WORST_CASE(?,O(n^3)) proof of input_Kt1b0umqFr.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). (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, 687 ms] (10) BOUNDS(1, n^3) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(0, y) -> 0 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0 if_minus(false, s(x), y) -> s(minus(x, y)) mod(0, y) -> 0 mod(s(x), 0) -> 0 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) if_mod(false, s(x), s(y)) -> 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^3). The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] mod(0, y) -> 0 [1] mod(s(x), 0) -> 0 [1] mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1] if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1] if_mod(false, s(x), s(y)) -> 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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] mod(0, y) -> 0 [1] mod(s(x), 0) -> 0 [1] mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1] if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1] if_mod(false, s(x), s(y)) -> s(x) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false minus :: 0:s -> 0:s -> 0:s if_minus :: true:false -> 0:s -> 0:s -> 0:s mod :: 0:s -> 0:s -> 0:s if_mod :: true:false -> 0:s -> 0:s -> 0:s 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: if_minus(v0, v1, v2) -> null_if_minus [0] if_mod(v0, v1, v2) -> null_if_mod [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] mod(v0, v1) -> null_mod [0] And the following fresh constants: null_if_minus, null_if_mod, null_le, null_minus, null_mod ---------------------------------------- (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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] mod(0, y) -> 0 [1] mod(s(x), 0) -> 0 [1] mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1] if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1] if_mod(false, s(x), s(y)) -> s(x) [1] if_minus(v0, v1, v2) -> null_if_minus [0] if_mod(v0, v1, v2) -> null_if_mod [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] mod(v0, v1) -> null_mod [0] The TRS has the following type information: le :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> true:false:null_le 0 :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod true :: true:false:null_le s :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod false :: true:false:null_le minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod if_minus :: true:false:null_le -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod if_mod :: true:false:null_le -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod null_if_minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod null_if_mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod null_le :: true:false:null_le null_minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod null_mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod 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 true => 2 false => 1 null_if_minus => 0 null_if_mod => 0 null_le => 0 null_minus => 0 null_mod => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0 