/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (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), 0 ms] (10) CpxRNTS (11) CompleteCoflocoProof [FINISHED, 1239 ms] (12) BOUNDS(1, n^2) (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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. The duplicating contexts are: f(true, [], y, z) f(true, x, [], z) f(true, x, y, []) The defined contexts are: f([], x1, x2, s(x3)) gt(x0, []) f([], x1, s(x2), x3) As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) S is empty. 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^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) [1] f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) [1] f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] The TRS has the following type information: f :: true:false -> s:0 -> s:0 -> s:0 -> f true :: true:false gt :: s:0 -> s:0 -> true:false plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 false :: true:false 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: f(v0, v1, v2, v3) -> null_f [0] And the following fresh constants: null_f ---------------------------------------- (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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) [1] f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] gt(0, v) -> false [1] gt(s(u), 0) -> true [1] gt(s(u), s(v)) -> gt(u, v) [1] f(v0, v1, v2, v3) -> null_f [0] The TRS has the following type information: f :: true:false -> s:0 -> s:0 -> s:0 -> null_f true :: true:false gt :: s:0 -> s:0 -> true:false plus :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 false :: true:false null_f :: null_f 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: true => 1 0 => 0 false => 0 null_f => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: f(z', z'', z1, z2) -{ 1 }-> f(gt(x, plus(y, z)), x, y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 f(z', z'', z1, z2) -{ 1 }-> f(gt(x, plus(y, z)), x, 1 + y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 f(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 gt(z', z'') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + u, z'' = 1 + v, u >= 0 gt(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + u, u >= 0 gt(z', z'') -{ 1 }-> 0 :|: z'' = v, v >= 0, z' = 0 plus(z', z'') -{ 1 }-> n :|: z'' = 0, n >= 0, z' = n plus(z', z'') -{ 1 }-> 1 + plus(n, m) :|: n >= 0, z'' = 1 + m, z' = n, m >= 0 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, V1, V6, V2),0,[f(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]). eq(start(V, V1, V6, V2),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V6, V2),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]). eq(f(V, V1, V6, V2, Out),1,[plus(V3, V5, Ret01),gt(V4, Ret01, Ret0),f(Ret0, V4, 1 + V3, V5, Ret)],[Out = Ret,V6 = V3,V5 >= 0,V2 = V5,V4 >= 0,V3 >= 0,V1 = V4,V = 1]). eq(f(V, V1, V6, V2, Out),1,[plus(V9, V8, Ret011),gt(V7, Ret011, Ret02),f(Ret02, V7, V9, 1 + V8, Ret1)],[Out = Ret1,V6 = V9,V8 >= 0,V2 = V8,V7 >= 0,V9 >= 0,V1 = V7,V = 1]). eq(plus(V, V1, Out),1,[],[Out = V10,V1 = 0,V10 >= 0,V = V10]). eq(plus(V, V1, Out),1,[plus(V11, V12, Ret11)],[Out = 1 + Ret11,V11 >= 0,V1 = 1 + V12,V = V11,V12 >= 0]). eq(gt(V, V1, Out),1,[],[Out = 0,V1 = V13,V13 >= 0,V = 0]). eq(gt(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V14,V14 >= 0]). eq(gt(V, V1, Out),1,[gt(V15, V16, Ret2)],[Out = Ret2,V16 >= 0,V = 1 + V15,V1 = 1 + V16,V15 >= 0]). eq(f(V, V1, V6, V2, Out),0,[],[Out = 0,V2 = V19,V18 >= 0,V6 = V20,V17 >= 0,V1 = V17,V20 >= 0,V19 >= 0,V = V18]). input_output_vars(f(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]). input_output_vars(plus(V,V1,Out),[V,V1],[Out]). input_output_vars(gt(V,V1,Out),[V,V1],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [gt/3] 1. recursive : [plus/3] 2. recursive : [f/5] 3. non_recursive : [start/4] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into gt/3 1. SCC is partially evaluated into plus/3 2. SCC is partially evaluated into f/5 3. SCC is partially evaluated into start/4 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations gt/3 * CE 11 is refined into CE [12] * CE 10 is refined into CE [13] * CE 9 is refined into CE [14] ### Cost equations --> "Loop" of gt/3 * CEs [13] --> Loop 10 * CEs [14] --> Loop 11 * CEs [12] --> Loop 12 ### Ranking functions of CR gt(V,V1,Out) * RF of phase [12]: [V,V1] #### Partial ranking functions of CR gt(V,V1,Out) * Partial RF of phase [12]: - RF of loop [12:1]: V V1 ### Specialization of cost equations plus/3 * CE 8 is refined into CE [15] * CE 7 is refined into CE [16] ### Cost equations --> "Loop" of plus/3 * CEs [16] --> Loop 13 * CEs [15] --> Loop 14 ### Ranking functions of CR plus(V,V1,Out) * RF of phase [14]: [V1] #### Partial ranking functions of CR plus(V,V1,Out) * Partial RF of phase [14]: - RF of loop [14:1]: V1 ### Specialization of cost equations f/5 * CE 6 is refined into CE [17] * CE 5 is refined into CE [18,19,20,21,22,23,24] * CE 4 is refined into CE [25,26,27,28,29,30,31] ### Cost equations --> "Loop" of f/5 * CEs [24] --> Loop 15 * CEs [31] --> Loop 16 * CEs [23] --> Loop 17 * CEs [30] --> Loop 18 * CEs [21] --> Loop 19 * CEs [28] --> Loop 20 * CEs [20] --> Loop 21 * CEs [27] --> Loop 22 * CEs [26] --> Loop 23 * CEs [19] --> Loop 24 * CEs [22] --> Loop 25 * CEs [29] --> Loop 26 * CEs [18] --> Loop 27 * CEs [25] --> Loop 28 * CEs [17] --> Loop 29 ### Ranking functions of CR f(V,V1,V6,V2,Out) * RF of phase [15,16]: [V1-V6-V2] * RF of phase [20]: [V1-V6] #### Partial ranking functions of CR f(V,V1,V6,V2,Out) * Partial RF of phase [15,16]: - RF of loop [15:1]: V1-V2 - RF of loop [15:1,16:1]: V1-V6-V2 - RF of loop [16:1]: V1-V6-1 * Partial RF of phase [20]: - RF of loop [20:1]: V1-V6 ### Specialization of cost equations start/4 * CE 1 is refined into CE [32,33,34,35,36,37,38,39] * CE 2 is refined into CE [40,41] * CE 3 is refined into CE [42,43,44,45] ### Cost equations --> "Loop" of start/4 * CEs [39] --> Loop 30 * CEs [38] --> Loop 31 * CEs [36] --> Loop 32 * CEs [37,41,44,45] --> Loop 33 * CEs [32,34,35] --> Loop 34 * CEs [33,40,43] --> Loop 35 * CEs [42] --> Loop 36 ### Ranking functions of CR start(V,V1,V6,V2) #### Partial ranking functions of CR start(V,V1,V6,V2) Computing Bounds ===================================== #### Cost of chains of gt(V,V1,Out): * Chain [[12],11]: 1*it(12)+1 Such that:it(12) =< V with precondition: [Out=0,V>=1,V1>=V] * Chain [[12],10]: 1*it(12)+1 Such that:it(12) =< V1 with precondition: [Out=1,V1>=1,V>=V1+1] * Chain [11]: 1 with precondition: [V=0,Out=0,V1>=0] * Chain [10]: 1 with precondition: [V1=0,Out=1,V>=1] #### Cost of chains of plus(V,V1,Out): * Chain [[14],13]: 1*it(14)+1 Such that:it(14) =< V1 with precondition: [V+V1=Out,V>=0,V1>=1] * Chain [13]: 1 with precondition: [V1=0,V=Out,V>=0] #### Cost of chains of f(V,V1,V6,V2,Out): * Chain [[20],29]: 3*it(20)+1*s(3)+0 Such that:aux(1) =< V1 it(20) =< V1-V6 s(3) =< it(20)*aux(1) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+1] * Chain [[20],22,29]: 3*it(20)+1*s(3)+1*s(4)+3 Such that:it(20) =< V1-V6 aux(2) =< V1 s(4) =< aux(2) s(3) =< it(20)*aux(2) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+1] * Chain [[20],21,29]: 3*it(20)+1*s(3)+1*s(5)+3 Such that:it(20) =< V1-V6 aux(3) =< V1 s(5) =< aux(3) s(3) =< it(20)*aux(3) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+1] * Chain [[20],19,[15,16],29]: 7*it(15)+3*it(20)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3 Such that:it(20) =< V1-V6 aux(13) =< V1 it(15) =< aux(13) aux(7) =< aux(13)-1 aux(6) =< aux(13) s(14) =< it(15)*aux(13) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) s(3) =< it(20)*aux(13) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+3] * Chain [[20],19,[15,16],18,29]: 9*it(15)+3*it(20)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+6 Such that:it(20) =< V1-V6 aux(18) =< V1 it(15) =< aux(18) aux(7) =< aux(18)-1 aux(6) =< aux(18) s(14) =< it(15)*aux(18) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) s(3) =< it(20)*aux(18) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+3] * Chain [[20],19,[15,16],17,29]: 5*it(15)+3*it(16)+3*it(20)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+1*s(21)+6 Such that:aux(20) =< V1+1 it(20) =< V1-V6+1 aux(23) =< V1 it(15) =< aux(23) it(16) =< aux(20) s(21) =< aux(20) it(16) =< aux(23) aux(7) =< aux(23)-1 aux(6) =< aux(23) s(14) =< it(15)*aux(23) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) s(3) =< it(20)*aux(23) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+3] * Chain [[20],19,29]: 3*it(20)+1*s(3)+1*s(18)+3 Such that:it(20) =< V1-V6 aux(24) =< V1 s(18) =< aux(24) s(3) =< it(20)*aux(24) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [[20],19,18,29]: 3*it(20)+1*s(3)+2*s(18)+1*s(19)+6 Such that:s(19) =< 1 it(20) =< V1-V6 aux(25) =< V1 s(18) =< aux(25) s(3) =< it(20)*aux(25) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [[20],19,17,29]: 3*it(20)+1*s(3)+2*s(18)+1*s(21)+6 Such that:s(21) =< 2 it(20) =< V1-V6 aux(26) =< V1 s(18) =< aux(26) s(3) =< it(20)*aux(26) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [[15,16],29]: 3*it(15)+3*it(16)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+0 Such that:aux(4) =< V1 aux(8) =< V1-V6 aux(11) =< V1-V6-V2 it(15) =< aux(11) it(16) =< aux(11) it(16) =< aux(8) aux(7) =< aux(8)-1 aux(6) =< aux(4) s(14) =< it(15)*aux(8) s(15) =< it(15)*aux(4) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,Out=0,V6>=0,V2>=1,V1>=V2+V6+1] * Chain [[15,16],18,29]: 3*it(15)+3*it(16)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+1*s(19)+1*s(20)+3 Such that:aux(14) =< V1 aux(15) =< V1-V6 aux(16) =< V1-V6-V2 s(20) =< aux(14) it(16) =< aux(15) s(19) =< aux(15) it(15) =< aux(16) it(16) =< aux(16) aux(7) =< aux(15)-1 aux(6) =< aux(14) s(14) =< it(15)*aux(15) s(15) =< it(15)*aux(14) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,Out=0,V6>=0,V2>=1,V1>=V2+V6+1] * Chain [[15,16],17,29]: 3*it(15)+3*it(16)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+1*s(21)+1*s(22)+3 Such that:aux(8) =< V1-V6 aux(19) =< V1 aux(20) =< V1-V6+1 aux(21) =< V1-V6-V2 s(22) =< aux(19) it(16) =< aux(20) s(21) =< aux(20) it(15) =< aux(21) it(16) =< aux(8) it(16) =< aux(21) aux(7) =< aux(8)-1 aux(6) =< aux(19) s(14) =< it(15)*aux(8) s(15) =< it(15)*aux(19) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,Out=0,V6>=0,V2>=1,V1>=V2+V6+1] * Chain [29]: 0 with precondition: [Out=0,V>=0,V1>=0,V6>=0,V2>=0] * Chain [28,29]: 3 with precondition: [V=1,V1=0,V2=0,Out=0,V6>=0] * Chain [27,29]: 3 with precondition: [V=1,V1=0,V2=0,Out=0,V6>=0] * Chain [26,29]: 1*s(23)+3 Such that:s(23) =< V2 with precondition: [V=1,V1=0,Out=0,V6>=0,V2>=1] * Chain [25,29]: 1*s(24)+3 Such that:s(24) =< V2+1 with precondition: [V=1,V1=0,Out=0,V6>=0,V2>=1] * Chain [24,[15,16],29]: 6*it(15)+2*s(14)+1*s(16)+1*s(17)+3 Such that:aux(27) =< V1 it(15) =< aux(27) aux(7) =< aux(27)-1 aux(6) =< aux(27) s(14) =< it(15)*aux(27) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [24,[15,16],18,29]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+6 Such that:aux(28) =< V1 it(15) =< aux(28) aux(7) =< aux(28)-1 aux(6) =< aux(28) s(14) =< it(15)*aux(28) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [24,[15,16],17,29]: 4*it(15)+3*it(16)+2*s(14)+1*s(16)+1*s(17)+1*s(21)+6 Such that:aux(20) =< V1+1 aux(29) =< V1 it(15) =< aux(29) it(16) =< aux(20) s(21) =< aux(20) it(16) =< aux(29) aux(7) =< aux(29)-1 aux(6) =< aux(29) s(14) =< it(15)*aux(29) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [24,29]: 3 with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] * Chain [24,18,29]: 2*s(19)+6 Such that:aux(30) =< 1 s(19) =< aux(30) with precondition: [V=1,V1=1,V6=0,V2=0,Out=0] * Chain [24,17,29]: 1*s(21)+1*s(22)+6 Such that:s(22) =< 1 s(21) =< 2 with precondition: [V=1,V1=1,V6=0,V2=0,Out=0] * Chain [23,[20],29]: 3*it(20)+1*s(3)+3 Such that:aux(31) =< V1 it(20) =< aux(31) s(3) =< it(20)*aux(31) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [23,[20],22,29]: 4*it(20)+1*s(3)+6 Such that:aux(32) =< V1 it(20) =< aux(32) s(3) =< it(20)*aux(32) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [23,[20],21,29]: 4*it(20)+1*s(3)+6 Such that:aux(33) =< V1 it(20) =< aux(33) s(3) =< it(20)*aux(33) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [23,[20],19,[15,16],29]: 10*it(15)+3*s(3)+1*s(16)+1*s(17)+6 Such that:aux(34) =< V1 it(15) =< aux(34) aux(7) =< aux(34)-1 aux(6) =< aux(34) s(3) =< it(15)*aux(34) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=4] * Chain [23,[20],19,[15,16],18,29]: 