883.58/291.55 WORST_CASE(Omega(n^1), O(n^2)) 883.58/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 883.58/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 883.58/291.57 883.58/291.57 883.58/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 883.58/291.57 883.58/291.57 (0) CpxTRS 883.58/291.57 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 883.58/291.57 (2) CpxTRS 883.58/291.57 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 883.58/291.57 (4) CpxWeightedTrs 883.58/291.57 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 883.58/291.57 (6) CpxTypedWeightedTrs 883.58/291.57 (7) CompletionProof [UPPER BOUND(ID), 1 ms] 883.58/291.57 (8) CpxTypedWeightedCompleteTrs 883.58/291.57 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 883.58/291.57 (10) CpxRNTS 883.58/291.57 (11) CompleteCoflocoProof [FINISHED, 238 ms] 883.58/291.57 (12) BOUNDS(1, n^2) 883.58/291.57 (13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 883.58/291.57 (14) CpxTRS 883.58/291.57 (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 883.58/291.57 (16) typed CpxTrs 883.58/291.57 (17) OrderProof [LOWER BOUND(ID), 0 ms] 883.58/291.57 (18) typed CpxTrs 883.58/291.57 (19) RewriteLemmaProof [LOWER BOUND(ID), 273 ms] 883.58/291.57 (20) BEST 883.58/291.57 (21) proven lower bound 883.58/291.57 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 883.58/291.57 (23) BOUNDS(n^1, INF) 883.58/291.57 (24) typed CpxTrs 883.58/291.57 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (0) 883.58/291.57 Obligation: 883.58/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 883.58/291.57 883.58/291.57 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0) -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 S is empty. 883.58/291.57 Rewrite Strategy: FULL 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 883.58/291.57 Converted rc-obligation to irc-obligation. 883.58/291.57 883.58/291.57 The duplicating contexts are: 883.58/291.57 f(t, [], y) 883.58/291.57 f(t, x, []) 883.58/291.57 883.58/291.57 883.58/291.57 The defined contexts are: 883.58/291.57 f([], x1, s(x2)) 883.58/291.57 883.58/291.57 883.58/291.57 [] just represents basic- or constructor-terms in the following defined contexts: 883.58/291.57 f([], x1, s(x2)) 883.58/291.57 883.58/291.57 883.58/291.57 As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (2) 883.58/291.57 Obligation: 883.58/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). 883.58/291.57 883.58/291.57 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0) -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 S is empty. 883.58/291.57 Rewrite Strategy: INNERMOST 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 883.58/291.57 Transformed relative TRS to weighted TRS 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (4) 883.58/291.57 Obligation: 883.58/291.57 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 883.58/291.57 883.58/291.57 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) [1] 883.58/291.57 g(s(x), 0) -> t [1] 883.58/291.57 g(s(x), s(y)) -> g(x, y) [1] 883.58/291.57 883.58/291.57 Rewrite Strategy: INNERMOST 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 883.58/291.57 Infered types. 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (6) 883.58/291.57 Obligation: 883.58/291.57 Runtime Complexity Weighted TRS with Types. 