if_mod(z, z', z'') -{ 1 }-> mod(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> if_minus(le(1 + x, y), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 mod(z, z') -{ 1 }-> if_mod(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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(V1, V, V11),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V, V11),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V, V11),0,[fun(V1, V, V11, Out)],[V1 >= 0,V >= 0,V11 >= 0]). eq(start(V1, V, V11),0,[mod(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V, V11),0,[fun1(V1, V, V11, Out)],[V1 >= 0,V >= 0,V11 >= 0]). eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]). eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]). eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). eq(minus(V1, V, Out),1,[],[Out = 0,V6 >= 0,V1 = 0,V = V6]). eq(minus(V1, V, Out),1,[le(1 + V7, V8, Ret0),fun(Ret0, 1 + V7, V8, Ret1)],[Out = Ret1,V7 >= 0,V8 >= 0,V1 = 1 + V7,V = V8]). eq(fun(V1, V, V11, Out),1,[],[Out = 0,V1 = 2,V = 1 + V9,V11 = V10,V9 >= 0,V10 >= 0]). eq(fun(V1, V, V11, Out),1,[minus(V12, V13, Ret11)],[Out = 1 + Ret11,V = 1 + V12,V11 = V13,V1 = 1,V12 >= 0,V13 >= 0]). eq(mod(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 0,V = V14]). eq(mod(V1, V, Out),1,[],[Out = 0,V15 >= 0,V1 = 1 + V15,V = 0]). eq(mod(V1, V, Out),1,[le(V16, V17, Ret01),fun1(Ret01, 1 + V17, 1 + V16, Ret2)],[Out = Ret2,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]). eq(fun1(V1, V, V11, Out),1,[minus(V18, V19, Ret02),mod(Ret02, 1 + V19, Ret3)],[Out = Ret3,V1 = 2,V = 1 + V18,V18 >= 0,V19 >= 0,V11 = 1 + V19]). eq(fun1(V1, V, V11, Out),1,[],[Out = 1 + V20,V = 1 + V20,V1 = 1,V20 >= 0,V21 >= 0,V11 = 1 + V21]). eq(fun(V1, V, V11, Out),0,[],[Out = 0,V23 >= 0,V11 = V24,V22 >= 0,V1 = V23,V = V22,V24 >= 0]). eq(fun1(V1, V, V11, Out),0,[],[Out = 0,V27 >= 0,V11 = V25,V26 >= 0,V1 = V27,V = V26,V25 >= 0]). eq(le(V1, V, Out),0,[],[Out = 0,V29 >= 0,V28 >= 0,V1 = V29,V = V28]). eq(minus(V1, V, Out),0,[],[Out = 0,V30 >= 0,V31 >= 0,V1 = V30,V = V31]). eq(mod(V1, V, Out),0,[],[Out = 0,V33 >= 0,V32 >= 0,V1 = V33,V = V32]). input_output_vars(le(V1,V,Out),[V1,V],[Out]). input_output_vars(minus(V1,V,Out),[V1,V],[Out]). input_output_vars(fun(V1,V,V11,Out),[V1,V,V11],[Out]). input_output_vars(mod(V1,V,Out),[V1,V],[Out]). input_output_vars(fun1(V1,V,V11,Out),[V1,V,V11],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [le/3] 1. recursive : [fun/4,minus/3] 2. recursive : [fun1/4,(mod)/3] 3. non_recursive : [start/3] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into le/3 1. SCC is partially evaluated into minus/3 2. SCC is partially evaluated into (mod)/3 3. SCC is partially evaluated into start/3 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations le/3 * CE 23 is refined into CE [24] * CE 21 is refined into CE [25] * CE 20 is refined into CE [26] * CE 22 is refined into CE [27] ### Cost equations --> "Loop" of le/3 * CEs [27] --> Loop 13 * CEs [24] --> Loop 14 * CEs [25] --> Loop 15 * CEs [26] --> Loop 16 ### Ranking functions of CR