14.76/4.80 WORST_CASE(Omega(n^1), O(n^1)) 14.76/4.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 14.76/4.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.76/4.82 14.76/4.82 14.76/4.82 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 14.76/4.82 14.76/4.82 (0) CpxTRS 14.76/4.82 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 14.76/4.82 (2) CpxWeightedTrs 14.76/4.82 (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 14.76/4.82 (4) CpxWeightedTrs 14.76/4.82 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 14.76/4.82 (6) CpxTypedWeightedTrs 14.76/4.82 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 14.76/4.82 (8) CpxTypedWeightedCompleteTrs 14.76/4.82 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 14.76/4.82 (10) CpxRNTS 14.76/4.82 (11) CompleteCoflocoProof [FINISHED, 39 ms] 14.76/4.82 (12) BOUNDS(1, n^1) 14.76/4.82 (13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 14.76/4.82 (14) CpxTRS 14.76/4.82 (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 14.76/4.82 (16) typed CpxTrs 14.76/4.82 (17) OrderProof [LOWER BOUND(ID), 0 ms] 14.76/4.82 (18) typed CpxTrs 14.76/4.82 (19) RewriteLemmaProof [LOWER BOUND(ID), 288 ms] 14.76/4.82 (20) BEST 14.76/4.82 (21) proven lower bound 14.76/4.82 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 14.76/4.82 (23) BOUNDS(n^1, INF) 14.76/4.82 (24) typed CpxTrs 14.76/4.82 (25) RewriteLemmaProof [LOWER BOUND(ID), 166 ms] 14.76/4.82 (26) BOUNDS(1, INF) 14.76/4.82 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (0) 14.76/4.82 Obligation: 14.76/4.82 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 14.76/4.82 14.76/4.82 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0) -> 0 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +(x, 0) -> x 14.76/4.82 +(x, s(y)) -> s(+(x, y)) 14.76/4.82 +(s(x), y) -> s(+(x, y)) 14.76/4.82 double(x) -> +(x, x) 14.76/4.82 14.76/4.82 S is empty. 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 14.76/4.82 Transformed relative TRS to weighted TRS 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (2) 14.76/4.82 Obligation: 14.76/4.82 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 14.76/4.82 14.76/4.82 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0) -> 0 [1] 14.76/4.82 double(s(x)) -> s(s(double(x))) [1] 14.76/4.82 +(x, 0) -> x [1] 14.76/4.82 +(x, s(y)) -> s(+(x, y)) [1] 14.76/4.82 +(s(x), y) -> s(+(x, y)) [1] 14.76/4.82 double(x) -> +(x, x) [1] 14.76/4.82 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) 14.76/4.82 Renamed defined symbols to avoid conflicts with arithmetic symbols: 14.76/4.82 14.76/4.82 + => plus 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (4) 14.76/4.82 Obligation: 14.76/4.82 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 14.76/4.82 14.76/4.82 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0) -> 0 [1] 14.76/4.82 double(s(x)) -> s(s(double(x))) [1] 14.76/4.82 plus(x, 0) -> x [1] 14.76/4.82 plus(x, s(y)) -> s(plus(x, y)) [1] 14.76/4.82 plus(s(x), y) -> s(plus(x, y)) [1] 14.76/4.82 double(x) -> plus(x, x) [1] 14.76/4.82 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 14.76/4.82 Infered types. 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (6) 14.76/4.82 Obligation: 14.76/4.82 Runtime Complexity Weighted TRS with Types. 