1099.37/291.48 WORST_CASE(Omega(n^1), O(n^2)) 1099.37/291.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1099.37/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1099.37/291.50 1099.37/291.50 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1099.37/291.50 1099.37/291.50 (0) CpxTRS 1099.37/291.50 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1099.37/291.50 (2) CpxWeightedTrs 1099.37/291.50 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1099.37/291.50 (4) CpxTypedWeightedTrs 1099.37/291.50 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 1099.37/291.50 (6) CpxTypedWeightedCompleteTrs 1099.37/291.50 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 1099.37/291.50 (8) CpxRNTS 1099.37/291.50 (9) CompleteCoflocoProof [FINISHED, 2318 ms] 1099.37/291.50 (10) BOUNDS(1, n^2) 1099.37/291.50 (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1099.37/291.50 (12) TRS for Loop Detection 1099.37/291.50 (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1099.37/291.50 (14) BEST 1099.37/291.50 (15) proven lower bound 1099.37/291.50 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1099.37/291.50 (17) BOUNDS(n^1, INF) 1099.37/291.50 (18) TRS for Loop Detection 1099.37/291.50 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (0) 1099.37/291.50 Obligation: 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1099.37/291.50 1099.37/291.50 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true 1099.37/291.50 le(s(x), 0) -> false 1099.37/291.50 le(s(x), s(y)) -> le(x, y) 1099.37/291.50 inc(0) -> 0 1099.37/291.50 inc(s(x)) -> s(inc(x)) 1099.37/291.50 minus(0, y) -> 0 1099.37/291.50 minus(x, 0) -> x 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) 1099.37/291.50 quot(0, s(y)) -> 0 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 1099.37/291.50 log(x) -> log2(x, 0) 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) 1099.37/291.50 if(true, b, x, y) -> log_undefined 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) 1099.37/291.50 if2(true, x, s(y)) -> y 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) 1099.37/291.50 1099.37/291.50 S is empty. 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1099.37/291.50 Transformed relative TRS to weighted TRS 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (2) 1099.37/291.50 Obligation: 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 1099.37/291.50 1099.37/291.50 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true [1] 1099.37/291.50 le(s(x), 0) -> false [1] 1099.37/291.50 le(s(x), s(y)) -> le(x, y) [1] 1099.37/291.50 inc(0) -> 0 [1] 1099.37/291.50 inc(s(x)) -> s(inc(x)) [1] 1099.37/291.50 minus(0, y) -> 0 [1] 1099.37/291.50 minus(x, 0) -> x [1] 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) [1] 1099.37/291.50 quot(0, s(y)) -> 0 [1] 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 1099.37/291.50 log(x) -> log2(x, 0) [1] 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) [1] 1099.37/291.50 if(true, b, x, y) -> log_undefined [1] 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) [1] 1099.37/291.50 if2(true, x, s(y)) -> y [1] 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) [1] 1099.37/291.50 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1099.37/291.50 Infered types. 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (4) 1099.37/291.50 Obligation: 1099.37/291.50 Runtime Complexity Weighted TRS with Types. 