893.53/291.52 WORST_CASE(Omega(n^1), O(n^2)) 893.53/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 893.53/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 893.53/291.55 893.53/291.55 893.53/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 893.53/291.55 893.53/291.55 (0) CpxTRS 893.53/291.55 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 893.53/291.55 (2) CpxWeightedTrs 893.53/291.55 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 893.53/291.55 (4) CpxTypedWeightedTrs 893.53/291.55 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 893.53/291.55 (6) CpxTypedWeightedCompleteTrs 893.53/291.55 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 893.53/291.55 (8) CpxRNTS 893.53/291.55 (9) CompleteCoflocoProof [FINISHED, 252 ms] 893.53/291.55 (10) BOUNDS(1, n^2) 893.53/291.55 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 893.53/291.55 (12) CpxTRS 893.53/291.55 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 893.53/291.55 (14) typed CpxTrs 893.53/291.55 (15) OrderProof [LOWER BOUND(ID), 0 ms] 893.53/291.55 (16) typed CpxTrs 893.53/291.55 (17) RewriteLemmaProof [LOWER BOUND(ID), 290 ms] 893.53/291.55 (18) BEST 893.53/291.55 (19) proven lower bound 893.53/291.55 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 893.53/291.55 (21) BOUNDS(n^1, INF) 893.53/291.55 (22) typed CpxTrs 893.53/291.55 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (0) 893.53/291.55 Obligation: 893.53/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 893.53/291.55 893.53/291.55 893.53/291.55 The TRS R consists of the following rules: 893.53/291.55 893.53/291.55 lt(0, s(x)) -> true 893.53/291.55 lt(x, 0) -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0 893.53/291.55 893.53/291.55 S is empty. 893.53/291.55 Rewrite Strategy: INNERMOST 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 893.53/291.55 Transformed relative TRS to weighted TRS 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (2) 893.53/291.55 Obligation: 893.53/291.55 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 893.53/291.55 893.53/291.55 893.53/291.55 The TRS R consists of the following rules: 893.53/291.55 893.53/291.55 lt(0, s(x)) -> true [1] 893.53/291.55 lt(x, 0) -> false [1] 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) [1] 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) [1] 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) [1] 893.53/291.55 help(false, x, y) -> 0 [1] 893.53/291.55 893.53/291.55 Rewrite Strategy: INNERMOST 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 893.53/291.55 Infered types. 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (4) 893.53/291.55 Obligation: 893.53/291.55 Runtime Complexity Weighted TRS with Types. 893.53/291.55 The TRS R consists of the following rules: 893.53/291.55 893.53/291.55 lt(0, s(x)) -> true [1] 893.53/291.55 lt(x, 0) -> false [1] 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) [1] 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) [1] 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) [1] 893.53/291.55 help(false, x, y) -> 0 [1] 893.53/291.55 893.53/291.55 The TRS has the following type information: 893.53/291.55 lt :: 0:s -> 0:s -> true:false 893.53/291.55 0 :: 0:s 893.53/291.55 s :: 0:s -> 0:s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0:s -> 0:s -> 0:s 893.53/291.55 help :: true:false -> 0:s -> 0:s -> 0:s 893.53/291.55 893.53/291.55 Rewrite Strategy: INNERMOST 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (5) CompletionProof (UPPER BOUND(ID)) 893.53/291.55 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 893.53/291.55 none 893.53/291.55 893.53/291.55 And the following fresh constants: none 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (6) 893.53/291.55 Obligation: 893.53/291.55 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 893.53/291.55 893.53/291.55 Runtime Complexity Weighted TRS with Types. 