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0) -> 0 [1] 14.76/4.82 double(s(x)) -> s(s(double(x))) [1] 14.76/4.82 plus(x, 0) -> x [1] 14.76/4.82 plus(x, s(y)) -> s(plus(x, y)) [1] 14.76/4.82 plus(s(x), y) -> s(plus(x, y)) [1] 14.76/4.82 double(x) -> plus(x, x) [1] 14.76/4.82 14.76/4.82 The TRS has the following type information: 14.76/4.82 double :: 0:s -> 0:s 14.76/4.82 0 :: 0:s 14.76/4.82 s :: 0:s -> 0:s 14.76/4.82 plus :: 0:s -> 0:s -> 0:s 14.76/4.82 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (7) CompletionProof (UPPER BOUND(ID)) 14.76/4.82 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 14.76/4.82 none 14.76/4.82 14.76/4.82 And the following fresh constants: none 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (8) 14.76/4.82 Obligation: 14.76/4.82 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 14.76/4.82 14.76/4.82 Runtime Complexity Weighted TRS with Types. 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0) -> 0 [1] 14.76/4.82 double(s(x)) -> s(s(double(x))) [1] 14.76/4.82 plus(x, 0) -> x [1] 14.76/4.82 plus(x, s(y)) -> s(plus(x, y)) [1] 14.76/4.82 plus(s(x), y) -> s(plus(x, y)) [1] 14.76/4.82 double(x) -> plus(x, x) [1] 14.76/4.82 14.76/4.82 The TRS has the following type information: 14.76/4.82 double :: 0:s -> 0:s 14.76/4.82 0 :: 0:s 14.76/4.82 s :: 0:s -> 0:s 14.76/4.82 plus :: 0:s -> 0:s -> 0:s 14.76/4.82 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 14.76/4.82 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 14.76/4.82 The constant constructors are abstracted as follows: 14.76/4.82 14.76/4.82 0 => 0 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (10) 14.76/4.82 Obligation: 14.76/4.82 Complexity RNTS consisting of the following rules: 14.76/4.82 14.76/4.82 double(z) -{ 1 }-> plus(x, x) :|: x >= 0, z = x 14.76/4.82 double(z) -{ 1 }-> 0 :|: z = 0 14.76/4.82 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x 14.76/4.82 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 14.76/4.82 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x 14.76/4.82 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 14.76/4.82 14.76/4.82 Only complete derivations are relevant for the runtime complexity. 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (11) CompleteCoflocoProof (FINISHED) 14.76/4.82 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 14.76/4.82 14.76/4.82 eq(start(V, V2),0,[double(V, Out)],[V >= 0]). 14.76/4.82 eq(start(V, V2),0,[plus(V, V2, Out)],[V >= 0,V2 >= 0]). 14.76/4.82 eq(double(V, Out),1,[],[Out = 0,V = 0]). 14.76/4.82 eq(double(V, Out),1,[double(V1, Ret11)],[Out = 2 + Ret11,V1 >= 0,V = 1 + V1]). 14.76/4.82 eq(plus(V, V2, Out),1,[],[Out = V3,V3 >= 0,V = V3,V2 = 0]). 14.76/4.82 eq(plus(V, V2, Out),1,[plus(V4, V5, Ret1)],[Out = 1 + Ret1,V2 = 1 + V5,V4 >= 0,V5 >= 0,V = V4]). 14.76/4.82 eq(plus(V, V2, Out),1,[plus(V6, V7, Ret12)],[Out = 1 + Ret12,V6 >= 0,V7 >= 0,V = 1 + V6,V2 = V7]). 14.76/4.82 eq(double(V, Out),1,[plus(V8, V8, Ret)],[Out = Ret,V8 >= 0,V = V8]). 14.76/4.82 input_output_vars(double(V,Out),[V],[Out]). 14.76/4.82 input_output_vars(plus(V,V2,Out),[V,V2],[Out]). 14.76/4.82 14.76/4.82 14.76/4.82 CoFloCo proof output: 14.76/4.82 Preprocessing Cost Relations 14.76/4.82 ===================================== 14.76/4.82 14.76/4.82 #### Computed strongly connected components 14.76/4.82 0. recursive : [plus/3] 14.76/4.82 1. recursive : [double/2] 14.76/4.82 2. non_recursive : [start/2] 14.76/4.82 14.76/4.82 #### Obtained direct recursion through partial evaluation 14.76/4.82 0. SCC is partially evaluated into plus/3 14.76/4.82 1. SCC is partially evaluated into double/2 14.76/4.82 2. SCC is partially evaluated into start/2 14.76/4.82 14.76/4.82 Control-Flow Refinement of Cost Relations 14.76/4.82 ===================================== 14.76/4.82 14.76/4.82 ### Specialization of cost equations plus/3 