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) [1] 883.58/291.57 g(s(x), 0) -> t [1] 883.58/291.57 g(s(x), s(y)) -> g(x, y) [1] 883.58/291.57 883.58/291.57 The TRS has the following type information: 883.58/291.57 f :: t -> s:0 -> s:0 -> f 883.58/291.57 t :: t 883.58/291.57 g :: s:0 -> s:0 -> t 883.58/291.57 s :: s:0 -> s:0 883.58/291.57 0 :: s:0 883.58/291.57 883.58/291.57 Rewrite Strategy: INNERMOST 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (7) CompletionProof (UPPER BOUND(ID)) 883.58/291.57 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 883.58/291.57 883.58/291.57 g(v0, v1) -> null_g [0] 883.58/291.57 f(v0, v1, v2) -> null_f [0] 883.58/291.57 883.58/291.57 And the following fresh constants: null_g, null_f 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (8) 883.58/291.57 Obligation: 883.58/291.57 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 883.58/291.57 883.58/291.57 Runtime Complexity Weighted TRS with Types. 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) [1] 883.58/291.57 g(s(x), 0) -> t [1] 883.58/291.57 g(s(x), s(y)) -> g(x, y) [1] 883.58/291.57 g(v0, v1) -> null_g [0] 883.58/291.57 f(v0, v1, v2) -> null_f [0] 883.58/291.57 883.58/291.57 The TRS has the following type information: 883.58/291.57 f :: t:null_g -> s:0 -> s:0 -> null_f 883.58/291.57 t :: t:null_g 883.58/291.57 g :: s:0 -> s:0 -> t:null_g 883.58/291.57 s :: s:0 -> s:0 883.58/291.57 0 :: s:0 883.58/291.57 null_g :: t:null_g 883.58/291.57 null_f :: null_f 883.58/291.57 883.58/291.57 Rewrite Strategy: INNERMOST 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 883.58/291.57 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 883.58/291.57 The constant constructors are abstracted as follows: 883.58/291.57 883.58/291.57 t => 1 883.58/291.57 0 => 0 883.58/291.57 null_g => 0 883.58/291.57 null_f => 0 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (10) 883.58/291.57 Obligation: 883.58/291.57 Complexity RNTS consisting of the following rules: 883.58/291.57 883.58/291.57 f(z, z', z'') -{ 1 }-> f(g(x, y), x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 883.58/291.57 f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 883.58/291.57 g(z, z') -{ 1 }-> g(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 883.58/291.57 g(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 883.58/291.57 g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 883.58/291.57 883.58/291.57 Only complete derivations are relevant for the runtime complexity. 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (11) CompleteCoflocoProof (FINISHED) 883.58/291.57 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 883.58/291.57 883.58/291.57 eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 883.58/291.57 eq(start(V1, V, V2),0,[g(V1, V, Out)],[V1 >= 0,V >= 0]). 883.58/291.57 eq(f(V1, V, V2, Out),1,[g(V4, V3, Ret0),f(Ret0, V4, 1 + V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]). 883.58/291.57 eq(g(V1, V, Out),1,[],[Out = 1,V5 >= 0,V1 = 1 + V5,V = 0]). 883.58/291.57 eq(g(V1, V, Out),1,[g(V6, V7, Ret1)],[Out = Ret1,V = 1 + V7,V6 >= 0,V7 >= 0,V1 = 1 + V6]). 883.58/291.57 eq(g(V1, V, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V9,V = V8]). 883.58/291.57 eq(f(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]). 883.58/291.57 input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]). 883.58/291.57 input_output_vars(g(V1,V,Out),[V1,V],[Out]). 883.58/291.57 883.58/291.57 883.58/291.57 CoFloCo proof output: 883.58/291.57 Preprocessing Cost Relations 883.58/291.57 ===================================== 883.58/291.57 883.58/291.57 #### Computed strongly connected components 883.58/291.57 0. recursive : [g/3] 883.58/291.57 1. recursive : [f/4] 883.58/291.57 2. non_recursive : [start/3] 883.58/291.57 883.58/291.57 #### Obtained direct recursion through partial evaluation 883.58/291.57 0. SCC is partially evaluated into g/3 883.58/291.57 1. SCC is partially evaluated into f/4 883.58/291.57 