 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (5) CompletionProof (UPPER BOUND(ID)) 878.01/291.51 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 878.01/291.51 878.01/291.51 f(v0, v1, v2) -> null_f [0] 878.01/291.51 878.01/291.51 And the following fresh constants: null_f 878.01/291.51 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (6) 878.01/291.51 Obligation: 878.01/291.51 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 878.01/291.51 878.01/291.51 Runtime Complexity Weighted TRS with Types. 878.01/291.51 The TRS R consists of the following rules: 878.01/291.51 878.01/291.51 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) [1] 878.01/291.51 gt(0, v) -> false [1] 878.01/291.51 gt(s(u), 0) -> true [1] 878.01/291.51 gt(s(u), s(v)) -> gt(u, v) [1] 878.01/291.51 f(v0, v1, v2) -> null_f [0] 878.01/291.51 878.01/291.51 The TRS has the following type information: 878.01/291.51 f :: true:false -> s:0 -> s:0 -> null_f 878.01/291.51 true :: true:false 878.01/291.51 gt :: s:0 -> s:0 -> true:false 878.01/291.51 s :: s:0 -> s:0 878.01/291.51 0 :: s:0 878.01/291.51 false :: true:false 878.01/291.51 null_f :: null_f 878.01/291.51 878.01/291.51 Rewrite Strategy: INNERMOST 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 878.01/291.51 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 878.01/291.51 The constant constructors are abstracted as follows: 878.01/291.51 878.01/291.51 true => 1 878.01/291.51 0 => 0 878.01/291.51 false => 0 878.01/291.51 null_f => 0 878.01/291.51 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (8) 878.01/291.51 Obligation: 878.01/291.51 Complexity RNTS consisting of the following rules: 878.01/291.51 878.01/291.51 f(z, z', z'') -{ 1 }-> f(gt(x, y), 1 + x, 1 + (1 + y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 878.01/291.51 f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 878.01/291.51 gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 878.01/291.51 gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 878.01/291.51 gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0 878.01/291.51 878.01/291.51 Only complete derivations are relevant for the runtime complexity. 878.01/291.51 878.01/291.51 ---------------------------------------- 878.01/291.51 878.01/291.51 (9) CompleteCoflocoProof (FINISHED) 878.01/291.51 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 878.01/291.51 878.01/291.51 eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 878.01/291.51 eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]). 878.01/291.51 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]). 878.01/291.51 eq(gt(V1, V, Out),1,[],[Out = 0,V5 >= 0,V = V5,V1 = 0]). 878.01/291.51 eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V6,V = 0,V6 >= 0]). 878.01/291.51 eq(gt(V1, V, Out),1,[gt(V7, V8, Ret1)],[Out = Ret1,V8 >= 0,V = 1 + V8,V1 = 1 + V7,V7 >= 0]). 878.01/291.51 eq(f(V1, V, V2, Out),0,[],[Out = 0,V10 >= 0,V2 = V11,V9 >= 0,V1 = V10,V = V9,V11 >= 0]). 878.01/291.51 input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]). 878.01/291.51 input_output_vars(gt(V1,V,Out),[V1,V],[Out]). 