14.76/4.82 * CE 7 is refined into CE [9] 14.76/4.82 * CE 8 is refined into CE [10] 14.76/4.82 * CE 6 is refined into CE [11] 14.76/4.82 14.76/4.82 14.76/4.82 ### Cost equations --> "Loop" of plus/3 14.76/4.82 * CEs [11] --> Loop 7 14.76/4.82 * CEs [9] --> Loop 8 14.76/4.82 * CEs [10] --> Loop 9 14.76/4.82 14.76/4.82 ### Ranking functions of CR plus(V,V2,Out) 14.76/4.82 * RF of phase [8,9]: [V+V2] 14.76/4.82 14.76/4.82 #### Partial ranking functions of CR plus(V,V2,Out) 14.76/4.82 * Partial RF of phase [8,9]: 14.76/4.82 - RF of loop [8:1]: 14.76/4.82 V2 14.76/4.82 - RF of loop [9:1]: 14.76/4.82 V 14.76/4.82 14.76/4.82 14.76/4.82 ### Specialization of cost equations double/2 14.76/4.82 * CE 3 is refined into CE [12] 14.76/4.82 * CE 5 is refined into CE [13,14] 14.76/4.82 * CE 4 is refined into CE [15] 14.76/4.82 14.76/4.82 14.76/4.82 ### Cost equations --> "Loop" of double/2 14.76/4.82 * CEs [15] --> Loop 10 14.76/4.82 * CEs [14] --> Loop 11 14.76/4.82 * CEs [12,13] --> Loop 12 14.76/4.82 14.76/4.82 ### Ranking functions of CR double(V,Out) 14.76/4.82 * RF of phase [10]: [V] 14.76/4.82 14.76/4.82 #### Partial ranking functions of CR double(V,Out) 14.76/4.82 * Partial RF of phase [10]: 14.76/4.82 - RF of loop [10:1]: 14.76/4.82 V 14.76/4.82 14.76/4.82 14.76/4.82 ### Specialization of cost equations start/2 14.76/4.82 * CE 1 is refined into CE [16,17] 14.76/4.82 * CE 2 is refined into CE [18,19] 14.76/4.82 14.76/4.82 14.76/4.82 ### Cost equations --> "Loop" of start/2 14.76/4.82 * CEs [17] --> Loop 13 14.76/4.82 * CEs [19] --> Loop 14 14.76/4.82 * CEs [18] --> Loop 15 14.76/4.82 * CEs [16] --> Loop 16 14.76/4.82 14.76/4.82 ### Ranking functions of CR start(V,V2) 14.76/4.82 14.76/4.82 #### Partial ranking functions of CR start(V,V2) 14.76/4.82 14.76/4.82 14.76/4.82 Computing Bounds 14.76/4.82 ===================================== 14.76/4.82 14.76/4.82 #### Cost of chains of plus(V,V2,Out): 14.76/4.82 * Chain [[8,9],7]: 1*it(8)+1*it(9)+1 14.76/4.82 Such that:it(9) =< -V2+Out 14.76/4.82 it(8) =< V2 14.76/4.82 aux(3) =< Out 14.76/4.82 it(8) =< aux(3) 14.76/4.82 it(9) =< aux(3) 14.76/4.82 14.76/4.82 with precondition: [V+V2=Out,V>=0,V2>=0,V+V2>=1] 14.76/4.82 14.76/4.82 * Chain [7]: 1 14.76/4.82 with precondition: [V2=0,V=Out,V>=0] 14.76/4.82 14.76/4.82 14.76/4.82 #### Cost of chains of double(V,Out): 14.76/4.82 * Chain [[10],12]: 1*it(10)+2 14.76/4.82 Such that:it(10) =< Out/2 14.76/4.82 14.76/4.82 with precondition: [2*V=Out,V>=1] 14.76/4.82 14.76/4.82 * Chain [[10],11]: 1*it(10)+2*s(1)+2 14.76/4.82 Such that:aux(4) =< Out/2 14.76/4.82 aux(5) =< Out 14.76/4.82 it(10) =< aux(5) 14.76/4.82 s(1) =< aux(4) 14.76/4.82 s(1) =< aux(5) 14.76/4.82 14.76/4.82 with precondition: [2*V=Out,2*V>=3] 14.76/4.82 14.76/4.82 * Chain [12]: 2 14.76/4.82 with precondition: [V=0,Out=0] 14.76/4.82 14.76/4.82 * Chain [11]: 2*s(1)+2 14.76/4.82 Such that:s(3) =< 2*V 14.76/4.82 aux(4) =< V 14.76/4.82 s(1) =< aux(4) 14.76/4.82 s(1) =< s(3) 14.76/4.82 14.76/4.82 with precondition: [2*V=Out,2*V>=1] 14.76/4.82 14.76/4.82 14.76/4.82 #### Cost of chains of start(V,V2): 14.76/4.82 * Chain [16]: 2 14.76/4.82 with precondition: [V=0] 14.76/4.82 14.76/4.82 * Chain [15]: 1 14.76/4.82 with precondition: [V2=0,V>=0] 14.76/4.82 14.76/4.82 * Chain [14]: 1*s(12)+1*s(13)+1 14.76/4.82 Such that:s(12) =< V 14.76/4.82 s(14) =< V+V2 14.76/4.82 s(13) =< V2 14.76/4.82 s(13) =< s(14) 14.76/4.82 s(12) =< s(14) 14.76/4.82 14.76/4.82 with precondition: [V>=0,V2>=0,V+V2>=1] 14.76/4.82 14.76/4.82 * Chain [13]: 1*s(19)+4*s(20)+1*s(21)+2 14.76/4.82 Such that:aux(7) =< V 