29.22/10.39 WORST_CASE(Omega(n^1), O(n^1)) 29.22/10.40 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 29.22/10.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.22/10.40 29.22/10.40 29.22/10.40 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.22/10.40 29.22/10.40 (0) CpxTRS 29.22/10.40 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 29.22/10.40 (2) CpxWeightedTrs 29.22/10.40 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 29.22/10.40 (4) CpxTypedWeightedTrs 29.22/10.40 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 29.22/10.40 (6) CpxTypedWeightedCompleteTrs 29.22/10.40 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 29.22/10.40 (8) CpxRNTS 29.22/10.40 (9) CompleteCoflocoProof [FINISHED, 1382 ms] 29.22/10.40 (10) BOUNDS(1, n^1) 29.22/10.40 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 29.22/10.40 (12) CpxTRS 29.22/10.40 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 29.22/10.40 (14) typed CpxTrs 29.22/10.40 (15) OrderProof [LOWER BOUND(ID), 0 ms] 29.22/10.40 (16) typed CpxTrs 29.22/10.40 (17) RewriteLemmaProof [LOWER BOUND(ID), 302 ms] 29.22/10.40 (18) BEST 29.22/10.40 (19) proven lower bound 29.22/10.40 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 29.22/10.40 (21) BOUNDS(n^1, INF) 29.22/10.40 (22) typed CpxTrs 29.22/10.40 (23) RewriteLemmaProof [LOWER BOUND(ID), 76 ms] 29.22/10.40 (24) typed CpxTrs 29.22/10.40 29.22/10.40 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (0) 29.22/10.40 Obligation: 29.22/10.40 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.22/10.40 29.22/10.40 29.22/10.40 The TRS R consists of the following rules: 29.22/10.40 29.22/10.40 le(0, y) -> true 29.22/10.40 le(s(x), 0) -> false 29.22/10.40 le(s(x), s(y)) -> le(x, y) 29.22/10.40 minus(x, 0) -> x 29.22/10.40 minus(s(x), s(y)) -> minus(x, y) 29.22/10.40 gcd(0, y) -> y 29.22/10.40 gcd(s(x), 0) -> s(x) 29.22/10.40 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.22/10.40 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.22/10.40 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.22/10.40 29.22/10.40 S is empty. 29.22/10.40 Rewrite Strategy: INNERMOST 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 29.22/10.40 Transformed relative TRS to weighted TRS 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (2) 29.22/10.40 Obligation: 29.22/10.40 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 29.22/10.40 29.22/10.40 29.22/10.40 The TRS R consists of the following rules: 29.22/10.40 29.22/10.40 le(0, y) -> true [1] 29.22/10.40 le(s(x), 0) -> false [1] 29.22/10.40 le(s(x), s(y)) -> le(x, y) [1] 29.22/10.40 minus(x, 0) -> x [1] 29.22/10.40 minus(s(x), s(y)) -> minus(x, y) [1] 29.22/10.40 gcd(0, y) -> y [1] 29.22/10.40 gcd(s(x), 0) -> s(x) [1] 29.22/10.40 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 29.22/10.40 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] 29.22/10.40 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] 29.22/10.40 29.22/10.40 Rewrite Strategy: INNERMOST 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 29.22/10.40 Infered types. 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (4) 29.22/10.40 Obligation: 29.22/10.40 Runtime Complexity Weighted TRS with Types. 29.22/10.40 The TRS R consists of the following rules: 29.22/10.40 29.22/10.40 le(0, y) -> true [1] 29.22/10.40 le(s(x), 0) -> false [1] 29.22/10.40 le(s(x), s(y)) -> le(x, y) [1] 29.22/10.40 minus(x, 0) -> x [1] 29.22/10.40 minus(s(x), s(y)) -> minus(x, y) [1] 29.22/10.40 gcd(0, y) -> y [1] 29.22/10.40 gcd(s(x), 0) -> s(x) [1] 29.22/10.40 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 29.22/10.40 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] 29.22/10.40 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] 29.22/10.40 29.22/10.40 The TRS has the following type information: 29.22/10.40 le :: 0:s -> 0:s -> true:false 29.22/10.40 0 :: 0:s 29.22/10.40 true :: true:false 29.22/10.40 s :: 0:s -> 0:s 29.22/10.40 false :: true:false 29.22/10.40 minus :: 0:s -> 0:s -> 0:s 29.22/10.40 gcd :: 0:s -> 0:s -> 0:s 29.22/10.40 if_gcd :: true:false -> 0:s -> 0:s -> 0:s 29.22/10.40 29.22/10.40 Rewrite Strategy: INNERMOST 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (5) CompletionProof (UPPER BOUND(ID)) 29.22/10.40 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 29.22/10.40 29.22/10.40 minus(v0, v1) -> null_minus [0] 29.22/10.40 if_gcd(v0, v1, v2) -> null_if_gcd [0] 29.22/10.40 le(v0, v1) -> null_le [0] 29.22/10.40 gcd(v0, v1) -> null_gcd [0] 29.22/10.40 29.22/10.40 And the following fresh constants: null_minus, null_if_gcd, null_le, null_gcd 29.22/10.40 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (6) 29.22/10.40 Obligation: 29.22/10.40 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 29.22/10.40 29.22/10.40 Runtime Complexity Weighted TRS with Types. 29.22/10.40 The TRS R consists of the following rules: 29.22/10.40 29.22/10.40 le(0, y) -> true [1] 29.22/10.40 le(s(x), 0) -> false [1] 29.22/10.40 le(s(x), s(y)) -> le(x, y) [1] 29.22/10.40 minus(x, 0) -> x [1] 29.22/10.40 minus(s(x), s(y)) -> minus(x, y) [1] 29.22/10.40 gcd(0, y) -> y [1] 29.22/10.40 gcd(s(x), 0) -> s(x) [1] 29.22/10.40 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] 29.22/10.40 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] 29.22/10.40 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] 29.22/10.40 minus(v0, v1) -> null_minus [0] 29.22/10.40 if_gcd(v0, v1, v2) -> null_if_gcd [0] 29.22/10.40 le(v0, v1) -> null_le [0] 29.22/10.40 gcd(v0, v1) -> null_gcd [0] 29.22/10.40 29.22/10.40 The TRS has the following type information: 29.22/10.40 le :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> true:false:null_le 29.22/10.40 0 :: 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 true :: true:false:null_le 29.22/10.40 s :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 false :: true:false:null_le 29.22/10.40 minus :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 gcd :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 if_gcd :: true:false:null_le -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 null_minus :: 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 null_if_gcd :: 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 null_le :: true:false:null_le 29.22/10.40 null_gcd :: 0:s:null_minus:null_if_gcd:null_gcd 29.22/10.40 29.22/10.40 Rewrite Strategy: INNERMOST 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 29.22/10.40 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 29.22/10.40 The constant constructors are abstracted as follows: 29.22/10.40 29.22/10.40 0 => 0 29.22/10.40 true => 2 29.22/10.40 false => 1 29.22/10.40 null_minus => 0 29.22/10.40 null_if_gcd => 0 29.22/10.40 null_le => 0 29.22/10.40 null_gcd => 0 29.22/10.40 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (8) 29.22/10.40 Obligation: 29.22/10.40 Complexity RNTS consisting of the following rules: 29.22/10.40 29.22/10.40 gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y 29.22/10.40 gcd(z, z') -{ 1 }-> if_gcd(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 29.22/10.40 gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 29.22/10.40 gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 29.22/10.40 if_gcd(z, z', z'') -{ 1 }-> gcd(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y 29.22/10.40 if_gcd(z, z', z'') -{ 1 }-> gcd(minus(y, x), 1 + x) :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y 29.22/10.40 if_gcd(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 29.22/10.40 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 29.22/10.40 le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y 29.22/10.40 le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 29.22/10.40 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 29.22/10.40 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 29.22/10.40 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 29.22/10.40 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 29.22/10.40 29.22/10.40 Only complete derivations are relevant for the runtime complexity. 29.22/10.40 29.22/10.40 ---------------------------------------- 29.22/10.40 29.22/10.40 (9) CompleteCoflocoProof (FINISHED) 29.22/10.40 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 29.22/10.40 29.22/10.40 eq(start(V1, V, V14),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). 29.22/10.40 eq(start(V1, V, V14),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 29.22/10.40 eq(start(V1, V, V14),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]). 29.22/10.40 eq(start(V1, V, V14),0,[fun(V1, V, V14, Out)],[V1 >= 0,V >= 0,V14 >= 0]). 29.22/10.40 eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]). 29.22/10.40 eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]). 29.22/10.40 eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). 29.22/10.40 eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]). 29.22/10.40 eq(minus(V1, V, Out),1,[minus(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]). 29.22/10.40 eq(gcd(V1, V, Out),1,[],[Out = V9,V9 >= 0,V1 = 0,V = V9]). 29.22/10.40 eq(gcd(V1, V, Out),1,[],[Out = 1 + V10,V10 >= 0,V1 = 1 + V10,V = 0]). 29.22/10.40 eq(gcd(V1, V, Out),1,[le(V11, V12, Ret0),fun(Ret0, 1 + V12, 1 + V11, Ret2)],[Out = Ret2,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]). 29.22/10.40 eq(fun(V1, V, V14, Out),1,[minus(V15, V13, Ret01),gcd(Ret01, 1 + V13, Ret3)],[Out = Ret3,V1 = 2,V = 1 + V15,V15 >= 0,V13 >= 0,V14 = 1 + V13]). 29.22/10.40 eq(fun(V1, V, V14, Out),1,[minus(V17, V16, Ret02),gcd(Ret02, 1 + V16, Ret4)],[Out = Ret4,V = 1 + V16,V1 = 1,V16 >= 0,V17 >= 0,V14 = 1 + V17]). 29.22/10.40 eq(minus(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]). 29.22/10.40 eq(fun(V1, V, V14, Out),0,[],[Out = 0,V21 >= 0,V14 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]). 29.22/10.40 eq(le(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]). 29.22/10.40 eq(gcd(V1, V, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V1 = V25,V = V26]). 29.22/10.40 input_output_vars(le(V1,V,Out),[V1,V],[Out]). 29.22/10.40 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 29.22/10.40 input_output_vars(gcd(V1,V,Out),[V1,V],[Out]). 29.22/10.40 input_output_vars(fun(V1,V,V14,Out),[V1,V,V14],[Out]). 29.22/10.40 29.22/10.40 29.22/10.40 CoFloCo proof output: 29.22/10.40 Preprocessing Cost Relations 29.22/10.40 ===================================== 29.22/10.40 29.22/10.40 #### Computed strongly connected components 29.22/10.40 0. recursive : [le/3] 29.22/10.40 1. recursive : [minus/3] 29.22/10.40 2. recursive : [fun/4,gcd/3] 29.22/10.40 3. non_recursive : [start/3] 29.22/10.40 29.22/10.40 #### Obtained direct recursion through partial evaluation 29.22/10.40 0. SCC is partially evaluated into le/3 29.22/10.40 1. SCC is partially evaluated into minus/3 29.22/10.40 2. SCC is partially evaluated into gcd/3 29.22/10.40 3. SCC is partially evaluated into start/3 29.22/10.40 29.22/10.40 Control-Flow Refinement of Cost Relations 29.22/10.40 ===================================== 29.22/10.40 29.22/10.40 ### Specialization of cost equations le/3 29.22/10.40 * CE 19 is refined into CE [20] 29.22/10.40 * CE 17 is refined into CE [21] 29.22/10.40 * CE 16 is refined into CE [22] 29.22/10.40 * CE 18 is refined into CE [23] 29.22/10.40 29.22/10.40 29.22/10.40 ### Cost equations --> "Loop" of le/3 29.22/10.40 * CEs [23] --> Loop 15 29.22/10.40 * CEs [20] --> Loop 16 29.22/10.40 * CEs [21] --> Loop 17 29.22/10.40 * CEs [22] --> Loop 18 29.22/10.40 29.22/10.40 ### Ranking functions of CR le(V1,V,Out) 29.22/10.40 * RF of phase [15]: [V,V1] 29.22/10.40 29.22/10.40 #### Partial ranking functions of CR le(V1,V,Out) 29.22/10.40 * Partial RF of phase [15]: 29.22/10.40 - RF of loop [15:1]: 29.22/10.40 V 29.22/10.40 V1 29.22/10.40 29.22/10.40 29.22/10.40 ### Specialization of cost equations minus/3 29.22/10.40 * CE 9 is refined into CE [24] 29.22/10.40 * CE 7 is refined into CE [25] 29.22/10.40 * CE 8 is refined into CE [26] 29.22/10.40 29.22/10.40 29.22/10.40 ### Cost equations --> "Loop" of minus/3 29.22/10.40 * CEs [26] --> Loop 19 29.22/10.40 * CEs [24] --> Loop 20 29.22/10.40 * CEs [25] --> Loop 21 29.22/10.40 29.22/10.40 ### Ranking functions of CR minus(V1,V,Out) 29.22/10.40 * RF of phase [19]: [V,V1] 29.22/10.40 29.22/10.40 #### Partial ranking functions of CR minus(V1,V,Out) 29.22/10.40 * Partial RF of phase [19]: 29.22/10.40 - RF of loop [19:1]: 29.22/10.40 V 29.22/10.40 V1 29.22/10.40 29.22/10.40 29.22/10.40 ### Specialization of cost equations gcd/3 29.22/10.40 * CE 10 is refined into CE [27,28,29,30,31] 29.22/10.40 * CE 15 is refined into CE [32] 29.22/10.40 * CE 14 is refined into CE [33] 29.22/10.40 * CE 13 is refined into CE [34] 29.22/10.40 * CE 12 is refined into CE [35,36,37,38] 29.22/10.40 * CE 11 is refined into CE [39,40,41,42] 29.22/10.40 29.22/10.40 29.22/10.40 ### Cost equations --> "Loop" of gcd/3 29.22/10.40 * CEs [42] --> Loop 22 29.22/10.40 * CEs [38] --> Loop 23 29.22/10.40 * CEs [41] --> Loop 24 29.22/10.40 * CEs [37] --> Loop 25 29.22/10.40 * CEs [35] --> Loop 26 29.22/10.40 * CEs [36] --> Loop 27 29.22/10.40 * CEs [39] --> Loop 28 29.22/10.40 * CEs [40] --> Loop 29 29.22/10.40 * CEs [27] --> Loop 30 29.22/10.40 * CEs [33] --> Loop 31 29.22/10.40 * CEs [28,29,30,31,32] --> Loop 32 29.22/10.40 * CEs [34] --> Loop 33 29.22/10.40 29.22/10.40 ### Ranking functions of CR gcd(V1,V,Out) 29.22/10.40 * RF of phase [22,23]: [V1+V-3] 29.22/10.40 * RF of phase [26]: [V1] 29.22/10.40 29.22/10.40 #### Partial ranking functions of CR gcd(V1,V,Out) 29.22/10.40 * Partial RF of phase [22,23]: 29.22/10.40 - RF of loop [22:1]: 29.22/10.40 V-2 29.22/10.40 V1/2+V/2-2 29.22/10.40 - RF of loop [23:1]: 29.22/10.40 V1-1 depends on loops [22:1] 29.22/10.40 V1-V+1 depends on loops [22:1] 29.22/10.40 * Partial RF of phase [26]: 29.22/10.40 - RF of loop [26:1]: 29.22/10.40 V1 29.22/10.40 29.22/10.40 29.22/10.40 ### Specialization of cost equations start/3 29.22/10.40 * CE 3 is refined into CE [43,44,45,46,47,48,49,50,51,52,53,54] 29.22/10.40 * CE 1 is refined into CE [55] 29.22/10.40 * CE 2 is refined into CE [56,57,58,59,60,61,62,63,64,65,66,67] 29.22/10.40 * CE 4 is refined into CE [68,69,70,71,72] 29.22/10.40 * CE 5 is refined into CE [73,74,75] 29.22/10.40 * CE 6 is refined into CE [76,77,78,79,80,81,82,83,84] 29.22/10.40 29.22/10.40 29.22/10.40 ### Cost equations --> "Loop" of start/3 29.22/10.40 * CEs [80,81] --> Loop 34 29.22/10.40 * CEs [69,73,79] --> Loop 35 29.22/10.40 * CEs [51] --> Loop 36 29.22/10.40 * CEs [49] --> Loop 37 29.22/10.40 * CEs [43,44,45,46,47,48,50,52,53,54] --> Loop 38 29.22/10.40 * CEs [62] --> Loop 39 29.22/10.40 * CEs [64,78] --> Loop 40 29.22/10.40 * CEs [56,57,58,59,60,61,63,65,66,67] --> Loop 41 29.22/10.40 * CEs [55,68,70,71,72,74,75,76,77,82,83,84] --> Loop 42 29.22/10.40 29.22/10.40 ### Ranking functions of CR start(V1,V,V14) 29.22/10.40 29.22/10.40 #### Partial ranking functions of CR start(V1,V,V14) 29.22/10.40 29.22/10.40 29.22/10.40 Computing Bounds 29.22/10.40 ===================================== 29.22/10.40 29.22/10.40 #### Cost of chains of le(V1,V,Out): 29.22/10.40 * Chain [[15],18]: 1*it(15)+1 29.22/10.40 Such that:it(15) =< V1 29.22/10.40 29.22/10.40 with precondition: [Out=2,V1>=1,V>=V1] 29.22/10.40 29.22/10.40 * Chain [[15],17]: 1*it(15)+1 29.22/10.40 Such that:it(15) =< V 29.22/10.40 29.22/10.40 with precondition: [Out=1,V>=1,V1>=V+1] 29.22/10.40 29.22/10.40 * Chain [[15],16]: 1*it(15)+0 29.22/10.40 Such that:it(15) =< V 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=1,V>=1] 29.22/10.40 29.22/10.40 * Chain [18]: 1 29.22/10.40 with precondition: [V1=0,Out=2,V>=0] 29.22/10.40 29.22/10.40 * Chain [17]: 1 29.22/10.40 with precondition: [V=0,Out=1,V1>=1] 29.22/10.40 29.22/10.40 * Chain [16]: 0 29.22/10.40 with precondition: [Out=0,V1>=0,V>=0] 29.22/10.40 29.22/10.40 29.22/10.40 #### Cost of chains of minus(V1,V,Out): 29.22/10.40 * Chain [[19],21]: 1*it(19)+1 29.22/10.40 Such that:it(19) =< V 29.22/10.40 29.22/10.40 with precondition: [V1=Out+V,V>=1,V1>=V] 29.22/10.40 29.22/10.40 * Chain [[19],20]: 1*it(19)+0 29.22/10.40 Such that:it(19) =< V 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=1,V>=1] 29.22/10.40 29.22/10.40 * Chain [21]: 1 29.22/10.40 with precondition: [V=0,V1=Out,V1>=0] 29.22/10.40 29.22/10.40 * Chain [20]: 0 29.22/10.40 with precondition: [Out=0,V1>=0,V>=0] 29.22/10.40 29.22/10.40 29.22/10.40 #### Cost of chains of gcd(V1,V,Out): 29.22/10.40 * Chain [[26],33]: 4*it(26)+1 29.22/10.40 Such that:it(26) =< V1 29.22/10.40 29.22/10.40 with precondition: [V=1,Out=1,V1>=1] 29.22/10.40 29.22/10.40 * Chain [[26],32]: 6*it(26)+1*s(5)+2 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(2) =< V1 29.22/10.40 it(26) =< aux(2) 29.22/10.40 29.22/10.40 with precondition: [V=1,Out=0,V1>=1] 29.22/10.40 29.22/10.40 * Chain [[26],30]: 4*it(26)+2 29.22/10.40 Such that:it(26) =< V1 29.22/10.40 29.22/10.40 with precondition: [V=1,Out=0,V1>=2] 29.22/10.40 29.22/10.40 * Chain [[26],27,33]: 4*it(26)+4 29.22/10.40 Such that:it(26) =< V1 29.22/10.40 29.22/10.40 with precondition: [V=1,Out=1,V1>=2] 29.22/10.40 29.22/10.40 * Chain [[26],27,32]: 4*it(26)+1*s(5)+5 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 it(26) =< V1 29.22/10.40 29.22/10.40 with precondition: [V=1,Out=0,V1>=2] 29.22/10.40 29.22/10.40 * Chain [[22,23],33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+1 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 aux(20) =< V1+V 29.22/10.40 aux(21) =< V1+V-Out 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(23) =< V1/2+V/2-Out/2 29.22/10.40 aux(24) =< V 29.22/10.40 aux(25) =< V-Out 29.22/10.40 aux(7) =< 2*V-2*Out 29.22/10.40 aux(26) =< V1 29.22/10.40 it(22) =< aux(20) 29.22/10.40 it(23) =< aux(20) 29.22/10.40 s(16) =< aux(20) 29.22/10.40 it(22) =< aux(21) 29.22/10.40 it(23) =< aux(21) 29.22/10.40 s(16) =< aux(21) 29.22/10.40 it(22) =< aux(23) 29.22/10.40 it(23) =< aux(23) 29.22/10.40 aux(5) =< aux(24) 29.22/10.40 it(22) =< aux(24) 29.22/10.40 aux(5) =< aux(25) 29.22/10.40 it(22) =< aux(25) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(5)+aux(26) 29.22/10.40 s(18) =< aux(5)+aux(26) 29.22/10.40 s(18) =< it(23)*aux(24) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< s(16) 29.22/10.40 29.22/10.40 with precondition: [Out>=2,V1>=Out,V>=Out] 29.22/10.40 29.22/10.40 * Chain [[22,23],32]: 