WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 127 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] (10) CpxRNTS (11) CompleteCoflocoProof [FINISHED, 708 ms] (12) BOUNDS(1, n^1) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) a(Z, y, z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) a(Z, y, z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) 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: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1] a(Z, y, z) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1] a(Z, y, z) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] The TRS has the following type information: a :: C:Z -> a -> a -> C:Z C :: C:Z -> C:Z -> C:Z Z :: C:Z eqZList :: C:Z -> C:Z -> False:True and :: False:True -> False:True -> False:True False :: False:True True :: False:True second :: C:Z -> C:Z first :: C:Z -> C:Z Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: and(v0, v1) -> null_and [0] second(v0) -> null_second [0] first(v0) -> null_first [0] a(v0, v1, v2) -> null_a [0] eqZList(v0, v1) -> null_eqZList [0] And the following fresh constants: null_and, null_second, null_first, null_a, null_eqZList, const ---------------------------------------- (8) 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: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1] a(Z, y, z) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] and(v0, v1) -> null_and [0] second(v0) -> null_second [0] first(v0) -> null_first [0] a(v0, v1, v2) -> null_a [0] eqZList(v0, v1) -> null_eqZList [0] The TRS has the following type information: a :: C:Z:null_second:null_first:null_a -> a -> a -> C:Z:null_second:null_first:null_a C :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a Z :: C:Z:null_second:null_first:null_a eqZList :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> False:True:null_and:null_eqZList and :: False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList False :: False:True:null_and:null_eqZList True :: False:True:null_and:null_eqZList second :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a first :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a null_and :: False:True:null_and:null_eqZList null_second :: C:Z:null_second:null_first:null_a null_first :: C:Z:null_second:null_first:null_a null_a :: C:Z:null_second:null_first:null_a null_eqZList :: False:True:null_and:null_eqZList const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Z => 0 False => 1 True => 2 null_and => 0 null_second => 0 null_first => 0 null_a => 0 null_eqZList => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: a(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0 a(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 a(z', z'', z1) -{ 1 }-> 1 + a(x1, y, z) + a(x2, y, y) :|: z' = 1 + x1 + x2, z1 = z, x1 >= 0, z >= 0, z'' = y, y >= 0, x2 >= 0 and(z', z'') -{ 0 }-> 2 :|: z' = 2, z'' = 2 and(z', z'') -{ 0 }-> 1 :|: z' = 1, z'' = 1 and(z', z'') -{ 0 }-> 1 :|: z' = 2, z'' = 1 and(z', z'') -{ 0 }-> 1 :|: z' = 1, z'' = 2 and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 eqZList(z', z'') -{ 1 }-> and(eqZList(x1, y1), eqZList(x2, y2)) :|: y1 >= 0, z' = 1 + x1 + x2, x1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, x2 >= 0 eqZList(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eqZList(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0 eqZList(z', z'') -{ 1 }-> 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0 eqZList(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 first(z') -{ 1 }-> x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0 first(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 second(z') -{ 1 }-> x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0 second(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (11) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V, V2, V6),0,[a(V, V2, V6, Out)],[V >= 0,V2 >= 0,V6 >= 0]). eq(start(V, V2, V6),0,[eqZList(V, V2, Out)],[V >= 0,V2 >= 0]). eq(start(V, V2, V6),0,[second(V, Out)],[V >= 0]). eq(start(V, V2, V6),0,[first(V, Out)],[V >= 0]). eq(start(V, V2, V6),0,[and(V, V2, Out)],[V >= 0,V2 >= 0]). eq(a(V, V2, V6, Out),1,[a(V4, V1, V5, Ret01),a(V3, V1, V1, Ret1)],[Out = 1 + Ret01 + Ret1,V = 1 + V3 + V4,V6 = V5,V4 >= 0,V5 >= 0,V2 = V1,V1 >= 0,V3 >= 0]). eq(a(V, V2, V6, Out),1,[],[Out = 0,V6 = V8,V8 >= 0,V2 = V7,V7 >= 0,V = 0]). eq(eqZList(V, V2, Out),1,[eqZList(V11, V12, Ret0),eqZList(V9, V10, Ret11),and(Ret0, Ret11, Ret)],[Out = Ret,V12 >= 0,V = 1 + V11 + V9,V11 >= 0,V2 = 1 + V10 + V12,V10 >= 0,V9 >= 0]). eq(eqZList(V, V2, Out),1,[],[Out = 1,V2 = 0,V = 1 + V13 + V14,V13 >= 0,V14 >= 0]). eq(eqZList(V, V2, Out),1,[],[Out = 1,V15 >= 0,V2 = 1 + V15 + V16,V16 >= 0,V = 0]). eq(eqZList(V, V2, Out),1,[],[Out = 2,V2 = 0,V = 0]). eq(second(V, Out),1,[],[Out = V17,V = 1 + V17 + V18,V18 >= 0,V17 >= 0]). eq(first(V, Out),1,[],[Out = V20,V = 1 + V19 + V20,V20 >= 0,V19 >= 0]). eq(and(V, V2, Out),0,[],[Out = 1,V = 1,V2 = 1]). eq(and(V, V2, Out),0,[],[Out = 1,V = 2,V2 = 1]). eq(and(V, V2, Out),0,[],[Out = 1,V = 1,V2 = 2]). eq(and(V, V2, Out),0,[],[Out = 2,V = 2,V2 = 2]). eq(and(V, V2, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V2 = V21,V = V22]). eq(second(V, Out),0,[],[Out = 0,V23 >= 0,V = V23]). eq(first(V, Out),0,[],[Out = 0,V24 >= 0,V = V24]). eq(a(V, V2, V6, Out),0,[],[Out = 0,V25 >= 0,V6 = V27,V26 >= 0,V2 = V26,V27 >= 0,V = V25]). eq(eqZList(V, V2, Out),0,[],[Out = 0,V28 >= 0,V29 >= 0,V2 = V29,V = V28]). input_output_vars(a(V,V2,V6,Out),[V,V2,V6],[Out]). input_output_vars(eqZList(V,V2,Out),[V,V2],[Out]). input_output_vars(second(V,Out),[V],[Out]). input_output_vars(first(V,Out),[V],[Out]). input_output_vars(and(V,V2,Out),[V,V2],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive [multiple] : [a/4] 1. non_recursive : [and/3] 2. recursive [non_tail,multiple] : [eqZList/3] 3. non_recursive : [first/2] 4. non_recursive : [second/2] 5. non_recursive : [start/3] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into a/4 1. SCC is partially evaluated into and/3 2. SCC is partially evaluated into eqZList/3 3. SCC is partially evaluated into first/2 4. SCC is partially evaluated into second/2 5. SCC is partially evaluated into start/3 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations a/4 * CE 7 is refined into CE [23] * CE 8 is refined into CE [24] * CE 6 is refined into CE [25] ### Cost equations --> "Loop" of a/4 * CEs [25] --> Loop 18 * CEs [23,24] --> Loop 19 ### Ranking functions of CR a(V,V2,V6,Out) * RF of phase [18]: [V] #### Partial ranking functions of CR a(V,V2,V6,Out) * Partial RF of phase [18]: - RF of loop [18:1,18:2]: V ### Specialization of cost equations and/3 * CE 22 is refined into CE [26] * CE 21 is refined into CE [27] * CE 19 is refined into CE [28] * CE 20 is refined into CE [29] * CE 18 is refined into CE [30] ### Cost equations --> "Loop" of and/3 * CEs [26] --> Loop 20 * CEs [27] --> Loop 21 * CEs [28] --> Loop 22 * CEs [29] --> Loop 23 * CEs [30] --> Loop 24 ### Ranking functions of CR and(V,V2,Out) #### Partial ranking functions of CR and(V,V2,Out) ### Specialization of cost equations eqZList/3 * CE 13 is refined into CE [31] * CE 10 is refined into CE [32] * CE 11 is refined into CE [33] * CE 12 is refined into CE [34] * CE 9 is refined into CE [35,36,37,38,39] ### Cost equations --> "Loop" of eqZList/3 * CEs [38] --> Loop 25 * CEs [37] --> Loop 26 * CEs [36] --> Loop 27 * CEs [35] --> Loop 28 * CEs [39] --> Loop 29 * CEs [31] --> Loop 30 * CEs [32] --> Loop 31 * CEs [33] --> Loop 32 * CEs [34] --> Loop 33 ### Ranking functions of CR eqZList(V,V2,Out) * RF of phase [25]: [V,V2] * RF of phase [26,27,28]: [V,V2] * RF of phase [29]: [V,V2] #### Partial ranking functions of CR eqZList(V,V2,Out) * Partial RF of phase [25]: - RF of loop [25:1,25:2]: V V2 * Partial RF of phase [26,27,28]: - RF of loop [26:1,26:2,27:1,27:2,28:1,28:2]: V V2 * Partial RF of phase [29]: - RF of loop [29:1,29:2]: V V2 ### Specialization of cost equations first/2 * CE 16 is refined into CE [40] * CE 17 is refined into CE [41] ### Cost equations --> "Loop" of first/2 * CEs [40] --> Loop 34 * CEs [41] --> Loop 35 ### Ranking functions of CR first(V,Out) #### Partial ranking functions of CR first(V,Out) ### Specialization of cost equations second/2 * CE 14 is refined into CE [42] * CE 15 is refined into CE [43] ### Cost equations --> "Loop" of second/2 * CEs [42] --> Loop 36 * CEs [43] --> Loop 37 ### Ranking functions of CR second(V,Out) #### Partial ranking functions of CR second(V,Out) ### Specialization of cost equations start/3 * CE 1 is refined into CE [44,45] * CE 2 is refined into CE [46,47,48,49,50,51] * CE 3 is refined into CE [52,53] * CE 4 is refined into CE [54,55] * CE 5 is refined into CE [56,57,58,59,60] ### Cost equations --> "Loop" of start/3 * CEs [48] --> Loop 38 * CEs [59] --> Loop 39 * CEs [58] --> Loop 40 * CEs [57] --> Loop 41 * CEs [51,56] --> Loop 42 * CEs [47] --> Loop 43 * CEs [44,45,46,49,50,52,53,54,55,60] --> Loop 44 ### Ranking functions of CR start(V,V2,V6) #### Partial ranking functions of CR start(V,V2,V6) Computing Bounds ===================================== #### Cost of chains of a(V,V2,V6,Out): * Chain [19]: 1 with precondition: [Out=0,V>=0,V2>=0,V6>=0] * Chain [multiple([18],[[19]])]: 1*it(18)+1*it([19])+0 Such that:it(18) =< V it([19]) =< V+1 with precondition: [V2>=0,V6>=0,Out>=1,V>=Out] #### Cost of chains of and(V,V2,Out): * Chain [24]: 0 with precondition: [V=1,V2=1,Out=1] * Chain [23]: 0 with precondition: [V=1,V2=2,Out=1] * Chain [22]: 0 with precondition: [V=2,V2=1,Out=1] * Chain [21]: 0 with precondition: [V=2,V2=2,Out=2] * Chain [20]: 0 with precondition: [Out=0,V>=0,V2>=0] #### Cost of chains of eqZList(V,V2,Out): * Chain [33]: 1 with precondition: [V=0,V2=0,Out=2] * Chain [32]: 1 with precondition: [V=0,Out=1,V2>=1] * Chain [31]: 1 with precondition: [V2=0,Out=1,V>=1] * Chain [30]: 0 with precondition: [Out=0,V>=0,V2>=0] * Chain [multiple([29],[[multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])],[multiple([25],[[33]])],[33],[32],[31],[30]])]: 1*it(29)+1*it([31])+1*it([32])+1*it([33])+5*s(1)+1*s(3)+1*s(4)+1*s(5)+1*s(6)+1*s(7)+0 Such that:aux(11) =< V aux(12) =< V+1 aux(13) =< V/2+1/2 aux(14) =< V/3+V2/3+2/3 aux(15) =< V2+1 aux(16) =< V2/2+1/2 it(29) =< aux(11) it([31]) =< aux(11) it([31]) =< aux(12) it([32]) =< aux(12) it([33]) =< aux(12) it([31]) =< aux(13) it([33]) =< aux(15) s(1) =< aux(15) it([32]) =< aux(16) s(11) =< aux(15)*(1/2) s(9) =< aux(14)*(6/5) s(6) =< aux(13) s(5) =< aux(14) s(3) =< s(9) s(4) =< s(9) s(5) =< s(9) s(4) =< aux(15) s(5) =< aux(15) s(7) =< aux(15) s(7) =< s(11) s(5) =< s(1)*(1/3)+aux(14) s(3) =< s(1)*(1/5)+s(9) s(4) =< s(1)*(1/5)+s(9) s(5) =< s(1)*(1/5)+s(9) with precondition: [Out=0,V>=1,V2>=1] * Chain [multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])]: 1*it(26)+1*it(27)+1*it(28)+1*it([31])+1*it([32])+3*it([33])+0 Such that:aux(3) =< V/2+1/2 aux(4) =< V/3+V2/3 aux(5) =< 2/5*V+2/5*V2 aux(6) =< V2 aux(7) =< V2+1 aux(8) =< V2/2+1/2 