14.76/4.82 aux(8) =< 2*V 14.76/4.82 s(19) =< aux(7) 14.76/4.82 s(20) =< aux(7) 14.76/4.82 s(20) =< aux(8) 14.76/4.82 s(21) =< aux(8) 14.76/4.82 14.76/4.82 with precondition: [2*V>=1] 14.76/4.82 14.76/4.82 14.76/4.82 Closed-form bounds of start(V,V2): 14.76/4.82 ------------------------------------- 14.76/4.82 * Chain [16] with precondition: [V=0] 14.76/4.82 - Upper bound: 2 14.76/4.82 - Complexity: constant 14.76/4.82 * Chain [15] with precondition: [V2=0,V>=0] 14.76/4.82 - Upper bound: 1 14.76/4.82 - Complexity: constant 14.76/4.82 * Chain [14] with precondition: [V>=0,V2>=0,V+V2>=1] 14.76/4.82 - Upper bound: V+V2+1 14.76/4.82 - Complexity: n 14.76/4.82 * Chain [13] with precondition: [2*V>=1] 14.76/4.82 - Upper bound: 7*V+2 14.76/4.82 - Complexity: n 14.76/4.82 14.76/4.82 ### Maximum cost of start(V,V2): max([1,max([6*V+1,nat(V2)])+V])+1 14.76/4.82 Asymptotic class: n 14.76/4.82 * Total analysis performed in 113 ms. 14.76/4.82 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (12) 14.76/4.82 BOUNDS(1, n^1) 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (13) RenamingProof (BOTH BOUNDS(ID, ID)) 14.76/4.82 Renamed function symbols to avoid clashes with predefined symbol. 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (14) 14.76/4.82 Obligation: 14.76/4.82 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 14.76/4.82 14.76/4.82 14.76/4.82 The TRS R consists of the following rules: 14.76/4.82 14.76/4.82 double(0') -> 0' 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +'(x, 0') -> x 14.76/4.82 +'(x, s(y)) -> s(+'(x, y)) 14.76/4.82 +'(s(x), y) -> s(+'(x, y)) 14.76/4.82 double(x) -> +'(x, x) 14.76/4.82 14.76/4.82 S is empty. 14.76/4.82 Rewrite Strategy: INNERMOST 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (15) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 14.76/4.82 Infered types. 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (16) 14.76/4.82 Obligation: 14.76/4.82 Innermost TRS: 14.76/4.82 Rules: 14.76/4.82 double(0') -> 0' 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +'(x, 0') -> x 14.76/4.82 +'(x, s(y)) -> s(+'(x, y)) 14.76/4.82 +'(s(x), y) -> s(+'(x, y)) 14.76/4.82 double(x) -> +'(x, x) 14.76/4.82 14.76/4.82 Types: 14.76/4.82 double :: 0':s -> 0':s 14.76/4.82 0' :: 0':s 14.76/4.82 s :: 0':s -> 0':s 14.76/4.82 +' :: 0':s -> 0':s -> 0':s 14.76/4.82 hole_0':s1_0 :: 0':s 14.76/4.82 gen_0':s2_0 :: Nat -> 0':s 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (17) OrderProof (LOWER BOUND(ID)) 14.76/4.82 Heuristically decided to analyse the following defined symbols: 14.76/4.82 double, +' 14.76/4.82 14.76/4.82 They will be analysed ascendingly in the following order: 14.76/4.82 +' < double 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (18) 14.76/4.82 Obligation: 14.76/4.82 Innermost TRS: 14.76/4.82 Rules: 14.76/4.82 double(0') -> 0' 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +'(x, 0') -> x 14.76/4.82 +'(x, s(y)) -> s(+'(x, y)) 14.76/4.82 +'(s(x), y) -> s(+'(x, y)) 14.76/4.82 double(x) -> +'(x, x) 14.76/4.82 14.76/4.82 Types: 14.76/4.82 double :: 0':s -> 0':s 14.76/4.82 0' :: 0':s 14.76/4.82 s :: 0':s -> 0':s 14.76/4.82 +' :: 0':s -> 0':s -> 0':s 14.76/4.82 hole_0':s1_0 :: 0':s 14.76/4.82 gen_0':s2_0 :: Nat -> 0':s 14.76/4.82 14.76/4.82 14.76/4.82 Generator Equations: 14.76/4.82 gen_0':s2_0(0) <=> 0' 14.76/4.82 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 14.76/4.82 14.76/4.82 14.76/4.82 The following defined symbols remain to be analysed: 14.76/4.82 +', double 14.76/4.82 14.76/4.82 They will be analysed ascendingly in the