14.14/4.61 WORST_CASE(Omega(n^2), O(n^2)) 14.14/4.62 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 14.14/4.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.14/4.62 14.14/4.62 14.14/4.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2). 14.14/4.62 14.14/4.62 (0) CpxTRS 14.14/4.62 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 14.14/4.62 (2) CpxWeightedTrs 14.14/4.62 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 14.14/4.62 (4) CpxTypedWeightedTrs 14.14/4.62 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 14.14/4.62 (6) CpxTypedWeightedCompleteTrs 14.14/4.62 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms] 14.14/4.62 (8) CpxRNTS 14.14/4.62 (9) CompleteCoflocoProof [FINISHED, 177 ms] 14.14/4.62 (10) BOUNDS(1, n^2) 14.14/4.62 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 14.14/4.62 (12) CpxTRS 14.14/4.62 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 14.14/4.62 (14) typed CpxTrs 14.14/4.62 (15) OrderProof [LOWER BOUND(ID), 0 ms] 14.14/4.62 (16) typed CpxTrs 14.14/4.62 (17) RewriteLemmaProof [LOWER BOUND(ID), 301 ms] 14.14/4.62 (18) BEST 14.14/4.62 (19) proven lower bound 14.14/4.62 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 14.14/4.62 (21) BOUNDS(n^1, INF) 14.14/4.62 (22) typed CpxTrs 14.14/4.62 (23) RewriteLemmaProof [LOWER BOUND(ID), 47 ms] 14.14/4.62 (24) proven lower bound 14.14/4.62 (25) LowerBoundPropagationProof [FINISHED, 0 ms] 14.14/4.62 (26) BOUNDS(n^2, INF) 14.14/4.62 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (0) 14.14/4.62 Obligation: 14.14/4.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2). 14.14/4.62 14.14/4.62 14.14/4.62 The TRS R consists of the following rules: 14.14/4.62 14.14/4.62 and(tt, X) -> activate(X) 14.14/4.62 plus(N, 0) -> N 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.62 x(N, 0) -> 0 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.62 activate(X) -> X 14.14/4.62 14.14/4.62 S is empty. 14.14/4.62 Rewrite Strategy: INNERMOST 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 14.14/4.62 Transformed relative TRS to weighted TRS 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (2) 14.14/4.62 Obligation: 14.14/4.62 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 14.14/4.62 14.14/4.62 14.14/4.62 The TRS R consists of the following rules: 14.14/4.62 14.14/4.62 and(tt, X) -> activate(X) [1] 14.14/4.62 plus(N, 0) -> N [1] 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) [1] 14.14/4.62 x(N, 0) -> 0 [1] 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) [1] 14.14/4.62 activate(X) -> X [1] 14.14/4.62 14.14/4.62 Rewrite Strategy: INNERMOST 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 14.14/4.62 Infered types. 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (4) 14.14/4.62 Obligation: 14.14/4.62 Runtime Complexity Weighted TRS with Types. 