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true [1] 1099.37/291.50 le(s(x), 0) -> false [1] 1099.37/291.50 le(s(x), s(y)) -> le(x, y) [1] 1099.37/291.50 inc(0) -> 0 [1] 1099.37/291.50 inc(s(x)) -> s(inc(x)) [1] 1099.37/291.50 minus(0, y) -> 0 [1] 1099.37/291.50 minus(x, 0) -> x [1] 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) [1] 1099.37/291.50 quot(0, s(y)) -> 0 [1] 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 1099.37/291.50 log(x) -> log2(x, 0) [1] 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) [1] 1099.37/291.50 if(true, b, x, y) -> log_undefined [1] 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) [1] 1099.37/291.50 if2(true, x, s(y)) -> y [1] 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) [1] 1099.37/291.50 1099.37/291.50 The TRS has the following type information: 1099.37/291.50 le :: 0:s:log_undefined -> 0:s:log_undefined -> true:false 1099.37/291.50 0 :: 0:s:log_undefined 1099.37/291.50 true :: true:false 1099.37/291.50 s :: 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 false :: true:false 1099.37/291.50 inc :: 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 minus :: 0:s:log_undefined -> 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 quot :: 0:s:log_undefined -> 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 log :: 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 log2 :: 0:s:log_undefined -> 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 if :: true:false -> true:false -> 0:s:log_undefined -> 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 log_undefined :: 0:s:log_undefined 1099.37/291.50 if2 :: true:false -> 0:s:log_undefined -> 0:s:log_undefined -> 0:s:log_undefined 1099.37/291.50 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (5) CompletionProof (UPPER BOUND(ID)) 1099.37/291.50 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 1099.37/291.50 1099.37/291.50 le(v0, v1) -> null_le [0] 1099.37/291.50 inc(v0) -> null_inc [0] 1099.37/291.50 minus(v0, v1) -> null_minus [0] 1099.37/291.50 quot(v0, v1) -> null_quot [0] 1099.37/291.50 if2(v0, v1, v2) -> null_if2 [0] 1099.37/291.50 if(v0, v1, v2, v3) -> null_if [0] 1099.37/291.50 1099.37/291.50 And the following fresh constants: null_le, null_inc, null_minus, null_quot, null_if2, null_if 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (6) 1099.37/291.50 Obligation: 1099.37/291.50 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 1099.37/291.50 1099.37/291.50 Runtime Complexity Weighted TRS with Types. 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true [1] 1099.37/291.50 le(s(x), 0) -> false [1] 1099.37/291.50 le(s(x), s(y)) -> le(x, y) [1] 1099.37/291.50 inc(0) -> 0 [1] 1099.37/291.50 inc(s(x)) -> s(inc(x)) [1] 1099.37/291.50 minus(0, y) -> 0 [1] 1099.37/291.50 minus(x, 0) -> x [1] 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) [1] 1099.37/291.50 quot(0, s(y)) -> 0 [1] 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] 1099.37/291.50 log(x) -> log2(x, 0) [1] 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) [1] 1099.37/291.50 if(true, b, x, y) -> log_undefined [1] 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) [1] 1099.37/291.50 if2(true, x, s(y)) -> y [1] 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) [1] 1099.37/291.50 le(v0, v1) -> null_le [0] 1099.37/291.50 inc(v0) -> null_inc [0] 1099.37/291.50 minus(v0, v1) -> null_minus [0] 1099.37/291.50 quot(v0, v1) -> null_quot [0] 1099.37/291.50 if2(v0, v1, v2) -> null_if2 [0] 1099.37/291.50 if(v0, v1, v2, v3) -> null_if [0] 1099.37/291.50 1099.37/291.50 The TRS has the following type information: 1099.37/291.50 le :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> true:false:null_le 1099.37/291.50 0 :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 true :: true:false:null_le 1099.37/291.50 s :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 false :: true:false:null_le 1099.37/291.50 inc :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 minus :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 quot :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 log :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 log2 :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 if :: true:false:null_le -> true:false:null_le -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 log_undefined :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 if2 :: true:false:null_le -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if -> 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 null_le :: true:false:null_le 1099.37/291.50 null_inc :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 null_minus :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 null_quot :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 null_if2 :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 null_if :: 0:s:log_undefined:null_inc:null_minus:null_quot:null_if2:null_if 1099.37/291.50 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1099.37/291.50 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1099.37/291.50 The constant constructors are abstracted as follows: 1099.37/291.50 1099.37/291.50 0 => 0 1099.37/291.50 true => 2 1099.37/291.50 false => 1 1099.37/291.50 log_undefined => 1 1099.37/291.50 null_le => 0 1099.37/291.50 null_inc => 0 1099.37/291.50 null_minus => 0 1099.37/291.50 null_quot => 0 1099.37/291.50 null_if2 => 0 1099.37/291.50 null_if => 0 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (8) 1099.37/291.50 Obligation: 1099.37/291.50 Complexity RNTS consisting of the following rules: 1099.37/291.50 1099.37/291.50 if(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x 1099.37/291.50 if(z, z', z'', z1) -{ 1 }-> 1 :|: z = 2, b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x 1099.37/291.50 if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 1099.37/291.50 if2(z, z', z'') -{ 1 }-> y :|: z = 2, z' = x, x >= 0, y >= 0, z'' = 1 + y 1099.37/291.50 if2(z, z', z'') -{ 1 }-> log2(quot(x, 1 + (1 + 0)), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 1099.37/291.50 if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1099.37/291.50 inc(z) -{ 1 }-> 0 :|: z = 0 1099.37/291.50 inc(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1099.37/291.50 inc(z) -{ 1 }-> 1 + inc(x) :|: x >= 0, z = 1 + x 1099.37/291.50 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 1099.37/291.50 le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y 1099.37/291.50 le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 1099.37/291.50 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1099.37/291.50 log(z) -{ 1 }-> log2(x, 0) :|: x >= 0, z = x 1099.37/291.50 log2(z, z') -{ 1 }-> if(le(x, 0), le(x, 1 + 0), x, inc(y)) :|: x >= 0, y >= 0, z = x, z' = y 1099.37/291.50 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 1099.37/291.50 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 1099.37/291.50 minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y 1099.37/291.50 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1099.37/291.50 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 1099.37/291.50 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1099.37/291.50 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 1099.37/291.50 1099.37/291.50 Only complete derivations are relevant for the runtime complexity. 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (9) CompleteCoflocoProof (FINISHED) 1099.37/291.50 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 1099.37/291.50 1099.37/291.50 eq(start(V1, V, V17, V21),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[inc(V1, Out)],[V1 >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[log(V1, Out)],[V1 >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[log2(V1, V, Out)],[V1 >= 0,V >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[if(V1, V, V17, V21, Out)],[V1 >= 0,V >= 0,V17 >= 0,V21 >= 0]). 