4*it(22)+4*it(23)+5*s(3)+2*s(17)+2 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(27) =< V1 29.22/10.40 aux(28) =< V1+V 29.22/10.40 aux(29) =< V 29.22/10.40 s(3) =< aux(28) 29.22/10.40 it(22) =< aux(28) 29.22/10.40 it(23) =< aux(28) 29.22/10.40 it(22) =< aux(29) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(29)+aux(27) 29.22/10.40 s(18) =< aux(29)+aux(27) 29.22/10.40 s(18) =< it(23)*aux(29) 29.22/10.40 s(17) =< s(18) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=2,V>=2] 29.22/10.40 29.22/10.40 * Chain [[22,23],29,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+4 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(30) =< V1 29.22/10.40 aux(31) =< V1+V 29.22/10.40 aux(32) =< V1/2+V/2 29.22/10.40 aux(33) =< V 29.22/10.40 it(22) =< aux(32) 29.22/10.40 it(22) =< aux(31) 29.22/10.40 it(23) =< aux(31) 29.22/10.40 it(23) =< aux(32) 29.22/10.40 it(22) =< aux(33) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(33)+aux(30) 29.22/10.40 s(18) =< aux(33)+aux(30) 29.22/10.40 s(18) =< it(23)*aux(33) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(31) 29.22/10.40 29.22/10.40 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],29,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+5 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(8) =< V1-V+1 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(34) =< V1 29.22/10.40 aux(35) =< V1+V 29.22/10.40 aux(36) =< V1/2+V/2 29.22/10.40 aux(37) =< V 29.22/10.40 it(22) =< aux(36) 29.22/10.40 it(22) =< aux(35) 29.22/10.40 it(23) =< aux(35) 29.22/10.40 it(23) =< aux(36) 29.22/10.40 it(22) =< aux(37) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(37)+aux(34) 29.22/10.40 s(18) =< aux(37)+aux(34) 29.22/10.40 s(18) =< it(23)*aux(37) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(35) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,[26],33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+5 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(38) =< V1 29.22/10.40 aux(39) =< V1+V 29.22/10.40 aux(40) =< V 29.22/10.40 it(26) =< aux(40) 29.22/10.40 it(22) =< aux(39) 29.22/10.40 it(23) =< aux(39) 29.22/10.40 it(22) =< aux(40) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(40)+aux(38) 29.22/10.40 s(18) =< aux(40)+aux(38) 29.22/10.40 s(18) =< it(23)*aux(40) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(39) 29.22/10.40 29.22/10.40 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,[26],32]: 4*it(22)+4*it(23)+6*it(26)+1*s(5)+2*s(15)+2*s(17)+6 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(41) =< V1 29.22/10.40 aux(42) =< V1+V 29.22/10.40 aux(43) =< V 29.22/10.40 aux(44) =< 2*V 29.22/10.40 it(26) =< aux(44) 29.22/10.40 it(22) =< aux(42) 29.22/10.40 it(23) =< aux(42) 29.22/10.40 it(22) =< aux(43) 29.22/10.40 it(23) =< aux(44)+aux(8) 29.22/10.40 it(23) =< aux(43)+aux(41) 29.22/10.40 s(18) =< aux(43)+aux(41) 29.22/10.40 s(18) =< it(23)*aux(43) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(42) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,[26],30]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+6 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(45) =< V1 29.22/10.40 aux(46) =< V1+V 29.22/10.40 aux(47) =< V 29.22/10.40 it(26) =< aux(47) 29.22/10.40 it(22) =< aux(46) 29.22/10.40 it(23) =< aux(46) 29.22/10.40 it(22) =< aux(47) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(47)+aux(45) 29.22/10.40 s(18) =< aux(47)+aux(45) 29.22/10.40 s(18) =< it(23)*aux(47) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(46) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,[26],27,33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+8 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(48) =< V1 29.22/10.40 aux(49) =< V1+V 29.22/10.40 aux(50) =< V 29.22/10.40 it(26) =< aux(50) 29.22/10.40 it(22) =< aux(49) 29.22/10.40 it(23) =< aux(49) 29.22/10.40 it(22) =< aux(50) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(50)+aux(48) 29.22/10.40 s(18) =< aux(50)+aux(48) 29.22/10.40 s(18) =< it(23)*aux(50) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(49) 29.22/10.40 29.22/10.40 with precondition: [Out=1,V1>=3,V>=3,V+V1>=7] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,[26],27,32]: 4*it(22)+4*it(23)+4*it(26)+1*s(5)+2*s(15)+2*s(17)+9 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(51) =< V1 29.22/10.40 aux(52) =< V1+V 29.22/10.40 aux(53) =< V 29.22/10.40 it(26) =< aux(53) 29.22/10.40 it(22) =< aux(52) 29.22/10.40 it(23) =< aux(52) 29.22/10.40 it(22) =< aux(53) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(53)+aux(51) 29.22/10.40 s(18) =< aux(53)+aux(51) 29.22/10.40 s(18) =< it(23)*aux(53) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(52) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,32]: 4*it(22)+4*it(23)+2*s(3)+1*s(5)+2*s(15)+2*s(17)+6 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(8) =< V1-V+1 29.22/10.40 it(22) =< V1/2+V/2 29.22/10.40 aux(54) =< V1 29.22/10.40 aux(55) =< V1+V 29.22/10.40 aux(56) =< V 29.22/10.40 aux(57) =< 2*V 29.22/10.40 s(3) =< aux(57) 29.22/10.40 it(22) =< aux(55) 29.22/10.40 it(23) =< aux(55) 29.22/10.40 it(22) =< aux(56) 29.22/10.40 it(23) =< aux(57)+aux(8) 29.22/10.40 it(23) =< aux(56)+aux(54) 29.22/10.40 s(18) =< aux(56)+aux(54) 29.22/10.40 s(18) =< it(23)*aux(56) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(55) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,30]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+6 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(58) =< V1 29.22/10.40 aux(59) =< V1+V 29.22/10.40 aux(60) =< V1/2+V/2 29.22/10.40 aux(61) =< V 29.22/10.40 it(22) =< aux(60) 29.22/10.40 it(22) =< aux(59) 29.22/10.40 it(23) =< aux(59) 29.22/10.40 it(23) =< aux(60) 29.22/10.40 it(22) =< aux(61) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(61)+aux(58) 29.22/10.40 s(18) =< aux(61)+aux(58) 29.22/10.40 s(18) =< it(23)*aux(61) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(59) 29.22/10.40 29.22/10.40 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,27,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+8 29.22/10.40 Such that:aux(8) =< V1-V+1 