878.01/291.51 878.01/291.51 878.01/291.51 CoFloCo proof output: 878.01/291.51 Preprocessing Cost Relations 878.01/291.51 ===================================== 878.01/291.51 878.01/291.51 #### Computed strongly connected components 878.01/291.51 0. recursive : [gt/3] 878.01/291.51 1. recursive : [f/4] 878.01/291.51 2. non_recursive : [start/3] 878.01/291.51 878.01/291.51 #### Obtained direct recursion through partial evaluation 878.01/291.51 0. SCC is partially evaluated into gt/3 878.01/291.51 1. SCC is partially evaluated into f/4 878.01/291.51 2. SCC is partially evaluated into start/3 878.01/291.51 878.01/291.51 Control-Flow Refinement of Cost Relations 878.01/291.51 ===================================== 878.01/291.51 878.01/291.51 ### Specialization of cost equations gt/3 878.01/291.51 * CE 7 is refined into CE [8] 878.01/291.51 * CE 6 is refined into CE [9] 878.01/291.51 * CE 5 is refined into CE [10] 878.01/291.51 878.01/291.51 878.01/291.51 ### Cost equations --> "Loop" of gt/3 878.01/291.51 * CEs [9] --> Loop 7 878.01/291.51 * CEs [10] --> Loop 8 878.01/291.51 * CEs [8] --> Loop 9 878.01/291.51 878.01/291.51 ### Ranking functions of CR gt(V1,V,Out) 878.01/291.51 * RF of phase [9]: [V,V1] 878.01/291.51 878.01/291.51 #### Partial ranking functions of CR gt(V1,V,Out) 878.01/291.51 * Partial RF of phase [9]: 878.01/291.51 - RF of loop [9:1]: 878.01/291.51 V 878.01/291.51 V1 878.01/291.51 878.01/291.51 878.01/291.51 ### Specialization of cost equations f/4 878.01/291.51 * CE 4 is refined into CE [11] 878.01/291.51 * CE 3 is refined into CE [12,13,14,15] 878.01/291.51 878.01/291.51 878.01/291.51 ### Cost equations --> "Loop" of f/4 878.01/291.51 * CEs [15] --> Loop 10 878.01/291.51 * CEs [14] --> Loop 11 878.01/291.51 * CEs [13] --> Loop 12 878.01/291.51 * CEs [12] --> Loop 13 878.01/291.51 * CEs [11] --> Loop 14 878.01/291.51 878.01/291.51 ### Ranking functions of CR f(V1,V,V2,Out) 878.01/291.51 * RF of phase [10]: [V-V2] 878.01/291.51 878.01/291.51 #### Partial ranking functions of CR f(V1,V,V2,Out) 878.01/291.51 * Partial RF of phase [10]: 878.01/291.51 - RF of loop [10:1]: 878.01/291.51 V-V2 878.01/291.51 878.01/291.51 878.01/291.51 ### Specialization of cost equations start/3 878.01/291.51 * CE 1 is refined into CE [16,17,18,19] 878.01/291.51 * CE 2 is refined into CE [20,21,22,23] 878.01/291.51 878.01/291.51 878.01/291.51 ### Cost equations --> "Loop" of start/3 878.01/291.51 * CEs [23] --> Loop 15 878.01/291.51 * CEs [21] --> Loop 16 878.01/291.51 * CEs [19] --> Loop 17 878.01/291.51 * CEs [18,22] --> Loop 18 878.01/291.51 * CEs [16,17] --> Loop 19 878.01/291.51 * CEs [20] --> Loop 20 878.01/291.51 878.01/291.51 ### Ranking functions of CR start(V1,V,V2) 878.01/291.51 878.01/291.51 #### Partial ranking functions of CR start(V1,V,V2) 878.01/291.51 878.01/291.51 878.01/291.51 Computing Bounds 878.01/291.51 ===================================== 878.01/291.51 878.01/291.51 #### Cost of chains of gt(V1,V,Out): 878.01/291.51 * Chain [[9],8]: 1*it(9)+1 878.01/291.51 Such that:it(9) =< V1 878.01/291.51 878.01/291.51 with precondition: [Out=0,V1>=1,V>=V1] 878.01/291.51 878.01/291.51 * Chain [[9],7]: 1*it(9)+1 878.01/291.51 Such that:it(9) =< V 878.01/291.51 878.01/291.51 with precondition: [Out=1,V>=1,V1>=V+1] 878.01/291.51 878.01/291.51 * Chain [8]: 1 878.01/291.51 with precondition: [V1=0,Out=0,V>=0] 878.01/291.51 878.01/291.51 * Chain [7]: 1 878.01/291.51 with precondition: [V=0,Out=1,V1>=1] 878.01/291.51 878.01/291.51 878.01/291.51 #### Cost of chains of f(V1,V,V2,Out): 878.01/291.51 * Chain [[10],14]: 2*it(10)+1*s(3)+0 878.01/291.51 Such that:it(10) =< V-V2 