12*it(15)+3*s(3)+1*s(16)+1*s(17)+9 Such that:aux(35) =< V1 it(15) =< aux(35) aux(7) =< aux(35)-1 aux(6) =< aux(35) s(3) =< it(15)*aux(35) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=4] * Chain [23,[20],19,[15,16],17,29]: 8*it(15)+3*it(16)+3*s(3)+1*s(16)+1*s(17)+1*s(21)+9 Such that:aux(20) =< V1+1 aux(36) =< V1 it(15) =< aux(36) it(16) =< aux(20) s(21) =< aux(20) it(16) =< aux(36) aux(7) =< aux(36)-1 aux(6) =< aux(36) s(3) =< it(15)*aux(36) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=4] * Chain [23,[20],19,29]: 4*it(20)+1*s(3)+6 Such that:aux(37) =< V1 it(20) =< aux(37) s(3) =< it(20)*aux(37) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,[20],19,18,29]: 5*it(20)+1*s(3)+1*s(19)+9 Such that:s(19) =< 1 aux(38) =< V1 it(20) =< aux(38) s(3) =< it(20)*aux(38) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,[20],19,17,29]: 5*it(20)+1*s(3)+1*s(21)+9 Such that:s(21) =< 2 aux(39) =< V1 it(20) =< aux(39) s(3) =< it(20)*aux(39) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,29]: 3 with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] * Chain [23,22,29]: 1*s(4)+6 Such that:s(4) =< 1 with precondition: [V=1,V1=1,V6=0,V2=0,Out=0] * Chain [23,21,29]: 1*s(5)+6 Such that:s(5) =< 1 with precondition: [V=1,V1=1,V6=0,V2=0,Out=0] * Chain [23,19,[15,16],29]: 6*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+6 Such that:s(18) =< 1 aux(40) =< V1 it(15) =< aux(40) aux(7) =< aux(40)-1 aux(6) =< aux(40) s(14) =< it(15)*aux(40) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,19,[15,16],18,29]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+9 Such that:s(18) =< 1 aux(41) =< V1 it(15) =< aux(41) aux(7) =< aux(41)-1 aux(6) =< aux(41) s(14) =< it(15)*aux(41) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,19,[15,16],17,29]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+9 Such that:s(18) =< 1 aux(42) =< V1 it(15) =< aux(42) aux(7) =< aux(42)-1 aux(6) =< aux(42) s(14) =< it(15)*aux(42) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V6=0,V2=0,Out=0,V1>=3] * Chain [23,19,29]: 1*s(18)+6 Such that:s(18) =< 1 with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] * Chain [23,19,18,29]: 2*s(18)+1*s(20)+9 Such that:s(20) =< 2 aux(43) =< 1 s(18) =< aux(43) with precondition: [V=1,V1=2,V6=0,V2=0,Out=0] * Chain [23,19,17,29]: 1*s(18)+2*s(21)+9 Such that:s(18) =< 1 aux(44) =< 2 s(21) =< aux(44) with precondition: [V=1,V1=2,V6=0,V2=0,Out=0] * Chain [22,29]: 1*s(4)+3 Such that:s(4) =< V1 with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=V1] * Chain [21,29]: 1*s(5)+3 Such that:s(5) =< V1 with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=V1] * Chain [19,[15,16],29]: 6*it(15)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+1*s(18)+3 Such that:aux(4) =< V1 s(18) =< V6 aux(12) =< V1-V6 it(15) =< aux(12) aux(7) =< aux(12)-1 aux(6) =< aux(4) s(14) =< it(15)*aux(12) s(15) =< it(15)*aux(4) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [19,[15,16],18,29]: 7*it(15)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+1*s(18)+1*s(20)+6 Such that:aux(14) =< V1 s(18) =< V6 aux(17) =< V1-V6 s(20) =< aux(14) it(15) =< aux(17) aux(7) =< aux(17)-1 