893.53/291.55 The TRS R consists of the following rules: 893.53/291.55 893.53/291.55 lt(0, s(x)) -> true [1] 893.53/291.55 lt(x, 0) -> false [1] 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) [1] 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) [1] 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) [1] 893.53/291.55 help(false, x, y) -> 0 [1] 893.53/291.55 893.53/291.55 The TRS has the following type information: 893.53/291.55 lt :: 0:s -> 0:s -> true:false 893.53/291.55 0 :: 0:s 893.53/291.55 s :: 0:s -> 0:s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0:s -> 0:s -> 0:s 893.53/291.55 help :: true:false -> 0:s -> 0:s -> 0:s 893.53/291.55 893.53/291.55 Rewrite Strategy: INNERMOST 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 893.53/291.55 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 893.53/291.55 The constant constructors are abstracted as follows: 893.53/291.55 893.53/291.55 0 => 0 893.53/291.55 true => 1 893.53/291.55 false => 0 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (8) 893.53/291.55 Obligation: 893.53/291.55 Complexity RNTS consisting of the following rules: 893.53/291.55 893.53/291.55 help(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 893.53/291.55 help(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 893.53/291.55 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 893.53/291.55 lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 893.53/291.55 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 893.53/291.55 minus(z, z') -{ 1 }-> help(lt(y, x), x, y) :|: x >= 0, y >= 0, z = x, z' = y 893.53/291.55 893.53/291.55 Only complete derivations are relevant for the runtime complexity. 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (9) CompleteCoflocoProof (FINISHED) 893.53/291.55 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 893.53/291.55 893.53/291.55 eq(start(V1, V, V10),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]). 893.53/291.55 eq(start(V1, V, V10),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 893.53/291.55 eq(start(V1, V, V10),0,[help(V1, V, V10, Out)],[V1 >= 0,V >= 0,V10 >= 0]). 893.53/291.55 eq(lt(V1, V, Out),1,[],[Out = 1,V = 1 + V2,V2 >= 0,V1 = 0]). 893.53/291.55 eq(lt(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = V3,V = 0]). 893.53/291.55 eq(lt(V1, V, Out),1,[lt(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). 893.53/291.55 eq(minus(V1, V, Out),1,[lt(V7, V6, Ret0),help(Ret0, V6, V7, Ret1)],[Out = Ret1,V6 >= 0,V7 >= 0,V1 = V6,V = V7]). 893.53/291.55 eq(help(V1, V, V10, Out),1,[minus(V8, 1 + V9, Ret11)],[Out = 1 + Ret11,V = V8,V10 = V9,V1 = 1,V8 >= 0,V9 >= 0]). 893.53/291.55 eq(help(V1, V, V10, Out),1,[],[Out = 0,V = V12,V10 = V11,V12 >= 0,V11 >= 0,V1 = 0]). 893.53/291.55 input_output_vars(lt(V1,V,Out),[V1,V],[Out]). 893.53/291.55 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 893.53/291.55 input_output_vars(help(V1,V,V10,Out),[V1,V,V10],[Out]). 893.53/291.55 893.53/291.55 893.53/291.55 CoFloCo proof output: 893.53/291.55 Preprocessing Cost Relations 893.53/291.55 ===================================== 893.53/291.55 893.53/291.55 #### Computed strongly connected components 893.53/291.55 0. recursive : [lt/3] 893.53/291.55 1. recursive : [help/4,minus/3] 893.53/291.55 2. non_recursive : [start/3] 893.53/291.55 893.53/291.55 #### Obtained direct recursion through partial evaluation 893.53/291.55 0. SCC is partially evaluated into lt/3 893.53/291.55 1. SCC is partially evaluated into minus/3 893.53/291.55 2. SCC is partially evaluated into start/3 893.53/291.55 893.53/291.55 Control-Flow Refinement of Cost Relations 893.53/291.55 ===================================== 893.53/291.55 893.53/291.55 ### Specialization of cost equations lt/3 893.53/291.55 * CE 9 is refined into CE [10] 893.53/291.55 * CE 8 is refined into CE [11] 893.53/291.55 * CE 7 is refined into CE [12] 893.53/291.55 893.53/291.55 893.53/291.55 ### Cost equations --> "Loop" of lt/3 893.53/291.55 * CEs [11] --> Loop 8 893.53/291.55 * CEs [12] --> Loop 9 893.53/291.55 * CEs [10] --> Loop 10 893.53/291.55 893.53/291.55 ### Ranking functions of CR lt(V1,V,Out) 893.53/291.55 * RF of phase [10]: [V,V1] 893.53/291.55 893.53/291.55 #### Partial ranking functions of CR lt(V1,V,Out) 893.53/291.55 * Partial RF of phase [10]: 893.53/291.55 - RF of loop [10:1]: 893.53/291.55 V 893.53/291.55 V1 893.53/291.55 893.53/291.55 893.53/291.55 ### Specialization of cost equations minus/3 893.53/291.55 * CE 6 is refined into CE [13,14] 893.53/291.55 * CE 5 is refined into CE [15,16] 893.53/291.55 893.53/291.55 893.53/291.55 ### Cost equations --> "Loop" of minus/3 893.53/291.55 * CEs [16] --> Loop 11 893.53/291.55 * CEs [15] --> Loop 12 893.53/291.55 * CEs [14] --> Loop 13 893.53/291.55 * CEs [13] --> Loop 14 893.53/291.55 893.53/291.55 ### Ranking functions of CR minus(V1,V,Out) 893.53/291.55 * RF of phase [13]: [V1-V] 893.53/291.55 893.53/291.55 #### Partial ranking functions of CR minus(V1,V,Out) 893.53/291.55 * Partial RF of phase [13]: 893.53/291.55 - RF of loop [13:1]: 893.53/291.55 V1-V 893.53/291.55 893.53/291.55 893.53/291.55 ### Specialization of cost equations start/3 893.53/291.55 * CE 2 is refined into CE [17,18,19] 893.53/291.55 * CE 1 is refined into CE [20] 893.53/291.55 * CE 3 is refined into CE [21,22,23,24] 893.53/291.55 * CE 4 is refined into CE [25,26,27,28,29] 893.53/291.55 893.53/291.55 893.53/291.55 ### Cost equations --> "Loop" of start/3 893.53/291.55 * CEs [23,29] --> Loop 15 893.53/291.55 * CEs [19] --> Loop 16 893.53/291.55 * CEs [18,24,28] --> Loop 17 893.53/291.55 * CEs [17,22,26,27] --> Loop 18 893.53/291.55 * CEs [20,21,25] --> Loop 19 893.53/291.55 893.53/291.55 ### Ranking functions of CR start(V1,V,V10) 893.53/291.55 893.53/291.55 #### Partial ranking functions of CR start(V1,V,V10) 893.53/291.55 893.53/291.55 893.53/291.55 Computing Bounds 893.53/291.55 ===================================== 893.53/291.55 893.53/291.55 #### Cost of chains of lt(V1,V,Out): 893.53/291.55 * Chain [[10],9]: 1*it(10)+1 893.53/291.55 Such that:it(10) =< V1 893.53/291.55 893.53/291.55 with precondition: [Out=1,V1>=1,V>=V1+1] 893.53/291.55 893.53/291.55 * Chain [[10],8]: 1*it(10)+1 893.53/291.55 Such that:it(10) =< V 893.53/291.55 893.53/291.55 with precondition: [Out=0,V>=1,V1>=V] 893.53/291.55 893.53/291.55 * Chain [9]: 1 893.53/291.55 with precondition: [V1=0,Out=1,V>=1] 893.53/291.55 893.53/291.55 * Chain [8]: 1 893.53/291.55 with precondition: [V=0,Out=0,V1>=0] 893.53/291.55 893.53/291.55 893.53/291.55 #### Cost of chains of minus(V1,V,Out): 893.53/291.55 * Chain [[13],11]: 3*it(13)+1*s(1)+1*s(4)+3 893.53/291.55 Such that:it(13) =< Out 893.53/291.55 aux(2) =< V+Out 893.53/291.55 s(1) =< aux(2) 893.53/291.55 s(4) =< it(13)*aux(2) 893.53/291.55 893.53/291.55 with precondition: [V1=Out+V,V>=1,V1>=V+1] 893.53/291.55 893.53/291.55 * Chain [14,[13],11]: 4*it(13)+1*s(4)+6 893.53/291.55 Such that:aux(3) =< Out 893.53/291.55 it(13) =< aux(3) 893.53/291.55 s(4) =< it(13)*aux(3) 893.53/291.55 893.53/291.55 with precondition: [V=0,V1=Out,V1>=2] 893.53/291.55 893.53/291.55 * Chain [14,11]: 1*s(1)+6 893.53/291.55 Such that:s(1) =< 1 893.53/291.55 893.53/291.55 with precondition: [V1=1,V=0,Out=1] 893.53/291.55 893.53/291.55 * Chain [12]: 3 893.53/291.55 with precondition: [V1=0,Out=0,V>=0] 893.53/291.55 893.53/291.55 * Chain [11]: 1*s(1)+3 893.53/291.55 Such that:s(1) =< V1 893.53/291.55 893.53/291.55 with precondition: [Out=0,V1>=1,V>=V1] 893.53/291.55 893.53/291.55 893.53/291.55 #### Cost of chains of start(V1,V,V10): 893.53/291.55 * Chain [19]: 3 893.53/291.55 with precondition: [V1=0,V>=0] 893.53/291.55 893.53/291.55 * Chain [18]: 1*s(5)+4*s(7)+1*s(8)+6 893.53/291.55 Such that:s(5) =< 1 893.53/291.55 s(6) =< V1 893.53/291.55 s(7) =< s(6) 893.53/291.55 s(8) =< s(7)*s(6) 893.53/291.55 893.53/291.55 with precondition: [V=0,V1>=0] 893.53/291.55 893.53/291.55 * Chain [17]: 