14.14/4.62 The TRS R consists of the following rules: 14.14/4.62 14.14/4.62 and(tt, X) -> activate(X) [1] 14.14/4.62 plus(N, 0) -> N [1] 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) [1] 14.14/4.62 x(N, 0) -> 0 [1] 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) [1] 14.14/4.62 activate(X) -> X [1] 14.14/4.62 14.14/4.62 The TRS has the following type information: 14.14/4.62 and :: tt -> and:activate -> and:activate 14.14/4.62 tt :: tt 14.14/4.62 activate :: and:activate -> and:activate 14.14/4.62 plus :: 0:s -> 0:s -> 0:s 14.14/4.62 0 :: 0:s 14.14/4.62 s :: 0:s -> 0:s 14.14/4.62 x :: 0:s -> 0:s -> 0:s 14.14/4.62 14.14/4.62 Rewrite Strategy: INNERMOST 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (5) CompletionProof (UPPER BOUND(ID)) 14.14/4.62 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 14.14/4.62 none 14.14/4.62 14.14/4.62 And the following fresh constants: const 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (6) 14.14/4.62 Obligation: 14.14/4.62 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 14.14/4.62 14.14/4.62 Runtime Complexity Weighted TRS with Types. 14.14/4.62 The TRS R consists of the following rules: 14.14/4.62 14.14/4.62 and(tt, X) -> activate(X) [1] 14.14/4.62 plus(N, 0) -> N [1] 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) [1] 14.14/4.62 x(N, 0) -> 0 [1] 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) [1] 14.14/4.62 activate(X) -> X [1] 14.14/4.62 14.14/4.62 The TRS has the following type information: 14.14/4.62 and :: tt -> and:activate -> and:activate 14.14/4.62 tt :: tt 14.14/4.62 activate :: and:activate -> and:activate 14.14/4.62 plus :: 0:s -> 0:s -> 0:s 14.14/4.62 0 :: 0:s 14.14/4.62 s :: 0:s -> 0:s 14.14/4.62 x :: 0:s -> 0:s -> 0:s 14.14/4.62 const :: and:activate 14.14/4.62 14.14/4.62 Rewrite Strategy: INNERMOST 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 14.14/4.62 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 14.14/4.62 The constant constructors are abstracted as follows: 14.14/4.62 14.14/4.62 tt => 0 14.14/4.62 0 => 0 14.14/4.62 const => 0 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (8) 14.14/4.62 Obligation: 14.14/4.62 Complexity RNTS consisting of the following rules: 14.14/4.62 14.14/4.62 activate(z) -{ 1 }-> X :|: X >= 0, z = X 14.14/4.62 and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0 14.14/4.62 plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0 14.14/4.62 plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0 14.14/4.62 x(z, z') -{ 1 }-> plus(x(N, M), N) :|: z' = 1 + M, z = N, M >= 0, N >= 0 14.14/4.62 x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 14.14/4.62 14.14/4.62 Only complete derivations are relevant for the runtime complexity. 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (9) CompleteCoflocoProof (FINISHED) 14.14/4.62 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 14.14/4.62 14.14/4.62 eq(start(V1, V),0,[and(V1, V, Out)],[V1 >= 0,V >= 0]). 14.14/4.62 eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]). 14.14/4.62 eq(start(V1, V),0,[x(V1, V, Out)],[V1 >= 0,V >= 0]). 14.14/4.62 eq(start(V1, V),0,[activate(V1, Out)],[V1 >= 0]). 14.14/4.62 eq(and(V1, V, Out),1,[activate(X1, Ret)],[Out = Ret,V = X1,X1 >= 0,V1 = 0]). 14.14/4.62 eq(plus(V1, V, Out),1,[],[Out = N1,V1 = N1,V = 0,N1 >= 0]). 14.14/4.62 eq(plus(V1, V, Out),1,[plus(N2, M1, Ret1)],[Out = 1 + Ret1,V = 1 + M1,V1 = N2,M1 >= 0,N2 >= 0]). 14.14/4.62 eq(x(V1, V, Out),1,[],[Out = 0,V1 = N3,V = 0,N3 >= 0]). 14.14/4.62 eq(x(V1, V, Out),1,[x(N4, M2, Ret0),plus(Ret0, N4, Ret2)],[Out = Ret2,V = 1 + M2,V1 = N4,M2 >= 0,N4 >= 0]). 14.14/4.62 eq(activate(V1, Out),1,[],[Out = X2,X2 >= 0,V1 = X2]). 