le(V1,V,Out) * RF of phase [13]: [V,V1] #### Partial ranking functions of CR le(V1,V,Out) * Partial RF of phase [13]: - RF of loop [13:1]: V V1 ### Specialization of cost equations minus/3 * CE 9 is refined into CE [28,29,30,31] * CE 11 is refined into CE [32] * CE 12 is refined into CE [33] * CE 13 is refined into CE [34] * CE 10 is refined into CE [35,36] ### Cost equations --> "Loop" of minus/3 * CEs [36] --> Loop 17 * CEs [35] --> Loop 18 * CEs [28] --> Loop 19 * CEs [29,30,31,32,33,34] --> Loop 20 ### Ranking functions of CR minus(V1,V,Out) * RF of phase [17]: [V1-1,V1-V] * RF of phase [18]: [V1] #### Partial ranking functions of CR minus(V1,V,Out) * Partial RF of phase [17]: - RF of loop [17:1]: V1-1 V1-V * Partial RF of phase [18]: - RF of loop [18:1]: V1 ### Specialization of cost equations (mod)/3 * CE 15 is refined into CE [37,38] * CE 18 is refined into CE [39] * CE 14 is refined into CE [40,41,42,43,44] * CE 17 is refined into CE [45] * CE 19 is refined into CE [46] * CE 16 is refined into CE [47,48,49,50] ### Cost equations --> "Loop" of (mod)/3 * CEs [50] --> Loop 21 * CEs [49] --> Loop 22 * CEs [48] --> Loop 23 * CEs [47] --> Loop 24 * CEs [38] --> Loop 25 * CEs [40] --> Loop 26 * CEs [39] --> Loop 27 * CEs [37] --> Loop 28 * CEs [41] --> Loop 29 * CEs [42,43,44,45,46] --> Loop 30 ### Ranking functions of CR mod(V1,V,Out) * RF of phase [21]: [V1/2-1,V1/2-V/2] * RF of phase [23]: [V1-1] #### Partial ranking functions of CR mod(V1,V,Out) * Partial RF of phase [21]: - RF of loop [21:1]: V1/2-1 V1/2-V/2 * Partial RF of phase [23]: - RF of loop [23:1]: V1-1 ### Specialization of cost equations start/3 * CE 3 is refined into CE [51,52,53,54,55,56,57,58] * CE 5 is refined into CE [59] * CE 1 is refined into CE [60] * CE 2 is refined into CE [61] * CE 4 is refined into CE [62,63,64] * CE 6 is refined into CE [65,66,67,68,69] * CE 7 is refined into CE [70,71,72] * CE 8 is refined into CE [73,74,75,76,77,78,79] ### Cost equations --> "Loop" of start/3 * CEs [76] --> Loop 31 * CEs [66,71,75] --> Loop 32 * CEs [51,52,53,54,55,56,57,58,59] --> Loop 33 * CEs [74] --> Loop 34 * CEs [61,62,63,64] --> Loop 35 * CEs [60,65,67,68,69,70,72,73,77,78,79] --> Loop 36 ### Ranking functions of CR start(V1,V,V11) #### Partial ranking functions of CR start(V1,V,V11) Computing Bounds ===================================== #### Cost of chains of le(V1,V,Out): * Chain [[13],16]: 1*it(13)+1 Such that:it(13) =< V1 with precondition: [Out=2,V1>=1,V>=V1] * Chain [[13],15]: 1*it(13)+1 Such that:it(13) =< V with precondition: [Out=1,V>=1,V1>=V+1] * Chain [[13],14]: 1*it(13)+0 Such that:it(13) =< V with precondition: [Out=0,V1>=1,V>=1] * Chain [16]: 1 with precondition: [V1=0,Out=2,V>=0] * Chain [15]: 1 with precondition: [V=0,Out=1,V1>=1] * Chain [14]: 0 with precondition: [Out=0,V1>=0,V>=0] #### Cost of chains of minus(V1,V,Out): * Chain [[18],20]: 3*it(18)+2*s(4)+3 Such that:aux(1) =< V1-Out it(18) =< Out s(4) =< aux(1) with precondition: [V=0,Out>=1,V1>=Out] * Chain [[18],19]: 3*it(18)+2 Such that:it(18) =< Out with precondition: [V=0,Out>=1,V1>=Out+1] * Chain [[17],20]: 3*it(17)+2*s(2)+2*s(4)+1*s(8)+3 Such that:aux(1) =< V1-Out it(17) =< Out aux(4) =< V s(4) =< aux(1) s(2) =< aux(4) s(8) =< it(17)*aux(4) with precondition: [V>=1,Out>=1,V1>=Out+V] * Chain [20]: 2*s(2)+2*s(4)+3 Such that:aux(1) =< V1 aux(2) =< V s(4) =< aux(1) s(2) =< aux(2) with precondition: [Out=0,V1>=0,V>=0] * Chain [19]: 2 with precondition: [V=0,Out=0,V1>=1] #### Cost of chains of mod(V1,V,Out): * Chain [[23],30]: 10*it(23)+1*s(19)+6*s(28)+2 Such that:s(19) =< 1 aux(10) =< V1 it(23) =< aux(10) s(31) =< it(23)*aux(10) s(28) =< s(31) with precondition: [V=1,Out=0,V1>=2] * Chain [[23],26]: 8*it(23)+6*s(28)+2 Such that:aux(11) =< V1 it(23) =< aux(11) s(31) =< it(23)*aux(11) s(28) =< s(31) with precondition: [V=1,Out=0,V1>=2] * Chain [[23],24,30]: 10*it(23)+1*s(19)+6*s(28)+8 Such that:s(19) =< 1 aux(12) =< V1 it(23) =< aux(12) s(31) =< it(23)*aux(12) s(28) =< s(31) with precondition: [V=1,Out=0,V1>=2] * Chain [[21],30]: 13*it(21)+1*s(19)+3*s(50)+1*s(52)+2 Such that:aux(14) =< V1-V aux(17) =< V1 aux(18) =< V it(21) =< aux(17) s(19) =< aux(18) s(50) =< it(21)*aux(14) s(52) =< s(50)*aux(18) with precondition: [Out=0,V>=2,V1>=V+1] * Chain [[21],29]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+2 Such that:aux(14) =< V1-V it(21) =< V1/2-V/2 s(45) =< V aux(19) =< V1 s(50) =< it(21)*aux(14) s(49) =< aux(19) s(52) =< s(50)*s(45) with precondition: [Out=0,V>=2,V1>=V+1] * Chain [[21],28]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+3 Such that:aux(14) =< V1-V it(21) =< V1/2-V/2 s(45) =< V aux(20) =< V1 s(50) =< it(21)*aux(14) s(49) =< aux(20) s(52) =< s(50)*s(45) with precondition: [Out=1,V>=2,V1>=V+1] * Chain [[21],25]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+1*s(55)+3 Such that:aux(15) =< V1 aux(14) =< V1-V aux(16) =< V1-Out it(21) =< V1/2-V/2 s(45) =< V s(55) =< Out s(53) =< aux(15) s(53) =< aux(16) s(50) =< it(21)*aux(14) s(49) =< s(53) s(52) =< s(50)*s(45) with precondition: [Out>=2,V>=Out+1,V1>=Out+V] * Chain [[21],22,30]: 13*it(21)+4*s(19)+3*s(50)+1*s(52)+8 Such that:aux(14) =< V1-V aux(23) =< V1 aux(24) =< V it(21) =< aux(23) s(19) =< aux(24) s(50) =< it(21)*aux(14) s(52) =< s(50)*aux(24) with precondition: [Out=0,V>=2,V1>=2*V] * Chain [30]: 2*s(17)+1*s(19)+2 Such that:s(19) =< V aux(6) =< V1 s(17) =< aux(6) with precondition: [Out=0,V1>=0,V>=0] * Chain [29]: 2 with precondition: [V1=1,Out=0,V>=2] * Chain [28]: 3 with precondition: [V1=1,Out=1,V>=2] * Chain [27]: 1 with precondition: [V=0,Out=0,V1>=1] * Chain [26]: 2 with precondition: [V=1,Out=0,V1>=1] * Chain [25]: 1*s(55)+3 Such that:s(55) =< V1 with precondition: [V1=Out,V1>=2,V>=V1+1] * Chain [24,30]: 1*s(19)+2*s(34)+8 Such that:s(19) =< 1 s(32) =< V1 s(34) =< s(32) with precondition: [V=1,Out=0,V1>=1] * Chain [22,30]: 4*s(19)+2*s(59)+8 Such that:s(57) =< V1 aux(22) =< V s(19) =< aux(22) s(59) =< s(57) with precondition: [Out=0,V>=2,V1>=V] #### Cost of chains of start(V1,V,V11): * Chain [36]: 16*s(106)+55*s(108)+1*s(118)+18*s(119)+6*s(125)+2*s(126)+9*s(127)+3*s(128)+8 Such that:aux(32) =< V1 aux(33) =< V1-V