878.01/291.51 aux(1) =< 2*V-V2 878.01/291.51 s(3) =< it(10)*aux(1) 878.01/291.51 878.01/291.51 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 878.01/291.51 878.01/291.51 * Chain [[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+2 878.01/291.51 Such that:it(10) =< V-V2 878.01/291.51 aux(1) =< 2*V-V2 878.01/291.51 s(4) =< 2*V-V2+1 878.01/291.51 s(3) =< it(10)*aux(1) 878.01/291.51 878.01/291.51 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 878.01/291.51 878.01/291.51 * Chain [14]: 0 878.01/291.51 with precondition: [Out=0,V1>=0,V>=0,V2>=0] 878.01/291.51 878.01/291.51 * Chain [13,14]: 2 878.01/291.51 with precondition: [V1=1,V=0,Out=0,V2>=0] 878.01/291.51 878.01/291.51 * Chain [12,[10],14]: 2*it(10)+1*s(3)+2 878.01/291.51 Such that:it(10) =< V 878.01/291.51 aux(1) =< 2*V 878.01/291.51 s(3) =< it(10)*aux(1) 878.01/291.51 878.01/291.51 with precondition: [V1=1,V2=0,Out=0,V>=2] 878.01/291.51 878.01/291.51 * Chain [12,[10],11,14]: 2*it(10)+1*s(3)+1*s(4)+4 878.01/291.51 Such that:it(10) =< V 878.01/291.51 aux(1) =< 2*V 878.01/291.51 s(4) =< 2*V+1 878.01/291.51 s(3) =< it(10)*aux(1) 878.01/291.51 878.01/291.51 with precondition: [V1=1,V2=0,Out=0,V>=2] 878.01/291.51 878.01/291.51 * Chain [12,14]: 2 878.01/291.51 with precondition: [V1=1,V2=0,Out=0,V>=1] 878.01/291.52 878.01/291.52 * Chain [12,11,14]: 1*s(4)+4 878.01/291.52 Such that:s(4) =< 3 878.01/291.52 878.01/291.52 with precondition: [V1=1,V=1,V2=0,Out=0] 878.01/291.52 878.01/291.52 * Chain [11,14]: 1*s(4)+2 878.01/291.52 Such that:s(4) =< V+1 878.01/291.52 878.01/291.52 with precondition: [V1=1,Out=0,V>=1,V2>=V] 878.01/291.52 878.01/291.52 878.01/291.52 #### Cost of chains of start(V1,V,V2): 878.01/291.52 * Chain [20]: 1 878.01/291.52 with precondition: [V1=0,V>=0] 878.01/291.52 878.01/291.52 * Chain [19]: 1*s(20)+1*s(21)+4*s(24)+2*s(25)+4 878.01/291.52 Such that:s(20) =< 3 878.01/291.52 s(22) =< V 878.01/291.52 s(23) =< 2*V 878.01/291.52 s(21) =< 2*V+1 878.01/291.52 s(24) =< s(22) 878.01/291.52 s(25) =< s(24)*s(23) 878.01/291.52 878.01/291.52 with precondition: [V1>=0,V>=0,V2>=0] 878.01/291.52 878.01/291.52 * Chain [18]: 1*s(26)+1*s(27)+2 878.01/291.52 Such that:s(27) =< V1 878.01/291.52 s(26) =< V+1 878.01/291.52 878.01/291.52 with precondition: [V1>=1,V>=V1] 878.01/291.52 878.01/291.52 * Chain [17]: 1*s(28)+4*s(31)+2*s(32)+2 878.01/291.52 Such that:s(29) =< V-V2 878.01/291.52 s(30) =< 2*V-V2 878.01/291.52 s(28) =< 2*V-V2+1 878.01/291.52 s(31) =< s(29) 878.01/291.52 s(32) =< s(31)*s(30) 878.01/291.52 878.01/291.52 with precondition: [V1=1,V2>=1,V>=V2+1] 878.01/291.52 878.01/291.52 * Chain [16]: 1 878.01/291.52 with precondition: [V=0,V1>=1] 878.01/291.52 878.01/291.52 * Chain [15]: 1*s(33)+1 878.01/291.52 Such that:s(33) =< V 878.01/291.52 878.01/291.52 with precondition: [V>=1,V1>=V+1] 878.01/291.52 878.01/291.52 878.01/291.52 Closed-form bounds of start(V1,V,V2): 878.01/291.52 ------------------------------------- 878.01/291.52 * Chain [20] with precondition: [V1=0,V>=0] 878.01/291.52 - Upper bound: 1 878.01/291.52 - Complexity: constant 878.01/291.52 * Chain [19] with precondition: [V1>=0,V>=0,V2>=0] 878.01/291.52 - Upper bound: 4*V+7+2*V*(2*V)+(2*V+1) 878.01/291.52 - Complexity: n^2 878.01/291.52 * Chain [18] with precondition: [V1>=1,V>=V1] 878.01/291.52 - Upper bound: V1+V+3 878.01/291.52 - Complexity: n 878.01/291.52 * Chain [17] with precondition: [V1=1,V2>=1,V>=V2+1] 878.01/291.52 - Upper bound: 6*V-5*V2+3+(2*V-2*V2)*(2*V-V2) 878.01/291.52 - Complexity: n^2 878.01/291.52 * Chain [16] with precondition: [V=0,V1>=1] 878.01/291.52 - Upper bound: 1 878.01/291.52 - Complexity: constant 878.01/291.52 * Chain [15] with precondition: [V>=1,V1>=V+1] 878.01/291.52 - Upper bound: V+1 878.01/291.52 - Complexity: n 878.01/291.52 878.01/291.52 ### 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])+1 878.01/291.52 Asymptotic class: n^2 878.01/291.52 * Total analysis performed in 206 ms. 878.01/291.52 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (10) 878.01/291.52 BOUNDS(1, n^2) 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 878.01/291.52 Renamed function symbols to avoid clashes with predefined symbol. 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (12) 878.01/291.52 Obligation: 878.01/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 878.01/291.52 878.01/291.52 878.01/291.52 The TRS R consists of the following rules: 878.01/291.52 878.01/291.52 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.52 gt(0', v) -> false 878.01/291.52 gt(s(u), 0') -> true 878.01/291.52 gt(s(u), s(v)) -> gt(u, v) 878.01/291.52 878.01/291.52 S is empty. 878.01/291.52 Rewrite Strategy: INNERMOST 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 878.01/291.52 Infered types. 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (14) 878.01/291.52 Obligation: 878.01/291.52 Innermost TRS: 878.01/291.52 Rules: 878.01/291.52 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.52 gt(0', v) -> false 878.01/291.52 gt(s(u), 0') -> true 878.01/291.52 gt(s(u), s(v)) -> gt(u, v) 878.01/291.52 878.01/291.52 Types: 878.01/291.52 f :: true:false -> s:0' -> s:0' -> f 878.01/291.52 true :: true:false 878.01/291.52 gt :: s:0' -> s:0' -> true:false 878.01/291.52 s :: s:0' -> s:0' 878.01/291.52 0' :: s:0' 878.01/291.52 false :: true:false 878.01/291.52 hole_f1_0 :: f 878.01/291.52 hole_true:false2_0 :: true:false 878.01/291.52 hole_s:0'3_0 :: s:0' 878.01/291.52 gen_s:0'4_0 :: Nat -> s:0' 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (15) OrderProof (LOWER BOUND(ID)) 878.01/291.52 Heuristically decided to analyse the following defined symbols: 878.01/291.52 f, gt 878.01/291.52 878.01/291.52 They will be analysed ascendingly in the following order: 878.01/291.52 gt < f 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (16) 878.01/291.52 Obligation: 878.01/291.52 Innermost TRS: 878.01/291.52 Rules: 878.01/291.52 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.52 gt(0', v) -> false 878.01/291.52 gt(s(u), 0') -> true 878.01/291.52 gt(s(u), s(v)) -> gt(u, v) 878.01/291.52 878.01/291.52 Types: 878.01/291.52 f :: true:false -> s:0' -> s:0' -> f 878.01/291.52 true :: true:false 878.01/291.52 gt :: s:0' -> s:0' -> true:false 878.01/291.52 s :: s:0' -> s:0' 878.01/291.52 0' :: s:0' 878.01/291.52 false :: true:false 878.01/291.52 hole_f1_0 :: f 878.01/291.52 hole_true:false2_0 :: true:false 878.01/291.52 hole_s:0'3_0 :: s:0' 878.01/291.52 gen_s:0'4_0 :: Nat -> s:0' 878.01/291.52 878.01/291.52 878.01/291.52 Generator Equations: 878.01/291.52 gen_s:0'4_0(0) <=> 0' 878.01/291.52 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 878.01/291.52 878.01/291.52 878.01/291.52 The following defined symbols remain to be analysed: 878.01/291.52 