14.14/4.62 input_output_vars(and(V1,V,Out),[V1,V],[Out]). 14.14/4.62 input_output_vars(plus(V1,V,Out),[V1,V],[Out]). 14.14/4.62 input_output_vars(x(V1,V,Out),[V1,V],[Out]). 14.14/4.62 input_output_vars(activate(V1,Out),[V1],[Out]). 14.14/4.62 14.14/4.62 14.14/4.62 CoFloCo proof output: 14.14/4.62 Preprocessing Cost Relations 14.14/4.62 ===================================== 14.14/4.62 14.14/4.62 #### Computed strongly connected components 14.14/4.62 0. non_recursive : [activate/2] 14.14/4.62 1. non_recursive : [and/3] 14.14/4.62 2. recursive : [plus/3] 14.14/4.62 3. recursive [non_tail] : [x/3] 14.14/4.62 4. non_recursive : [start/2] 14.14/4.62 14.14/4.62 #### Obtained direct recursion through partial evaluation 14.14/4.62 0. SCC is completely evaluated into other SCCs 14.14/4.62 1. SCC is completely evaluated into other SCCs 14.14/4.62 2. SCC is partially evaluated into plus/3 14.14/4.62 3. SCC is partially evaluated into x/3 14.14/4.62 4. SCC is partially evaluated into start/2 14.14/4.62 14.14/4.62 Control-Flow Refinement of Cost Relations 14.14/4.62 ===================================== 14.14/4.62 14.14/4.62 ### Specialization of cost equations plus/3 14.14/4.62 * CE 6 is refined into CE [9] 14.14/4.62 * CE 5 is refined into CE [10] 14.14/4.62 14.14/4.62 14.14/4.62 ### Cost equations --> "Loop" of plus/3 14.14/4.62 * CEs [10] --> Loop 6 14.14/4.62 * CEs [9] --> Loop 7 14.14/4.62 14.14/4.62 ### Ranking functions of CR plus(V1,V,Out) 14.14/4.62 * RF of phase [7]: [V] 14.14/4.62 14.14/4.62 #### Partial ranking functions of CR plus(V1,V,Out) 14.14/4.62 * Partial RF of phase [7]: 14.14/4.62 - RF of loop [7:1]: 14.14/4.62 V 14.14/4.62 14.14/4.62 14.14/4.62 ### Specialization of cost equations x/3 14.14/4.62 * CE 8 is refined into CE [11,12] 14.14/4.62 * CE 7 is refined into CE [13] 14.14/4.62 14.14/4.62 14.14/4.62 ### Cost equations --> "Loop" of x/3 14.14/4.62 * CEs [13] --> Loop 8 14.14/4.62 * CEs [12] --> Loop 9 14.14/4.62 * CEs [11] --> Loop 10 14.14/4.62 14.14/4.62 ### Ranking functions of CR x(V1,V,Out) 14.14/4.62 * RF of phase [9]: [V] 14.14/4.62 * RF of phase [10]: [V] 14.14/4.62 14.14/4.62 #### Partial ranking functions of CR x(V1,V,Out) 14.14/4.62 * Partial RF of phase [9]: 14.14/4.62 - RF of loop [9:1]: 14.14/4.62 V 14.14/4.62 * Partial RF of phase [10]: 14.14/4.62 - RF of loop [10:1]: 14.14/4.62 V 14.14/4.62 14.14/4.62 14.14/4.62 ### Specialization of cost equations start/2 14.14/4.62 * CE 1 is refined into CE [14] 14.14/4.62 * CE 2 is refined into CE [15,16] 14.14/4.62 * CE 3 is refined into CE [17,18,19] 14.14/4.62 * CE 4 is refined into CE [20] 14.14/4.62 14.14/4.62 14.14/4.62 ### Cost equations --> "Loop" of start/2 14.14/4.62 * CEs [15,18] --> Loop 11 14.14/4.62 * CEs [14,16,17,19,20] --> Loop 12 14.14/4.62 14.14/4.62 ### Ranking functions of CR start(V1,V) 14.14/4.62 14.14/4.62 #### Partial ranking functions of CR start(V1,V) 14.14/4.62 14.14/4.62 14.14/4.62 Computing Bounds 14.14/4.62 ===================================== 14.14/4.62 14.14/4.62 #### Cost of chains of plus(V1,V,Out): 14.14/4.62 * Chain [[7],6]: 1*it(7)+1 14.14/4.62 Such that:it(7) =< V 14.14/4.62 14.14/4.62 with precondition: [V+V1=Out,V1>=0,V>=1] 14.14/4.62 14.14/4.62 * Chain [6]: 1 14.14/4.62 with precondition: [V=0,V1=Out,V1>=0] 14.14/4.62 14.14/4.62 