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(62) =< V1 29.22/10.40 aux(63) =< V1+V 29.22/10.40 aux(64) =< V1/2+V/2 29.22/10.40 aux(65) =< V 29.22/10.40 it(22) =< aux(64) 29.22/10.40 it(22) =< aux(63) 29.22/10.40 it(23) =< aux(63) 29.22/10.40 it(23) =< aux(64) 29.22/10.40 it(22) =< aux(65) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(65)+aux(62) 29.22/10.40 s(18) =< aux(65)+aux(62) 29.22/10.40 s(18) =< it(23)*aux(65) 29.22/10.40 s(17) =< s(18) 29.22/10.40 s(15) =< aux(63) 29.22/10.40 29.22/10.40 with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 29.22/10.40 29.22/10.40 * Chain [[22,23],28,27,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+9 29.22/10.40 Such that:s(5) =< 1 29.22/10.40 aux(8) =< V1-V+1 29.22/10.40 aux(7) =< 2*V 29.22/10.40 aux(66) =< V1 29.22/10.40 aux(67) =< V1+V 29.22/10.40 aux(68) =< V1/2+V/2 29.22/10.40 aux(69) =< V 29.22/10.40 it(22) =< aux(68) 29.22/10.40 it(22) =< aux(67) 29.22/10.40 it(23) =< aux(67) 29.22/10.40 it(23) =< aux(68) 29.22/10.40 it(22) =< aux(69) 29.22/10.40 it(23) =< aux(7)+aux(8) 29.22/10.40 it(23) =< aux(69)+aux(66) 29.22/10.40 s(18) =< aux(69)+aux(66) 29.22/10.41 s(18) =< it(23)*aux(69) 29.22/10.41 s(17) =< s(18) 29.22/10.41 s(15) =< aux(67) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 29.22/10.41 29.22/10.41 * Chain [[22,23],25,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(20)+4 29.22/10.41 Such that:aux(19) =< V1 29.22/10.41 aux(8) =< V1-V+1 29.22/10.41 aux(20) =< V1+V 29.22/10.41 aux(21) =< V1+V-2*Out 29.22/10.41 aux(22) =< V1-Out 29.22/10.41 it(22) =< V1/2+V/2 29.22/10.41 aux(23) =< V1/2+V/2-Out 29.22/10.41 aux(24) =< V 29.22/10.41 aux(25) =< V-Out 29.22/10.41 aux(7) =< 2*V-2*Out 29.22/10.41 aux(70) =< Out 29.22/10.41 s(20) =< aux(70) 29.22/10.41 it(22) =< aux(20) 29.22/10.41 it(23) =< aux(20) 29.22/10.41 s(16) =< aux(20) 29.22/10.41 it(22) =< aux(21) 29.22/10.41 it(23) =< aux(21) 29.22/10.41 s(16) =< aux(21) 29.22/10.41 it(22) =< aux(23) 29.22/10.41 it(23) =< aux(23) 29.22/10.41 aux(5) =< aux(24) 29.22/10.41 it(22) =< aux(24) 29.22/10.41 aux(5) =< aux(25) 29.22/10.41 it(22) =< aux(25) 29.22/10.41 it(23) =< aux(7)+aux(8) 29.22/10.41 it(23) =< aux(5)+aux(19) 29.22/10.41 s(18) =< aux(5)+aux(22) 29.22/10.41 s(18) =< aux(5)+aux(19) 29.22/10.41 it(23) =< aux(5)+aux(22) 29.22/10.41 s(18) =< it(23)*aux(24) 29.22/10.41 s(17) =< s(18) 29.22/10.41 s(15) =< s(16) 29.22/10.41 29.22/10.41 with precondition: [Out>=2,V1>=Out,V>=Out,V+V1>=3*Out] 29.22/10.41 29.22/10.41 * Chain [[22,23],25,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5 29.22/10.41 Such that:aux(8) =< V1-V+1 29.22/10.41 aux(7) =< 2*V 29.22/10.41 aux(72) =< V1 29.22/10.41 aux(73) =< V1+V 29.22/10.41 aux(74) =< V1/2+V/2 29.22/10.41 aux(75) =< V 29.22/10.41 it(22) =< aux(74) 29.22/10.41 s(5) =< aux(75) 29.22/10.41 it(22) =< aux(73) 29.22/10.41 it(23) =< aux(73) 29.22/10.41 it(23) =< aux(74) 29.22/10.41 it(22) =< aux(75) 29.22/10.41 it(23) =< aux(7)+aux(8) 29.22/10.41 it(23) =< aux(75)+aux(72) 29.22/10.41 s(18) =< aux(75)+aux(72) 29.22/10.41 s(18) =< it(23)*aux(75) 29.22/10.41 s(17) =< s(18) 29.22/10.41 s(15) =< aux(73) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V1>=2,V>=2,V+V1>=6] 29.22/10.41 29.22/10.41 * Chain [[22,23],24,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(22)+4 29.22/10.41 Such that:aux(19) =< V1 29.22/10.41 aux(8) =< V1-V+1 29.22/10.41 aux(20) =< V1+V 29.22/10.41 aux(21) =< V1+V-2*Out 29.22/10.41 aux(22) =< V1-Out 29.22/10.41 it(22) =< V1/2+V/2 29.22/10.41 aux(23) =< V1/2+V/2-Out 29.22/10.41 aux(24) =< V 29.22/10.41 aux(25) =< V-Out 29.22/10.41 aux(7) =< 2*V-2*Out 29.22/10.41 aux(76) =< Out 29.22/10.41 s(22) =< aux(76) 29.22/10.41 it(22) =< aux(20) 29.22/10.41 it(23) =< aux(20) 29.22/10.41 s(16) =< aux(20) 29.22/10.41 it(22) =< aux(21) 29.22/10.41 it(23) =< aux(21) 29.22/10.41 s(16) =< aux(21) 29.22/10.41 it(22) =< aux(23) 29.22/10.41 it(23) =< aux(23) 29.22/10.41 aux(5) =< aux(24) 29.22/10.41 it(22) =< aux(24) 29.22/10.41 aux(5) =< aux(25) 29.22/10.41 it(22) =< aux(25) 29.22/10.41 it(23) =< aux(7)+aux(8) 29.22/10.41 it(23) =< aux(5)+aux(19) 29.22/10.41 s(18) =< aux(5)+aux(22) 29.22/10.41 s(18) =< aux(5)+aux(19) 29.22/10.41 it(23) =< aux(5)+aux(22) 29.22/10.41 s(18) =< it(23)*aux(24) 29.22/10.41 s(17) =< s(18) 29.22/10.41 s(15) =< s(16) 29.22/10.41 29.22/10.41 with precondition: [Out>=2,V1>=Out+1,V>=Out+1,V+V1>=3*Out+2] 29.22/10.41 29.22/10.41 * Chain [[22,23],24,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5 29.22/10.41 Such that:aux(8) =< V1-V+1 29.22/10.41 aux(7) =< 2*V 29.22/10.41 aux(78) =< V1 29.22/10.41 aux(79) =< V1+V 29.22/10.41 aux(80) =< V1/2+V/2 29.22/10.41 aux(81) =< V 29.22/10.41 it(22) =< aux(80) 29.22/10.41 s(5) =< aux(78) 29.22/10.41 it(22) =< aux(79) 29.22/10.41 it(23) =< aux(79) 29.22/10.41 it(23) =< aux(80) 29.22/10.41 it(22) =< aux(81) 29.22/10.41 it(23) =< aux(7)+aux(8) 29.22/10.41 it(23) =< aux(81)+aux(78) 29.22/10.41 s(18) =< aux(81)+aux(78) 29.22/10.41 s(18) =< it(23)*aux(81) 29.22/10.41 s(17) =< s(18) 29.22/10.41 s(15) =< aux(79) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 29.22/10.41 29.22/10.41 * Chain [33]: 1 29.22/10.41 with precondition: [V1=0,V=Out,V>=0] 29.22/10.41 29.22/10.41 * Chain [32]: 2*s(3)+1*s(5)+2 29.22/10.41 Such that:s(5) =< V 29.22/10.41 aux(1) =< V1 29.22/10.41 s(3) =< aux(1) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V1>=0,V>=0] 29.22/10.41 29.22/10.41 * Chain [31]: 1 29.22/10.41 with precondition: [V=0,V1=Out,V1>=1] 29.22/10.41 29.22/10.41 * Chain [30]: 2 29.22/10.41 with precondition: [V=1,Out=0,V1>=1] 29.22/10.41 29.22/10.41 * Chain [29,33]: 4 29.22/10.41 with precondition: [V1=1,Out=1,V>=2] 29.22/10.41 29.22/10.41 * Chain [29,32]: 1*s(5)+5 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,[26],33]: 4*it(26)+5 29.22/10.41 Such that:it(26) =< V 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=1,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,[26],32]: 6*it(26)+1*s(5)+6 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 aux(2) =< V 29.22/10.41 it(26) =< aux(2) 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,[26],30]: 4*it(26)+6 29.22/10.41 Such that:it(26) =< V 