1099.37/291.50 eq(start(V1, V, V17, V21),0,[if2(V1, V, V17, Out)],[V1 >= 0,V >= 0,V17 >= 0]). 1099.37/291.50 eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]). 1099.37/291.50 eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]). 1099.37/291.50 eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). 1099.37/291.50 eq(inc(V1, Out),1,[],[Out = 0,V1 = 0]). 1099.37/291.50 eq(inc(V1, Out),1,[inc(V6, Ret1)],[Out = 1 + Ret1,V6 >= 0,V1 = 1 + V6]). 1099.37/291.50 eq(minus(V1, V, Out),1,[],[Out = 0,V7 >= 0,V1 = 0,V = V7]). 1099.37/291.50 eq(minus(V1, V, Out),1,[],[Out = V8,V8 >= 0,V1 = V8,V = 0]). 1099.37/291.50 eq(minus(V1, V, Out),1,[minus(V9, V10, Ret2)],[Out = Ret2,V = 1 + V10,V9 >= 0,V10 >= 0,V1 = 1 + V9]). 1099.37/291.50 eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V11,V11 >= 0,V1 = 0]). 1099.37/291.50 eq(quot(V1, V, Out),1,[minus(V13, V12, Ret10),quot(Ret10, 1 + V12, Ret11)],[Out = 1 + Ret11,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]). 1099.37/291.50 eq(log(V1, Out),1,[log2(V14, 0, Ret3)],[Out = Ret3,V14 >= 0,V1 = V14]). 1099.37/291.50 eq(log2(V1, V, Out),1,[le(V15, 0, Ret0),le(V15, 1 + 0, Ret12),inc(V16, Ret31),if(Ret0, Ret12, V15, Ret31, Ret4)],[Out = Ret4,V15 >= 0,V16 >= 0,V1 = V15,V = V16]). 1099.37/291.50 eq(if(V1, V, V17, V21, Out),1,[],[Out = 1,V1 = 2,V19 >= 0,V21 = V20,V18 >= 0,V20 >= 0,V = V19,V17 = V18]). 1099.37/291.50 eq(if(V1, V, V17, V21, Out),1,[if2(V22, V24, V23, Ret5)],[Out = Ret5,V22 >= 0,V21 = V23,V1 = 1,V24 >= 0,V23 >= 0,V = V22,V17 = V24]). 1099.37/291.50 eq(if2(V1, V, V17, Out),1,[],[Out = V25,V1 = 2,V = V26,V26 >= 0,V25 >= 0,V17 = 1 + V25]). 1099.37/291.50 eq(if2(V1, V, V17, Out),1,[quot(V27, 1 + (1 + 0), Ret01),log2(Ret01, V28, Ret6)],[Out = Ret6,V = V27,V17 = V28,V1 = 1,V27 >= 0,V28 >= 0]). 1099.37/291.50 eq(le(V1, V, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V30,V = V29]). 1099.37/291.50 eq(inc(V1, Out),0,[],[Out = 0,V31 >= 0,V1 = V31]). 1099.37/291.50 eq(minus(V1, V, Out),0,[],[Out = 0,V33 >= 0,V32 >= 0,V1 = V33,V = V32]). 1099.37/291.50 eq(quot(V1, V, Out),0,[],[Out = 0,V34 >= 0,V35 >= 0,V1 = V34,V = V35]). 1099.37/291.50 eq(if2(V1, V, V17, Out),0,[],[Out = 0,V36 >= 0,V17 = V38,V37 >= 0,V1 = V36,V = V37,V38 >= 0]). 1099.37/291.50 eq(if(V1, V, V17, V21, Out),0,[],[Out = 0,V21 = V42,V40 >= 0,V17 = V41,V39 >= 0,V1 = V40,V = V39,V41 >= 0,V42 >= 0]). 1099.37/291.50 input_output_vars(le(V1,V,Out),[V1,V],[Out]). 1099.37/291.50 input_output_vars(inc(V1,Out),[V1],[Out]). 1099.37/291.50 input_output_vars(minus(V1,V,Out),[V1,V],[Out]). 1099.37/291.50 input_output_vars(quot(V1,V,Out),[V1,V],[Out]). 1099.37/291.50 input_output_vars(log(V1,Out),[V1],[Out]). 1099.37/291.50 input_output_vars(log2(V1,V,Out),[V1,V],[Out]). 1099.37/291.50 input_output_vars(if(V1,V,V17,V21,Out),[V1,V,V17,V21],[Out]). 1099.37/291.50 input_output_vars(if2(V1,V,V17,Out),[V1,V,V17],[Out]). 1099.37/291.50 1099.37/291.50 1099.37/291.50 CoFloCo proof output: 1099.37/291.50 Preprocessing Cost Relations 1099.37/291.50 ===================================== 1099.37/291.50 1099.37/291.50 #### Computed strongly connected components 1099.37/291.50 0. recursive : [inc/2] 1099.37/291.50 1. recursive : [le/3] 1099.37/291.50 2. recursive : [minus/3] 1099.37/291.50 3. recursive : [quot/3] 1099.37/291.50 4. recursive : [if/5,if2/4,log2/3] 1099.37/291.50 5. non_recursive : [log/2] 1099.37/291.50 6. non_recursive : [start/4] 1099.37/291.50 1099.37/291.50 #### Obtained direct recursion through partial evaluation 1099.37/291.50 0. SCC is partially evaluated into inc/2 1099.37/291.50 1. SCC is partially evaluated into le/3 1099.37/291.50 2. SCC is partially evaluated into minus/3 1099.37/291.50 3. SCC is partially evaluated into quot/3 1099.37/291.50 4. SCC is