aux(6) =< aux(14) s(14) =< it(15)*aux(17) s(15) =< it(15)*aux(14) s(16) =< it(15)*aux(7) s(17) =< it(15)*aux(6) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [19,[15,16],17,29]: 3*it(15)+3*it(16)+1*s(14)+1*s(15)+1*s(16)+1*s(17)+1*s(18)+1*s(21)+1*s(22)+6 Such that:aux(19) =< V1 aux(20) =< V1-V6+1 s(18) =< V6 aux(22) =< V1-V6 s(22) =< aux(19) it(16) =< aux(20) s(21) =< aux(20) it(15) =< aux(22) it(16) =< aux(22) aux(7) =< aux(22)-1 aux(6) =< aux(19) s(14) =< it(15)*aux(22) s(15) =< it(15)*aux(19) s(16) =< it(16)*aux(7) s(17) =< it(16)*aux(6) with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+2] * Chain [19,29]: 1*s(18)+3 Such that:s(18) =< V6 with precondition: [V=1,V2=0,Out=0,V6>=1,V1>=V6+1] * Chain [19,18,29]: 1*s(18)+1*s(19)+1*s(20)+6 Such that:s(19) =< 1 s(18) =< V6 s(20) =< V6+1 with precondition: [V=1,V2=0,Out=0,V1=V6+1,V1>=2] * Chain [19,17,29]: 1*s(18)+1*s(21)+1*s(22)+6 Such that:s(21) =< 2 s(18) =< V6 s(22) =< V6+1 with precondition: [V=1,V2=0,Out=0,V1=V6+1,V1>=2] * Chain [18,29]: 1*s(19)+1*s(20)+3 Such that:s(20) =< V1 s(19) =< V2 with precondition: [V=1,Out=0,V1>=1,V6>=0,V2>=1,V2+V6>=V1] * Chain [17,29]: 1*s(21)+1*s(22)+3 Such that:s(22) =< V1 s(21) =< V2+1 with precondition: [V=1,Out=0,V1>=1,V6>=0,V2>=1,V2+V6>=V1] #### Cost of chains of start(V,V1,V6,V2): * Chain [36]: 1 with precondition: [V=0,V1>=0] * Chain [35]: 1*s(273)+1*s(274)+3 Such that:s(273) =< V2 s(274) =< V2+1 with precondition: [V1=0,V>=0] * Chain [34]: 10*s(279)+2*s(280)+95*s(281)+27*s(284)+7*s(285)+7*s(286)+6*s(287)+2*s(288)+2*s(289)+2*s(290)+18 Such that:s(275) =< 1 s(276) =< 2 s(277) =< V1 s(278) =< V1+1 s(279) =< s(275) s(280) =< s(276) s(281) =< s(277) s(282) =< s(277)-1 s(283) =< s(277) s(284) =< s(281)*s(277) s(285) =< s(281)*s(282) s(286) =< s(281)*s(283) s(287) =< s(278) s(288) =< s(278) s(287) =< s(277) s(289) =< s(287)*s(282) s(290) =< s(287)*s(283) with precondition: [V>=0,V1>=0,V6>=0,V2>=0] * Chain [33]: 4*s(292)+1*s(294)+3 Such that:s(294) =< V aux(63) =< V1 s(292) =< aux(63) with precondition: [V>=0,V1>=1] * Chain [32]: 2*s(304)+2*s(305)+40*s(306)+4*s(307)+6*s(308)+2*s(309)+30*s(310)+11*s(311)+3*s(312)+3*s(315)+1*s(316)+1*s(317)+2*s(318)+2*s(319)+3*s(320)+1*s(321)+6*s(323)+1*s(324)+1*s(325)+1*s(326)+2*s(327)+2*s(328)+6 Such that:s(297) =< 1 s(298) =< 2 s(299) =< V1 s(296) =< V1+1 s(300) =< V1-V6 s(301) =< V1-V6+1 s(302) =< V6 s(303) =< V6+1 s(304) =< s(297) s(305) =< s(298) s(306) =< s(300) s(307) =< s(301) s(308) =< s(302) s(309) =< s(303) s(310) =< s(299) s(311) =< s(306)*s(299) s(312) =< s(301) s(312) =< s(300) s(313) =< s(300)-1 s(314) =< s(299) s(315) =< s(306)*s(300) s(316) =< s(312)*s(313) s(317) =< s(312)*s(314) s(318) =< s(306)*s(313) s(319) =< s(306)*s(314) s(320) =< s(296) s(321) =< s(296) s(320) =< s(299) s(322) =< s(299)-1 s(323) =< s(310)*s(299) s(324) =< s(320)*s(322) s(325) =< s(320)*s(314) s(326) =< s(307)*s(299) s(327) =< s(310)*s(322) s(328) =< s(310)*s(314) with precondition: [V=1,V2=0,V6>=1,V1>=V6+1] * Chain [31]: 1*s(329)+1*s(330)+2*s(332)+3 Such that:s(331) =< V1 s(329) =< V2 s(330) =< V2+1 s(332) =< s(331) with precondition: [V=1,V1>=1,V6>=0,V2>=1,V2+V6>=V1] * Chain [30]: 2*s(337)+6*s(338)+1*s(339)+9*s(340)+3*s(343)+3*s(344)+2*s(345)+2*s(346)+3*s(347)+1*s(348)+1*s(349)+1*s(350)+3 Such that:s(334) =< V1 s(335) =< V1-V6 s(333) =< V1-V6+1 s(336) =< V1-V6-V2 s(337) =< s(334) s(338) =< s(335) s(339) =< s(335) s(340) =< s(336) s(338) =< s(336) s(341) =< s(335)-1 s(342) =< s(334) s(343) =< s(340)*s(335) s(344) =< s(340)*s(334) s(345) =< s(338)*s(341) s(346) =< s(338)*s(342) s(347) =< s(333) s(348) =< s(333) s(347) =< s(335) s(347) =< s(336) s(349) =< s(347)*s(341) s(350) =< s(347)*s(342) with precondition: [V=1,V6>=0,V2>=1,V1>=V2+V6+1] Closed-form bounds of start(V,V1,V6,V2): ------------------------------------- * Chain [36] with precondition: [V=0,V1>=0] - Upper bound: 1 - Complexity: constant * Chain [35] with precondition: [V1=0,V>=0] - Upper bound: nat(V2)+3+nat(V2+1) - Complexity: n * Chain [34] with precondition: [V>=0,V1>=0,V6>=0,V2>=0] - Upper bound: 95*V1+32+34*V1*V1+7*V1*nat(V1-1)+(V1+1)*(2*V1)+(V1+1)*(nat(V1-1)*2)+(8*V1+8) - Complexity: n^2 * Chain [33] with precondition: [V>=0,V1>=1] - Upper bound: V+4*V1+3 - Complexity: n * Chain [32] with precondition: [V=1,V2=0,V6>=1,V1>=V6+1] - Upper bound: 40*V1-40*V6+(7*V1-7*V6+7+(30*V1+12+8*V1*V1+(V1-1)*(2*V1)+(V1+1)*V1+(V1-V6+1)*(2*V1)+(V1-V6)*(13*V1)+6*V6+(V1-1)*(V1+1)+(V1-V6-1)*(V1-V6+1)+(2*V1-2*V6-2)*(V1-V6)+(4*V1+4)+(2*V6+2)))+(3*V1-3*V6)*(V1-V6) - Complexity: n^2 * Chain [31] with precondition: [V=1,V1>=1,V6>=0,V2>=1,V6+V2>=V1] - Upper bound: 2*V1+2*V2+4 - Complexity: n * Chain [30] with precondition: [V=1,V6>=0,V2>=1,V1>=V6+V2+1] - Upper bound: 9*V1-9*V6-9*V2+(7*V1-7*V6+(4*V1-4*V6+4+(2*V1+3+(V1-V6+1)*V1+(V1-V6)*(2*V1)+(V1-V6-V2)*(3*V1)+(V1-V6-1)*(V1-V6+1)+(2*V1-2*V6-2)*(V1-V6)))+(V1-V6-V2)*(3*V1-3*V6)) - Complexity: n^2 ### Maximum cost of start(V,V1,V6,V2): max([nat(V2)+2+nat(V2+1),2*V1+2+max([max([nat(V2+1)+nat(V2),2*V1*nat(V1-V6)+nat(V1-V6+1)*V1+3*V1*nat(V1-V6-V2)+nat(V1-V6+1)*nat(nat(V1-V6)+ -1)+nat(nat(V1-V6)+ -1)*2*nat(V1-V6)+nat(V1-V6+1)*4+nat(V1-V6)*7+nat(V1-V6)*3*nat(V1-V6-V2)+nat(V1-V6-V2)*9]),2*V1+max([V,26*V1+9+8*V1*V1+2*V1*nat(V1-1)+(V1+1)*V1+(V1+1)*nat(V1-1)+(4*V1+4)+max([65*V1+20+26*V1*V1+5*V1*nat(V1-1)+(V1+1)*V1+(V1+1)*nat(V1-1)+(4*V1+4),13*V1*nat(V1-V6)+2*V1*nat(V1-V6+1)+nat(V6)*6+nat(V1-V6+1)*nat(nat(V1-V6)+ -1)+nat(nat(V1-V6)+ -1)*2*nat(V1-V6)+nat(V6+1)*2+nat(V1-V6+1)*7+nat(V1-V6)*40+nat(V1-V6)*3*nat(V1-V6)])])])])+1 Asymptotic class: n^2 * Total analysis performed in 1120 ms. ---------------------------------------- (12) BOUNDS(1, n^2) ---------------------------------------- (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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) S is empty. Rewrite Strategy: FULL ---------------------------------------- (15) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence gt(s(u), s(v)) ->^+ gt(u, v) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [u / s(u), v / s(v)]. The result substitution is [ ]. ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) S is empty. Rewrite Strategy: FULL ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) S is empty. Rewrite Strategy: FULL