2. SCC is partially evaluated into start/3 883.58/291.57 883.58/291.57 Control-Flow Refinement of Cost Relations 883.58/291.57 ===================================== 883.58/291.57 883.58/291.57 ### Specialization of cost equations g/3 883.58/291.57 * CE 7 is refined into CE [8] 883.58/291.57 * CE 5 is refined into CE [9] 883.58/291.57 * CE 6 is refined into CE [10] 883.58/291.57 883.58/291.57 883.58/291.57 ### Cost equations --> "Loop" of g/3 883.58/291.57 * CEs [10] --> Loop 7 883.58/291.57 * CEs [8] --> Loop 8 883.58/291.57 * CEs [9] --> Loop 9 883.58/291.57 883.58/291.57 ### Ranking functions of CR g(V1,V,Out) 883.58/291.57 * RF of phase [7]: [V,V1] 883.58/291.57 883.58/291.57 #### Partial ranking functions of CR g(V1,V,Out) 883.58/291.57 * Partial RF of phase [7]: 883.58/291.57 - RF of loop [7:1]: 883.58/291.57 V 883.58/291.57 V1 883.58/291.57 883.58/291.57 883.58/291.57 ### Specialization of cost equations f/4 883.58/291.57 * CE 4 is refined into CE [11] 883.58/291.57 * CE 3 is refined into CE [12,13,14] 883.58/291.57 883.58/291.57 883.58/291.57 ### Cost equations --> "Loop" of f/4 883.58/291.57 * CEs [14] --> Loop 10 883.58/291.57 * CEs [13] --> Loop 11 883.58/291.57 * CEs [12] --> Loop 12 883.58/291.57 * CEs [11] --> Loop 13 883.58/291.57 883.58/291.57 ### Ranking functions of CR f(V1,V,V2,Out) 883.58/291.57 * RF of phase [10]: [V-V2] 883.58/291.57 883.58/291.57 #### Partial ranking functions of CR f(V1,V,V2,Out) 883.58/291.57 * Partial RF of phase [10]: 883.58/291.57 - RF of loop [10:1]: 883.58/291.57 V-V2 883.58/291.57 883.58/291.57 883.58/291.57 ### Specialization of cost equations start/3 883.58/291.57 * CE 1 is refined into CE [15] 883.58/291.57 * CE 2 is refined into CE [16,17,18] 883.58/291.57 883.58/291.57 883.58/291.57 ### Cost equations --> "Loop" of start/3 883.58/291.57 * CEs [15,16,17,18] --> Loop 14 883.58/291.57 883.58/291.57 ### Ranking functions of CR start(V1,V,V2) 883.58/291.57 883.58/291.57 #### Partial ranking functions of CR start(V1,V,V2) 883.58/291.57 883.58/291.57 883.58/291.57 Computing Bounds 883.58/291.57 ===================================== 883.58/291.57 883.58/291.57 #### Cost of chains of g(V1,V,Out): 883.58/291.57 * Chain [[7],9]: 1*it(7)+1 883.58/291.57 Such that:it(7) =< V 883.58/291.57 883.58/291.57 with precondition: [Out=1,V>=1,V1>=V+1] 883.58/291.57 883.58/291.57 * Chain [[7],8]: 1*it(7)+0 883.58/291.57 Such that:it(7) =< V 883.58/291.57 883.58/291.57 with precondition: [Out=0,V1>=1,V>=1] 883.58/291.57 883.58/291.57 * Chain [9]: 1 883.58/291.57 with precondition: [V=0,Out=1,V1>=1] 883.58/291.57 883.58/291.57 * Chain [8]: 0 883.58/291.57 with precondition: [Out=0,V1>=0,V>=0] 883.58/291.57 883.58/291.57 883.58/291.57 #### Cost of chains of f(V1,V,V2,Out): 883.58/291.57 * Chain [[10],13]: 2*it(10)+1*s(4)+0 883.58/291.57 Such that:aux(1) =< V 883.58/291.57 it(10) =< V-V2 883.58/291.57 s(4) =< it(10)*aux(1) 883.58/291.57 883.58/291.57 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 883.58/291.57 883.58/291.57 * Chain [[10],11,13]: 2*it(10)+1*s(4)+1*s(5)+1 883.58/291.57 Such that:aux(1) =< V 883.58/291.57 s(5) =< V+1 883.58/291.57 it(10) =< V-V2 883.58/291.57 s(4) =< it(10)*aux(1) 883.58/291.57 883.58/291.57 with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 883.58/291.57 883.58/291.57 * Chain [13]: 0 883.58/291.57 with precondition: [Out=0,V1>=0,V>=0,V2>=0] 883.58/291.57 883.58/291.57 * Chain [12,[10],13]: 2*it(10)+1*s(4)+2 883.58/291.57 Such that:aux(2) =< V 883.58/291.57 it(10) =< aux(2) 883.58/291.57 s(4) =< it(10)*aux(2) 883.58/291.57 883.58/291.57 with precondition: [V1=1,V2=0,Out=0,V>=2] 883.58/291.57 883.58/291.57 * Chain [12,[10],11,13]: 