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=3] 29.22/10.41 29.22/10.41 * Chain [28,[26],27,33]: 4*it(26)+8 29.22/10.41 Such that:it(26) =< V 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=1,V>=3] 29.22/10.41 29.22/10.41 * Chain [28,[26],27,32]: 4*it(26)+1*s(5)+9 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 it(26) =< V 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=3] 29.22/10.41 29.22/10.41 * Chain [28,32]: 2*s(3)+1*s(5)+6 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 aux(1) =< V 29.22/10.41 s(3) =< aux(1) 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,30]: 6 29.22/10.41 with precondition: [V1=1,Out=0,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,27,33]: 8 29.22/10.41 with precondition: [V1=1,Out=1,V>=2] 29.22/10.41 29.22/10.41 * Chain [28,27,32]: 1*s(5)+9 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 29.22/10.41 with precondition: [V1=1,Out=0,V>=2] 29.22/10.41 29.22/10.41 * Chain [27,33]: 4 29.22/10.41 with precondition: [V=1,Out=1,V1>=1] 29.22/10.41 29.22/10.41 * Chain [27,32]: 1*s(5)+5 29.22/10.41 Such that:s(5) =< 1 29.22/10.41 29.22/10.41 with precondition: [V=1,Out=0,V1>=1] 29.22/10.41 29.22/10.41 * Chain [25,33]: 2*s(20)+4 29.22/10.41 Such that:aux(70) =< Out 29.22/10.41 s(20) =< aux(70) 29.22/10.41 29.22/10.41 with precondition: [V=Out,V>=2,V1>=V] 29.22/10.41 29.22/10.41 * Chain [25,32]: 3*s(5)+5 29.22/10.41 Such that:aux(71) =< V 29.22/10.41 s(5) =< aux(71) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V>=2,V1>=V] 29.22/10.41 29.22/10.41 * Chain [24,33]: 2*s(22)+4 29.22/10.41 Such that:aux(76) =< Out 29.22/10.41 s(22) =< aux(76) 29.22/10.41 29.22/10.41 with precondition: [V1=Out,V1>=2,V>=V1+1] 29.22/10.41 29.22/10.41 * Chain [24,32]: 3*s(5)+5 29.22/10.41 Such that:aux(77) =< V1 29.22/10.41 s(5) =< aux(77) 29.22/10.41 29.22/10.41 with precondition: [Out=0,V1>=2,V>=V1+1] 29.22/10.41 29.22/10.41 29.22/10.41 #### Cost of chains of start(V1,V,V14): 29.22/10.41 * Chain [42]: 45*s(266)+9*s(268)+10*s(278)+68*s(279)+36*s(281)+18*s(283)+43*s(284)+8*s(285)+32*s(286)+16*s(288)+9 29.22/10.41 Such that:s(271) =< 1 29.22/10.41 aux(114) =< V1 29.22/10.41 aux(115) =< V1-V+1 29.22/10.41 aux(116) =< V1+V 29.22/10.41 aux(117) =< V1/2+V/2 29.22/10.41 aux(118) =< V 29.22/10.41 aux(119) =< 2*V 29.22/10.41 s(268) =< aux(114) 29.22/10.41 s(266) =< aux(118) 29.22/10.41 s(278) =< s(271) 29.22/10.41 s(279) =< aux(117) 29.22/10.41 s(279) =< aux(116) 29.22/10.41 s(281) =< aux(116) 29.22/10.41 s(281) =< aux(117) 29.22/10.41 s(279) =< aux(118) 29.22/10.41 s(281) =< aux(119)+aux(115) 29.22/10.41 s(281) =< aux(118)+aux(114) 29.22/10.41 s(282) =< aux(118)+aux(114) 29.22/10.41 s(282) =< s(281)*aux(118) 29.22/10.41 s(283) =< s(282) 29.22/10.41 s(284) =< aux(116) 29.22/10.41 s(285) =< aux(119) 29.22/10.41 s(286) =< aux(116) 29.22/10.41 s(286) =< aux(119)+aux(115) 29.22/10.41 s(286) =< aux(118)+aux(114) 29.70/10.41 s(287) =< aux(118)+aux(114) 29.70/10.41 s(287) =< s(286)*aux(118) 29.70/10.41 s(288) =< s(287) 29.70/10.41 29.70/10.41 with precondition: [V1>=0,V>=0] 29.70/10.41 29.70/10.41 * Chain [41]: 64*s(341)+40*s(342)+20*s(344)+10*s(346)+96*s(347)+8*s(348)+20*s(349)+10*s(351)+101*s(359)+40*s(369)+20*s(371)+10*s(373)+16*s(375)+20*s(376)+10*s(378)+68*s(389)+36*s(391)+18*s(393)+32*s(396)+16*s(398)+8*s(399)+11 29.70/10.41 Such that:s(340) =< 2 29.70/10.41 s(363) =< -V+1 29.70/10.41 s(365) =< V/2 29.70/10.41 aux(130) =< 1 29.70/10.41 aux(131) =< -2*V+V14+1 29.70/10.41 aux(132) =< -V+V14 29.70/10.41 aux(133) =< V 29.70/10.41 aux(134) =< 2*V 29.70/10.41 aux(135) =< V14 29.70/10.41 aux(136) =< V14/2 29.70/10.41 s(359) =< aux(133) 29.70/10.41 s(341) =< aux(130) 29.70/10.41 s(347) =< aux(135) 29.70/10.41 s(369) =< s(365) 29.70/10.41 s(369) =< aux(133) 29.70/10.41 s(371) =< aux(133) 29.70/10.41 s(371) =< s(365) 29.70/10.41 s(371) =< aux(134)+s(363) 29.70/10.41 s(372) =< aux(133) 29.70/10.41 s(372) =< s(371)*aux(133) 29.70/10.41 s(373) =< s(372) 29.70/10.41 s(375) =< aux(134) 29.70/10.41 s(376) =< aux(133) 29.70/10.41 s(376) =< aux(134)+s(363) 29.70/10.41 s(377) =< aux(133) 29.70/10.41 s(377) =< s(376)*aux(133) 29.70/10.41 s(378) =< s(377) 29.70/10.41 s(389) =< aux(136) 29.70/10.41 s(389) =< aux(135) 29.70/10.41 s(391) =< aux(135) 29.70/10.41 s(391) =< aux(136) 29.70/10.41 s(389) =< aux(133) 29.70/10.41 s(391) =< aux(134)+aux(131) 29.70/10.41 s(391) =< aux(133)+aux(132) 29.70/10.41 s(392) =< aux(133)+aux(132) 29.70/10.41 s(392) =< s(391)*aux(133) 29.70/10.41 s(393) =< s(392) 29.70/10.41 s(396) =< aux(135) 29.70/10.41 s(396) =< aux(134)+aux(131) 29.70/10.41 s(396) =< aux(133)+aux(132) 29.70/10.41 s(397) =< aux(133)+aux(132) 29.70/10.41 s(397) =< s(396)*aux(133) 29.70/10.41 s(398) =< s(397) 29.70/10.41 s(399) =< aux(132) 29.70/10.41 s(342) =< aux(136) 29.70/10.41 s(342) =< aux(135) 29.70/10.41 s(344) =< aux(135) 29.70/10.41 s(344) =< aux(136) 29.70/10.41 s(342) =< aux(130) 29.70/10.41 s(344) =< s(340)+aux(135) 29.70/10.41 s(344) =< aux(130)+aux(135) 29.70/10.41 s(345) =< aux(130)+aux(135) 29.70/10.41 s(345) =< s(344)*aux(130) 29.70/10.41 s(346) =< s(345) 29.70/10.41 s(348) =< s(340) 29.70/10.41 s(349) =< aux(135) 29.70/10.41 s(349) =< s(340)+aux(135) 29.70/10.41 s(349) =< aux(130)+aux(135) 29.70/10.41 s(350) =< aux(130)+aux(135) 29.70/10.41 s(350) =< s(349)*aux(130) 29.70/10.41 s(351) =< s(350) 29.70/10.41 29.70/10.41 with precondition: [V1=1,V>=1,V14>=1] 29.70/10.41 29.70/10.41 * Chain [40]: 9*s(447)+8*s(451)+10 29.70/10.41 Such that:s(450) =< V 29.70/10.41 aux(137) =< V14 29.70/10.41 s(451) =< s(450) 29.70/10.41 s(447) =< aux(137) 29.70/10.41 29.70/10.41 with precondition: [V1=1,V>=2] 29.70/10.41 29.70/10.41 * Chain [39]: 1*s(452)+3 29.70/10.41 Such that:s(452) =< V14 29.70/10.41 29.70/10.41 with precondition: [V1=1,V=V14,V>=2] 29.70/10.41 29.70/10.41 * Chain [38]: 64*s(460)+40*s(461)+20*s(463)+10*s(465)+96*s(466)+8*s(467)+20*s(468)+10*s(470)+101*s(478)+40*s(488)+20*s(490)+10*s(492)+16*s(494)+20*s(495)+10*s(497)+68*s(508)+36*s(510)+18*s(512)+32*s(515)+16*s(517)+8*s(518)+11 29.70/10.41 Such that:s(459) =< 2 29.70/10.41 s(482) =< -V14+1 29.70/10.41 s(484) =< V14/2 