aux(34) =< V1/2-V/2 aux(35) =< V s(108) =< aux(32) s(119) =< aux(34) s(106) =< aux(35) s(125) =< s(108)*aux(33) s(126) =< s(125)*aux(35) s(127) =< s(119)*aux(33) s(128) =< s(127)*aux(35) s(118) =< s(108)*aux(35) with precondition: [V1>=0,V>=0] * Chain [35]: 15*s(149)+4*s(150)+1*s(160)+4 Such that:aux(38) =< V aux(39) =< V11 s(149) =< aux(38) s(150) =< aux(39) s(160) =< s(149)*aux(39) with precondition: [V1=1,V>=1,V11>=0] * Chain [34]: 3 with precondition: [V1=1,V>=2] * Chain [33]: 158*s(163)+32*s(164)+6*s(165)+3*s(173)+1*s(174)+16*s(183)+9*s(185)+3*s(186)+18*s(198)+4*s(204)+6*s(205)+6*s(211)+2*s(212)+3*s(213)+1*s(214)+1*s(220)+12*s(228)+6*s(231)+2*s(233)+12 Such that:s(207) =< V-2*V11 s(205) =< V/2-V11 s(167) =< -V11 s(165) =< -V11/2 aux(52) =< 1 aux(53) =< V aux(54) =< V-V11 aux(55) =< V11 s(183) =< aux(52) s(163) =< aux(53) s(185) =< s(163)*aux(53) s(186) =< s(185)*aux(52) s(197) =< s(163)*aux(53) s(198) =< s(197) s(164) =< aux(55) s(220) =< s(183)*aux(55) s(228) =< aux(54) s(231) =< s(228)*aux(54) s(233) =< s(231)*aux(55) s(204) =< s(163)*aux(55) s(173) =< s(165)*s(167) s(174) =< s(173)*aux(55) s(211) =< s(163)*s(207) s(212) =< s(211)*aux(55) s(213) =< s(205)*s(207) s(214) =< s(213)*aux(55) with precondition: [V1=2,V>=1,V11>=0] * Chain [32]: 8*s(259)+3 Such that:aux(56) =< V1 s(259) =< aux(56) with precondition: [V=0,V1>=1] * Chain [31]: 3*s(263)+30*s(264)+18*s(266)+8 Such that:s(261) =< 1 s(262) =< V1 s(263) =< s(261) s(264) =< s(262) s(265) =< s(264)*s(262) s(266) =< s(265) with precondition: [V=1,V1>=1] Closed-form bounds of start(V1,V,V11): ------------------------------------- * Chain [36] with precondition: [V1>=0,V>=0] - Upper bound: 55*V1+8+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+16*V+3*V*nat(V1-V)*nat(V1/2-V/2)+nat(V1-V)*9*nat(V1/2-V/2)+nat(V1/2-V/2)*18 - Complexity: n^3 * Chain [35] with precondition: [V1=1,V>=1,V11>=0] - Upper bound: 15*V+4+V11*V+4*V11 - Complexity: n^2 * Chain [34] with precondition: [V1=1,V>=2] - Upper bound: 3 - Complexity: constant * Chain [33] with precondition: [V1=2,V>=1,V11>=0] - Upper bound: 158*V+28+30*V*V+4*V*V11+2*V*V11*nat(V-2*V11)+6*V*nat(V-2*V11)+33*V11+2*V11*nat(V-V11)*nat(V-V11)+nat(V-2*V11)*V11*nat(V/2-V11)+nat(V-V11)*12+nat(V-V11)*6*nat(V-V11)+nat(V-2*V11)*3*nat(V/2-V11)+nat(V/2-V11)*6 - Complexity: n^3 * Chain [32] with precondition: [V=0,V1>=1] - Upper bound: 8*V1+3 - Complexity: n * Chain [31] with precondition: [V=1,V1>=1] - Upper bound: 30*V1+11+18*V1*V1 - Complexity: n^2 ### Maximum cost of start(V1,V,V11): max([22*V1+5+max([18*V1*V1+3,25*V1+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+16*V+3*V*nat(V1-V)*nat(V1/2-V/2)+nat(V1-V)*9*nat(V1/2-V/2)+nat(V1/2-V/2)*18])+8*V1,143*V+24+30*V*V+3*V*nat(V11)+2*V*nat(V11)*nat(V-2*V11)+6*V*nat(V-2*V11)+nat(V11)*29+nat(V11)*2*nat(V-V11)*nat(V-V11)+nat(V-2*V11)*nat(V11)*nat(V/2-V11)+nat(V-V11)*12+nat(V-V11)*6*nat(V-V11)+nat(V-2*V11)*3*nat(V/2-V11)+nat(V/2-V11)*6+(15*V+1+nat(V11)*V+nat(V11)*4)])+3 Asymptotic class: n^3 * Total analysis performed in 587 ms. ---------------------------------------- (10) BOUNDS(1, n^3)