14.14/4.62 #### Cost of chains of x(V1,V,Out): 14.14/4.62 * Chain [[10],8]: 2*it(10)+1 14.14/4.62 Such that:it(10) =< V 14.14/4.62 14.14/4.62 with precondition: [V1=0,Out=0,V>=1] 14.14/4.62 14.14/4.62 * Chain [[9],8]: 2*it(9)+1*s(3)+1 14.14/4.62 Such that:aux(1) =< V1 14.14/4.62 it(9) =< V 14.14/4.62 s(3) =< it(9)*aux(1) 14.14/4.62 14.14/4.62 with precondition: [V1>=1,V>=1,Out+1>=V+V1] 14.14/4.62 14.14/4.62 * Chain [8]: 1 14.14/4.62 with precondition: [V=0,Out=0,V1>=0] 14.14/4.62 14.14/4.62 14.14/4.62 #### Cost of chains of start(V1,V): 14.14/4.62 * Chain [12]: 5*s(4)+1*s(8)+2 14.14/4.62 Such that:s(6) =< V1 14.14/4.62 aux(2) =< V 14.14/4.62 s(4) =< aux(2) 14.14/4.62 s(8) =< s(4)*s(6) 14.14/4.62 14.14/4.62 with precondition: [V1>=0] 14.14/4.62 14.14/4.62 * Chain [11]: 1 14.14/4.62 with precondition: [V=0,V1>=0] 14.14/4.62 14.14/4.62 14.14/4.62 Closed-form bounds of start(V1,V): 14.14/4.62 ------------------------------------- 14.14/4.62 * Chain [12] with precondition: [V1>=0] 14.14/4.62 - Upper bound: nat(V)*V1+2+nat(V)*5 14.14/4.62 - Complexity: n^2 14.14/4.62 * Chain [11] with precondition: [V=0,V1>=0] 14.14/4.62 - Upper bound: 1 14.14/4.62 - Complexity: constant 14.14/4.62 14.14/4.62 ### Maximum cost of start(V1,V): nat(V)*V1+1+nat(V)*5+1 14.14/4.62 Asymptotic class: n^2 14.14/4.62 * Total analysis performed in 111 ms. 14.14/4.62 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (10) 14.14/4.62 BOUNDS(1, n^2) 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 14.14/4.62 Renamed function symbols to avoid clashes with predefined symbol. 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (12) 14.14/4.62 Obligation: 14.14/4.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 14.14/4.62 14.14/4.62 14.14/4.62 The TRS R consists of the following rules: 14.14/4.62 14.14/4.62 and(tt, X) -> activate(X) 14.14/4.62 plus(N, 0') -> N 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.62 x(N, 0') -> 0' 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.62 activate(X) -> X 14.14/4.62 14.14/4.62 S is empty. 14.14/4.62 Rewrite Strategy: INNERMOST 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 14.14/4.62 Infered types. 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (14) 14.14/4.62 Obligation: 14.14/4.62 Innermost TRS: 14.14/4.62 Rules: 14.14/4.62 and(tt, X) -> activate(X) 14.14/4.62 plus(N, 0') -> N 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.62 x(N, 0') -> 0' 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.62 activate(X) -> X 14.14/4.62 14.14/4.62 Types: 14.14/4.62 and :: tt -> and:activate -> and:activate 14.14/4.62 tt :: tt 14.14/4.62 activate :: and:activate -> and:activate 14.14/4.62 plus :: 0':s -> 0':s -> 0':s 14.14/4.62 0' :: 0':s 14.14/4.62 s :: 0':s -> 0':s 14.14/4.62 x :: 0':s -> 0':s -> 0':s 14.14/4.62 hole_and:activate1_0 :: and:activate 14.14/4.62 hole_tt2_0 :: tt 14.14/4.62 hole_0':s3_0 :: 0':s 14.14/4.62 gen_0':s4_0 :: Nat -> 0':s 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (15) OrderProof (LOWER BOUND(ID)) 14.14/4.62 Heuristically decided to analyse the following defined symbols: 14.14/4.62 plus, x 14.14/4.62 14.14/4.62 They will be analysed ascendingly in the following order: 14.14/4.62 plus < x 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (16) 