30.58/8.74 WORST_CASE(Omega(n^1), O(n^1)) 30.58/8.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.58/8.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.58/8.75 30.58/8.75 30.58/8.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.58/8.75 30.58/8.75 (0) CpxTRS 30.58/8.75 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 30.58/8.75 (2) CpxWeightedTrs 30.58/8.75 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 30.58/8.75 (4) CpxTypedWeightedTrs 30.58/8.75 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 30.58/8.75 (6) CpxTypedWeightedCompleteTrs 30.58/8.75 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 30.58/8.75 (8) CpxRNTS 30.58/8.75 (9) CompleteCoflocoProof [FINISHED, 954 ms] 30.58/8.75 (10) BOUNDS(1, n^1) 30.58/8.75 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 30.58/8.75 (12) TRS for Loop Detection 30.58/8.75 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 30.58/8.75 (14) BEST 30.58/8.75 (15) proven lower bound 30.58/8.75 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 30.58/8.75 (17) BOUNDS(n^1, INF) 30.58/8.75 (18) TRS for Loop Detection 30.58/8.75 30.58/8.75 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (0) 30.58/8.75 Obligation: 30.58/8.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.58/8.75 30.58/8.75 30.58/8.75 The TRS R consists of the following rules: 30.58/8.75 30.58/8.75 minus(X, s(Y)) -> pred(minus(X, Y)) 30.58/8.75 minus(X, 0) -> X 30.58/8.75 pred(s(X)) -> X 30.58/8.75 le(s(X), s(Y)) -> le(X, Y) 30.58/8.75 le(s(X), 0) -> false 30.58/8.75 le(0, Y) -> true 30.58/8.75 gcd(0, Y) -> 0 30.58/8.75 gcd(s(X), 0) -> s(X) 30.58/8.75 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 30.58/8.75 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 30.58/8.75 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 30.58/8.75 30.58/8.75 S is empty. 30.58/8.75 Rewrite Strategy: INNERMOST 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 30.58/8.75 Transformed relative TRS to weighted TRS 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (2) 30.58/8.75 Obligation: 30.58/8.75 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 30.58/8.75 30.58/8.75 30.58/8.75 The TRS R consists of the following rules: 30.58/8.75 30.58/8.75 minus(X, s(Y)) -> pred(minus(X, Y)) [1] 30.58/8.75 minus(X, 0) -> X [1] 30.58/8.75 pred(s(X)) -> X [1] 30.58/8.75 le(s(X), s(Y)) -> le(X, Y) [1] 30.58/8.75 le(s(X), 0) -> false [1] 30.58/8.75 le(0, Y) -> true [1] 30.58/8.75 gcd(0, Y) -> 0 [1] 30.58/8.75 gcd(s(X), 0) -> s(X) [1] 30.58/8.75 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1] 30.58/8.75 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1] 30.58/8.75 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1] 30.58/8.75 30.58/8.75 Rewrite Strategy: INNERMOST 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 30.58/8.75 Infered types. 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (4) 30.58/8.75 Obligation: 30.58/8.75 Runtime Complexity Weighted TRS with Types. 30.58/8.75 The TRS R consists of the following rules: 30.58/8.75 30.58/8.75 minus(X, s(Y)) -> pred(minus(X, Y)) [1] 30.58/8.75 minus(X, 0) -> X [1] 30.58/8.75 pred(s(X)) -> X [1] 30.58/8.75 le(s(X), s(Y)) -> le(X, Y) [1] 30.58/8.75 le(s(X), 0) -> false [1] 30.58/8.75 le(0, Y) -> true [1] 30.58/8.75 gcd(0, Y) -> 0 [1] 30.58/8.75 gcd(s(X), 0) -> s(X) [1] 30.58/8.75 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1] 30.58/8.75 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1] 30.58/8.75 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1] 30.58/8.75 30.58/8.75 The TRS has the following type information: 30.58/8.75 minus :: s:0 -> s:0 -> s:0 30.58/8.75 s :: s:0 -> s:0 30.58/8.75 pred :: s:0 -> s:0 30.58/8.75 0 :: s:0 30.58/8.75 le :: s:0 -> s:0 -> false:true 30.58/8.75 false :: false:true 30.58/8.75 true :: false:true 30.58/8.75 gcd :: s:0 -> s:0 -> s:0 30.58/8.75 if :: false:true -> s:0 -> s:0 -> s:0 30.58/8.75 30.58/8.75 Rewrite Strategy: INNERMOST 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (5) CompletionProof (UPPER BOUND(ID)) 30.58/8.75 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 30.58/8.75 30.58/8.75 pred(v0) -> null_pred [0] 30.58/8.75 if(v0, v1, v2) -> null_if [0] 30.58/8.75 minus(v0, v1) -> null_minus [0] 30.58/8.75 le(v0, v1) -> null_le [0] 30.58/8.75 gcd(v0, v1) -> null_gcd [0] 30.58/8.75 30.58/8.75 And the following fresh constants: null_pred, null_if, null_minus, null_le, null_gcd 30.58/8.75 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (6) 30.58/8.75 Obligation: 30.58/8.75 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 30.58/8.75 30.58/8.75 Runtime Complexity Weighted TRS with Types. 