WORST_CASE(?,O(n^1)) proof of input_Kzx8Ur0JaA.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (8) CpxRNTS (9) CompleteCoflocoProof [FINISHED, 148 ms] (10) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: admit(x, nil) -> nil admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) cond(true, y) -> y S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: admit(x, nil) -> nil [1] admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1] cond(true, y) -> y [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: admit(x, nil) -> nil [1] admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1] cond(true, y) -> y [1] The TRS has the following type information: admit :: carry -> nil:. -> nil:. nil :: nil:. . :: w -> nil:. -> nil:. w :: w cond :: =:true -> nil:. -> nil:. = :: sum -> w -> =:true sum :: carry -> w -> w -> sum carry :: carry -> w -> w -> carry true :: =:true Rewrite Strategy: INNERMOST ---------------------------------------- (5) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: admit(v0, v1) -> null_admit [0] cond(v0, v1) -> null_cond [0] And the following fresh constants: null_admit, null_cond, const, const1 ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: admit(x, nil) -> nil [1] admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1] cond(true, y) -> y [1] admit(v0, v1) -> null_admit [0] cond(v0, v1) -> null_cond [0] The TRS has the following type information: admit :: carry -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond nil :: nil:.:null_admit:null_cond . :: w -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond w :: w cond :: =:true -> nil:.:null_admit:null_cond -> nil:.:null_admit:null_cond = :: sum -> w -> =:true sum :: carry -> w -> w -> sum carry :: carry -> w -> w -> carry true :: =:true null_admit :: nil:.:null_admit:null_cond null_cond :: nil:.:null_admit:null_cond const :: carry const1 :: sum Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 w => 0 true => 0 null_admit => 0 null_cond => 0 const => 0 const1 => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: admit(z', z'') -{ 1 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + admit(1 + x + u + v, z)))) :|: v >= 0, z >= 0, z' = x, x >= 0, z'' = 1 + u + (1 + v + (1 + 0 + z)), u >= 0 admit(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0 admit(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 cond(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 cond(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (9) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V, V1),0,[admit(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1),0,[cond(V, V1, Out)],[V >= 0,V1 >= 0]). eq(admit(V, V1, Out),1,[],[Out = 0,V1 = 0,V = V2,V2 >= 0]). eq(admit(V, V1, Out),1,[admit(1 + V5 + V3 + V4, V6, Ret1111),cond(1 + (1 + V5 + V3 + V4) + 0, 1 + V3 + (1 + V4 + (1 + 0 + Ret1111)), Ret)],[Out = Ret,V4 >= 0,V6 >= 0,V = V5,V5 >= 0,V1 = 3 + V3 + V4 + V6,V3 >= 0]). eq(cond(V, V1, Out),1,[],[Out = V7,V1 = V7,V7 >= 0,V = 0]). eq(admit(V, V1, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V8,V = V9]). eq(cond(V, V1, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V10,V = V11]). input_output_vars(admit(V,V1,Out),[V,V1],[Out]). input_output_vars(cond(V,V1,Out),[V,V1],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. non_recursive : [cond/3] 1. recursive [non_tail] : [admit/3] 2. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into cond/3 1. SCC is partially evaluated into admit/3 2. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations cond/3 * CE 7 is refined into CE [8] * CE 6 is refined into CE [9] ### Cost equations --> "Loop" of cond/3 * CEs [8] --> Loop 6 * CEs [9] --> Loop 7 ### Ranking functions of CR cond(V,V1,Out) #### Partial ranking functions of CR cond(V,V1,Out) ### Specialization of cost equations admit/3 * CE 3 is refined into CE [10] * CE 5 is refined into CE [11] * CE 4 is refined into CE [12] ### Cost equations --> "Loop" of admit/3 * CEs [12] --> Loop 8 * CEs [10,11] --> Loop 9 ### Ranking functions of CR admit(V,V1,Out) * RF of phase [8]: [V/2+V1/2-1,V1/3-2/3] #### Partial ranking functions of CR admit(V,V1,Out) * Partial RF of phase [8]: - RF of loop [8:1]: V/2+V1/2-1 V1/3-2/3 ### Specialization of cost equations start/2 * CE 1 is refined into CE [13] * CE 2 is refined into CE [14,15] ### Cost equations --> "Loop" of start/2 * CEs [13,14,15] --> Loop 10 ### Ranking functions of CR start(V,V1) #### Partial ranking functions of CR start(V,V1) Computing Bounds ===================================== #### Cost of chains of cond(V,V1,Out): * Chain [7]: 1 with precondition: [V=0,V1=Out,V1>=0] * Chain [6]: 0 with precondition: [Out=0,V>=0,V1>=0] #### Cost of chains of admit(V,V1,Out): * Chain [[8],9]: 1*it(8)+1 Such that:it(8) =< V1/3 with precondition: [Out=0,V>=0,V1>=3] * Chain [9]: 1 with precondition: [Out=0,V>=0,V1>=0] #### Cost of chains of start(V,V1): * Chain [10]: 1*s(2)+1 Such that:s(2) =< V1/3 with precondition: [V>=0,V1>=0] Closed-form bounds of start(V,V1): ------------------------------------- * Chain [10] with precondition: [V>=0,V1>=0] - Upper bound: V1/3+1 - Complexity: n ### Maximum cost of start(V,V1): V1/3+1 Asymptotic class: n * Total analysis performed in 64 ms. ---------------------------------------- (10) BOUNDS(1, n^1)