14.14/4.62 Obligation: 14.14/4.62 Innermost TRS: 14.14/4.62 Rules: 14.14/4.62 and(tt, X) -> activate(X) 14.14/4.62 plus(N, 0') -> N 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.62 x(N, 0') -> 0' 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.62 activate(X) -> X 14.14/4.62 14.14/4.62 Types: 14.14/4.62 and :: tt -> and:activate -> and:activate 14.14/4.62 tt :: tt 14.14/4.62 activate :: and:activate -> and:activate 14.14/4.62 plus :: 0':s -> 0':s -> 0':s 14.14/4.62 0' :: 0':s 14.14/4.62 s :: 0':s -> 0':s 14.14/4.62 x :: 0':s -> 0':s -> 0':s 14.14/4.62 hole_and:activate1_0 :: and:activate 14.14/4.62 hole_tt2_0 :: tt 14.14/4.62 hole_0':s3_0 :: 0':s 14.14/4.62 gen_0':s4_0 :: Nat -> 0':s 14.14/4.62 14.14/4.62 14.14/4.62 Generator Equations: 14.14/4.62 gen_0':s4_0(0) <=> 0' 14.14/4.62 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 14.14/4.62 14.14/4.62 14.14/4.62 The following defined symbols remain to be analysed: 14.14/4.62 plus, x 14.14/4.62 14.14/4.62 They will be analysed ascendingly in the following order: 14.14/4.62 plus < x 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (17) RewriteLemmaProof (LOWER BOUND(ID)) 14.14/4.62 Proved the following rewrite lemma: 14.14/4.62 plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0) 14.14/4.62 14.14/4.62 Induction Base: 14.14/4.62 plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1) 14.14/4.62 gen_0':s4_0(a) 14.14/4.62 14.14/4.62 Induction Step: 14.14/4.62 plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1) 14.14/4.62 s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) ->_IH 14.14/4.62 s(gen_0':s4_0(+(a, c7_0))) 14.14/4.62 14.14/4.62 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (18) 14.14/4.62 Complex Obligation (BEST) 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.62 14.14/4.62 (19) 14.14/4.62 Obligation: 14.14/4.62 Proved the lower bound n^1 for the following obligation: 14.14/4.62 14.14/4.62 Innermost TRS: 14.14/4.62 Rules: 14.14/4.62 and(tt, X) -> activate(X) 14.14/4.62 plus(N, 0') -> N 14.14/4.62 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.62 x(N, 0') -> 0' 14.14/4.62 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.62 activate(X) -> X 14.14/4.62 14.14/4.62 Types: 14.14/4.62 and :: tt -> and:activate -> and:activate 14.14/4.62 tt :: tt 14.14/4.62 activate :: and:activate -> and:activate 14.14/4.62 plus :: 0':s -> 0':s -> 0':s 14.14/4.62 0' :: 0':s 14.14/4.62 s :: 0':s -> 0':s 14.14/4.62 x :: 0':s -> 0':s -> 0':s 14.14/4.62 hole_and:activate1_0 :: and:activate 14.14/4.62 hole_tt2_0 :: tt 14.14/4.62 hole_0':s3_0 :: 0':s 14.14/4.62 gen_0':s4_0 :: Nat -> 0':s 14.14/4.62 14.14/4.62 14.14/4.62 Generator Equations: 14.14/4.62 gen_0':s4_0(0) <=> 0' 14.14/4.62 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 14.14/4.62 14.14/4.62 14.14/4.62 The following defined symbols remain to be analysed: 14.14/4.62 plus, x 14.14/4.62 14.14/4.62 They will be analysed ascendingly in the following order: 14.14/4.62 plus < x 14.14/4.62 14.14/4.62 ---------------------------------------- 14.14/4.63 14.14/4.63 (20) LowerBoundPropagationProof (FINISHED) 14.14/4.63 Propagated lower bound. 