2*it(10)+1*s(4)+1*s(5)+3 883.58/291.57 Such that:s(5) =< V+1 883.58/291.57 aux(3) =< V 883.58/291.57 it(10) =< aux(3) 883.58/291.57 s(4) =< it(10)*aux(3) 883.58/291.57 883.58/291.57 with precondition: [V1=1,V2=0,Out=0,V>=2] 883.58/291.57 883.58/291.57 * Chain [12,13]: 2 883.58/291.57 with precondition: [V1=1,V2=0,Out=0,V>=1] 883.58/291.57 883.58/291.57 * Chain [12,11,13]: 1*s(5)+3 883.58/291.57 Such that:s(5) =< 2 883.58/291.57 883.58/291.57 with precondition: [V1=1,V2=0,Out=0,V>=1] 883.58/291.57 883.58/291.57 * Chain [11,13]: 1*s(5)+1 883.58/291.57 Such that:s(5) =< V2+1 883.58/291.57 883.58/291.57 with precondition: [V1=1,Out=0,V>=0,V2>=0] 883.58/291.57 883.58/291.57 883.58/291.57 #### Cost of chains of start(V1,V,V2): 883.58/291.57 * Chain [14]: 1*s(22)+1*s(23)+2*s(27)+4*s(28)+6*s(29)+2*s(30)+2*s(31)+3 883.58/291.57 Such that:s(22) =< 2 883.58/291.57 s(25) =< V+1 883.58/291.57 s(26) =< V-V2 883.58/291.57 s(23) =< V2+1 883.58/291.57 aux(7) =< V 883.58/291.57 s(29) =< aux(7) 883.58/291.57 s(27) =< s(25) 883.58/291.57 s(28) =< s(26) 883.58/291.57 s(30) =< s(29)*aux(7) 883.58/291.57 s(31) =< s(28)*aux(7) 883.58/291.57 883.58/291.57 with precondition: [V1>=0,V>=0] 883.58/291.57 883.58/291.57 883.58/291.57 Closed-form bounds of start(V1,V,V2): 883.58/291.57 ------------------------------------- 883.58/291.57 * Chain [14] with precondition: [V1>=0,V>=0] 883.58/291.57 - Upper bound: 6*V+5+2*V*V+2*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*4 883.58/291.57 - Complexity: n^2 883.58/291.57 883.58/291.57 ### Maximum cost of start(V1,V,V2): 6*V+5+2*V*V+2*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*4 883.58/291.57 Asymptotic class: n^2 883.58/291.57 * Total analysis performed in 168 ms. 883.58/291.57 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (12) 883.58/291.57 BOUNDS(1, n^2) 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (13) RenamingProof (BOTH BOUNDS(ID, ID)) 883.58/291.57 Renamed function symbols to avoid clashes with predefined symbol. 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (14) 883.58/291.57 Obligation: 883.58/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 883.58/291.57 883.58/291.57 883.58/291.57 The TRS R consists of the following rules: 883.58/291.57 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0') -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 S is empty. 883.58/291.57 Rewrite Strategy: FULL 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (15) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 883.58/291.57 Infered types. 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (16) 883.58/291.57 Obligation: 883.58/291.57 TRS: 883.58/291.57 Rules: 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0') -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 Types: 883.58/291.57 f :: t -> s:0' -> s:0' -> f 883.58/291.57 t :: t 883.58/291.57 g :: s:0' -> s:0' -> t 883.58/291.57 s :: s:0' -> s:0' 883.58/291.57 0' :: s:0' 883.58/291.57 hole_f1_0 :: f 883.58/291.57 hole_t2_0 :: t 883.58/291.57 hole_s:0'3_0 :: s:0' 883.58/291.57 gen_s:0'4_0 :: Nat -> s:0' 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (17) OrderProof (LOWER BOUND(ID)) 883.58/291.57 Heuristically decided to analyse the following defined symbols: 883.58/291.57 f, g 883.58/291.57 883.58/291.57 They will be analysed ascendingly in the following order: 883.58/291.57 g < f 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (18) 883.58/291.57 Obligation: 883.58/291.57 TRS: 883.58/291.57 Rules: 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0') -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 Types: 883.58/291.57 f :: t -> s:0' -> s:0' -> f 