30.58/8.75 The TRS R consists of the following rules: 30.58/8.75 30.58/8.75 minus(X, s(Y)) -> pred(minus(X, Y)) [1] 30.58/8.75 minus(X, 0) -> X [1] 30.58/8.75 pred(s(X)) -> X [1] 30.58/8.75 le(s(X), s(Y)) -> le(X, Y) [1] 30.58/8.75 le(s(X), 0) -> false [1] 30.58/8.75 le(0, Y) -> true [1] 30.58/8.75 gcd(0, Y) -> 0 [1] 30.58/8.75 gcd(s(X), 0) -> s(X) [1] 30.58/8.75 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) [1] 30.58/8.75 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) [1] 30.58/8.75 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) [1] 30.58/8.75 pred(v0) -> null_pred [0] 30.58/8.75 if(v0, v1, v2) -> null_if [0] 30.58/8.75 minus(v0, v1) -> null_minus [0] 30.58/8.75 le(v0, v1) -> null_le [0] 30.58/8.75 gcd(v0, v1) -> null_gcd [0] 30.58/8.75 30.58/8.75 The TRS has the following type information: 30.58/8.75 minus :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 s :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 pred :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 0 :: s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 le :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> false:true:null_le 30.58/8.75 false :: false:true:null_le 30.58/8.75 true :: false:true:null_le 30.58/8.75 gcd :: s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 if :: false:true:null_le -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd -> s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 null_pred :: s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 null_if :: s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 null_minus :: s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 null_le :: false:true:null_le 30.58/8.75 null_gcd :: s:0:null_pred:null_if:null_minus:null_gcd 30.58/8.75 30.58/8.75 Rewrite Strategy: INNERMOST 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 30.58/8.75 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 30.58/8.75 The constant constructors are abstracted as follows: 30.58/8.75 30.58/8.75 0 => 0 30.58/8.75 false => 1 30.58/8.75 true => 2 30.58/8.75 null_pred => 0 30.58/8.75 null_if => 0 30.58/8.75 null_minus => 0 30.58/8.75 null_le => 0 30.58/8.75 null_gcd => 0 30.58/8.75 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (8) 30.58/8.75 Obligation: 30.58/8.75 Complexity RNTS consisting of the following rules: 30.58/8.75 30.58/8.75 gcd(z, z') -{ 1 }-> if(le(Y, X), 1 + X, 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 30.58/8.75 gcd(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0 30.58/8.75 gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 30.58/8.75 gcd(z, z') -{ 1 }-> 1 + X :|: z = 1 + X, X >= 0, z' = 0 30.58/8.75 if(z, z', z'') -{ 1 }-> gcd(minus(X, Y), 1 + Y) :|: z = 2, z'' = 1 + Y, Y >= 0, z' = 1 + X, X >= 0 30.58/8.75 if(z, z', z'') -{ 1 }-> gcd(minus(Y, X), 1 + X) :|: z'' = 1 + Y, Y >= 0, z = 1, z' = 1 + X, X >= 0 30.58/8.75 if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 30.58/8.75 le(z, z') -{ 1 }-> le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 30.58/8.75 le(z, z') -{ 1 }-> 2 :|: z' = Y, Y >= 0, z = 0 30.58/8.75 le(z, z') -{ 1 }-> 1 :|: z = 1 + X, X >= 0, z' = 0 30.58/8.75 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 30.58/8.75 minus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 30.58/8.75 minus(z, z') -{ 1 }-> pred(minus(X, Y)) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X 30.58/8.75 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 30.58/8.75 pred(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 30.58/8.75 pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 30.58/8.75 30.58/8.75 Only complete derivations are relevant for the runtime complexity. 30.58/8.75 30.58/8.75 ---------------------------------------- 30.58/8.75 30.58/8.75 (9) CompleteCoflocoProof (FINISHED) 30.58/8.75 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 30.58/8.75 30.58/8.75 eq(start(V1, V, V2),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 30.58/8.75 eq(start(V1, V, V2),0,[pred(V1, Out)],[V1 >= 0]). 30.58/8.75 eq(start(V1, V, V2),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). 30.58/8.75 eq(start(V1, V, V2),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]). 30.58/8.75 eq(start(V1, V, V2),0,[if(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]). 30.58/8.75 eq(minus(V1, V, Out),1,[minus(X1, Y1, Ret0),pred(Ret0, Ret)],[Out = Ret,Y1 >= 0,V = 1 + Y1,X1 >= 0,V1 = X1]). 30.58/8.75 eq(minus(V1, V, Out),1,[],[Out = X2,X2 >= 0,V1 = X2,V = 0]). 30.58/8.75 eq(pred(V1, Out),1,[],[Out = X3,V1 = 1 + X3,X3 >= 0]). 30.58/8.75 eq(le(V1, V, Out),1,[le(X4, Y2, Ret1)],[Out = Ret1,V1 = 1 + X4,Y2 >= 0,V = 1 + Y2,X4 >= 0]). 