partially evaluated into log2/3 1099.37/291.50 5. SCC is completely evaluated into other SCCs 1099.37/291.50 6. SCC is partially evaluated into start/4 1099.37/291.50 1099.37/291.50 Control-Flow Refinement of Cost Relations 1099.37/291.50 ===================================== 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations inc/2 1099.37/291.50 * CE 26 is refined into CE [33] 1099.37/291.50 * CE 28 is refined into CE [34] 1099.37/291.50 * CE 27 is refined into CE [35] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of inc/2 1099.37/291.50 * CEs [35] --> Loop 19 1099.37/291.50 * CEs [33,34] --> Loop 20 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR inc(V1,Out) 1099.37/291.50 * RF of phase [19]: [V1] 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR inc(V1,Out) 1099.37/291.50 * Partial RF of phase [19]: 1099.37/291.50 - RF of loop [19:1]: 1099.37/291.50 V1 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations le/3 1099.37/291.50 * CE 25 is refined into CE [36] 1099.37/291.50 * CE 23 is refined into CE [37] 1099.37/291.50 * CE 22 is refined into CE [38] 1099.37/291.50 * CE 24 is refined into CE [39] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of le/3 1099.37/291.50 * CEs [39] --> Loop 21 1099.37/291.50 * CEs [36] --> Loop 22 1099.37/291.50 * CEs [37] --> Loop 23 1099.37/291.50 * CEs [38] --> Loop 24 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR le(V1,V,Out) 1099.37/291.50 * RF of phase [21]: [V,V1] 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR le(V1,V,Out) 1099.37/291.50 * Partial RF of phase [21]: 1099.37/291.50 - RF of loop [21:1]: 1099.37/291.50 V 1099.37/291.50 V1 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations minus/3 1099.37/291.50 * CE 30 is refined into CE [40] 1099.37/291.50 * CE 29 is refined into CE [41] 1099.37/291.50 * CE 32 is refined into CE [42] 1099.37/291.50 * CE 31 is refined into CE [43] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of minus/3 1099.37/291.50 * CEs [43] --> Loop 25 1099.37/291.50 * CEs [40] --> Loop 26 1099.37/291.50 * CEs [41,42] --> Loop 27 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR minus(V1,V,Out) 1099.37/291.50 * RF of phase [25]: [V,V1] 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR minus(V1,V,Out) 1099.37/291.50 * Partial RF of phase [25]: 1099.37/291.50 - RF of loop [25:1]: 1099.37/291.50 V 1099.37/291.50 V1 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations quot/3 1099.37/291.50 * CE 14 is refined into CE [44] 1099.37/291.50 * CE 16 is refined into CE [45] 1099.37/291.50 * CE 15 is refined into CE [46,47,48] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of quot/3 1099.37/291.50 * CEs [48] --> Loop 28 1099.37/291.50 * CEs [47] --> Loop 29 1099.37/291.50 * CEs [46] --> Loop 30 1099.37/291.50 * CEs [44,45] --> Loop 31 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR quot(V1,V,Out) 1099.37/291.50 * RF of phase [28]: [V1-1,V1-V+1] 1099.37/291.50 * RF of phase [30]: [V1] 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR quot(V1,V,Out) 1099.37/291.50 * Partial RF of phase [28]: 1099.37/291.50 - RF of loop [28:1]: 1099.37/291.50 V1-1 1099.37/291.50 V1-V+1 1099.37/291.50 * Partial RF of phase [30]: 1099.37/291.50 - RF of loop [30:1]: 1099.37/291.50 V1 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations log2/3 1099.37/291.50 * CE 21 is refined into CE [49,50,51,52] 1099.37/291.50 * CE 17 is refined into CE [53,54,55,56,57,58] 1099.37/291.50 * CE 19 is refined into CE [59] 1099.37/291.50 * CE 20 is refined into CE [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77] 1099.37/291.50 * CE 18 is refined into CE [78,79,80,81,82,83,84,85] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of log2/3 1099.37/291.50 * CEs [85] --> Loop 