29.70/10.41 aux(148) =< 1 29.70/10.41 aux(149) =< V 29.70/10.41 aux(150) =< V-2*V14+1 29.70/10.41 aux(151) =< V-V14 29.70/10.41 aux(152) =< V/2 29.70/10.41 aux(153) =< V14 29.70/10.41 aux(154) =< 2*V14 29.70/10.41 s(478) =< aux(153) 29.70/10.41 s(460) =< aux(148) 29.70/10.41 s(466) =< aux(149) 29.70/10.41 s(488) =< s(484) 29.70/10.41 s(488) =< aux(153) 29.70/10.41 s(490) =< aux(153) 29.70/10.41 s(490) =< s(484) 29.70/10.41 s(490) =< aux(154)+s(482) 29.70/10.41 s(491) =< aux(153) 29.70/10.41 s(491) =< s(490)*aux(153) 29.70/10.41 s(492) =< s(491) 29.70/10.41 s(494) =< aux(154) 29.70/10.41 s(495) =< aux(153) 29.70/10.41 s(495) =< aux(154)+s(482) 29.70/10.41 s(496) =< aux(153) 29.70/10.41 s(496) =< s(495)*aux(153) 29.70/10.41 s(497) =< s(496) 29.70/10.41 s(508) =< aux(152) 29.70/10.41 s(508) =< aux(149) 29.70/10.41 s(510) =< aux(149) 29.70/10.41 s(510) =< aux(152) 29.70/10.41 s(508) =< aux(153) 29.70/10.41 s(510) =< aux(154)+aux(150) 29.70/10.41 s(510) =< aux(153)+aux(151) 29.70/10.41 s(511) =< aux(153)+aux(151) 29.70/10.41 s(511) =< s(510)*aux(153) 29.70/10.41 s(512) =< s(511) 29.70/10.41 s(515) =< aux(149) 29.70/10.41 s(515) =< aux(154)+aux(150) 29.70/10.41 s(515) =< aux(153)+aux(151) 29.70/10.41 s(516) =< aux(153)+aux(151) 29.70/10.41 s(516) =< s(515)*aux(153) 29.70/10.41 s(517) =< s(516) 29.70/10.41 s(518) =< aux(151) 29.70/10.41 s(461) =< aux(152) 29.70/10.41 s(461) =< aux(149) 29.70/10.41 s(463) =< aux(149) 29.70/10.41 s(463) =< aux(152) 29.70/10.41 s(461) =< aux(148) 29.70/10.41 s(463) =< s(459)+aux(149) 29.70/10.41 s(463) =< aux(148)+aux(149) 29.70/10.41 s(464) =< aux(148)+aux(149) 29.70/10.41 s(464) =< s(463)*aux(148) 29.70/10.41 s(465) =< s(464) 29.70/10.41 s(467) =< s(459) 29.70/10.41 s(468) =< aux(149) 29.70/10.41 s(468) =< s(459)+aux(149) 29.70/10.41 s(468) =< aux(148)+aux(149) 29.70/10.41 s(469) =< aux(148)+aux(149) 29.70/10.41 s(469) =< s(468)*aux(148) 29.70/10.41 s(470) =< s(469) 29.70/10.41 29.70/10.41 with precondition: [V1=2,V>=1,V14>=1] 29.70/10.41 29.70/10.41 * Chain [37]: 1*s(566)+3 29.70/10.41 Such that:s(566) =< V14 29.70/10.41 29.70/10.41 with precondition: [V1=2,V=V14,V>=2] 29.70/10.41 29.70/10.41 * Chain [36]: 9*s(567)+10 29.70/10.41 Such that:aux(155) =< V14 29.70/10.41 s(567) =< aux(155) 29.70/10.41 29.70/10.41 with precondition: [V1=2,V=V14+1,V>=3] 29.70/10.41 29.70/10.41 * Chain [35]: 1 29.70/10.41 with precondition: [V=0,V1>=0] 29.70/10.41 29.70/10.41 * Chain [34]: 3*s(572)+22*s(573)+5 29.70/10.41 Such that:s(570) =< 1 29.70/10.41 aux(156) =< V1 29.70/10.41 s(572) =< s(570) 29.70/10.41 s(573) =< aux(156) 29.70/10.41 29.70/10.41 with precondition: [V=1,V1>=1] 29.70/10.41 29.70/10.41 29.70/10.41 Closed-form bounds of start(V1,V,V14): 29.70/10.41 ------------------------------------- 29.70/10.41 * Chain [42] with precondition: [V1>=0,V>=0] 29.70/10.41 - Upper bound: 188*V1+240*V+19 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [41] with precondition: [V1=1,V>=1,V14>=1] 29.70/10.41 - Upper bound: 227*V+224*V14+151+nat(-V+V14)*42+20*V+34*V14 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [40] with precondition: [V1=1,V>=2] 29.70/10.41 - Upper bound: 8*V+10+nat(V14)*9 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [39] with precondition: [V1=1,V=V14,V>=2] 29.70/10.41 - Upper bound: V14+3 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [38] with precondition: [V1=2,V>=1,V14>=1] 29.70/10.41 - Upper bound: 224*V+227*V14+151+nat(V-V14)*42+34*V+20*V14 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [37] with precondition: [V1=2,V=V14,V>=2] 29.70/10.41 - Upper bound: V14+3 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [36] with precondition: [V1=2,V=V14+1,V>=3] 29.70/10.41 - Upper bound: 9*V14+10 29.70/10.41 - Complexity: n 29.70/10.41 * Chain [35] with precondition: [V=0,V1>=0] 29.70/10.41 - Upper bound: 1 29.70/10.41 - Complexity: constant 29.70/10.41 * Chain [34] with precondition: [V=1,V1>=1] 29.70/10.41 - Upper bound: 22*V1+8 29.70/10.41 - Complexity: n 29.70/10.41 29.70/10.41 ### Maximum cost of start(V1,V,V14): max([188*V1+240*V+18,187*V+141+nat(V14)*186+20*V+nat(V14/2)*40+max([nat(V14)*29+32*V+nat(-V+V14)*42+nat(V14/2)*28,nat(2*V14)*16+29*V+nat(V-V14)*42+14*V])+8*V+(nat(V14)*8+7)+(nat(V14)+2)])+1 29.70/10.41 Asymptotic class: n 29.70/10.41 * Total analysis performed in 1235 ms. 29.70/10.41 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (10) 29.70/10.41 BOUNDS(1, n^1) 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 29.70/10.41 Renamed function symbols to avoid clashes with predefined symbol. 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (12) 29.70/10.41 Obligation: 29.70/10.41 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 29.70/10.41 29.70/10.41 29.70/10.41 The TRS R consists of the following rules: 29.70/10.41 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 S is empty. 29.70/10.41 Rewrite Strategy: INNERMOST 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 29.70/10.41 Infered types. 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (14) 29.70/10.41 Obligation: 29.70/10.41 Innermost TRS: 29.70/10.41 Rules: 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 Types: 29.70/10.41 le :: 0':s -> 0':s -> true:false 29.70/10.41 0' :: 0':s 29.70/10.41 true :: true:false 29.70/10.41 s :: 0':s -> 0':s 29.70/10.41 false :: true:false 29.70/10.41 minus :: 0':s -> 0':s -> 0':s 29.70/10.41 gcd :: 0':s -> 0':s -> 0':s 29.70/10.41 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 29.70/10.41 hole_true:false1_0 :: true:false 29.70/10.41 hole_0':s2_0 :: 0':s 29.70/10.41 gen_0':s3_0 :: Nat -> 0':s 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (15) OrderProof (LOWER BOUND(ID)) 29.70/10.41 Heuristically decided to analyse the following defined symbols: 29.70/10.41 le, minus, gcd 29.70/10.41 29.70/10.41 They will be analysed ascendingly in the following order: 29.70/10.41 le < gcd 29.70/10.41 minus < gcd 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (16) 29.70/10.41 Obligation: 29.70/10.41 Innermost TRS: 29.70/10.41 Rules: 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 Types: 29.70/10.41 le :: 0':s -> 0':s -> true:false 29.70/10.41 0' :: 0':s 29.70/10.41 true :: true:false 29.70/10.41 s :: 0':s -> 0':s 29.70/10.41 false :: true:false 29.70/10.41 minus :: 0':s -> 0':s -> 0':s 29.70/10.41 gcd :: 0':s -> 0':s -> 0':s 29.70/10.41 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 29.70/10.41 hole_true:false1_0 :: true:false 29.70/10.41 hole_0':s2_0 :: 0':s 29.70/10.41 gen_0':s3_0 :: Nat -> 0':s 29.70/10.41 29.70/10.41 29.70/10.41 Generator Equations: 29.70/10.41 gen_0':s3_0(0) <=> 0' 29.70/10.41 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 29.70/10.41 29.70/10.41 29.70/10.41 The following defined symbols remain to be analysed: 29.70/10.41 le, minus, gcd 29.70/10.41 29.70/10.41 They will be analysed ascendingly in the following order: 29.70/10.41 le < gcd 29.70/10.41 minus < gcd 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (17) RewriteLemmaProof (LOWER BOUND(ID)) 29.70/10.41 Proved the following rewrite lemma: 29.70/10.41 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 29.70/10.41 29.70/10.41 Induction Base: 29.70/10.41 le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 29.70/10.41 true 29.70/10.41 29.70/10.41 Induction Step: 29.70/10.41 le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 29.70/10.41 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 29.70/10.41 true 29.70/10.41 29.70/10.41 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (18) 29.70/10.41 Complex Obligation (BEST) 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (19) 29.70/10.41 Obligation: 29.70/10.41 Proved the lower bound n^1 for the following obligation: 29.70/10.41 29.70/10.41 Innermost TRS: 29.70/10.41 Rules: 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 Types: 29.70/10.41 le :: 0':s -> 0':s -> true:false 29.70/10.41 0' :: 0':s 29.70/10.41 true :: true:false 29.70/10.41 s :: 0':s -> 0':s 29.70/10.41 false :: true:false 29.70/10.41 minus :: 0':s -> 0':s -> 0':s 29.70/10.41 gcd :: 0':s -> 0':s -> 0':s 29.70/10.41 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 29.70/10.41 hole_true:false1_0 :: true:false 29.70/10.41 hole_0':s2_0 :: 0':s 29.70/10.41 gen_0':s3_0 :: Nat -> 0':s 29.70/10.41 29.70/10.41 29.70/10.41 Generator Equations: 29.70/10.41 gen_0':s3_0(0) <=> 0' 29.70/10.41 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 29.70/10.41 29.70/10.41 29.70/10.41 The following defined symbols remain to be analysed: 29.70/10.41 le, minus, gcd 29.70/10.41 29.70/10.41 They will be analysed ascendingly in the following order: 29.70/10.41 le < gcd 29.70/10.41 minus < gcd 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (20) LowerBoundPropagationProof (FINISHED) 29.70/10.41 Propagated lower bound. 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (21) 29.70/10.41 BOUNDS(n^1, INF) 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (22) 29.70/10.41 Obligation: 29.70/10.41 Innermost TRS: 29.70/10.41 Rules: 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 Types: 29.70/10.41 le :: 0':s -> 0':s -> true:false 29.70/10.41 0' :: 0':s 29.70/10.41 true :: true:false 29.70/10.41 s :: 0':s -> 0':s 29.70/10.41 false :: true:false 29.70/10.41 minus :: 0':s -> 0':s -> 0':s 29.70/10.41 gcd :: 0':s -> 0':s -> 0':s 29.70/10.41 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 29.70/10.41 hole_true:false1_0 :: true:false 29.70/10.41 hole_0':s2_0 :: 0':s 29.70/10.41 gen_0':s3_0 :: Nat -> 0':s 29.70/10.41 29.70/10.41 29.70/10.41 Lemmas: 29.70/10.41 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 29.70/10.41 29.70/10.41 29.70/10.41 Generator Equations: 29.70/10.41 gen_0':s3_0(0) <=> 0' 29.70/10.41 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 29.70/10.41 29.70/10.41 29.70/10.41 The following defined symbols remain to be analysed: 29.70/10.41 minus, gcd 29.70/10.41 29.70/10.41 They will be analysed ascendingly in the following order: 29.70/10.41 minus < gcd 29.70/10.41 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (23) RewriteLemmaProof (LOWER BOUND(ID)) 29.70/10.41 Proved the following rewrite lemma: 29.70/10.41 minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) -> gen_0':s3_0(0), rt in Omega(1 + n294_0) 29.70/10.41 29.70/10.41 Induction Base: 29.70/10.41 minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 29.70/10.41 gen_0':s3_0(0) 29.70/10.41 29.70/10.41 Induction Step: 29.70/10.41 minus(gen_0':s3_0(+(n294_0, 1)), gen_0':s3_0(+(n294_0, 1))) ->_R^Omega(1) 29.70/10.41 minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) ->_IH 29.70/10.41 gen_0':s3_0(0) 29.70/10.41 29.70/10.41 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 29.70/10.41 ---------------------------------------- 29.70/10.41 29.70/10.41 (24) 29.70/10.41 Obligation: 29.70/10.41 Innermost TRS: 29.70/10.41 Rules: 29.70/10.41 le(0', y) -> true 29.70/10.41 le(s(x), 0') -> false 29.70/10.41 le(s(x), s(y)) -> le(x, y) 29.70/10.41 minus(x, 0') -> x 29.70/10.41 minus(s(x), s(y)) -> minus(x, y) 29.70/10.41 gcd(0', y) -> y 29.70/10.41 gcd(s(x), 0') -> s(x) 29.70/10.41 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 29.70/10.41 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 29.70/10.41 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 29.70/10.41 29.70/10.41 Types: 29.70/10.41 le :: 0':s -> 0':s -> true:false 29.70/10.41 0' :: 0':s 29.70/10.41 true :: true:false 29.70/10.41 s :: 0':s -> 0':s 29.70/10.41 false :: true:false 29.70/10.41 minus :: 0':s -> 0':s -> 0':s 29.70/10.41 gcd :: 0':s -> 0':s -> 0':s 29.70/10.41 if_gcd :: true:false -> 0':s -> 0':s -> 0':s 29.70/10.41 hole_true:false1_0 :: true:false 29.70/10.41 hole_0':s2_0 :: 0':s 29.70/10.41 gen_0':s3_0 :: Nat -> 0':s 29.70/10.41 29.70/10.41 29.70/10.41 Lemmas: 29.70/10.41 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 29.70/10.41 minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) -> gen_0':s3_0(0), rt in Omega(1 + n294_0) 29.70/10.41 29.70/10.41 29.70/10.41 Generator Equations: 29.70/10.41 gen_0':s3_0(0) <=> 0' 29.70/10.41 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 29.70/10.41 29.70/10.41 29.70/10.41 The following defined symbols remain to be analysed: 29.70/10.41 gcd 29.76/10.46 EOF