1*s(9)+2*s(10)+4 893.53/291.55 Such that:s(9) =< V 893.53/291.55 aux(4) =< V1 893.53/291.55 s(10) =< aux(4) 893.53/291.55 893.53/291.55 with precondition: [V1>=1,V>=V1] 893.53/291.55 893.53/291.55 * Chain [16]: 3*s(12)+1*s(14)+1*s(15)+4 893.53/291.55 Such that:s(13) =< V 893.53/291.55 s(12) =< V-V10 893.53/291.55 s(14) =< s(13) 893.53/291.55 s(15) =< s(12)*s(13) 893.53/291.55 893.53/291.55 with precondition: [V1=1,V10>=0,V>=V10+2] 893.53/291.55 893.53/291.55 * Chain [15]: 1*s(16)+3*s(17)+1*s(19)+1*s(20)+3 893.53/291.55 Such that:s(18) =< V1 893.53/291.55 s(17) =< V1-V 893.53/291.55 s(16) =< V 893.53/291.55 s(19) =< s(18) 893.53/291.55 s(20) =< s(17)*s(18) 893.53/291.55 893.53/291.55 with precondition: [V>=1,V1>=V] 893.53/291.55 893.53/291.55 893.53/291.55 Closed-form bounds of start(V1,V,V10): 893.53/291.55 ------------------------------------- 893.53/291.55 * Chain [19] with precondition: [V1=0,V>=0] 893.53/291.55 - Upper bound: 3 893.53/291.55 - Complexity: constant 893.53/291.55 * Chain [18] with precondition: [V=0,V1>=0] 893.53/291.55 - Upper bound: 4*V1+7+V1*V1 893.53/291.55 - Complexity: n^2 893.53/291.55 * Chain [17] with precondition: [V1>=1,V>=V1] 893.53/291.55 - Upper bound: 2*V1+V+4 893.53/291.55 - Complexity: n 893.53/291.55 * Chain [16] with precondition: [V1=1,V10>=0,V>=V10+2] 893.53/291.55 - Upper bound: 3*V-3*V10+(V+4+(V-V10)*V) 893.53/291.55 - Complexity: n^2 893.53/291.55 * Chain [15] with precondition: [V>=1,V1>=V] 893.53/291.55 - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1+V) 893.53/291.55 - Complexity: n^2 893.53/291.55 893.53/291.55 ### Maximum cost of start(V1,V,V10): max([max([V1+1+max([V,2*V1+3+V1*V1]),nat(V1-V)*V1+V+nat(V1-V)*3])+V1,V+1+nat(V-V10)*V+nat(V-V10)*3])+3 893.53/291.55 Asymptotic class: n^2 893.53/291.55 * Total analysis performed in 169 ms. 893.53/291.55 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (10) 893.53/291.55 BOUNDS(1, n^2) 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 893.53/291.55 Renamed function symbols to avoid clashes with predefined symbol. 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (12) 893.53/291.55 Obligation: 893.53/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.53/291.55 893.53/291.55 893.53/291.55 The TRS R consists of the following rules: 893.53/291.55 893.53/291.55 lt(0', s(x)) -> true 893.53/291.55 lt(x, 0') -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0' 893.53/291.55 893.53/291.55 S is empty. 893.53/291.55 Rewrite Strategy: INNERMOST 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 893.53/291.55 Infered types. 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (14) 893.53/291.55 Obligation: 893.53/291.55 Innermost TRS: 893.53/291.55 Rules: 893.53/291.55 lt(0', s(x)) -> true 893.53/291.55 lt(x, 0') -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0' 893.53/291.55 893.53/291.55 Types: 893.53/291.55 lt :: 0':s -> 0':s -> true:false 893.53/291.55 0' :: 0':s 893.53/291.55 s :: 0':s -> 0':s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0':s -> 0':s -> 0':s 893.53/291.55 help :: true:false -> 0':s -> 0':s -> 0':s 893.53/291.55 hole_true:false1_0 :: true:false 893.53/291.55 hole_0':s2_0 :: 0':s 893.53/291.55 gen_0':s3_0 :: Nat -> 0':s 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (15) OrderProof (LOWER BOUND(ID)) 893.53/291.55 Heuristically decided to analyse the following defined symbols: 893.53/291.55 lt, minus 893.53/291.55 893.53/291.55 They will be analysed ascendingly in the following order: 893.53/291.55 lt < minus 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (16) 893.53/291.55 Obligation: 893.53/291.55 Innermost TRS: 893.53/291.55 Rules: 893.53/291.55 lt(0', s(x)) -> true 893.53/291.55 lt(x, 0') -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0' 893.53/291.55 893.53/291.55 Types: 893.53/291.55 lt :: 0':s -> 0':s -> true:false 893.53/291.55 0' :: 0':s 893.53/291.55 s :: 0':s -> 0':s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0':s -> 0':s -> 0':s 893.53/291.55 help :: true:false -> 0':s -> 0':s -> 0':s 893.53/291.55 hole_true:false1_0 :: true:false 893.53/291.55 hole_0':s2_0 :: 0':s 893.53/291.55 gen_0':s3_0 :: Nat -> 0':s 893.53/291.55 893.53/291.55 893.53/291.55 Generator Equations: 893.53/291.55 gen_0':s3_0(0) <=> 0' 893.53/291.55 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 893.53/291.55 893.53/291.55 893.53/291.55 The following defined symbols remain to be analysed: 893.53/291.55 lt, minus 893.53/291.55 893.53/291.55 They will be analysed ascendingly in the following order: 893.53/291.55 lt < minus 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (17) RewriteLemmaProof (LOWER BOUND(ID)) 893.53/291.55 Proved the following rewrite lemma: 893.53/291.55 lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0) 893.53/291.55 893.53/291.55 Induction Base: 893.53/291.55 lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) ->_R^Omega(1) 893.53/291.55 true 893.53/291.55 893.53/291.55 Induction Step: 893.53/291.55 lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) 893.53/291.55 lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) ->_IH 893.53/291.55 true 893.53/291.55 893.53/291.55 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (18) 893.53/291.55 Complex Obligation (BEST) 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (19) 893.53/291.55 Obligation: 893.53/291.55 Proved the lower bound n^1 for the following obligation: 893.53/291.55 893.53/291.55 Innermost TRS: 893.53/291.55 Rules: 893.53/291.55 lt(0', s(x)) -> true 893.53/291.55 lt(x, 0') -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0' 893.53/291.55 893.53/291.55 Types: 893.53/291.55 lt :: 0':s -> 0':s -> true:false 893.53/291.55 0' :: 0':s 893.53/291.55 s :: 0':s -> 0':s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0':s -> 0':s -> 0':s 893.53/291.55 help :: true:false -> 0':s -> 0':s -> 0':s 893.53/291.55 hole_true:false1_0 :: true:false 893.53/291.55 hole_0':s2_0 :: 0':s 893.53/291.55 gen_0':s3_0 :: Nat -> 0':s 893.53/291.55 893.53/291.55 893.53/291.55 Generator Equations: 893.53/291.55 gen_0':s3_0(0) <=> 0' 893.53/291.55 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 893.53/291.55 893.53/291.55 893.53/291.55 The following defined symbols remain to be analysed: 893.53/291.55 lt, minus 893.53/291.55 893.53/291.55 They will be analysed ascendingly in the following order: 893.53/291.55 lt < minus 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (20) LowerBoundPropagationProof (FINISHED) 893.53/291.55 Propagated lower bound. 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (21) 893.53/291.55 BOUNDS(n^1, INF) 893.53/291.55 893.53/291.55 ---------------------------------------- 893.53/291.55 893.53/291.55 (22) 893.53/291.55 Obligation: 893.53/291.55 Innermost TRS: 893.53/291.55 Rules: 893.53/291.55 lt(0', s(x)) -> true 893.53/291.55 lt(x, 0') -> false 893.53/291.55 lt(s(x), s(y)) -> lt(x, y) 893.53/291.55 minus(x, y) -> help(lt(y, x), x, y) 893.53/291.55 help(true, x, y) -> s(minus(x, s(y))) 893.53/291.55 help(false, x, y) -> 0' 893.53/291.55 893.53/291.55 Types: 893.53/291.55 lt :: 0':s -> 0':s -> true:false 893.53/291.55 0' :: 0':s 893.53/291.55 s :: 0':s -> 0':s 893.53/291.55 true :: true:false 893.53/291.55 false :: true:false 893.53/291.55 minus :: 0':s -> 0':s -> 0':s 893.53/291.55 help :: true:false -> 0':s -> 0':s -> 0':s 893.53/291.55 hole_true:false1_0 :: true:false 893.53/291.55 hole_0':s2_0 :: 0':s 893.53/291.55 gen_0':s3_0 :: Nat -> 0':s 893.53/291.55 893.53/291.55 893.53/291.55 Lemmas: 893.53/291.55 lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0) 893.53/291.55 893.53/291.55 893.53/291.55 Generator Equations: 893.53/291.55 gen_0':s3_0(0) <=> 0' 893.53/291.55 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 893.53/291.55 893.53/291.55 893.53/291.55 The following defined symbols remain to be analysed: 893.53/291.55 minus 893.82/291.59 EOF