gt, f 878.01/291.52 878.01/291.52 They will be analysed ascendingly in the following order: 878.01/291.52 gt < f 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (17) RewriteLemmaProof (LOWER BOUND(ID)) 878.01/291.52 Proved the following rewrite lemma: 878.01/291.52 gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 878.01/291.52 878.01/291.52 Induction Base: 878.01/291.52 gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1) 878.01/291.52 false 878.01/291.52 878.01/291.52 Induction Step: 878.01/291.52 gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1) 878.01/291.52 gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH 878.01/291.52 false 878.01/291.52 878.01/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (18) 878.01/291.52 Complex Obligation (BEST) 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (19) 878.01/291.52 Obligation: 878.01/291.52 Proved the lower bound n^1 for the following obligation: 878.01/291.52 878.01/291.52 Innermost TRS: 878.01/291.52 Rules: 878.01/291.52 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.52 gt(0', v) -> false 878.01/291.52 gt(s(u), 0') -> true 878.01/291.52 gt(s(u), s(v)) -> gt(u, v) 878.01/291.52 878.01/291.52 Types: 878.01/291.52 f :: true:false -> s:0' -> s:0' -> f 878.01/291.52 true :: true:false 878.01/291.52 gt :: s:0' -> s:0' -> true:false 878.01/291.52 s :: s:0' -> s:0' 878.01/291.52 0' :: s:0' 878.01/291.52 false :: true:false 878.01/291.52 hole_f1_0 :: f 878.01/291.52 hole_true:false2_0 :: true:false 878.01/291.52 hole_s:0'3_0 :: s:0' 878.01/291.52 gen_s:0'4_0 :: Nat -> s:0' 878.01/291.52 878.01/291.52 878.01/291.52 Generator Equations: 878.01/291.52 gen_s:0'4_0(0) <=> 0' 878.01/291.52 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 878.01/291.52 878.01/291.52 878.01/291.52 The following defined symbols remain to be analysed: 878.01/291.52 gt, f 878.01/291.52 878.01/291.52 They will be analysed ascendingly in the following order: 878.01/291.52 gt < f 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (20) LowerBoundPropagationProof (FINISHED) 878.01/291.52 Propagated lower bound. 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (21) 878.01/291.52 BOUNDS(n^1, INF) 878.01/291.52 878.01/291.52 ---------------------------------------- 878.01/291.52 878.01/291.52 (22) 878.01/291.52 Obligation: 878.01/291.52 Innermost TRS: 878.01/291.52 Rules: 878.01/291.52 f(true, x, y) -> f(gt(x, y), s(x), s(s(y))) 878.01/291.52 gt(0', v) -> false 878.01/291.52 gt(s(u), 0') -> true 878.01/291.52 gt(s(u), s(v)) -> gt(u, v) 878.01/291.52 878.01/291.52 Types: 878.01/291.52 f :: true:false -> s:0' -> s:0' -> f 878.01/291.52 true :: true:false 878.01/291.52 gt :: s:0' -> s:0' -> true:false 878.01/291.52 s :: s:0' -> s:0' 878.01/291.52 0' :: s:0' 878.01/291.52 false :: true:false 878.01/291.52 hole_f1_0 :: f 878.01/291.52 hole_true:false2_0 :: true:false 878.01/291.52 hole_s:0'3_0 :: s:0' 878.01/291.52 gen_s:0'4_0 :: Nat -> s:0' 878.01/291.52 878.01/291.52 878.01/291.52 Lemmas: 878.01/291.52 gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 878.01/291.52 878.01/291.52 878.01/291.52 Generator Equations: 878.01/291.52 gen_s:0'4_0(0) <=> 0' 878.01/291.52 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 878.01/291.52 878.01/291.52 878.01/291.52 The following defined symbols remain to be analysed: 878.01/291.52 f 878.33/291.57 EOF