32 1099.37/291.50 * CEs [81] --> Loop 33 1099.37/291.50 * CEs [83,84] --> Loop 34 1099.37/291.50 * CEs [79,80] --> Loop 35 1099.37/291.50 * CEs [82] --> Loop 36 1099.37/291.50 * CEs [78] --> Loop 37 1099.37/291.50 * CEs [59] --> Loop 38 1099.37/291.50 * CEs [57,58,68,69,76,77] --> Loop 39 1099.37/291.50 * CEs [49,50,51,52] --> Loop 40 1099.37/291.50 * CEs [53,54,55,56,60,61,62,63,64,65,66,67,70,71,72,73,74,75] --> Loop 41 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR log2(V1,V,Out) 1099.37/291.50 * RF of phase [32,34]: [V1-1] 1099.37/291.50 * RF of phase [33,35]: [V1-1] 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR log2(V1,V,Out) 1099.37/291.50 * Partial RF of phase [32,34]: 1099.37/291.50 - RF of loop [32:1]: 1099.37/291.50 V1-2 1099.37/291.50 - RF of loop [34:1]: 1099.37/291.50 V1-1 1099.37/291.50 * Partial RF of phase [33,35]: 1099.37/291.50 - RF of loop [33:1]: 1099.37/291.50 V1-2 1099.37/291.50 - RF of loop [35:1]: 1099.37/291.50 V1-1 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Specialization of cost equations start/4 1099.37/291.50 * CE 6 is refined into CE [86] 1099.37/291.50 * CE 7 is refined into CE [87] 1099.37/291.50 * CE 5 is refined into CE [88] 1099.37/291.50 * CE 1 is refined into CE [89] 1099.37/291.50 * CE 2 is refined into CE [90] 1099.37/291.50 * CE 3 is refined into CE [91,92,93,94,95,96,97,98,99,100,101] 1099.37/291.50 * CE 4 is refined into CE [102,103,104,105,106,107,108,109,110,111,112] 1099.37/291.50 * CE 8 is refined into CE [113,114,115,116,117] 1099.37/291.50 * CE 9 is refined into CE [118,119] 1099.37/291.50 * CE 10 is refined into CE [120,121,122] 1099.37/291.50 * CE 11 is refined into CE [123,124,125,126,127] 1099.37/291.50 * CE 12 is refined into CE [128,129,130] 1099.37/291.50 * CE 13 is refined into CE [131,132,133,134,135] 1099.37/291.50 1099.37/291.50 1099.37/291.50 ### Cost equations --> "Loop" of start/4 1099.37/291.50 * CEs [123] --> Loop 42 1099.37/291.50 * CEs [114,120] --> Loop 43 1099.37/291.50 * CEs [86] --> Loop 44 1099.37/291.50 * CEs [87] --> Loop 45 1099.37/291.50 * CEs [88,133] --> Loop 46 1099.37/291.50 * CEs [89,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112] --> Loop 47 1099.37/291.50 * CEs [90,113,115,116,117,118,119,121,122,124,125,126,127,128,129,130,131,132,134,135] --> Loop 48 1099.37/291.50 1099.37/291.50 ### Ranking functions of CR start(V1,V,V17,V21) 1099.37/291.50 1099.37/291.50 #### Partial ranking functions of CR start(V1,V,V17,V21) 1099.37/291.50 1099.37/291.50 1099.37/291.50 Computing Bounds 1099.37/291.50 ===================================== 1099.37/291.50 1099.37/291.50 #### Cost of chains of inc(V1,Out): 1099.37/291.50 * Chain [[19],20]: 1*it(19)+1 1099.37/291.50 Such that:it(19) =< Out 1099.37/291.50 1099.37/291.50 with precondition: [Out>=1,V1>=Out] 1099.37/291.50 1099.37/291.50 * Chain [20]: 1 1099.37/291.50 with precondition: [Out=0,V1>=0] 1099.37/291.50 1099.37/291.50 1099.37/291.50 #### Cost of chains of le(V1,V,Out): 1099.37/291.50 * Chain [[21],24]: 1*it(21)+1 1099.37/291.50 Such that:it(21) =< V1 1099.37/291.50 1099.37/291.50 with precondition: [Out=2,V1>=1,V>=V1] 1099.37/291.50 1099.37/291.50 * Chain [[21],23]: 1*it(21)+1 1099.37/291.50 Such that:it(21) =< V 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V>=1,V1>=V+1] 1099.37/291.50 1099.37/291.50 * Chain [[21],22]: 1*it(21)+0 1099.37/291.50 Such that:it(21) =< V 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=1,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [24]: 1 1099.37/291.50 with precondition: [V1=0,Out=2,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [23]: 1 1099.37/291.50 with precondition: [V=0,Out=1,V1>=1] 1099.37/291.50 1099.37/291.50 * Chain [22]: 0 1099.37/291.50 with precondition: [Out=0,V1>=0,V>=0] 1099.37/291.50 1099.37/291.50 1099.37/291.50 #### Cost of chains of minus(V1,V,Out): 1099.37/291.50 * Chain [[25],27]: 1*it(25)+1 1099.37/291.50 Such that:it(25) =< V 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=1,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[25],26]: 1*it(25)+1 1099.37/291.50 Such that:it(25) =< V 1099.37/291.50 1099.37/291.50 with precondition: [V1=Out+V,V>=1,V1>=V] 1099.37/291.50 1099.37/291.50 * Chain [27]: 1 1099.37/291.50 with precondition: [Out=0,V1>=0,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [26]: 1 1099.37/291.50 with precondition: [V=0,V1=Out,V1>=0] 1099.37/291.50 1099.37/291.50 1099.37/291.50 #### Cost of chains of quot(V1,V,Out): 1099.37/291.50 * Chain [[30],31]: 2*it(30)+1 1099.37/291.50 Such that:it(30) =< Out 1099.37/291.50 1099.37/291.50 with precondition: [V=1,Out>=1,V1>=Out] 1099.37/291.50 1099.37/291.50 * Chain [[30],29,31]: 2*it(30)+1*s(3)+3 1099.37/291.50 Such that:s(3) =< 1 1099.37/291.50 it(30) =< Out 1099.37/291.50 1099.37/291.50 with precondition: [V=1,Out>=2,V1>=Out] 1099.37/291.50 1099.37/291.50 * Chain [[28],31]: 2*it(28)+1*s(6)+1 1099.37/291.50 Such that:it(28) =< V1-V+1 1099.37/291.50 aux(3) =< V1 1099.37/291.50 it(28) =< aux(3) 1099.37/291.50 s(6) =< aux(3) 1099.37/291.50 1099.37/291.50 with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 1099.37/291.50 1099.37/291.50 * Chain [[28],29,31]: 2*it(28)+1*s(3)+1*s(6)+3 1099.37/291.50 Such that:it(28) =< V1-V+1 1099.37/291.50 s(3) =< V 1099.37/291.50 aux(4) =< V1 1099.37/291.50 it(28) =< aux(4) 1099.37/291.50 s(6) =< aux(4) 1099.37/291.50 1099.37/291.50 with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 1099.37/291.50 1099.37/291.50 * Chain [31]: 1 1099.37/291.50 with precondition: [Out=0,V1>=0,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [29,31]: 1*s(3)+3 1099.37/291.50 Such that:s(3) =< V 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=1,V>=1] 1099.37/291.50 1099.37/291.50 1099.37/291.50 #### Cost of chains of log2(V1,V,Out): 1099.37/291.50 * Chain [[33,35],41]: 12*it(33)+9*it(35)+14*s(10)+3*s(60)+2*s(61)+3*s(65)+5 1099.37/291.50 Such that:aux(6) =< 1 1099.37/291.50 s(66) =< 2*V1 1099.37/291.50 aux(18) =< V1 1099.37/291.50 aux(19) =< 3*V1 1099.37/291.50 s(10) =< aux(6) 1099.37/291.50 it(33) =< aux(18) 1099.37/291.50 it(35) =< aux(18) 1099.37/291.50 it(35) =< aux(19) 1099.37/291.50 s(61) =< aux(18)*2 1099.37/291.50 s(65) =< s(66) 1099.37/291.50 s(60) =< aux(19) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [[33,35],39]: 12*it(33)+9*it(35)+3*s(60)+2*s(61)+3*s(65)+6*s(68)+5 1099.37/291.50 Such that:aux(20) =< 1 1099.37/291.50 s(66) =< 2*V1 1099.37/291.50 aux(22) =< V1 1099.37/291.50 aux(23) =< 3*V1 1099.37/291.50 s(68) =< aux(20) 1099.37/291.50 it(33) =< aux(22) 1099.37/291.50 it(35) =< aux(22) 1099.37/291.50 it(35) =< aux(23) 1099.37/291.50 s(61) =< aux(22)*2 1099.37/291.50 s(65) =< s(66) 1099.37/291.50 s(60) =< aux(23) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [[33,35],37,41]: 12*it(33)+9*it(35)+15*s(10)+3*s(60)+2*s(61)+3*s(65)+12 1099.37/291.50 Such that:aux(24) =< 1 1099.37/291.50 s(66) =< 2*V1 1099.37/291.50 aux(25) =< V1 1099.37/291.50 aux(26) =< 3*V1 1099.37/291.50 s(10) =< aux(24) 1099.37/291.50 it(33) =< aux(25) 1099.37/291.50 it(35) =< aux(25) 1099.37/291.50 it(35) =< aux(26) 1099.37/291.50 s(61) =< aux(25)*2 1099.37/291.50 s(65) =< s(66) 1099.37/291.50 s(60) =< aux(26) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=3,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [[33,35],37,40]: 12*it(33)+9*it(35)+3*s(60)+2*s(61)+3*s(65)+3*s(79)+12 1099.37/291.50 Such that:aux(29) =< 1 1099.37/291.50 s(66) =< 2*V1 1099.37/291.50 aux(30) =< V1 1099.37/291.50 aux(31) =< 3*V1 1099.37/291.50 s(79) =< aux(29) 