WORST_CASE(?,O(n^1)) proof of input_0h7axLT1hz.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 151 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: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) 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: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [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: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [1] The TRS has the following type information: p :: s:0:times -> s:0:times s :: s:0:times -> s:0:times fac :: s:0:times -> s:0:times 0 :: s:0:times times :: s:0:times -> s:0:times -> s:0:times 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: p(v0) -> null_p [0] fac(v0) -> null_fac [0] And the following fresh constants: null_p, null_fac ---------------------------------------- (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: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [1] p(v0) -> null_p [0] fac(v0) -> null_fac [0] The TRS has the following type information: p :: s:0:times:null_p:null_fac -> s:0:times:null_p:null_fac s :: s:0:times:null_p:null_fac -> s:0:times:null_p:null_fac fac :: s:0:times:null_p:null_fac -> s:0:times:null_p:null_fac 0 :: s:0:times:null_p:null_fac times :: s:0:times:null_p:null_fac -> s:0:times:null_p:null_fac -> s:0:times:null_p:null_fac null_p :: s:0:times:null_p:null_fac null_fac :: s:0:times:null_p:null_fac 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: 0 => 0 null_p => 0 null_fac => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + x) + fac(p(1 + x)) :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, 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),0,[p(V, Out)],[V >= 0]). eq(start(V),0,[fac(V, Out)],[V >= 0]). eq(p(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]). eq(fac(V, Out),1,[],[Out = 1,V = 0]). eq(fac(V, Out),1,[p(1 + V2, Ret10),fac(Ret10, Ret1)],[Out = 2 + Ret1 + V2,V2 >= 0,V = 1 + V2]). eq(p(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]). eq(fac(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]). input_output_vars(p(V,Out),[V],[Out]). input_output_vars(fac(V,Out),[V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. non_recursive : [p/2] 1. recursive : [fac/2] 2. non_recursive : [start/1] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into p/2 1. SCC is partially evaluated into fac/2 2. SCC is partially evaluated into start/1 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations p/2 * CE 3 is refined into CE [8] * CE 4 is refined into CE [9] ### Cost equations --> "Loop" of p/2 * CEs [8] --> Loop 7 * CEs [9] --> Loop 8 ### Ranking functions of CR p(V,Out) #### Partial ranking functions of CR p(V,Out) ### Specialization of cost equations fac/2 * CE 7 is refined into CE [10] * CE 5 is refined into CE [11] * CE 6 is refined into CE [12,13] ### Cost equations --> "Loop" of fac/2 * CEs [13] --> Loop 9 * CEs [12] --> Loop 10 * CEs [10] --> Loop 11 * CEs [11] --> Loop 12 ### Ranking functions of CR fac(V,Out) * RF of phase [9]: [V] #### Partial ranking functions of CR fac(V,Out) * Partial RF of phase [9]: - RF of loop [9:1]: V ### Specialization of cost equations start/1 * CE 1 is refined into CE [14,15] * CE 2 is refined into CE [16,17,18,19] ### Cost equations --> "Loop" of start/1 * CEs [14,15,16,17,18,19] --> Loop 13 ### Ranking functions of CR start(V) #### Partial ranking functions of CR start(V) Computing Bounds ===================================== #### Cost of chains of p(V,Out): * Chain [8]: 0 with precondition: [Out=0,V>=0] * Chain [7]: 1 with precondition: [V=Out+1,V>=1] #### Cost of chains of fac(V,Out): * Chain [[9],12]: 2*it(9)+1 Such that:it(9) =< V with precondition: [V>=1] * Chain [[9],11]: 2*it(9)+0 Such that:it(9) =< V with precondition: [V>=1,Out>=V+1] * Chain [[9],10,12]: 2*it(9)+2 Such that:it(9) =< V with precondition: [V>=1,Out>=2*V+2] * Chain [[9],10,11]: 2*it(9)+1 Such that:it(9) =< V with precondition: [V>=1,Out>=2*V+1] * Chain [12]: 1 with precondition: [V=0,Out=1] * Chain [11]: 0 with precondition: [Out=0,V>=0] * Chain [10,12]: 2 with precondition: [V+2=Out,V>=1] * Chain [10,11]: 1 with precondition: [V+1=Out,V>=1] #### Cost of chains of start(V): * Chain [13]: 8*s(6)+2 Such that:s(5) =< V s(6) =< s(5) with precondition: [V>=0] Closed-form bounds of start(V): ------------------------------------- * Chain [13] with precondition: [V>=0] - Upper bound: 8*V+2 - Complexity: n ### Maximum cost of start(V): 8*V+2 Asymptotic class: n * Total analysis performed in 71 ms. ---------------------------------------- (10) BOUNDS(1, n^1)