following order: 14.76/4.82 +' < double 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (19) RewriteLemmaProof (LOWER BOUND(ID)) 14.76/4.82 Proved the following rewrite lemma: 14.76/4.82 +'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 14.76/4.82 14.76/4.82 Induction Base: 14.76/4.82 +'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1) 14.76/4.82 gen_0':s2_0(a) 14.76/4.82 14.76/4.82 Induction Step: 14.76/4.82 +'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1) 14.76/4.82 s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH 14.76/4.82 s(gen_0':s2_0(+(a, c5_0))) 14.76/4.82 14.76/4.82 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (20) 14.76/4.82 Complex Obligation (BEST) 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (21) 14.76/4.82 Obligation: 14.76/4.82 Proved the lower bound n^1 for the following obligation: 14.76/4.82 14.76/4.82 Innermost TRS: 14.76/4.82 Rules: 14.76/4.82 double(0') -> 0' 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +'(x, 0') -> x 14.76/4.82 +'(x, s(y)) -> s(+'(x, y)) 14.76/4.82 +'(s(x), y) -> s(+'(x, y)) 14.76/4.82 double(x) -> +'(x, x) 14.76/4.82 14.76/4.82 Types: 14.76/4.82 double :: 0':s -> 0':s 14.76/4.82 0' :: 0':s 14.76/4.82 s :: 0':s -> 0':s 14.76/4.82 +' :: 0':s -> 0':s -> 0':s 14.76/4.82 hole_0':s1_0 :: 0':s 14.76/4.82 gen_0':s2_0 :: Nat -> 0':s 14.76/4.82 14.76/4.82 14.76/4.82 Generator Equations: 14.76/4.82 gen_0':s2_0(0) <=> 0' 14.76/4.82 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 14.76/4.82 14.76/4.82 14.76/4.82 The following defined symbols remain to be analysed: 14.76/4.82 +', double 14.76/4.82 14.76/4.82 They will be analysed ascendingly in the following order: 14.76/4.82 +' < double 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (22) LowerBoundPropagationProof (FINISHED) 14.76/4.82 Propagated lower bound. 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (23) 14.76/4.82 BOUNDS(n^1, INF) 14.76/4.82 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (24) 14.76/4.82 Obligation: 14.76/4.82 Innermost TRS: 14.76/4.82 Rules: 14.76/4.82 double(0') -> 0' 14.76/4.82 double(s(x)) -> s(s(double(x))) 14.76/4.82 +'(x, 0') -> x 14.76/4.82 +'(x, s(y)) -> s(+'(x, y)) 14.76/4.82 +'(s(x), y) -> s(+'(x, y)) 14.76/4.82 double(x) -> +'(x, x) 14.76/4.82 14.76/4.82 Types: 14.76/4.82 double :: 0':s -> 0':s 14.76/4.82 0' :: 0':s 14.76/4.82 s :: 0':s -> 0':s 14.76/4.82 +' :: 0':s -> 0':s -> 0':s 14.76/4.82 hole_0':s1_0 :: 0':s 14.76/4.82 gen_0':s2_0 :: Nat -> 0':s 14.76/4.82 14.76/4.82 14.76/4.82 Lemmas: 14.76/4.82 +'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 14.76/4.82 14.76/4.82 14.76/4.82 Generator Equations: 14.76/4.82 gen_0':s2_0(0) <=> 0' 14.76/4.82 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 14.76/4.82 14.76/4.82 14.76/4.82 The following defined symbols remain to be analysed: 14.76/4.82 double 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (25) RewriteLemmaProof (LOWER BOUND(ID)) 14.76/4.82 Proved the following rewrite lemma: 14.76/4.82 double(gen_0':s2_0(n544_0)) -> gen_0':s2_0(*(2, n544_0)), rt in Omega(1 + n544_0) 14.76/4.82 14.76/4.82 Induction Base: 14.76/4.82 double(gen_0':s2_0(0)) ->_R^Omega(1) 14.76/4.82 0' 14.76/4.82 14.76/4.82 Induction Step: 14.76/4.82 double(gen_0':s2_0(+(n544_0, 1))) ->_R^Omega(1) 14.76/4.82 s(s(double(gen_0':s2_0(n544_0)))) ->_IH 14.76/4.82 s(s(gen_0':s2_0(*(2, c545_0)))) 14.76/4.82 14.76/4.82 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 14.76/4.82 ---------------------------------------- 14.76/4.82 14.76/4.82 (26) 14.76/4.82 BOUNDS(1, INF) 15.11/6.92 EOF