30.58/8.75 eq(le(V1, V, Out),1,[],[Out = 1,V1 = 1 + X5,X5 >= 0,V = 0]). 30.58/8.75 eq(le(V1, V, Out),1,[],[Out = 2,V = Y3,Y3 >= 0,V1 = 0]). 30.58/8.75 eq(gcd(V1, V, Out),1,[],[Out = 0,V = Y4,Y4 >= 0,V1 = 0]). 30.58/8.75 eq(gcd(V1, V, Out),1,[],[Out = 1 + X6,V1 = 1 + X6,X6 >= 0,V = 0]). 30.58/8.75 eq(gcd(V1, V, Out),1,[le(Y5, X7, Ret01),if(Ret01, 1 + X7, 1 + Y5, Ret2)],[Out = Ret2,V1 = 1 + X7,Y5 >= 0,V = 1 + Y5,X7 >= 0]). 30.58/8.75 eq(if(V1, V, V2, Out),1,[minus(X8, Y6, Ret02),gcd(Ret02, 1 + Y6, Ret3)],[Out = Ret3,V1 = 2,V2 = 1 + Y6,Y6 >= 0,V = 1 + X8,X8 >= 0]). 30.58/8.75 eq(if(V1, V, V2, Out),1,[minus(Y7, X9, Ret03),gcd(Ret03, 1 + X9, Ret4)],[Out = Ret4,V2 = 1 + Y7,Y7 >= 0,V1 = 1,V = 1 + X9,X9 >= 0]). 30.58/8.75 eq(pred(V1, Out),0,[],[Out = 0,V3 >= 0,V1 = V3]). 30.58/8.75 eq(if(V1, V, V2, Out),0,[],[Out = 0,V5 >= 0,V2 = V6,V4 >= 0,V1 = V5,V = V4,V6 >= 0]). 30.58/8.75 eq(minus(V1, V, Out),0,[],[Out = 0,V8 >= 0,V7 >= 0,V1 = V8,V = V7]). 30.58/8.75 eq(le(V1, V, Out),0,[],[Out = 0,V9 >= 0,V10 >= 0,V1 = V9,V = V10]). 30.58/8.75 eq(gcd(V1, V, Out),0,[],[Out = 0,V11 >= 0,V12 >= 0,V1 = V11,V = V12]). 30.58/8.75 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 30.58/8.75 input_output_vars(pred(V1,Out),[V1],[Out]). 30.58/8.75 input_output_vars(le(V1,V,Out),[V1,V],[Out]). 30.58/8.75 input_output_vars(gcd(V1,V,Out),[V1,V],[Out]). 30.58/8.75 input_output_vars(if(V1,V,V2,Out),[V1,V,V2],[Out]). 30.58/8.75 30.58/8.75 30.58/8.75 CoFloCo proof output: 30.58/8.75 Preprocessing Cost Relations 30.58/8.75 ===================================== 30.58/8.75 30.58/8.75 #### Computed strongly connected components 30.58/8.75 0. non_recursive : [pred/2] 30.58/8.75 1. recursive [non_tail] : [minus/3] 30.58/8.75 2. recursive : [le/3] 30.58/8.75 3. recursive : [gcd/3,if/4] 30.58/8.75 4. non_recursive : [start/3] 30.58/8.75 30.58/8.75 #### Obtained direct recursion through partial evaluation 30.58/8.75 0. SCC is partially evaluated into pred/2 30.58/8.75 1. SCC is partially evaluated into minus/3 30.58/8.75 2. SCC is partially evaluated into le/3 30.58/8.75 3. SCC is partially evaluated into gcd/3 30.58/8.75 4. SCC is partially evaluated into start/3 30.58/8.75 30.58/8.75 Control-Flow Refinement of Cost Relations 30.58/8.75 ===================================== 30.58/8.75 30.58/8.75 ### Specialization of cost equations pred/2 30.58/8.75 * CE 17 is refined into CE [23] 30.58/8.75 * CE 18 is refined into CE [24] 30.58/8.75 30.58/8.75 30.58/8.75 ### Cost equations --> "Loop" of pred/2 30.58/8.75 * CEs [23] --> Loop 16 30.58/8.75 * CEs [24] --> Loop 17 30.58/8.75 30.58/8.75 ### Ranking functions of CR pred(V1,Out) 30.58/8.75 30.58/8.75 #### Partial ranking functions of CR pred(V1,Out) 30.58/8.75 30.58/8.75 30.58/8.75 ### Specialization of cost equations minus/3 30.58/8.75 * CE 10 is refined into CE [25] 30.58/8.75 * CE 9 is refined into CE [26] 30.58/8.75 * CE 8 is refined into CE [27,28] 30.58/8.75 30.58/8.75 30.58/8.75 ### Cost equations --> "Loop" of minus/3 30.58/8.75 * CEs [28] --> Loop 18 30.58/8.75 * CEs [27] --> Loop 19 30.58/8.75 * CEs [25] --> Loop 20 30.58/8.75 * CEs [26] --> Loop 21 30.58/8.75 30.58/8.75 ### Ranking functions of CR minus(V1,V,Out) 30.58/8.75 * RF of phase [18]: [V] 30.58/8.75 * RF of phase [19]: [V] 30.58/8.75 30.58/8.75 #### Partial ranking functions of CR minus(V1,V,Out) 30.58/8.75 * Partial RF of phase [18]: 30.58/8.75 - RF of loop [18:1]: 30.58/8.75 V 30.58/8.75 * Partial RF of phase [19]: 30.58/8.75 - RF of loop [19:1]: 30.58/8.75 V 30.58/8.75 30.58/8.75 30.58/8.75 ### Specialization of cost equations le/3 30.58/8.75 * CE 22 is refined into CE [29] 30.58/8.75 * CE 20 is refined into CE [30] 30.58/8.75 * CE 21 is refined into CE [31] 30.58/8.75 * CE 19 is refined into CE [32] 30.58/8.75 30.58/8.75 30.58/8.75 ### Cost equations --> "Loop" of le/3 30.58/8.75 * CEs [32] --> Loop 22 30.58/8.75 * CEs [29] --> Loop 23 30.58/8.75 * CEs [30] --> Loop 24 30.58/8.75 * CEs [31] --> Loop 25 30.58/8.75 30.58/8.75 ### Ranking functions of CR le(V1,V,Out) 30.58/8.75 * RF of phase [22]: [V,V1] 30.58/8.75 30.58/8.75 #### Partial ranking functions of CR le(V1,V,Out) 30.58/8.75 * Partial RF of phase [22]: 30.58/8.75 - RF of loop [22:1]: 30.58/8.75 V 30.58/8.75 V1 30.58/8.75 30.58/8.75 30.58/8.75 ### Specialization of cost equations gcd/3 30.58/8.75 * CE 15 is refined into CE [33] 30.58/8.75 * CE 11 is refined into CE [34,35,36,37,38] 30.58/8.75 * CE 14 is refined into CE [39] 30.58/8.75 * CE 16 is refined into CE [40] 30.58/8.75 * CE 13 is refined into CE [41,42,43,44] 30.58/8.75 * CE 12 is refined into CE [45,46,47,48] 30.58/8.75 