883.58/291.57 t :: t 883.58/291.57 g :: s:0' -> s:0' -> t 883.58/291.57 s :: s:0' -> s:0' 883.58/291.57 0' :: s:0' 883.58/291.57 hole_f1_0 :: f 883.58/291.57 hole_t2_0 :: t 883.58/291.57 hole_s:0'3_0 :: s:0' 883.58/291.57 gen_s:0'4_0 :: Nat -> s:0' 883.58/291.57 883.58/291.57 883.58/291.57 Generator Equations: 883.58/291.57 gen_s:0'4_0(0) <=> 0' 883.58/291.57 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 883.58/291.57 883.58/291.57 883.58/291.57 The following defined symbols remain to be analysed: 883.58/291.57 g, f 883.58/291.57 883.58/291.57 They will be analysed ascendingly in the following order: 883.58/291.57 g < f 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (19) RewriteLemmaProof (LOWER BOUND(ID)) 883.58/291.57 Proved the following rewrite lemma: 883.58/291.57 g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) -> t, rt in Omega(1 + n6_0) 883.58/291.57 883.58/291.57 Induction Base: 883.58/291.57 g(gen_s:0'4_0(+(1, 0)), gen_s:0'4_0(0)) ->_R^Omega(1) 883.58/291.57 t 883.58/291.57 883.58/291.57 Induction Step: 883.58/291.57 g(gen_s:0'4_0(+(1, +(n6_0, 1))), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1) 883.58/291.57 g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) ->_IH 883.58/291.57 t 883.58/291.57 883.58/291.57 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (20) 883.58/291.57 Complex Obligation (BEST) 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (21) 883.58/291.57 Obligation: 883.58/291.57 Proved the lower bound n^1 for the following obligation: 883.58/291.57 883.58/291.57 TRS: 883.58/291.57 Rules: 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0') -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 Types: 883.58/291.57 f :: t -> s:0' -> s:0' -> f 883.58/291.57 t :: t 883.58/291.57 g :: s:0' -> s:0' -> t 883.58/291.57 s :: s:0' -> s:0' 883.58/291.57 0' :: s:0' 883.58/291.57 hole_f1_0 :: f 883.58/291.57 hole_t2_0 :: t 883.58/291.57 hole_s:0'3_0 :: s:0' 883.58/291.57 gen_s:0'4_0 :: Nat -> s:0' 883.58/291.57 883.58/291.57 883.58/291.57 Generator Equations: 883.58/291.57 gen_s:0'4_0(0) <=> 0' 883.58/291.57 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 883.58/291.57 883.58/291.57 883.58/291.57 The following defined symbols remain to be analysed: 883.58/291.57 g, f 883.58/291.57 883.58/291.57 They will be analysed ascendingly in the following order: 883.58/291.57 g < f 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (22) LowerBoundPropagationProof (FINISHED) 883.58/291.57 Propagated lower bound. 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (23) 883.58/291.57 BOUNDS(n^1, INF) 883.58/291.57 883.58/291.57 ---------------------------------------- 883.58/291.57 883.58/291.57 (24) 883.58/291.57 Obligation: 883.58/291.57 TRS: 883.58/291.57 Rules: 883.58/291.57 f(t, x, y) -> f(g(x, y), x, s(y)) 883.58/291.57 g(s(x), 0') -> t 883.58/291.57 g(s(x), s(y)) -> g(x, y) 883.58/291.57 883.58/291.57 Types: 883.58/291.57 f :: t -> s:0' -> s:0' -> f 883.58/291.57 t :: t 883.58/291.57 g :: s:0' -> s:0' -> t 883.58/291.57 s :: s:0' -> s:0' 883.58/291.57 0' :: s:0' 883.58/291.57 hole_f1_0 :: f 883.58/291.57 hole_t2_0 :: t 883.58/291.57 hole_s:0'3_0 :: s:0' 883.58/291.57 gen_s:0'4_0 :: Nat -> s:0' 883.58/291.57 883.58/291.57 883.58/291.57 Lemmas: 883.58/291.57 g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) -> t, rt in Omega(1 + n6_0) 883.58/291.57 883.58/291.57 883.58/291.57 Generator Equations: 883.58/291.57 gen_s:0'4_0(0) <=> 0' 883.58/291.57 gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x)) 883.58/291.57 883.58/291.57 883.58/291.57 The following defined symbols remain to be analysed: 883.58/291.57 f 883.79/291.61 EOF