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (21) 14.14/4.63 BOUNDS(n^1, INF) 14.14/4.63 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (22) 14.14/4.63 Obligation: 14.14/4.63 Innermost TRS: 14.14/4.63 Rules: 14.14/4.63 and(tt, X) -> activate(X) 14.14/4.63 plus(N, 0') -> N 14.14/4.63 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.63 x(N, 0') -> 0' 14.14/4.63 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.63 activate(X) -> X 14.14/4.63 14.14/4.63 Types: 14.14/4.63 and :: tt -> and:activate -> and:activate 14.14/4.63 tt :: tt 14.14/4.63 activate :: and:activate -> and:activate 14.14/4.63 plus :: 0':s -> 0':s -> 0':s 14.14/4.63 0' :: 0':s 14.14/4.63 s :: 0':s -> 0':s 14.14/4.63 x :: 0':s -> 0':s -> 0':s 14.14/4.63 hole_and:activate1_0 :: and:activate 14.14/4.63 hole_tt2_0 :: tt 14.14/4.63 hole_0':s3_0 :: 0':s 14.14/4.63 gen_0':s4_0 :: Nat -> 0':s 14.14/4.63 14.14/4.63 14.14/4.63 Lemmas: 14.14/4.63 plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0) 14.14/4.63 14.14/4.63 14.14/4.63 Generator Equations: 14.14/4.63 gen_0':s4_0(0) <=> 0' 14.14/4.63 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 14.14/4.63 14.14/4.63 14.14/4.63 The following defined symbols remain to be analysed: 14.14/4.63 x 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (23) RewriteLemmaProof (LOWER BOUND(ID)) 14.14/4.63 Proved the following rewrite lemma: 14.14/4.63 x(gen_0':s4_0(a), gen_0':s4_0(n473_0)) -> gen_0':s4_0(*(n473_0, a)), rt in Omega(1 + a*n473_0 + n473_0) 14.14/4.63 14.14/4.63 Induction Base: 14.14/4.63 x(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1) 14.14/4.63 0' 14.14/4.63 14.14/4.63 Induction Step: 14.14/4.63 x(gen_0':s4_0(a), gen_0':s4_0(+(n473_0, 1))) ->_R^Omega(1) 14.14/4.63 plus(x(gen_0':s4_0(a), gen_0':s4_0(n473_0)), gen_0':s4_0(a)) ->_IH 14.14/4.63 plus(gen_0':s4_0(*(c474_0, a)), gen_0':s4_0(a)) ->_L^Omega(1 + a) 14.14/4.63 gen_0':s4_0(+(a, *(n473_0, a))) 14.14/4.63 14.14/4.63 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (24) 14.14/4.63 Obligation: 14.14/4.63 Proved the lower bound n^2 for the following obligation: 14.14/4.63 14.14/4.63 Innermost TRS: 14.14/4.63 Rules: 14.14/4.63 and(tt, X) -> activate(X) 14.14/4.63 plus(N, 0') -> N 14.14/4.63 plus(N, s(M)) -> s(plus(N, M)) 14.14/4.63 x(N, 0') -> 0' 14.14/4.63 x(N, s(M)) -> plus(x(N, M), N) 14.14/4.63 activate(X) -> X 14.14/4.63 14.14/4.63 Types: 14.14/4.63 and :: tt -> and:activate -> and:activate 14.14/4.63 tt :: tt 14.14/4.63 activate :: and:activate -> and:activate 14.14/4.63 plus :: 0':s -> 0':s -> 0':s 14.14/4.63 0' :: 0':s 14.14/4.63 s :: 0':s -> 0':s 14.14/4.63 x :: 0':s -> 0':s -> 0':s 14.14/4.63 hole_and:activate1_0 :: and:activate 14.14/4.63 hole_tt2_0 :: tt 14.14/4.63 hole_0':s3_0 :: 0':s 14.14/4.63 gen_0':s4_0 :: Nat -> 0':s 14.14/4.63 14.14/4.63 14.14/4.63 Lemmas: 14.14/4.63 plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0) 14.14/4.63 14.14/4.63 14.14/4.63 Generator Equations: 14.14/4.63 gen_0':s4_0(0) <=> 0' 14.14/4.63 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 14.14/4.63 14.14/4.63 14.14/4.63 The following defined symbols remain to be analysed: 14.14/4.63 x 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (25) LowerBoundPropagationProof (FINISHED) 14.14/4.63 Propagated lower bound. 14.14/4.63 ---------------------------------------- 14.14/4.63 14.14/4.63 (26) 14.14/4.63 BOUNDS(n^2, INF) 14.20/4.69 EOF