30.58/8.75 30.58/8.75 ### Cost equations --> "Loop" of gcd/3 30.58/8.75 * CEs [48] --> Loop 26 30.58/8.75 * CEs [44] --> Loop 27 30.58/8.75 * CEs [47] --> Loop 28 30.58/8.75 * CEs [43] --> Loop 29 30.58/8.75 * CEs [41] --> Loop 30 30.58/8.75 * CEs [42] --> Loop 31 30.58/8.75 * CEs [45] --> Loop 32 30.58/8.75 * CEs [46] --> Loop 33 30.58/8.75 * CEs [34] --> Loop 34 30.58/8.75 * CEs [33] --> Loop 35 30.58/8.75 * CEs [35] --> Loop 36 30.58/8.75 * CEs [36,37,38,39,40] --> Loop 37 30.58/8.75 30.58/8.75 ### Ranking functions of CR gcd(V1,V,Out) 30.58/8.75 * RF of phase [26,27]: [V1+V-3] 30.58/8.75 * RF of phase [30]: [V1] 30.58/8.75 30.58/8.75 #### Partial ranking functions of CR gcd(V1,V,Out) 30.58/8.75 * Partial RF of phase [26,27]: 30.58/8.75 - RF of loop [26:1]: 30.58/8.75 V-2 30.58/8.75 V1/2+V/2-2 30.58/8.75 - RF of loop [27:1]: 30.58/8.75 V1-1 depends on loops [26:1] 30.58/8.75 V1-V+1 depends on loops [26:1] 30.58/8.75 * Partial RF of phase [30]: 30.58/8.75 - RF of loop [30:1]: 30.58/8.75 V1 30.58/8.75 30.58/8.75 30.58/8.75 ### Specialization of cost equations start/3 30.58/8.75 * CE 3 is refined into CE [49,50,51,52] 30.58/8.75 * CE 1 is refined into CE [53] 30.58/8.75 * CE 2 is refined into CE [54,55,56,57] 30.58/8.75 * CE 4 is refined into CE [58,59,60] 30.58/8.75 * CE 5 is refined into CE [61,62] 30.58/8.75 * CE 6 is refined into CE [63,64,65,66,67] 30.58/8.75 * CE 7 is refined into CE [68,69,70] 30.58/8.75 30.58/8.75 30.58/8.75 ### Cost equations --> "Loop" of start/3 30.58/8.75 * CEs [70] --> Loop 38 30.58/8.75 * CEs [58,64,69] --> Loop 39 30.58/8.75 * CEs [49,50,51,52] --> Loop 40 30.58/8.75 * CEs [54,55,56,57] --> Loop 41 30.58/8.75 * CEs [53,59,60,61,62,63,65,66,67,68] --> Loop 42 30.58/8.75 30.58/8.75 ### Ranking functions of CR start(V1,V,V2) 30.58/8.75 30.58/8.75 #### Partial ranking functions of CR start(V1,V,V2) 30.58/8.75 30.58/8.75 30.58/8.75 Computing Bounds 30.58/8.75 ===================================== 30.58/8.75 30.58/8.75 #### Cost of chains of pred(V1,Out): 30.58/8.75 * Chain [17]: 0 30.58/8.75 with precondition: [Out=0,V1>=0] 30.58/8.75 30.58/8.75 * Chain [16]: 1 30.58/8.75 with precondition: [V1=Out+1,V1>=1] 30.58/8.75 30.58/8.75 30.58/8.75 #### Cost of chains of minus(V1,V,Out): 30.58/8.75 * Chain [[19],[18],21]: 3*it(18)+1 30.58/8.75 Such that:aux(1) =< V 30.58/8.75 it(18) =< aux(1) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=1,V>=2] 30.58/8.75 30.58/8.75 * Chain [[19],21]: 1*it(19)+1 30.58/8.75 Such that:it(19) =< V 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=0,V>=1] 30.58/8.75 30.58/8.75 * Chain [[19],20]: 1*it(19)+0 30.58/8.75 Such that:it(19) =< V 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=0,V>=1] 30.58/8.75 30.58/8.75 * Chain [[18],21]: 2*it(18)+1 30.58/8.75 Such that:it(18) =< V 30.58/8.75 30.58/8.75 with precondition: [V1=Out+V,V>=1,V1>=V] 30.58/8.75 30.58/8.75 * Chain [21]: 1 30.58/8.75 with precondition: [V=0,V1=Out,V1>=0] 30.58/8.75 30.58/8.75 * Chain [20]: 0 30.58/8.75 with precondition: [Out=0,V1>=0,V>=0] 30.58/8.75 30.58/8.75 30.58/8.75 #### Cost of chains of le(V1,V,Out): 30.58/8.75 * Chain [[22],25]: 1*it(22)+1 30.58/8.75 Such that:it(22) =< V1 30.58/8.75 30.58/8.75 with precondition: [Out=2,V1>=1,V>=V1] 30.58/8.75 30.58/8.75 * Chain [[22],24]: 1*it(22)+1 30.58/8.75 Such that:it(22) =< V 30.58/8.75 30.58/8.75 with precondition: [Out=1,V>=1,V1>=V+1] 30.58/8.75 30.58/8.75 * Chain [[22],23]: 1*it(22)+0 30.58/8.75 Such that:it(22) =< V 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=1,V>=1] 30.58/8.75 30.58/8.75 * Chain [25]: 1 30.58/8.75 with precondition: [V1=0,Out=2,V>=0] 30.58/8.75 30.58/8.75 * Chain [24]: 1 30.58/8.75 with precondition: [V=0,Out=1,V1>=1] 30.58/8.75 30.58/8.75 * Chain [23]: 0 30.58/8.75 with precondition: [Out=0,V1>=0,V>=0] 30.58/8.75 30.58/8.75 30.58/8.75 #### Cost of chains of gcd(V1,V,Out): 30.58/8.75 * Chain [[30],37]: 6*it(30)+1*s(8)+2 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(4) =< V1 30.58/8.75 it(30) =< aux(4) 30.58/8.75 30.58/8.75 with precondition: [V=1,Out=0,V1>=1] 30.58/8.75 30.58/8.75 * Chain [[30],34]: 4*it(30)+2 30.58/8.75 Such that:it(30) =< V1 30.58/8.75 30.58/8.75 with precondition: [V=1,Out=0,V1>=2] 30.58/8.75 30.58/8.75 * Chain [[30],31,37]: 4*it(30)+1*s(8)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 it(30) =< V1 30.58/8.75 30.58/8.75 with precondition: [V=1,Out=0,V1>=2] 30.58/8.75 30.58/8.75 * Chain [[26,27],37]: 4*it(26)+4*it(27)+6*s(6)+3*s(21)+2 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 it(26) =< V1/2+V/2 30.58/8.75 aux(28) =< V1 30.58/8.75 aux(29) =< V1+V 30.58/8.75 aux(30) =< V 30.58/8.75 s(6) =< aux(29) 30.58/8.75 it(26) =< aux(29) 30.58/8.75 it(27) =< aux(29) 30.58/8.75 it(26) =< aux(30) 30.58/8.75 it(27) =< aux(30)+aux(10) 30.58/8.75 it(27) =< aux(30)+aux(28) 30.58/8.75 s(22) =< aux(30)+aux(28) 30.58/8.75 s(22) =< it(27)*aux(30) 30.58/8.75 s(21) =< s(22) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2] 30.58/8.75 30.58/8.75 * Chain [[26,27],36]: 4*it(26)+4*it(27)+3*s(19)+3*s(21)+2 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 aux(31) =< V1 30.58/8.75 aux(32) =< V1+V 30.58/8.75 aux(33) =< V1/2+V/2 30.58/8.75 aux(34) =< V 30.58/8.75 it(26) =< aux(33) 30.58/8.75 it(26) =< aux(32) 30.58/8.75 it(27) =< aux(32) 30.58/8.75 it(27) =< aux(33) 30.58/8.75 it(26) =< aux(34) 30.58/8.75 it(27) =< aux(34)+aux(10) 30.58/8.75 it(27) =< aux(34)+aux(31) 30.58/8.75 s(22) =< aux(34)+aux(31) 30.58/8.75 s(22) =< it(27)*aux(34) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(32) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],33,37]: 4*it(26)+4*it(27)+1*s(8)+3*s(19)+3*s(21)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(10) =< V1-V+1 30.58/8.75 aux(35) =< V1 30.58/8.75 aux(36) =< V1+V 30.58/8.75 aux(37) =< V1/2+V/2 30.58/8.75 aux(38) =< V 30.58/8.75 it(26) =< aux(37) 30.58/8.75 it(26) =< aux(36) 30.58/8.75 it(27) =< aux(36) 30.58/8.75 it(27) =< aux(37) 30.58/8.75 it(26) =< aux(38) 30.58/8.75 it(27) =< aux(38)+aux(10) 30.58/8.75 it(27) =< aux(38)+aux(35) 30.58/8.75 s(22) =< aux(38)+aux(35) 30.58/8.75 s(22) =< it(27)*aux(38) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(36) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,[30],37]: 4*it(26)+4*it(27)+9*it(30)+1*s(8)+3*s(21)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(10) =< V1-V+1 30.58/8.75 it(26) =< V1/2+V/2 30.58/8.75 aux(39) =< V1 30.58/8.75 aux(40) =< V1+V 30.58/8.75 aux(41) =< V 30.58/8.75 it(30) =< aux(40) 30.58/8.75 it(26) =< aux(40) 30.58/8.75 it(27) =< aux(40) 30.58/8.75 it(26) =< aux(41) 30.58/8.75 it(27) =< aux(41)+aux(10) 30.58/8.75 it(27) =< aux(41)+aux(39) 30.58/8.75 s(22) =< aux(41)+aux(39) 30.58/8.75 s(22) =< it(27)*aux(41) 30.58/8.75 s(21) =< s(22) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,[30],34]: 4*it(26)+4*it(27)+7*it(30)+3*s(21)+6 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 it(26) =< V1/2+V/2 30.58/8.75 aux(42) =< V1 30.58/8.75 aux(43) =< V1+V 30.58/8.75 aux(44) =< V 30.58/8.75 it(30) =< aux(43) 30.58/8.75 it(26) =< aux(43) 30.58/8.75 it(27) =< aux(43) 30.58/8.75 it(26) =< aux(44) 30.58/8.75 it(27) =< aux(44)+aux(10) 30.58/8.75 it(27) =< aux(44)+aux(42) 30.58/8.75 s(22) =< aux(44)+aux(42) 30.58/8.75 s(22) =< it(27)*aux(44) 30.58/8.75 s(21) =< s(22) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,[30],31,37]: 4*it(26)+4*it(27)+7*it(30)+1*s(8)+3*s(21)+10 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(10) =< V1-V+1 30.58/8.75 it(26) =< V1/2+V/2 30.58/8.75 aux(45) =< V1 30.58/8.75 aux(46) =< V1+V 30.58/8.75 aux(47) =< V 30.58/8.75 it(30) =< aux(46) 30.58/8.75 it(26) =< aux(46) 30.58/8.75 it(27) =< aux(46) 30.58/8.75 it(26) =< aux(47) 30.58/8.75 it(27) =< aux(47)+aux(10) 30.58/8.75 it(27) =< aux(47)+aux(45) 30.58/8.75 s(22) =< aux(47)+aux(45) 30.58/8.75 s(22) =< it(27)*aux(47) 30.58/8.75 s(21) =< s(22) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,37]: 4*it(26)+4*it(27)+5*s(6)+1*s(8)+3*s(21)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(10) =< V1-V+1 30.58/8.75 it(26) =< V1/2+V/2 30.58/8.75 aux(48) =< V1 30.58/8.75 aux(49) =< V1+V 30.58/8.75 aux(50) =< V 30.58/8.75 s(6) =< aux(49) 30.58/8.75 it(26) =< aux(49) 30.58/8.75 it(27) =< aux(49) 30.58/8.75 it(26) =< aux(50) 30.58/8.75 it(27) =< aux(50)+aux(10) 30.58/8.75 it(27) =< aux(50)+aux(48) 30.58/8.75 s(22) =< aux(50)+aux(48) 30.58/8.75 s(22) =< it(27)*aux(50) 30.58/8.75 s(21) =< s(22) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,34]: 4*it(26)+4*it(27)+3*s(19)+3*s(21)+6 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 aux(51) =< V1 30.58/8.75 aux(52) =< V1+V 30.58/8.75 aux(53) =< V1/2+V/2 30.58/8.75 aux(54) =< V 30.58/8.75 it(26) =< aux(53) 30.58/8.75 it(26) =< aux(52) 30.58/8.75 it(27) =< aux(52) 30.58/8.75 it(27) =< aux(53) 30.58/8.75 it(26) =< aux(54) 30.58/8.75 it(27) =< aux(54)+aux(10) 30.58/8.75 it(27) =< aux(54)+aux(51) 30.58/8.75 s(22) =< aux(54)+aux(51) 30.58/8.75 s(22) =< it(27)*aux(54) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(52) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],32,31,37]: 4*it(26)+4*it(27)+1*s(8)+3*s(19)+3*s(21)+10 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(10) =< V1-V+1 30.58/8.75 aux(55) =< V1 30.58/8.75 aux(56) =< V1+V 30.58/8.75 aux(57) =< V1/2+V/2 30.58/8.75 aux(58) =< V 30.58/8.75 it(26) =< aux(57) 30.58/8.75 it(26) =< aux(56) 30.58/8.75 it(27) =< aux(56) 30.58/8.75 it(27) =< aux(57) 30.58/8.75 it(26) =< aux(58) 30.58/8.75 it(27) =< aux(58)+aux(10) 30.58/8.75 it(27) =< aux(58)+aux(55) 30.58/8.75 s(22) =< aux(58)+aux(55) 30.58/8.75 s(22) =< it(27)*aux(58) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(56) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 30.58/8.75 30.58/8.75 * Chain [[26,27],29,37]: 4*it(26)+4*it(27)+7*s(8)+3*s(19)+3*s(21)+6 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 aux(61) =< V1 30.58/8.75 aux(62) =< V1+V 30.58/8.75 aux(63) =< V1/2+V/2 30.58/8.75 aux(64) =< V 30.58/8.75 it(26) =< aux(63) 30.58/8.75 s(8) =< aux(63) 30.58/8.75 it(26) =< aux(62) 30.58/8.75 it(27) =< aux(62) 30.58/8.75 it(27) =< aux(63) 30.58/8.75 it(26) =< aux(64) 30.58/8.75 it(27) =< aux(64)+aux(10) 30.58/8.75 it(27) =< aux(64)+aux(61) 30.58/8.75 s(22) =< aux(64)+aux(61) 30.58/8.75 s(22) =< it(27)*aux(64) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(62) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=2,V>=2,V+V1>=6] 30.58/8.75 30.58/8.75 * Chain [[26,27],28,37]: 4*it(26)+4*it(27)+7*s(8)+3*s(19)+3*s(21)+6 30.58/8.75 Such that:aux(10) =< V1-V+1 30.58/8.75 aux(67) =< V1 30.58/8.75 aux(68) =< V1+V 30.58/8.75 aux(69) =< V1/2+V/2 30.58/8.75 aux(70) =< V 30.58/8.75 it(26) =< aux(69) 30.58/8.75 s(8) =< aux(67) 30.58/8.75 it(26) =< aux(68) 30.58/8.75 it(27) =< aux(68) 30.58/8.75 it(27) =< aux(69) 30.58/8.75 it(26) =< aux(70) 30.58/8.75 it(27) =< aux(70)+aux(10) 30.58/8.75 it(27) =< aux(70)+aux(67) 30.58/8.75 s(22) =< aux(70)+aux(67) 30.58/8.75 s(22) =< it(27)*aux(70) 30.58/8.75 s(21) =< s(22) 30.58/8.75 s(19) =< aux(68) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 30.58/8.75 30.58/8.75 * Chain [37]: 2*s(6)+1*s(8)+2 30.58/8.75 Such that:s(8) =< V 30.58/8.75 aux(3) =< V1 30.58/8.75 s(6) =< aux(3) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V1>=0,V>=0] 30.58/8.75 30.58/8.75 * Chain [36]: 2 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [35]: 1 30.58/8.75 with precondition: [V=0,V1=Out,V1>=1] 30.58/8.75 30.58/8.75 * Chain [34]: 2 30.58/8.75 with precondition: [V=1,Out=0,V1>=1] 30.58/8.75 30.58/8.75 * Chain [33,37]: 1*s(8)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [32,[30],37]: 6*it(30)+1*s(8)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(4) =< V 30.58/8.75 it(30) =< aux(4) 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [32,[30],34]: 4*it(30)+6 30.58/8.75 Such that:it(30) =< V 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=3] 30.58/8.75 30.58/8.75 * Chain [32,[30],31,37]: 4*it(30)+1*s(8)+10 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 it(30) =< V 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=3] 30.58/8.75 30.58/8.75 * Chain [32,37]: 2*s(6)+1*s(8)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 aux(3) =< V 30.58/8.75 s(6) =< aux(3) 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [32,34]: 6 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [32,31,37]: 1*s(8)+10 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 30.58/8.75 with precondition: [V1=1,Out=0,V>=2] 30.58/8.75 30.58/8.75 * Chain [31,37]: 1*s(8)+6 30.58/8.75 Such that:s(8) =< 1 30.58/8.75 30.58/8.75 with precondition: [V=1,Out=0,V1>=1] 30.58/8.75 30.58/8.75 * Chain [29,37]: 7*s(8)+6 30.58/8.75 Such that:aux(60) =< V 30.58/8.75 s(8) =< aux(60) 30.58/8.75 30.58/8.75 with precondition: [Out=0,V>=2,V1>=V] 30.58/8.75 30.58/8.75 * Chain [28,37]: 7*s(8)+6 30.58/8.75 Such that:aux(66) =< V1 30.58/8.76 s(8) =< aux(66) 30.58/8.76 30.58/8.76 with precondition: [Out=0,V1>=2,V>=V1+1] 30.58/8.76 30.58/8.76 30.58/8.76 #### Cost of chains of start(V1,V,V2): 30.58/8.76 * Chain [42]: 33*s(169)+17*s(173)+10*s(180)+44*s(181)+24*s(183)+18*s(185)+52*s(186)+20*s(187)+15*s(189)+7*s(191)+10 30.58/8.76 Such that:s(174) =< 1 30.58/8.76 s(176) =< V1-V+1 30.58/8.76 s(177) =< V1+V 30.58/8.76 s(178) =< V1/2+V/2 30.58/8.76 aux(79) =< V1 30.58/8.76 aux(80) =< V 30.58/8.76 s(173) =< aux(79) 30.58/8.76 s(169) =< aux(80) 30.58/8.76 s(180) =< s(174) 30.58/8.76 s(181) =< s(178) 30.58/8.76 s(181) =< s(177) 30.58/8.76 s(183) =< s(177) 30.58/8.76 s(183) =< s(178) 30.58/8.76 s(181) =< aux(80) 30.58/8.76 s(183) =< aux(80)+s(176) 30.58/8.76 s(183) =< aux(80)+aux(79) 30.58/8.76 s(184) =< aux(80)+aux(79) 30.58/8.76 s(184) =< s(183)*aux(80) 30.58/8.76 s(185) =< s(184) 30.58/8.76 s(186) =< s(177) 30.58/8.76 s(187) =< s(177) 30.58/8.76 s(187) =< aux(80)+s(176) 30.58/8.76 s(187) =< aux(80)+aux(79) 30.58/8.76 s(188) =< aux(80)+aux(79) 30.58/8.76 s(188) =< s(187)*aux(80) 30.58/8.76 s(189) =< s(188) 30.58/8.76 s(191) =< s(178) 30.58/8.76 30.58/8.76 with precondition: [V1>=0] 30.58/8.76 30.58/8.76 * Chain [41]: 57*s(198)+44*s(199)+24*s(201)+18*s(203)+134*s(204)+20*s(205)+15*s(207)+14*s(209)+107*s(215)+44*s(223)+24*s(225)+18*s(227)+20*s(229)+15*s(231)+7*s(233)+44*s(242)+24*s(244)+18*s(246)+20*s(248)+15*s(250)+16*s(251)+12 30.58/8.76 Such that:s(237) =< -2*V+V2+1 30.58/8.76 s(218) =< -V+1 30.58/8.76 s(236) =< -V+V2 30.58/8.76 s(220) =< V/2 30.58/8.76 aux(85) =< 1 30.58/8.76 aux(86) =< V 30.58/8.76 aux(87) =< V2 30.58/8.76 aux(88) =< V2/2 30.58/8.76 s(198) =< aux(85) 30.58/8.76 s(204) =< aux(87) 30.58/8.76 s(223) =< s(220) 30.58/8.76 s(215) =< aux(86) 30.58/8.76 s(223) =< aux(86) 30.58/8.76 s(225) =< aux(86) 30.58/8.76 s(225) =< s(220) 30.58/8.76 s(225) =< aux(86)+s(218) 30.58/8.76 s(226) =< aux(86) 30.58/8.76 s(226) =< s(225)*aux(86) 30.58/8.76 s(227) =< s(226) 30.58/8.76 s(229) =< aux(86) 30.58/8.76 s(229) =< aux(86)+s(218) 30.58/8.76 s(230) =< aux(86) 30.58/8.76 s(230) =< s(229)*aux(86) 30.58/8.76 s(231) =< s(230) 30.58/8.76 s(233) =< s(220) 30.58/8.76 s(242) =< aux(88) 30.58/8.76 s(242) =< aux(87) 30.58/8.76 s(244) =< aux(87) 30.58/8.76 s(244) =< aux(88) 30.58/8.76 s(242) =< aux(86) 30.58/8.76 s(244) =< aux(86)+s(237) 30.58/8.76 s(244) =< aux(86)+s(236) 30.58/8.76 s(245) =< aux(86)+s(236) 30.58/8.76 s(245) =< s(244)*aux(86) 30.58/8.76 s(246) =< s(245) 30.58/8.76 s(248) =< aux(87) 30.58/8.76 s(248) =< aux(86)+s(237) 30.58/8.76 s(248) =< aux(86)+s(236) 30.58/8.76 s(249) =< aux(86)+s(236) 30.58/8.76 s(249) =< s(248)*aux(86) 30.58/8.76 s(250) =< s(249) 30.58/8.76 s(251) =< s(236) 30.58/8.76 s(209) =< aux(88) 30.58/8.76 s(199) =< aux(88) 30.58/8.76 s(199) =< aux(87) 30.58/8.76 s(201) =< aux(87) 30.58/8.76 s(201) =< aux(88) 30.58/8.76 s(199) =< aux(85) 30.58/8.76 s(201) =< aux(85)+aux(87) 30.58/8.76 s(202) =< aux(85)+aux(87) 30.58/8.76 s(202) =< s(201)*aux(85) 30.58/8.76 s(203) =< s(202) 30.58/8.76 s(205) =< aux(87) 30.58/8.76 s(205) =< aux(85)+aux(87) 30.58/8.76 s(206) =< aux(85)+aux(87) 30.58/8.76 s(206) =< s(205)*aux(85) 30.58/8.76 s(207) =< s(206) 30.58/8.76 30.58/8.76 with precondition: [V1=1,V>=1,V2>=1] 30.58/8.76 30.58/8.76 * Chain [40]: 57*s(259)+44*s(260)+24*s(262)+18*s(264)+134*s(265)+20*s(266)+15*s(268)+14*s(270)+107*s(276)+44*s(284)+24*s(286)+18*s(288)+20*s(290)+15*s(292)+7*s(294)+44*s(303)+24*s(305)+18*s(307)+20*s(309)+15*s(311)+16*s(312)+12 30.58/8.76 Such that:s(298) =< V-2*V2+1 30.58/8.76 s(297) =< V-V2 30.58/8.76 s(279) =< -V2+1 30.58/8.76 s(281) =< V2/2 30.58/8.76 aux(93) =< 1 30.58/8.76 aux(94) =< V 30.58/8.76 aux(95) =< V/2 30.58/8.76 aux(96) =< V2 30.58/8.76 s(259) =< aux(93) 30.58/8.76 s(265) =< aux(94) 30.58/8.76 s(284) =< s(281) 30.58/8.76 s(276) =< aux(96) 30.58/8.76 s(284) =< aux(96) 30.58/8.76 s(286) =< aux(96) 30.58/8.76 s(286) =< s(281) 30.58/8.76 s(286) =< aux(96)+s(279) 30.58/8.76 s(287) =< aux(96) 30.58/8.76 s(287) =< s(286)*aux(96) 30.58/8.76 s(288) =< s(287) 30.58/8.76 s(290) =< aux(96) 30.58/8.76 s(290) =< aux(96)+s(279) 30.58/8.76 s(291) =< aux(96) 30.58/8.76 s(291) =< s(290)*aux(96) 30.58/8.76 s(292) =< s(291) 30.58/8.76 s(294) =< s(281) 30.58/8.76 s(303) =< aux(95) 30.58/8.76 s(303) =< aux(94) 30.58/8.76 s(305) =< aux(94) 30.58/8.76 s(305) =< aux(95) 30.58/8.76 s(303) =< aux(96) 30.58/8.76 s(305) =< aux(96)+s(298) 30.58/8.76 s(305) =< aux(96)+s(297) 30.58/8.76 s(306) =< aux(96)+s(297) 30.58/8.76 s(306) =< s(305)*aux(96) 30.58/8.76 s(307) =< s(306) 30.58/8.76 s(309) =< aux(94) 30.58/8.76 s(309) =< aux(96)+s(298) 30.58/8.76 s(309) =< aux(96)+s(297) 30.58/8.76 s(310) =< aux(96)+s(297) 30.58/8.76 s(310) =< s(309)*aux(96) 30.58/8.76 s(311) =< s(310) 30.58/8.76 s(312) =< s(297) 30.58/8.76 s(270) =< aux(95) 30.58/8.76 s(260) =< aux(95) 30.58/8.76 s(260) =< aux(94) 30.58/8.76 s(262) =< aux(94) 30.58/8.76 s(262) =< aux(95) 30.58/8.76 s(260) =< aux(93) 30.58/8.76 s(262) =< aux(93)+aux(94) 30.58/8.76 s(263) =< aux(93)+aux(94) 30.58/8.76 s(263) =< s(262)*aux(93) 30.58/8.76 s(264) =< s(263) 30.58/8.76 s(266) =< aux(94) 30.58/8.76 s(266) =< aux(93)+aux(94) 30.58/8.76 s(267) =< aux(93)+aux(94) 30.58/8.76 s(267) =< s(266)*aux(93) 30.58/8.76 s(268) =< s(267) 30.58/8.76 30.58/8.76 with precondition: [V1=2,V>=1,V2>=1] 30.58/8.76 30.58/8.76 * Chain [39]: 1 30.58/8.76 with precondition: [V=0,V1>=0] 30.58/8.76 30.58/8.76 * Chain [38]: 3*s(316)+14*s(317)+6 30.58/8.76 Such that:s(314) =< 1 30.58/8.76 s(315) =< V1 30.58/8.76 s(316) =< s(314) 30.58/8.76 s(317) =< s(315) 30.58/8.76 30.58/8.76 with precondition: [V=1,V1>=1] 30.58/8.76 30.58/8.76 30.58/8.76 Closed-form bounds of start(V1,V,V2): 30.58/8.76 ------------------------------------- 30.58/8.76 * Chain [42] with precondition: [V1>=0] 30.58/8.76 - Upper bound: 50*V1+20+nat(V)*66+nat(V1+V)*96+nat(V1/2+V/2)*51 30.58/8.76 - Complexity: n 30.58/8.76 * Chain [41] with precondition: [V1=1,V>=1,V2>=1] 30.58/8.76 - Upper bound: 217*V+255*V2+146+nat(-V+V2)*49+51/2*V+29*V2 30.58/8.76 - Complexity: n 30.58/8.76 * Chain [40] with precondition: [V1=2,V>=1,V2>=1] 30.58/8.76 - Upper bound: 255*V+217*V2+146+nat(V-V2)*49+29*V+51/2*V2 30.58/8.76 - Complexity: n 30.58/8.76 * Chain [39] with precondition: [V=0,V1>=0] 30.58/8.76 - Upper bound: 1 30.58/8.76 - Complexity: constant 30.58/8.76 * Chain [38] with precondition: [V=1,V1>=1] 30.58/8.76 - Upper bound: 14*V1+9 30.58/8.76 - Complexity: n 30.58/8.76 30.58/8.76 ### Maximum cost of start(V1,V,V2): max([14*V1+8,nat(V)*66+19+max([nat(V1+V)*96+50*V1+nat(V1/2+V/2)*51,nat(V)*151+126+nat(V2)*217+nat(V/2)*51+nat(V2/2)*51+max([nat(-V+V2)*49+nat(V2)*38+nat(V2/2)*7,nat(V-V2)*49+nat(V)*38+nat(V/2)*7])])])+1 30.58/8.76 Asymptotic class: n 30.58/8.76 * Total analysis performed in 829 ms. 30.58/8.76 30.58/8.76 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (10) 30.58/8.76 BOUNDS(1, n^1) 30.58/8.76 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 30.58/8.76 Transformed a relative TRS into a decreasing-loop problem. 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (12) 30.58/8.76 Obligation: 30.58/8.76 Analyzing the following TRS for decreasing loops: 30.58/8.76 30.58/8.76 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.58/8.76 30.58/8.76 30.58/8.76 The TRS R consists of the following rules: 30.58/8.76 30.58/8.76 minus(X, s(Y)) -> pred(minus(X, Y)) 30.58/8.76 minus(X, 0) -> X 30.58/8.76 pred(s(X)) -> X 30.58/8.76 le(s(X), s(Y)) -> le(X, Y) 30.58/8.76 le(s(X), 0) -> false 30.58/8.76 le(0, Y) -> true 30.58/8.76 gcd(0, Y) -> 0 30.58/8.76 gcd(s(X), 0) -> s(X) 30.58/8.76 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 30.58/8.76 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 30.58/8.76 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 30.58/8.76 30.58/8.76 S is empty. 30.58/8.76 Rewrite Strategy: INNERMOST 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (13) DecreasingLoopProof (LOWER BOUND(ID)) 30.58/8.76 The following loop(s) give(s) rise to the lower bound Omega(n^1): 30.58/8.76 30.58/8.76 The rewrite sequence 30.58/8.76 30.58/8.76 minus(X, s(Y)) ->^+ pred(minus(X, Y)) 30.58/8.76 30.58/8.76 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 30.58/8.76 30.58/8.76 The pumping substitution is [Y / s(Y)]. 30.58/8.76 30.58/8.76 The result substitution is [ ]. 30.58/8.76 30.58/8.76 30.58/8.76 30.58/8.76 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (14) 30.58/8.76 Complex Obligation (BEST) 30.58/8.76 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (15) 30.58/8.76 Obligation: 30.58/8.76 Proved the lower bound n^1 for the following obligation: 30.58/8.76 30.58/8.76 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.58/8.76 30.58/8.76 30.58/8.76 The TRS R consists of the following rules: 30.58/8.76 30.58/8.76 minus(X, s(Y)) -> pred(minus(X, Y)) 30.58/8.76 minus(X, 0) -> X 30.58/8.76 pred(s(X)) -> X 30.58/8.76 le(s(X), s(Y)) -> le(X, Y) 30.58/8.76 le(s(X), 0) -> false 30.58/8.76 le(0, Y) -> true 30.58/8.76 gcd(0, Y) -> 0 30.58/8.76 gcd(s(X), 0) -> s(X) 30.58/8.76 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 30.58/8.76 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 30.58/8.76 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 30.58/8.76 30.58/8.76 S is empty. 30.58/8.76 Rewrite Strategy: INNERMOST 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (16) LowerBoundPropagationProof (FINISHED) 30.58/8.76 Propagated lower bound. 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (17) 30.58/8.76 BOUNDS(n^1, INF) 30.58/8.76 30.58/8.76 ---------------------------------------- 30.58/8.76 30.58/8.76 (18) 30.58/8.76 Obligation: 30.58/8.76 Analyzing the following TRS for decreasing loops: 30.58/8.76 30.58/8.76 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.58/8.76 30.58/8.76 30.58/8.76 The TRS R consists of the following rules: 30.58/8.76 30.58/8.76 minus(X, s(Y)) -> pred(minus(X, Y)) 30.58/8.76 minus(X, 0) -> X 30.58/8.76 pred(s(X)) -> X 30.58/8.76 le(s(X), s(Y)) -> le(X, Y) 30.58/8.76 le(s(X), 0) -> false 30.58/8.76 le(0, Y) -> true 30.58/8.76 gcd(0, Y) -> 0 30.58/8.76 gcd(s(X), 0) -> s(X) 30.58/8.76 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 30.58/8.76 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 30.58/8.76 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 30.58/8.76 30.58/8.76 S is empty. 30.58/8.76 Rewrite Strategy: INNERMOST 30.58/8.80 EOF