it([31]) =< aux(3) it(28) =< aux(4) it(26) =< aux(5) it(27) =< aux(5) it(28) =< aux(5) it(27) =< aux(6) it(28) =< aux(6) it([32]) =< aux(6) it([33]) =< aux(7) it([32]) =< aux(8) it(28) =< it([33])*(1/3)+aux(4) it(26) =< it([33])*(1/5)+aux(5) it(27) =< it([33])*(1/5)+aux(5) it(28) =< it([33])*(1/5)+aux(5) with precondition: [Out=1,V>=1,V2>=1,V+V2>=3] * Chain [multiple([25],[[33]])]: 1*it(25)+1*it([33])+0 Such that:it(25) =< V2 it([33]) =< V2+1 with precondition: [Out=2,V=V2,V>=1] #### Cost of chains of first(V,Out): * Chain [35]: 0 with precondition: [Out=0,V>=0] * Chain [34]: 1 with precondition: [Out>=0,V>=Out+1] #### Cost of chains of second(V,Out): * Chain [37]: 0 with precondition: [Out=0,V>=0] * Chain [36]: 1 with precondition: [Out>=0,V>=Out+1] #### Cost of chains of start(V,V2,V6): * Chain [44]: 2*s(33)+1*s(34)+1*s(42)+1*s(43)+1*s(44)+8*s(45)+2*s(48)+1*s(49)+1*s(50)+1*s(51)+1*s(52)+1*s(60)+1*s(61)+1*s(62)+1*s(63)+1 Such that:s(54) =< V/3+V2/3 s(38) =< V/3+V2/3+2/3 s(55) =< 2/5*V+2/5*V2 s(56) =< V2 aux(17) =< V aux(18) =< V+1 aux(19) =< V/2+1/2 aux(20) =< V2+1 aux(21) =< V2/2+1/2 s(33) =< aux(17) s(34) =< aux(18) s(42) =< aux(17) s(42) =< aux(18) s(43) =< aux(18) s(44) =< aux(18) s(42) =< aux(19) s(44) =< aux(20) s(45) =< aux(20) s(43) =< aux(21) s(46) =< aux(20)*(1/2) s(47) =< s(38)*(6/5) s(48) =< aux(19) s(49) =< s(38) s(50) =< s(47) s(51) =< s(47) s(49) =< s(47) s(51) =< aux(20) s(49) =< aux(20) s(52) =< aux(20) s(52) =< s(46) s(49) =< s(45)*(1/3)+s(38) s(50) =< s(45)*(1/5)+s(47) s(51) =< s(45)*(1/5)+s(47) s(49) =< s(45)*(1/5)+s(47) s(60) =< s(54) s(61) =< s(55) s(62) =< s(55) s(60) =< s(55) s(62) =< s(56) s(60) =< s(56) s(63) =< s(56) s(63) =< aux(21) s(60) =< s(45)*(1/3)+s(54) s(61) =< s(45)*(1/5)+s(55) s(62) =< s(45)*(1/5)+s(55) s(60) =< s(45)*(1/5)+s(55) with precondition: [V>=0] * Chain [43]: 1 with precondition: [V=0,V2>=1] * Chain [42]: 1*s(65)+1*s(66)+0 Such that:s(65) =< V2 s(66) =< V2+1 with precondition: [V=V2,V>=1] * Chain [41]: 0 with precondition: [V=1,V2=2] * Chain [40]: 0 with precondition: [V=2,V2=1] * Chain [39]: 0 with precondition: [V=2,V2=2] * Chain [38]: 1 with precondition: [V2=0,V>=1] Closed-form bounds of start(V,V2,V6): ------------------------------------- * Chain [44] with precondition: [V>=0] - Upper bound: 3*V+1+nat(V2)+(3*V+3)+nat(V2+1)*9+nat(2/5*V+2/5*V2)*2+17/5*nat(V/3+V2/3+2/3)+(V+1)+nat(V/3+V2/3) - Complexity: n * Chain [43] with precondition: [V=0,V2>=1] - Upper bound: 1 - Complexity: constant * Chain [42] with precondition: [V=V2,V>=1] - Upper bound: 2*V2+1 - Complexity: n * Chain [41] with precondition: [V=1,V2=2] - Upper bound: 0 - Complexity: constant * Chain [40] with precondition: [V=2,V2=1] - Upper bound: 0 - Complexity: constant * Chain [39] with precondition: [V=2,V2=2] - Upper bound: 0 - Complexity: constant * Chain [38] with precondition: [V2=0,V>=1] - Upper bound: 1 - Complexity: constant ### Maximum cost of start(V,V2,V6): max([1,6*V+4+nat(V2+1)*8+nat(2/5*V+2/5*V2)*2+17/5*nat(V/3+V2/3+2/3)+(V+1)+nat(V/3+V2/3)+(nat(V2+1)+nat(V2))]) Asymptotic class: n * Total analysis performed in 560 ms. ---------------------------------------- (12) BOUNDS(1, n^1) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) a(Z, y, z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (15) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence a(C(x1, x2), y, z) ->^+ C(a(x1, y, z), a(x2, y, y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x1 / C(x1, x2)]. The result substitution is [ ]. ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) a(Z, y, z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) a(Z, y, z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST