100.29/26.48 WORST_CASE(Omega(n^1), O(n^1)) 100.29/26.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 100.29/26.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 100.29/26.49 100.29/26.49 100.29/26.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 100.29/26.49 100.29/26.49 (0) CpxTRS 100.29/26.49 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 100.29/26.49 (2) CpxWeightedTrs 100.29/26.49 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 100.29/26.49 (4) CpxTypedWeightedTrs 100.29/26.49 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 100.29/26.49 (6) CpxTypedWeightedCompleteTrs 100.29/26.49 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 100.29/26.49 (8) CpxRNTS 100.29/26.49 (9) CompleteCoflocoProof [FINISHED, 1366 ms] 100.29/26.49 (10) BOUNDS(1, n^1) 100.29/26.49 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 100.29/26.49 (12) CpxTRS 100.29/26.49 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 100.29/26.49 (14) typed CpxTrs 100.29/26.49 (15) OrderProof [LOWER BOUND(ID), 0 ms] 100.29/26.49 (16) typed CpxTrs 100.29/26.49 (17) RewriteLemmaProof [LOWER BOUND(ID), 316 ms] 100.29/26.49 (18) BEST 100.29/26.49 (19) proven lower bound 100.29/26.49 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 100.29/26.49 (21) BOUNDS(n^1, INF) 100.29/26.49 (22) typed CpxTrs 100.29/26.49 (23) RewriteLemmaProof [LOWER BOUND(ID), 835 ms] 100.29/26.49 (24) typed CpxTrs 100.29/26.49 100.29/26.49 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (0) 100.29/26.49 Obligation: 100.29/26.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 100.29/26.49 100.29/26.49 100.29/26.49 The TRS R consists of the following rules: 100.29/26.49 100.29/26.49 le(0, y) -> true 100.29/26.49 le(s(x), 0) -> false 100.29/26.49 le(s(x), s(y)) -> le(x, y) 100.29/26.49 pred(s(x)) -> x 100.29/26.49 minus(x, 0) -> x 100.29/26.49 minus(x, s(y)) -> pred(minus(x, y)) 100.29/26.49 gcd(0, y) -> y 100.29/26.49 gcd(s(x), 0) -> s(x) 100.29/26.49 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.29/26.49 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.29/26.49 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.29/26.49 100.29/26.49 S is empty. 100.29/26.49 Rewrite Strategy: INNERMOST 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 100.29/26.49 Transformed relative TRS to weighted TRS 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (2) 100.29/26.49 Obligation: 100.29/26.49 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 100.29/26.49 100.29/26.49 100.29/26.49 The TRS R consists of the following rules: 100.29/26.49 100.29/26.49 le(0, y) -> true [1] 100.29/26.49 le(s(x), 0) -> false [1] 100.29/26.49 le(s(x), s(y)) -> le(x, y) [1] 100.29/26.49 pred(s(x)) -> x [1] 100.29/26.49 minus(x, 0) -> x [1] 100.29/26.49 minus(x, s(y)) -> pred(minus(x, y)) [1] 100.29/26.49 gcd(0, y) -> y [1] 100.29/26.49 gcd(s(x), 0) -> s(x) [1] 100.29/26.49 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 100.29/26.49 if_gcd(true, x, y) -> gcd(minus(x, y), y) [1] 100.29/26.49 if_gcd(false, x, y) -> gcd(minus(y, x), x) [1] 100.29/26.49 100.29/26.49 Rewrite Strategy: INNERMOST 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 100.29/26.49 Infered types. 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (4) 100.29/26.49 Obligation: 100.29/26.49 Runtime Complexity Weighted TRS with Types. 100.29/26.49 The TRS R consists of the following rules: 100.29/26.49 100.29/26.49 le(0, y) -> true [1] 100.29/26.49 le(s(x), 0) -> false [1] 100.29/26.49 le(s(x), s(y)) -> le(x, y) [1] 100.29/26.49 pred(s(x)) -> x [1] 100.29/26.49 minus(x, 0) -> x [1] 100.29/26.49 minus(x, s(y)) -> pred(minus(x, y)) [1] 100.29/26.49 gcd(0, y) -> y [1] 100.29/26.49 gcd(s(x), 0) -> s(x) [1] 100.29/26.49 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 100.29/26.49 if_gcd(true, x, y) -> gcd(minus(x, y), y) [1] 100.29/26.49 if_gcd(false, x, y) -> gcd(minus(y, x), x) [1] 100.29/26.49 100.29/26.49 The TRS has the following type information: 100.29/26.49 le :: 0:s -> 0:s -> true:false 100.29/26.49 0 :: 0:s 100.29/26.49 true :: true:false 100.29/26.49 s :: 0:s -> 0:s 100.29/26.49 false :: true:false 100.29/26.49 pred :: 0:s -> 0:s 100.29/26.49 minus :: 0:s -> 0:s -> 0:s 100.29/26.49 gcd :: 0:s -> 0:s -> 0:s 100.29/26.49 if_gcd :: true:false -> 0:s -> 0:s -> 0:s 100.29/26.49 100.29/26.49 Rewrite Strategy: INNERMOST 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (5) CompletionProof (UPPER BOUND(ID)) 100.29/26.49 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 100.29/26.49 100.29/26.49 pred(v0) -> null_pred [0] 100.29/26.49 le(v0, v1) -> null_le [0] 100.29/26.49 minus(v0, v1) -> null_minus [0] 100.29/26.49 gcd(v0, v1) -> null_gcd [0] 100.29/26.49 if_gcd(v0, v1, v2) -> null_if_gcd [0] 100.29/26.49 100.29/26.49 And the following fresh constants: null_pred, null_le, null_minus, null_gcd, null_if_gcd 100.29/26.49 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (6) 100.29/26.49 Obligation: 100.29/26.49 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 100.29/26.49 100.29/26.49 Runtime Complexity Weighted TRS with Types. 100.29/26.49 The TRS R consists of the following rules: 100.29/26.49 100.29/26.49 le(0, y) -> true [1] 100.29/26.49 le(s(x), 0) -> false [1] 100.29/26.49 le(s(x), s(y)) -> le(x, y) [1] 100.29/26.49 pred(s(x)) -> x [1] 100.29/26.49 minus(x, 0) -> x [1] 100.29/26.49 minus(x, s(y)) -> pred(minus(x, y)) [1] 100.29/26.49 gcd(0, y) -> y [1] 100.29/26.49 gcd(s(x), 0) -> s(x) [1] 100.29/26.49 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 100.29/26.49 if_gcd(true, x, y) -> gcd(minus(x, y), y) [1] 100.29/26.49 if_gcd(false, x, y) -> gcd(minus(y, x), x) [1] 100.29/26.49 pred(v0) -> null_pred [0] 100.29/26.49 le(v0, v1) -> null_le [0] 100.29/26.49 minus(v0, v1) -> null_minus [0] 100.29/26.49 gcd(v0, v1) -> null_gcd [0] 100.29/26.49 if_gcd(v0, v1, v2) -> null_if_gcd [0] 100.29/26.49 100.29/26.49 The TRS has the following type information: 100.29/26.49 le :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> true:false:null_le 100.29/26.49 0 :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 true :: true:false:null_le 100.29/26.49 s :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 false :: true:false:null_le 100.29/26.49 pred :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 minus :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 gcd :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 if_gcd :: true:false:null_le -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd -> 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 null_pred :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 null_le :: true:false:null_le 100.29/26.49 null_minus :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 null_gcd :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 null_if_gcd :: 0:s:null_pred:null_minus:null_gcd:null_if_gcd 100.29/26.49 100.29/26.49 Rewrite Strategy: INNERMOST 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 100.29/26.49 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 100.29/26.49 The constant constructors are abstracted as follows: 100.29/26.49 100.29/26.49 0 => 0 100.29/26.49 true => 2 100.29/26.49 false => 1 100.29/26.49 null_pred => 0 100.29/26.49 null_le => 0 100.29/26.49 null_minus => 0 100.29/26.49 null_gcd => 0 100.29/26.49 null_if_gcd => 0 100.29/26.49 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (8) 100.29/26.49 Obligation: 100.29/26.49 Complexity RNTS consisting of the following rules: 100.29/26.49 100.29/26.49 gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y 100.29/26.49 gcd(z, z') -{ 1 }-> if_gcd(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 100.29/26.49 gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 100.29/26.49 gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 100.29/26.49 if_gcd(z, z', z'') -{ 1 }-> gcd(minus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 100.29/26.49 if_gcd(z, z', z'') -{ 1 }-> gcd(minus(y, x), x) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 100.29/26.49 if_gcd(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 100.29/26.49 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 100.29/26.49 le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y 100.29/26.49 le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 100.29/26.49 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 100.29/26.49 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 100.29/26.49 minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x 100.29/26.49 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 100.29/26.49 pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 100.29/26.49 pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 100.29/26.49 100.29/26.49 Only complete derivations are relevant for the runtime complexity. 100.29/26.49 100.29/26.49 ---------------------------------------- 100.29/26.49 100.29/26.49 (9) CompleteCoflocoProof (FINISHED) 100.29/26.49 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 100.29/26.49 100.29/26.49 eq(start(V1, V, V15),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). 100.29/26.49 eq(start(V1, V, V15),0,[pred(V1, Out)],[V1 >= 0]). 100.29/26.49 eq(start(V1, V, V15),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 100.29/26.49 eq(start(V1, V, V15),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]). 100.29/26.49 eq(start(V1, V, V15),0,[fun(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 0]). 100.29/26.49 eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]). 100.29/26.49 eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]). 100.29/26.49 eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). 100.29/26.49 eq(pred(V1, Out),1,[],[Out = V6,V6 >= 0,V1 = 1 + V6]). 100.29/26.49 eq(minus(V1, V, Out),1,[],[Out = V7,V7 >= 0,V1 = V7,V = 0]). 100.29/26.49 eq(minus(V1, V, Out),1,[minus(V8, V9, Ret0),pred(Ret0, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = V8]). 100.29/26.49 eq(gcd(V1, V, Out),1,[],[Out = V10,V10 >= 0,V1 = 0,V = V10]). 100.29/26.49 eq(gcd(V1, V, Out),1,[],[Out = 1 + V11,V11 >= 0,V1 = 1 + V11,V = 0]). 100.29/26.49 eq(gcd(V1, V, Out),1,[le(V12, V13, Ret01),fun(Ret01, 1 + V13, 1 + V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]). 100.29/26.49 eq(fun(V1, V, V15, Out),1,[minus(V16, V14, Ret02),gcd(Ret02, V14, Ret3)],[Out = Ret3,V1 = 2,V = V16,V15 = V14,V16 >= 0,V14 >= 0]). 100.29/26.49 eq(fun(V1, V, V15, Out),1,[minus(V18, V17, Ret03),gcd(Ret03, V17, Ret4)],[Out = Ret4,V = V17,V15 = V18,V1 = 1,V17 >= 0,V18 >= 0]). 100.29/26.49 eq(pred(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]). 100.29/26.49 eq(le(V1, V, Out),0,[],[Out = 0,V21 >= 0,V20 >= 0,V1 = V21,V = V20]). 100.29/26.49 eq(minus(V1, V, Out),0,[],[Out = 0,V23 >= 0,V22 >= 0,V1 = V23,V = V22]). 100.29/26.49 eq(gcd(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]). 100.29/26.49 eq(fun(V1, V, V15, Out),0,[],[Out = 0,V26 >= 0,V15 = V28,V27 >= 0,V1 = V26,V = V27,V28 >= 0]). 100.29/26.49 input_output_vars(le(V1,V,Out),[V1,V],[Out]). 100.29/26.49 input_output_vars(pred(V1,Out),[V1],[Out]). 100.29/26.49 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 100.29/26.49 input_output_vars(gcd(V1,V,Out),[V1,V],[Out]). 100.29/26.49 input_output_vars(fun(V1,V,V15,Out),[V1,V,V15],[Out]). 100.29/26.49 100.29/26.49 100.29/26.49 CoFloCo proof output: 100.29/26.49 Preprocessing Cost Relations 100.29/26.49 ===================================== 100.29/26.49 100.29/26.49 #### Computed strongly connected components 100.29/26.49 0. recursive : [le/3] 100.29/26.49 1. non_recursive : [pred/2] 100.29/26.49 2. recursive [non_tail] : [minus/3] 100.29/26.49 3. recursive : [fun/4,gcd/3] 100.29/26.49 4. non_recursive : [start/3] 100.29/26.49 100.29/26.49 #### Obtained direct recursion through partial evaluation 100.29/26.49 0. SCC is partially evaluated into le/3 100.29/26.49 1. SCC is partially evaluated into pred/2 100.29/26.49 2. SCC is partially evaluated into minus/3 100.29/26.49 3. SCC is partially evaluated into gcd/3 100.29/26.49 4. SCC is partially evaluated into start/3 100.29/26.49 100.29/26.49 Control-Flow Refinement of Cost Relations 100.29/26.49 ===================================== 100.29/26.49 100.29/26.49 ### Specialization of cost equations le/3 100.29/26.49 * CE 20 is refined into CE [23] 100.29/26.49 * CE 18 is refined into CE [24] 100.29/26.49 * CE 17 is refined into CE [25] 100.29/26.49 * CE 19 is refined into CE [26] 100.29/26.49 100.29/26.49 100.29/26.49 ### Cost equations --> "Loop" of le/3 100.29/26.49 * CEs [26] --> Loop 17 100.29/26.49 * CEs [23] --> Loop 18 100.29/26.49 * CEs [24] --> Loop 19 100.29/26.49 * CEs [25] --> Loop 20 100.29/26.49 100.29/26.49 ### Ranking functions of CR le(V1,V,Out) 100.29/26.49 * RF of phase [17]: [V,V1] 100.29/26.49 100.29/26.49 #### Partial ranking functions of CR le(V1,V,Out) 100.29/26.49 * Partial RF of phase [17]: 100.29/26.49 - RF of loop [17:1]: 100.29/26.49 V 100.29/26.49 V1 100.29/26.49 100.29/26.49 100.29/26.49 ### Specialization of cost equations pred/2 100.29/26.49 * CE 21 is refined into CE [27] 100.29/26.49 * CE 22 is refined into CE [28] 100.29/26.49 100.29/26.49 100.29/26.49 ### Cost equations --> "Loop" of pred/2 100.29/26.49 * CEs [27] --> Loop 21 100.29/26.49 * CEs [28] --> Loop 22 100.29/26.49 100.29/26.49 ### Ranking functions of CR pred(V1,Out) 100.29/26.49 100.29/26.49 #### Partial ranking functions of CR pred(V1,Out) 100.29/26.49 100.29/26.49 100.29/26.49 ### Specialization of cost equations minus/3 100.29/26.49 * CE 10 is refined into CE [29] 100.29/26.49 * CE 8 is refined into CE [30] 100.29/26.49 * CE 9 is refined into CE [31,32] 100.29/26.49 100.29/26.49 100.29/26.49 ### Cost equations --> "Loop" of minus/3 100.29/26.49 * CEs [32] --> Loop 23 100.29/26.49 * CEs [31] --> Loop 24 100.29/26.49 * CEs [29] --> Loop 25 100.29/26.49 * CEs [30] --> Loop 26 100.29/26.49 100.29/26.49 ### Ranking functions of CR minus(V1,V,Out) 100.29/26.49 * RF of phase [23]: [V] 100.29/26.49 * RF of phase [24]: [V] 100.29/26.49 100.29/26.49 #### Partial ranking functions of CR minus(V1,V,Out) 100.29/26.49 * Partial RF of phase [23]: 100.29/26.49 - RF of loop [23:1]: 100.29/26.49 V 100.29/26.49 * Partial RF of phase [24]: 100.29/26.49 - RF of loop [24:1]: 100.29/26.49 V 100.29/26.49 100.29/26.49 100.29/26.49 ### Specialization of cost equations gcd/3 100.29/26.49 * CE 11 is refined into CE [33,34,35,36,37] 100.29/26.49 * CE 16 is refined into CE [38] 100.29/26.49 * CE 15 is refined into CE [39] 100.29/26.49 * CE 14 is refined into CE [40] 100.29/26.49 * CE 13 is refined into CE [41,42,43,44] 100.29/26.49 * CE 12 is refined into CE [45,46,47,48] 100.29/26.49 100.29/26.49 100.29/26.49 ### Cost equations --> "Loop" of gcd/3 100.29/26.49 * CEs [44] --> Loop 27 100.29/26.49 * CEs [48] --> Loop 28 100.29/26.49 * CEs [43] --> Loop 29 100.29/26.49 * CEs [47] --> Loop 30 100.29/26.49 * CEs [42] --> Loop 31 100.29/26.49 * CEs [41] --> Loop 32 100.29/26.49 * CEs [46] --> Loop 33 100.29/26.49 * CEs [45] --> Loop 34 100.29/26.49 * CEs [33] --> Loop 35 100.29/26.49 * CEs [39] --> Loop 36 100.29/26.49 * CEs [34,35,36,37,38] --> Loop 37 100.29/26.49 * CEs [40] --> Loop 38 100.29/26.49 100.29/26.49 ### Ranking functions of CR gcd(V1,V,Out) 100.29/26.49 * RF of phase [27,28]: [V1+V-3] 100.29/26.49 * RF of phase [31]: [V1] 100.29/26.49 100.29/26.49 #### Partial ranking functions of CR gcd(V1,V,Out) 100.29/26.49 * Partial RF of phase [27,28]: 100.29/26.49 - RF of loop [27:1]: 100.29/26.49 V1-1 depends on loops [28:1] 100.29/26.49 V1-V+1 depends on loops [28:1] 100.29/26.49 - RF of loop [28:1]: 100.29/26.49 V-2 100.29/26.49 V1/2+V/2-2 100.29/26.49 * Partial RF of phase [31]: 100.29/26.49 - RF of loop [31:1]: 100.29/26.49 V1 100.29/26.49 100.29/26.49 100.29/26.49 ### Specialization of cost equations start/3 100.29/26.49 * CE 3 is refined into CE [49,50,51,52,53,54,55,56,57,58,59,60,61] 100.29/26.49 * CE 1 is refined into CE [62] 100.29/26.49 * CE 2 is refined into CE [63,64,65,66,67,68,69,70,71,72,73,74,75] 100.29/26.49 * CE 4 is refined into CE [76,77,78,79,80] 100.29/26.49 * CE 5 is refined into CE [81,82] 100.29/26.49 * CE 6 is refined into CE [83,84,85] 100.29/26.49 * CE 7 is refined into CE [86,87,88,89,90,91,92,93,94] 100.29/26.49 100.29/26.49 100.29/26.49 ### Cost equations --> "Loop" of start/3 100.29/26.49 * CEs [77,83,89] --> Loop 39 100.29/26.49 * CEs [56] --> Loop 40 100.29/26.49 * CEs [54] --> Loop 41 100.29/26.49 * CEs [57,58] --> Loop 42 100.29/26.49 * CEs [49,50,51,52,53,55,59,60,61] --> Loop 43 100.29/26.49 * CEs [68] --> Loop 44 100.29/26.49 * CEs [70,88] --> Loop 45 100.29/26.49 * CEs [71,72,90,91] --> Loop 46 100.29/26.49 * CEs [63,64,65,66,67,69,73,74,75] --> Loop 47 100.29/26.49 * CEs [62,76,78,79,80,81,82,84,85,86,87,92,93,94] --> Loop 48 100.29/26.49 100.29/26.49 ### Ranking functions of CR start(V1,V,V15) 100.29/26.49 100.29/26.49 #### Partial ranking functions of CR start(V1,V,V15) 100.29/26.49 100.29/26.49 100.29/26.49 Computing Bounds 100.29/26.49 ===================================== 100.29/26.49 100.29/26.49 #### Cost of chains of le(V1,V,Out): 100.29/26.49 * Chain [[17],20]: 1*it(17)+1 100.29/26.49 Such that:it(17) =< V1 100.29/26.49 100.29/26.49 with precondition: [Out=2,V1>=1,V>=V1] 100.29/26.49 100.29/26.49 * Chain [[17],19]: 1*it(17)+1 100.29/26.49 Such that:it(17) =< V 100.29/26.49 100.29/26.49 with precondition: [Out=1,V>=1,V1>=V+1] 100.29/26.49 100.29/26.49 * Chain [[17],18]: 1*it(17)+0 100.29/26.49 Such that:it(17) =< V 100.29/26.49 100.29/26.49 with precondition: [Out=0,V1>=1,V>=1] 100.29/26.49 100.29/26.49 * Chain [20]: 1 100.29/26.49 with precondition: [V1=0,Out=2,V>=0] 100.29/26.49 100.29/26.49 * Chain [19]: 1 100.29/26.49 with precondition: [V=0,Out=1,V1>=1] 100.29/26.49 100.29/26.49 * Chain [18]: 0 100.29/26.49 with precondition: [Out=0,V1>=0,V>=0] 100.29/26.49 100.29/26.49 100.29/26.49 #### Cost of chains of pred(V1,Out): 100.29/26.49 * Chain [22]: 0 100.29/26.49 with precondition: [Out=0,V1>=0] 100.29/26.49 100.29/26.49 * Chain [21]: 1 100.29/26.49 with precondition: [V1=Out+1,V1>=1] 100.29/26.49 100.29/26.49 100.29/26.49 #### Cost of chains of minus(V1,V,Out): 100.29/26.49 * Chain [[24],[23],26]: 3*it(23)+1 100.29/26.49 Such that:aux(1) =< V 100.29/26.49 it(23) =< aux(1) 100.29/26.49 100.29/26.49 with precondition: [Out=0,V1>=1,V>=2] 100.29/26.49 100.29/26.49 * Chain [[24],26]: 1*it(24)+1 100.29/26.49 Such that:it(24) =< V 100.29/26.49 100.29/26.49 with precondition: [Out=0,V1>=0,V>=1] 100.29/26.49 100.29/26.49 * Chain [[24],25]: 1*it(24)+0 100.29/26.49 Such that:it(24) =< V 100.29/26.49 100.29/26.49 with precondition: [Out=0,V1>=0,V>=1] 100.29/26.49 100.29/26.49 * Chain [[23],26]: 2*it(23)+1 100.29/26.49 Such that:it(23) =< V 100.29/26.49 100.29/26.49 with precondition: [V1=Out+V,V>=1,V1>=V] 100.29/26.49 100.29/26.49 * Chain [26]: 1 100.29/26.49 with precondition: [V=0,V1=Out,V1>=0] 100.29/26.49 100.29/26.49 * Chain [25]: 0 100.29/26.49 with precondition: [Out=0,V1>=0,V>=0] 100.29/26.49 100.29/26.49 100.29/26.49 #### Cost of chains of gcd(V1,V,Out): 100.29/26.49 * Chain [[31],38]: 6*it(31)+1 100.29/26.49 Such that:aux(6) =< V1 100.29/26.49 it(31) =< aux(6) 100.29/26.49 100.29/26.49 with precondition: [V=1,Out=1,V1>=1] 100.29/26.49 100.29/26.49 * Chain [[31],37]: 8*it(31)+1*s(11)+2 100.29/26.49 Such that:s(11) =< 1 100.29/26.49 aux(8) =< V1 100.29/26.49 it(31) =< aux(8) 100.29/26.49 100.29/26.49 with precondition: [V=1,Out=0,V1>=1] 100.29/26.49 100.29/26.49 * Chain [[31],35]: 6*it(31)+2 100.29/26.49 Such that:aux(9) =< V1 100.29/26.49 it(31) =< aux(9) 100.29/26.49 100.29/26.49 with precondition: [V=1,Out=0,V1>=2] 100.29/26.49 100.29/26.49 * Chain [[31],32,38]: 6*it(31)+5*s(13)+5 100.29/26.49 Such that:s(12) =< 1 100.29/26.49 aux(10) =< V1 100.29/26.49 s(13) =< s(12) 100.29/26.49 it(31) =< aux(10) 100.29/26.49 100.29/26.49 with precondition: [V=1,Out=1,V1>=2] 100.29/26.49 100.29/26.49 * Chain [[31],32,37]: 6*it(31)+6*s(11)+6 100.29/26.49 Such that:aux(11) =< 1 100.29/26.49 aux(12) =< V1 100.29/26.49 s(11) =< aux(11) 100.29/26.49 it(31) =< aux(12) 100.29/26.49 100.29/26.49 with precondition: [V=1,Out=0,V1>=2] 100.29/26.49 100.29/26.49 * Chain [[27,28],38]: 4*it(27)+4*it(28)+3*s(22)+3*s(24)+1 100.29/26.49 Such that:aux(18) =< V1-V+1 100.29/26.49 aux(30) =< V1+V 100.29/26.49 aux(31) =< V1+V-Out 100.29/26.49 it(28) =< V1/2+V/2-Out/2 100.29/26.49 aux(33) =< V 100.29/26.49 aux(34) =< V-Out 100.29/26.49 aux(17) =< 2*V-2*Out 100.29/26.49 aux(35) =< V1 100.29/26.49 it(27) =< aux(30) 100.29/26.49 it(28) =< aux(30) 100.29/26.49 s(25) =< aux(30) 100.29/26.49 it(27) =< aux(31) 100.29/26.49 it(28) =< aux(31) 100.29/26.49 s(25) =< aux(31) 100.29/26.49 aux(15) =< aux(33) 100.29/26.49 it(28) =< aux(33) 100.29/26.49 aux(15) =< aux(34) 100.29/26.49 it(28) =< aux(34) 100.29/26.49 it(27) =< aux(17)+aux(18) 100.29/26.49 it(27) =< aux(15)+aux(35) 100.29/26.49 s(23) =< aux(15)+aux(35) 100.29/26.49 s(23) =< it(27)*aux(33) 100.29/26.49 s(24) =< s(25) 100.29/26.49 s(22) =< s(23) 100.29/26.49 100.29/26.49 with precondition: [Out>=2,V1>=Out,V>=Out] 100.29/26.49 100.29/26.49 * Chain [[27,28],37]: 4*it(27)+4*it(28)+6*s(9)+3*s(22)+2 100.29/26.49 Such that:aux(18) =< V1-V+1 100.29/26.49 aux(17) =< 2*V 100.29/26.49 aux(36) =< V1 100.29/26.49 aux(37) =< V1+V 100.29/26.49 aux(38) =< V 100.29/26.49 it(28) =< aux(37) 100.29/26.49 s(9) =< aux(37) 100.29/26.49 it(27) =< aux(37) 100.29/26.49 it(28) =< aux(38) 100.29/26.49 it(27) =< aux(17)+aux(18) 100.29/26.49 it(27) =< aux(38)+aux(36) 100.29/26.49 s(23) =< aux(38)+aux(36) 100.29/26.49 s(23) =< it(27)*aux(38) 100.29/26.49 s(22) =< s(23) 100.29/26.49 100.29/26.49 with precondition: [Out=0,V1>=2,V>=2] 100.39/26.50 100.39/26.50 * Chain [[27,28],34,38]: 4*it(27)+4*it(28)+3*s(22)+3*s(24)+5*s(27)+5 100.39/26.50 Such that:s(26) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(39) =< V1 100.39/26.50 aux(40) =< V1+V 100.39/26.50 aux(41) =< V 100.39/26.50 s(27) =< s(26) 100.39/26.50 it(27) =< aux(40) 100.39/26.50 it(28) =< aux(40) 100.39/26.50 it(28) =< aux(41) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(41)+aux(39) 100.39/26.50 s(23) =< aux(41)+aux(39) 100.39/26.50 s(23) =< it(27)*aux(41) 100.39/26.50 s(24) =< aux(40) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],34,37]: 4*it(27)+4*it(28)+6*s(11)+3*s(22)+3*s(24)+6 100.39/26.50 Such that:aux(42) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(43) =< V1 100.39/26.50 aux(44) =< V1+V 100.39/26.50 aux(45) =< V 100.39/26.50 s(11) =< aux(42) 100.39/26.50 it(27) =< aux(44) 100.39/26.50 it(28) =< aux(44) 100.39/26.50 it(28) =< aux(45) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(45)+aux(43) 100.39/26.50 s(23) =< aux(45)+aux(43) 100.39/26.50 s(23) =< it(27)*aux(45) 100.39/26.50 s(24) =< aux(44) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,[31],38]: 4*it(27)+4*it(28)+6*it(31)+3*s(22)+3*s(24)+2*s(28)+5 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(46) =< V1 100.39/26.50 aux(47) =< V1+V 100.39/26.50 aux(48) =< V 100.39/26.50 aux(49) =< 2*V 100.39/26.50 it(28) =< aux(47) 100.39/26.50 it(31) =< aux(49) 100.39/26.50 it(27) =< aux(47) 100.39/26.50 it(28) =< aux(48) 100.39/26.50 it(27) =< aux(49)+aux(18) 100.39/26.50 it(27) =< aux(48)+aux(46) 100.39/26.50 s(23) =< aux(48)+aux(46) 100.39/26.50 s(23) =< it(27)*aux(48) 100.39/26.50 s(24) =< aux(47) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,[31],37]: 4*it(27)+4*it(28)+8*it(31)+3*s(11)+3*s(22)+3*s(24)+6 100.39/26.50 Such that:aux(50) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(51) =< V1 100.39/26.50 aux(52) =< V1+V 100.39/26.50 aux(53) =< V 100.39/26.50 aux(54) =< 2*V 100.39/26.50 it(28) =< aux(52) 100.39/26.50 s(11) =< aux(50) 100.39/26.50 it(31) =< aux(54) 100.39/26.50 it(27) =< aux(52) 100.39/26.50 it(28) =< aux(53) 100.39/26.50 it(27) =< aux(54)+aux(18) 100.39/26.50 it(27) =< aux(53)+aux(51) 100.39/26.50 s(23) =< aux(53)+aux(51) 100.39/26.50 s(23) =< it(27)*aux(53) 100.39/26.50 s(24) =< aux(52) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,[31],35]: 4*it(27)+4*it(28)+6*it(31)+3*s(22)+3*s(24)+2*s(28)+6 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(55) =< V1 100.39/26.50 aux(56) =< V1+V 100.39/26.50 aux(57) =< V 100.39/26.50 aux(58) =< 2*V 100.39/26.50 it(28) =< aux(56) 100.39/26.50 it(31) =< aux(58) 100.39/26.50 it(27) =< aux(56) 100.39/26.50 it(28) =< aux(57) 100.39/26.50 it(27) =< aux(58)+aux(18) 100.39/26.50 it(27) =< aux(57)+aux(55) 100.39/26.50 s(23) =< aux(57)+aux(55) 100.39/26.50 s(23) =< it(27)*aux(57) 100.39/26.50 s(24) =< aux(56) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,[31],32,38]: 4*it(27)+4*it(28)+6*it(31)+7*s(13)+3*s(22)+3*s(24)+9 100.39/26.50 Such that:aux(59) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(60) =< V1 100.39/26.50 aux(61) =< V1+V 100.39/26.50 aux(62) =< V 100.39/26.50 aux(63) =< 2*V 100.39/26.50 it(28) =< aux(61) 100.39/26.50 s(13) =< aux(59) 100.39/26.50 it(31) =< aux(63) 100.39/26.50 it(27) =< aux(61) 100.39/26.50 it(28) =< aux(62) 100.39/26.50 it(27) =< aux(63)+aux(18) 100.39/26.50 it(27) =< aux(62)+aux(60) 100.39/26.50 s(23) =< aux(62)+aux(60) 100.39/26.50 s(23) =< it(27)*aux(62) 100.39/26.50 s(24) =< aux(61) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=1,V1>=3,V>=3,V+V1>=7] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,[31],32,37]: 4*it(27)+4*it(28)+6*it(31)+8*s(11)+3*s(22)+3*s(24)+10 100.39/26.50 Such that:aux(64) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(65) =< V1 100.39/26.50 aux(66) =< V1+V 100.39/26.50 aux(67) =< V 100.39/26.50 aux(68) =< 2*V 100.39/26.50 it(28) =< aux(66) 100.39/26.50 s(11) =< aux(64) 100.39/26.50 it(31) =< aux(68) 100.39/26.50 it(27) =< aux(66) 100.39/26.50 it(28) =< aux(67) 100.39/26.50 it(27) =< aux(68)+aux(18) 100.39/26.50 it(27) =< aux(67)+aux(65) 100.39/26.50 s(23) =< aux(67)+aux(65) 100.39/26.50 s(23) =< it(27)*aux(67) 100.39/26.50 s(24) =< aux(66) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,37]: 4*it(27)+4*it(28)+2*s(9)+3*s(11)+3*s(22)+3*s(24)+6 100.39/26.50 Such that:aux(69) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(70) =< V1 100.39/26.50 aux(71) =< V1+V 100.39/26.50 aux(72) =< V 100.39/26.50 aux(73) =< 2*V 100.39/26.50 it(28) =< aux(71) 100.39/26.50 s(11) =< aux(69) 100.39/26.50 s(9) =< aux(73) 100.39/26.50 it(27) =< aux(71) 100.39/26.50 it(28) =< aux(72) 100.39/26.50 it(27) =< aux(73)+aux(18) 100.39/26.50 it(27) =< aux(72)+aux(70) 100.39/26.50 s(23) =< aux(72)+aux(70) 100.39/26.50 s(23) =< it(27)*aux(72) 100.39/26.50 s(24) =< aux(71) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,35]: 4*it(27)+4*it(28)+3*s(22)+3*s(24)+2*s(28)+6 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(74) =< V1 100.39/26.50 aux(75) =< V1+V 100.39/26.50 aux(76) =< V 100.39/26.50 it(27) =< aux(75) 100.39/26.50 it(28) =< aux(75) 100.39/26.50 it(28) =< aux(76) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(76)+aux(74) 100.39/26.50 s(23) =< aux(76)+aux(74) 100.39/26.50 s(23) =< it(27)*aux(76) 100.39/26.50 s(24) =< aux(75) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,32,38]: 4*it(27)+4*it(28)+7*s(13)+3*s(22)+3*s(24)+9 100.39/26.50 Such that:aux(77) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(78) =< V1 100.39/26.50 aux(79) =< V1+V 100.39/26.50 aux(80) =< V 100.39/26.50 s(13) =< aux(77) 100.39/26.50 it(27) =< aux(79) 100.39/26.50 it(28) =< aux(79) 100.39/26.50 it(28) =< aux(80) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(80)+aux(78) 100.39/26.50 s(23) =< aux(80)+aux(78) 100.39/26.50 s(23) =< it(27)*aux(80) 100.39/26.50 s(24) =< aux(79) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],33,32,37]: 4*it(27)+4*it(28)+8*s(11)+3*s(22)+3*s(24)+10 100.39/26.50 Such that:aux(81) =< 1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(82) =< V1 100.39/26.50 aux(83) =< V1+V 100.39/26.50 aux(84) =< V 100.39/26.50 s(11) =< aux(81) 100.39/26.50 it(27) =< aux(83) 100.39/26.50 it(28) =< aux(83) 100.39/26.50 it(28) =< aux(84) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(84)+aux(82) 100.39/26.50 s(23) =< aux(84)+aux(82) 100.39/26.50 s(23) =< it(27)*aux(84) 100.39/26.50 s(24) =< aux(83) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 100.39/26.50 100.39/26.50 * Chain [[27,28],30,38]: 4*it(27)+4*it(28)+3*s(22)+3*s(24)+6*s(29)+5 100.39/26.50 Such that:aux(29) =< V1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(30) =< V1+V 100.39/26.50 aux(31) =< V1+V-2*Out 100.39/26.50 aux(32) =< V1-Out 100.39/26.50 it(28) =< V1/2+V/2-Out 100.39/26.50 aux(33) =< V 100.39/26.50 aux(34) =< V-Out 100.39/26.50 aux(17) =< 2*V-2*Out 100.39/26.50 aux(85) =< Out 100.39/26.50 s(29) =< aux(85) 100.39/26.50 it(27) =< aux(30) 100.39/26.50 it(28) =< aux(30) 100.39/26.50 s(25) =< aux(30) 100.39/26.50 it(27) =< aux(31) 100.39/26.50 it(28) =< aux(31) 100.39/26.50 s(25) =< aux(31) 100.39/26.50 aux(15) =< aux(33) 100.39/26.50 it(28) =< aux(33) 100.39/26.50 aux(15) =< aux(34) 100.39/26.50 it(28) =< aux(34) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(15)+aux(29) 100.39/26.50 s(23) =< aux(15)+aux(32) 100.39/26.50 s(23) =< aux(15)+aux(29) 100.39/26.50 it(27) =< aux(15)+aux(32) 100.39/26.50 s(23) =< it(27)*aux(33) 100.39/26.50 s(24) =< s(25) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out>=2,V1>=Out+1,V>=Out+1,V+V1>=3*Out+2] 100.39/26.50 100.39/26.50 * Chain [[27,28],30,37]: 4*it(27)+4*it(28)+7*s(11)+3*s(22)+3*s(24)+6 100.39/26.50 Such that:aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(17) =< 2*V 100.39/26.50 aux(87) =< V1 100.39/26.50 aux(88) =< V1+V 100.39/26.50 aux(89) =< V 100.39/26.50 s(11) =< aux(87) 100.39/26.50 it(27) =< aux(88) 100.39/26.50 it(28) =< aux(88) 100.39/26.50 it(28) =< aux(89) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(89)+aux(87) 100.39/26.50 s(23) =< aux(89)+aux(87) 100.39/26.50 s(23) =< it(27)*aux(89) 100.39/26.50 s(24) =< aux(88) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 100.39/26.50 100.39/26.50 * Chain [[27,28],29,38]: 4*it(27)+4*it(28)+3*s(22)+3*s(24)+6*s(32)+5 100.39/26.50 Such that:aux(29) =< V1 100.39/26.50 aux(18) =< V1-V+1 100.39/26.50 aux(30) =< V1+V 100.39/26.50 aux(31) =< V1+V-2*Out 100.39/26.50 aux(32) =< V1-Out 100.39/26.50 it(28) =< V1/2+V/2-Out 100.39/26.50 aux(33) =< V 100.39/26.50 aux(34) =< V-Out 100.39/26.50 aux(17) =< 2*V-2*Out 100.39/26.50 aux(90) =< Out 100.39/26.50 s(32) =< aux(90) 100.39/26.50 it(27) =< aux(30) 100.39/26.50 it(28) =< aux(30) 100.39/26.50 s(25) =< aux(30) 100.39/26.50 it(27) =< aux(31) 100.39/26.50 it(28) =< aux(31) 100.39/26.50 s(25) =< aux(31) 100.39/26.50 aux(15) =< aux(33) 100.39/26.50 it(28) =< aux(33) 100.39/26.50 aux(15) =< aux(34) 100.39/26.50 it(28) =< aux(34) 100.39/26.50 it(27) =< aux(17)+aux(18) 100.39/26.50 it(27) =< aux(15)+aux(29) 100.39/26.50 s(23) =< aux(15)+aux(32) 100.39/26.50 s(23) =< aux(15)+aux(29) 100.39/26.50 it(27) =< aux(15)+aux(32) 100.39/26.50 s(23) =< it(27)*aux(33) 100.39/26.50 s(24) =< s(25) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out>=2,V1>=Out,V>=Out,V+V1>=3*Out] 100.39/26.50 100.39/26.50 * Chain [[27,28],29,37]: 4*it(27)+4*it(28)+7*s(11)+3*s(22)+3*s(24)+6 100.39/26.50 Such that:aux(18) =< V1-V+1 100.39/26.50 it(28) =< V1/2+V/2 100.39/26.50 aux(92) =< V1 100.39/26.50 aux(93) =< V1+V 100.39/26.50 aux(94) =< V 100.39/26.50 aux(95) =< 2*V 100.39/26.50 s(11) =< aux(95) 100.39/26.50 it(27) =< aux(93) 100.39/26.50 it(28) =< aux(93) 100.39/26.50 it(28) =< aux(94) 100.39/26.50 it(27) =< aux(95)+aux(18) 100.39/26.50 it(27) =< aux(94)+aux(92) 100.39/26.50 s(23) =< aux(94)+aux(92) 100.39/26.50 s(23) =< it(27)*aux(94) 100.39/26.50 s(24) =< aux(93) 100.39/26.50 s(22) =< s(23) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=2,V+V1>=6] 100.39/26.50 100.39/26.50 * Chain [38]: 1 100.39/26.50 with precondition: [V1=0,V=Out,V>=0] 100.39/26.50 100.39/26.50 * Chain [37]: 2*s(9)+1*s(11)+2 100.39/26.50 Such that:s(11) =< V 100.39/26.50 aux(7) =< V1 100.39/26.50 s(9) =< aux(7) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=0,V>=0] 100.39/26.50 100.39/26.50 * Chain [36]: 1 100.39/26.50 with precondition: [V=0,V1=Out,V1>=1] 100.39/26.50 100.39/26.50 * Chain [35]: 2 100.39/26.50 with precondition: [V=1,Out=0,V1>=1] 100.39/26.50 100.39/26.50 * Chain [34,38]: 5*s(27)+5 100.39/26.50 Such that:s(26) =< 1 100.39/26.50 s(27) =< s(26) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=1,V>=2] 100.39/26.50 100.39/26.50 * Chain [34,37]: 6*s(11)+6 100.39/26.50 Such that:aux(42) =< 1 100.39/26.50 s(11) =< aux(42) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,[31],38]: 6*it(31)+2*s(28)+5 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 aux(6) =< V 100.39/26.50 it(31) =< aux(6) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=1,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,[31],37]: 8*it(31)+3*s(11)+6 100.39/26.50 Such that:aux(8) =< V 100.39/26.50 aux(50) =< 1 100.39/26.50 s(11) =< aux(50) 100.39/26.50 it(31) =< aux(8) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,[31],35]: 6*it(31)+2*s(28)+6 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 aux(9) =< V 100.39/26.50 it(31) =< aux(9) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=3] 100.39/26.50 100.39/26.50 * Chain [33,[31],32,38]: 6*it(31)+7*s(13)+9 100.39/26.50 Such that:aux(10) =< V 100.39/26.50 aux(59) =< 1 100.39/26.50 s(13) =< aux(59) 100.39/26.50 it(31) =< aux(10) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=1,V>=3] 100.39/26.50 100.39/26.50 * Chain [33,[31],32,37]: 6*it(31)+8*s(11)+10 100.39/26.50 Such that:aux(12) =< V 100.39/26.50 aux(64) =< 1 100.39/26.50 s(11) =< aux(64) 100.39/26.50 it(31) =< aux(12) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=3] 100.39/26.50 100.39/26.50 * Chain [33,37]: 2*s(9)+3*s(11)+6 100.39/26.50 Such that:aux(7) =< V 100.39/26.50 aux(69) =< 1 100.39/26.50 s(11) =< aux(69) 100.39/26.50 s(9) =< aux(7) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,35]: 2*s(28)+6 100.39/26.50 Such that:s(28) =< 1 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,32,38]: 7*s(13)+9 100.39/26.50 Such that:aux(77) =< 1 100.39/26.50 s(13) =< aux(77) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=1,V>=2] 100.39/26.50 100.39/26.50 * Chain [33,32,37]: 8*s(11)+10 100.39/26.50 Such that:aux(81) =< 1 100.39/26.50 s(11) =< aux(81) 100.39/26.50 100.39/26.50 with precondition: [V1=1,Out=0,V>=2] 100.39/26.50 100.39/26.50 * Chain [32,38]: 5*s(13)+5 100.39/26.50 Such that:s(12) =< 1 100.39/26.50 s(13) =< s(12) 100.39/26.50 100.39/26.50 with precondition: [V=1,Out=1,V1>=1] 100.39/26.50 100.39/26.50 * Chain [32,37]: 6*s(11)+6 100.39/26.50 Such that:aux(11) =< 1 100.39/26.50 s(11) =< aux(11) 100.39/26.50 100.39/26.50 with precondition: [V=1,Out=0,V1>=1] 100.39/26.50 100.39/26.50 * Chain [30,38]: 6*s(29)+5 100.39/26.50 Such that:aux(85) =< Out 100.39/26.50 s(29) =< aux(85) 100.39/26.50 100.39/26.50 with precondition: [V1=Out,V1>=2,V>=V1+1] 100.39/26.50 100.39/26.50 * Chain [30,37]: 7*s(11)+6 100.39/26.50 Such that:aux(86) =< V1 100.39/26.50 s(11) =< aux(86) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V1>=2,V>=V1+1] 100.39/26.50 100.39/26.50 * Chain [29,38]: 6*s(32)+5 100.39/26.50 Such that:aux(90) =< Out 100.39/26.50 s(32) =< aux(90) 100.39/26.50 100.39/26.50 with precondition: [V=Out,V>=2,V1>=V] 100.39/26.50 100.39/26.50 * Chain [29,37]: 7*s(11)+6 100.39/26.50 Such that:aux(91) =< V 100.39/26.50 s(11) =< aux(91) 100.39/26.50 100.39/26.50 with precondition: [Out=0,V>=2,V1>=V] 100.39/26.50 100.39/26.50 100.39/26.50 #### Cost of chains of start(V1,V,V15): 100.39/26.50 * Chain [48]: 45*s(309)+17*s(311)+85*s(322)+36*s(323)+68*s(325)+72*s(327)+51*s(328)+32*s(329)+41*s(330)+10 100.39/26.50 Such that:aux(129) =< 1 100.39/26.50 aux(130) =< V1 100.39/26.50 aux(131) =< V1-V+1 100.39/26.50 aux(132) =< V1+V 100.39/26.50 aux(133) =< V1/2+V/2 100.39/26.50 aux(134) =< V 100.39/26.50 aux(135) =< 2*V 100.39/26.50 s(311) =< aux(130) 100.39/26.50 s(309) =< aux(134) 100.39/26.50 s(322) =< aux(129) 100.39/26.50 s(323) =< aux(133) 100.39/26.50 s(325) =< aux(132) 100.39/26.50 s(323) =< aux(132) 100.39/26.50 s(323) =< aux(134) 100.39/26.50 s(325) =< aux(135)+aux(131) 100.39/26.50 s(325) =< aux(134)+aux(130) 100.39/26.50 s(326) =< aux(134)+aux(130) 100.39/26.50 s(326) =< s(325)*aux(134) 100.39/26.50 s(327) =< aux(132) 100.39/26.50 s(328) =< s(326) 100.39/26.50 s(329) =< aux(132) 100.39/26.50 s(330) =< aux(135) 100.39/26.50 s(329) =< aux(134) 100.39/26.50 100.39/26.50 with precondition: [V1>=0] 100.39/26.50 100.39/26.50 * Chain [47]: 213*s(381)+40*s(384)+121*s(386)+137*s(392)+20*s(403)+40*s(405)+30*s(408)+70*s(410)+36*s(421)+68*s(423)+51*s(426)+32*s(427)+16*s(429)+12 100.39/26.50 Such that:s(397) =< -V+1 100.39/26.50 s(399) =< V/2 100.39/26.50 s(376) =< V15+1 100.39/26.50 aux(145) =< 1 100.39/26.50 aux(146) =< -2*V+V15+1 100.39/26.50 aux(147) =< -V+V15 100.39/26.50 aux(148) =< V 100.39/26.50 aux(149) =< 2*V 100.39/26.50 aux(150) =< V15 100.39/26.50 aux(151) =< V15/2 100.39/26.50 s(381) =< aux(145) 100.39/26.50 s(384) =< aux(150) 100.39/26.50 s(384) =< s(376) 100.39/26.50 s(386) =< aux(150) 100.39/26.50 s(403) =< s(399) 100.39/26.50 s(392) =< aux(148) 100.39/26.50 s(405) =< aux(148) 100.39/26.50 s(403) =< aux(148) 100.39/26.50 s(405) =< aux(149)+s(397) 100.39/26.50 s(406) =< aux(148) 100.39/26.50 s(406) =< s(405)*aux(148) 100.39/26.50 s(408) =< s(406) 100.39/26.50 s(410) =< aux(149) 100.39/26.50 s(421) =< aux(151) 100.39/26.50 s(423) =< aux(150) 100.39/26.50 s(421) =< aux(150) 100.39/26.50 s(421) =< aux(148) 100.39/26.50 s(423) =< aux(149)+aux(146) 100.39/26.50 s(423) =< aux(148)+aux(147) 100.39/26.50 s(424) =< aux(148)+aux(147) 100.39/26.50 s(424) =< s(423)*aux(148) 100.39/26.50 s(426) =< s(424) 100.39/26.50 s(427) =< aux(150) 100.39/26.50 s(427) =< aux(148) 100.39/26.50 s(429) =< aux(147) 100.39/26.50 100.39/26.50 with precondition: [V1=1,V>=0,V15>=0] 100.39/26.50 100.39/26.50 * Chain [46]: 50*s(475)+32*s(479)+32*s(488)+8 100.39/26.50 Such that:aux(154) =< 1 100.39/26.50 aux(155) =< V1 100.39/26.50 aux(156) =< V15 100.39/26.50 s(475) =< aux(154) 100.39/26.50 s(488) =< aux(155) 100.39/26.50 s(479) =< aux(156) 100.39/26.50 100.39/26.50 with precondition: [V=1,V1>=1] 100.39/26.50 100.39/26.50 * Chain [45]: 26*s(493)+42*s(496)+11 100.39/26.50 Such that:aux(158) =< 1 100.39/26.50 aux(159) =< V 100.39/26.50 s(493) =< aux(159) 100.39/26.50 s(496) =< aux(158) 100.39/26.50 100.39/26.50 with precondition: [V1=1,V>=2] 100.39/26.50 100.39/26.50 * Chain [44]: 2*s(502)+3 100.39/26.50 Such that:s(502) =< V15 100.39/26.50 100.39/26.50 with precondition: [V1=1,V=V15,V>=1] 100.39/26.50 100.39/26.50 * Chain [43]: 213*s(510)+40*s(513)+121*s(515)+137*s(521)+20*s(532)+40*s(534)+30*s(537)+70*s(539)+36*s(550)+68*s(552)+51*s(555)+32*s(556)+16*s(558)+12 100.39/26.50 Such that:s(505) =< V+1 100.39/26.50 s(526) =< -V15+1 100.39/26.50 s(528) =< V15/2 100.39/26.50 aux(169) =< 1 100.39/26.50 aux(170) =< V 100.39/26.50 aux(171) =< V-2*V15+1 100.39/26.50 aux(172) =< V-V15 100.39/26.50 aux(173) =< V/2 100.39/26.50 aux(174) =< V15 100.39/26.50 aux(175) =< 2*V15 100.39/26.50 s(510) =< aux(169) 100.39/26.50 s(513) =< aux(170) 100.39/26.50 s(513) =< s(505) 100.39/26.50 s(515) =< aux(170) 100.39/26.50 s(532) =< s(528) 100.39/26.50 s(521) =< aux(174) 100.39/26.50 s(534) =< aux(174) 100.39/26.50 s(532) =< aux(174) 100.39/26.50 s(534) =< aux(175)+s(526) 100.39/26.50 s(535) =< aux(174) 100.39/26.50 s(535) =< s(534)*aux(174) 100.39/26.50 s(537) =< s(535) 100.39/26.50 s(539) =< aux(175) 100.39/26.50 s(550) =< aux(173) 100.39/26.50 s(552) =< aux(170) 100.39/26.50 s(550) =< aux(170) 100.39/26.50 s(550) =< aux(174) 100.39/26.50 s(552) =< aux(175)+aux(171) 100.39/26.50 s(552) =< aux(174)+aux(172) 100.39/26.50 s(553) =< aux(174)+aux(172) 100.39/26.50 s(553) =< s(552)*aux(174) 100.39/26.50 s(555) =< s(553) 100.39/26.50 s(556) =< aux(170) 100.39/26.50 s(556) =< aux(174) 100.39/26.50 s(558) =< aux(172) 100.39/26.50 100.39/26.50 with precondition: [V1=2,V>=0,V15>=0] 100.39/26.50 100.39/26.50 * Chain [42]: 27*s(604)+32*s(608)+8 100.39/26.50 Such that:aux(178) =< 1 100.39/26.50 aux(179) =< V 100.39/26.50 s(604) =< aux(178) 100.39/26.50 s(608) =< aux(179) 100.39/26.50 100.39/26.50 with precondition: [V1=2,V15=1,V>=2] 100.39/26.50 100.39/26.50 * Chain [41]: 2*s(614)+3 100.39/26.50 Such that:s(614) =< V15 100.39/26.50 100.39/26.50 with precondition: [V1=2,V=V15,V>=1] 100.39/26.50 100.39/26.50 * Chain [40]: 14*s(615)+21*s(618)+11 100.39/26.50 Such that:s(616) =< 1 100.39/26.50 aux(180) =< V 100.39/26.50 s(615) =< aux(180) 100.39/26.50 s(618) =< s(616) 100.39/26.50 100.39/26.50 with precondition: [V1=2,V=V15+1,V>=3] 100.39/26.50 100.39/26.50 * Chain [39]: 1 100.39/26.50 with precondition: [V=0,V1>=0] 100.39/26.50 100.39/26.50 100.39/26.50 Closed-form bounds of start(V1,V,V15): 100.39/26.50 ------------------------------------- 100.39/26.50 * Chain [48] with precondition: [V1>=0] 100.39/26.50 - Upper bound: 68*V1+95+nat(V)*96+nat(2*V)*41+nat(V1+V)*172+nat(V1/2+V/2)*36 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [47] with precondition: [V1=1,V>=0,V15>=0] 100.39/26.50 - Upper bound: 398*V+261*V15+225+nat(-V+V15)*67+10*V+18*V15 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [46] with precondition: [V=1,V1>=1] 100.39/26.50 - Upper bound: 32*V1+58+nat(V15)*32 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [45] with precondition: [V1=1,V>=2] 100.39/26.50 - Upper bound: 26*V+53 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [44] with precondition: [V1=1,V=V15,V>=1] 100.39/26.50 - Upper bound: 2*V15+3 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [43] with precondition: [V1=2,V>=0,V15>=0] 100.39/26.50 - Upper bound: 261*V+398*V15+225+nat(V-V15)*67+18*V+10*V15 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [42] with precondition: [V1=2,V15=1,V>=2] 100.39/26.50 - Upper bound: 32*V+35 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [41] with precondition: [V1=2,V=V15,V>=1] 100.39/26.50 - Upper bound: 2*V15+3 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [40] with precondition: [V1=2,V=V15+1,V>=3] 100.39/26.50 - Upper bound: 14*V+32 100.39/26.50 - Complexity: n 100.39/26.50 * Chain [39] with precondition: [V=0,V1>=0] 100.39/26.50 - Upper bound: 1 100.39/26.50 - Complexity: constant 100.39/26.50 100.39/26.50 ### Maximum cost of start(V1,V,V15): max([32*V1+55+nat(V15)*30+(nat(V15)*2+2),nat(V)*12+3+max([18,nat(V)*64+60+max([nat(2*V)*41+68*V1+nat(V1+V)*172+nat(V1/2+V/2)*36,nat(V)*162+130+nat(V15)*258+nat(V/2)*20+nat(V15/2)*20+max([nat(2*V)*70+nat(V15)*3+nat(-V+V15)*67+nat(V15/2)*16,nat(2*V15)*70+nat(V)*3+nat(V-V15)*67+nat(V/2)*16])])+nat(V)*6])+(nat(V)*14+31)])+1 100.39/26.50 Asymptotic class: n 100.39/26.50 * Total analysis performed in 1225 ms. 100.39/26.50 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (10) 100.39/26.50 BOUNDS(1, n^1) 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 100.39/26.50 Renamed function symbols to avoid clashes with predefined symbol. 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (12) 100.39/26.50 Obligation: 100.39/26.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 100.39/26.50 100.39/26.50 100.39/26.50 The TRS R consists of the following rules: 100.39/26.50 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 S is empty. 100.39/26.50 Rewrite Strategy: INNERMOST 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 100.39/26.50 Infered types. 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (14) 100.39/26.50 Obligation: 100.39/26.50 Innermost TRS: 100.39/26.50 Rules: 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 Types: 100.39/26.50 le :: 0':s -> 0':s -> true:false 100.39/26.50 0' :: 0':s 100.39/26.50 true :: true:false 100.39/26.50 s :: 0':s -> 0':s 100.39/26.50 false :: true:false 100.39/26.50 pred :: 0':s -> 0':s 100.39/26.50 minus :: 0':s -> 0':s -> 0':s 100.39/26.50 gcd :: 0':s -> 0':s -> 0':s 100.39/26.50 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 100.39/26.50 hole_true:false1_0 :: true:false 100.39/26.50 hole_0':s2_0 :: 0':s 100.39/26.50 gen_0':s3_0 :: Nat -> 0':s 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (15) OrderProof (LOWER BOUND(ID)) 100.39/26.50 Heuristically decided to analyse the following defined symbols: 100.39/26.50 le, minus, gcd 100.39/26.50 100.39/26.50 They will be analysed ascendingly in the following order: 100.39/26.50 le < gcd 100.39/26.50 minus < gcd 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (16) 100.39/26.50 Obligation: 100.39/26.50 Innermost TRS: 100.39/26.50 Rules: 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 Types: 100.39/26.50 le :: 0':s -> 0':s -> true:false 100.39/26.50 0' :: 0':s 100.39/26.50 true :: true:false 100.39/26.50 s :: 0':s -> 0':s 100.39/26.50 false :: true:false 100.39/26.50 pred :: 0':s -> 0':s 100.39/26.50 minus :: 0':s -> 0':s -> 0':s 100.39/26.50 gcd :: 0':s -> 0':s -> 0':s 100.39/26.50 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 100.39/26.50 hole_true:false1_0 :: true:false 100.39/26.50 hole_0':s2_0 :: 0':s 100.39/26.50 gen_0':s3_0 :: Nat -> 0':s 100.39/26.50 100.39/26.50 100.39/26.50 Generator Equations: 100.39/26.50 gen_0':s3_0(0) <=> 0' 100.39/26.50 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 100.39/26.50 100.39/26.50 100.39/26.50 The following defined symbols remain to be analysed: 100.39/26.50 le, minus, gcd 100.39/26.50 100.39/26.50 They will be analysed ascendingly in the following order: 100.39/26.50 le < gcd 100.39/26.50 minus < gcd 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (17) RewriteLemmaProof (LOWER BOUND(ID)) 100.39/26.50 Proved the following rewrite lemma: 100.39/26.50 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 100.39/26.50 100.39/26.50 Induction Base: 100.39/26.50 le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 100.39/26.50 true 100.39/26.50 100.39/26.50 Induction Step: 100.39/26.50 le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 100.39/26.50 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 100.39/26.50 true 100.39/26.50 100.39/26.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (18) 100.39/26.50 Complex Obligation (BEST) 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (19) 100.39/26.50 Obligation: 100.39/26.50 Proved the lower bound n^1 for the following obligation: 100.39/26.50 100.39/26.50 Innermost TRS: 100.39/26.50 Rules: 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 Types: 100.39/26.50 le :: 0':s -> 0':s -> true:false 100.39/26.50 0' :: 0':s 100.39/26.50 true :: true:false 100.39/26.50 s :: 0':s -> 0':s 100.39/26.50 false :: true:false 100.39/26.50 pred :: 0':s -> 0':s 100.39/26.50 minus :: 0':s -> 0':s -> 0':s 100.39/26.50 gcd :: 0':s -> 0':s -> 0':s 100.39/26.50 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 100.39/26.50 hole_true:false1_0 :: true:false 100.39/26.50 hole_0':s2_0 :: 0':s 100.39/26.50 gen_0':s3_0 :: Nat -> 0':s 100.39/26.50 100.39/26.50 100.39/26.50 Generator Equations: 100.39/26.50 gen_0':s3_0(0) <=> 0' 100.39/26.50 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 100.39/26.50 100.39/26.50 100.39/26.50 The following defined symbols remain to be analysed: 100.39/26.50 le, minus, gcd 100.39/26.50 100.39/26.50 They will be analysed ascendingly in the following order: 100.39/26.50 le < gcd 100.39/26.50 minus < gcd 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (20) LowerBoundPropagationProof (FINISHED) 100.39/26.50 Propagated lower bound. 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (21) 100.39/26.50 BOUNDS(n^1, INF) 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (22) 100.39/26.50 Obligation: 100.39/26.50 Innermost TRS: 100.39/26.50 Rules: 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 Types: 100.39/26.50 le :: 0':s -> 0':s -> true:false 100.39/26.50 0' :: 0':s 100.39/26.50 true :: true:false 100.39/26.50 s :: 0':s -> 0':s 100.39/26.50 false :: true:false 100.39/26.50 pred :: 0':s -> 0':s 100.39/26.50 minus :: 0':s -> 0':s -> 0':s 100.39/26.50 gcd :: 0':s -> 0':s -> 0':s 100.39/26.50 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 100.39/26.50 hole_true:false1_0 :: true:false 100.39/26.50 hole_0':s2_0 :: 0':s 100.39/26.50 gen_0':s3_0 :: Nat -> 0':s 100.39/26.50 100.39/26.50 100.39/26.50 Lemmas: 100.39/26.50 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 100.39/26.50 100.39/26.50 100.39/26.50 Generator Equations: 100.39/26.50 gen_0':s3_0(0) <=> 0' 100.39/26.50 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 100.39/26.50 100.39/26.50 100.39/26.50 The following defined symbols remain to be analysed: 100.39/26.50 minus, gcd 100.39/26.50 100.39/26.50 They will be analysed ascendingly in the following order: 100.39/26.50 minus < gcd 100.39/26.50 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (23) RewriteLemmaProof (LOWER BOUND(ID)) 100.39/26.50 Proved the following rewrite lemma: 100.39/26.50 minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n306_0))) -> *4_0, rt in Omega(n306_0) 100.39/26.50 100.39/26.50 Induction Base: 100.39/26.50 minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0))) 100.39/26.50 100.39/26.50 Induction Step: 100.39/26.50 minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n306_0, 1)))) ->_R^Omega(1) 100.39/26.50 pred(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n306_0)))) ->_IH 100.39/26.50 pred(*4_0) 100.39/26.50 100.39/26.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 100.39/26.50 ---------------------------------------- 100.39/26.50 100.39/26.50 (24) 100.39/26.50 Obligation: 100.39/26.50 Innermost TRS: 100.39/26.50 Rules: 100.39/26.50 le(0', y) -> true 100.39/26.50 le(s(x), 0') -> false 100.39/26.50 le(s(x), s(y)) -> le(x, y) 100.39/26.50 pred(s(x)) -> x 100.39/26.50 minus(x, 0') -> x 100.39/26.50 minus(x, s(y)) -> pred(minus(x, y)) 100.39/26.50 gcd(0', y) -> y 100.39/26.50 gcd(s(x), 0') -> s(x) 100.39/26.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 100.39/26.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 100.39/26.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 100.39/26.50 100.39/26.50 Types: 100.39/26.50 le :: 0':s -> 0':s -> true:false 100.39/26.50 0' :: 0':s 100.39/26.50 true :: true:false 100.39/26.50 s :: 0':s -> 0':s 100.39/26.50 false :: true:false 100.39/26.50 pred :: 0':s -> 0':s 100.39/26.50 minus :: 0':s -> 0':s -> 0':s 100.39/26.50 gcd :: 0':s -> 0':s -> 0':s 100.39/26.50 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 100.39/26.50 hole_true:false1_0 :: true:false 100.39/26.50 hole_0':s2_0 :: 0':s 100.39/26.50 gen_0':s3_0 :: Nat -> 0':s 100.39/26.50 100.39/26.50 100.39/26.50 Lemmas: 100.39/26.50 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 100.39/26.50 minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n306_0))) -> *4_0, rt in Omega(n306_0) 100.39/26.50 100.39/26.50 100.39/26.50 Generator Equations: 100.39/26.50 gen_0':s3_0(0) <=> 0' 100.39/26.50 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 100.39/26.50 100.39/26.50 100.39/26.50 The following defined symbols remain to be analysed: 100.39/26.50 gcd 100.44/26.56 EOF