1099.37/291.50 it(33) =< aux(30) 1099.37/291.50 it(35) =< aux(30) 1099.37/291.50 it(35) =< aux(31) 1099.37/291.50 s(61) =< aux(30)*2 1099.37/291.50 s(65) =< s(66) 1099.37/291.50 s(60) =< aux(31) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=3,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],[33,35],41]: 18*it(32)+18*it(33)+9*it(35)+14*s(10)+3*s(60)+2*s(61)+3*s(110)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+5 1099.37/291.50 Such that:aux(6) =< 1 1099.37/291.50 aux(42) =< V1 1099.37/291.50 aux(19) =< 6*V1+12*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(45) =< 2*V1+4*V 1099.37/291.50 aux(46) =< 3*V1 1099.37/291.50 s(10) =< aux(6) 1099.37/291.50 it(33) =< aux(45) 1099.37/291.50 it(35) =< aux(45) 1099.37/291.50 it(35) =< aux(19) 1099.37/291.50 s(61) =< aux(45)*2 1099.37/291.50 s(60) =< aux(19) 1099.37/291.50 aux(36) =< aux(42) 1099.37/291.50 it(32) =< aux(42) 1099.37/291.50 aux(36) =< aux(46) 1099.37/291.50 it(32) =< aux(46) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(36)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(110) =< aux(36) 1099.37/291.50 s(120) =< it(32)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(46) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],[33,35],39]: 18*it(32)+18*it(33)+9*it(35)+3*s(60)+2*s(61)+6*s(68)+3*s(110)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+5 1099.37/291.50 Such that:aux(20) =< 1 1099.37/291.50 aux(42) =< V1 1099.37/291.50 aux(23) =< 6*V1+12*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(47) =< 2*V1+4*V 1099.37/291.50 aux(48) =< 3*V1 1099.37/291.50 s(68) =< aux(20) 1099.37/291.50 it(33) =< aux(47) 1099.37/291.50 it(35) =< aux(47) 1099.37/291.50 it(35) =< aux(23) 1099.37/291.50 s(61) =< aux(47)*2 1099.37/291.50 s(60) =< aux(23) 1099.37/291.50 aux(36) =< aux(42) 1099.37/291.50 it(32) =< aux(42) 1099.37/291.50 aux(36) =< aux(48) 1099.37/291.50 it(32) =< aux(48) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(36)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(110) =< aux(36) 1099.37/291.50 s(120) =< it(32)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(48) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],[33,35],37,41]: 18*it(32)+18*it(33)+9*it(35)+15*s(10)+3*s(60)+2*s(61)+3*s(110)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+12 1099.37/291.50 Such that:aux(24) =< 1 1099.37/291.50 aux(42) =< V1 1099.37/291.50 aux(26) =< 6*V1+12*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(49) =< 2*V1+4*V 1099.37/291.50 aux(50) =< 3*V1 1099.37/291.50 s(10) =< aux(24) 1099.37/291.50 it(33) =< aux(49) 1099.37/291.50 it(35) =< aux(49) 1099.37/291.50 it(35) =< aux(26) 1099.37/291.50 s(61) =< aux(49)*2 1099.37/291.50 s(60) =< aux(26) 1099.37/291.50 aux(36) =< aux(42) 1099.37/291.50 it(32) =< aux(42) 1099.37/291.50 aux(36) =< aux(50) 1099.37/291.50 it(32) =< aux(50) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(36)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(110) =< aux(36) 1099.37/291.50 s(120) =< it(32)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(50) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=5,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],[33,35],37,40]: 18*it(32)+18*it(33)+9*it(35)+3*s(60)+2*s(61)+3*s(79)+3*s(110)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+12 1099.37/291.50 Such that:aux(29) =< 1 1099.37/291.50 aux(42) =< V1 1099.37/291.50 aux(31) =< 6*V1+12*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(51) =< 2*V1+4*V 1099.37/291.50 aux(52) =< 3*V1 1099.37/291.50 s(79) =< aux(29) 1099.37/291.50 it(33) =< aux(51) 1099.37/291.50 it(35) =< aux(51) 1099.37/291.50 it(35) =< aux(31) 1099.37/291.50 s(61) =< aux(51)*2 1099.37/291.50 s(60) =< aux(31) 1099.37/291.50 aux(36) =< aux(42) 1099.37/291.50 it(32) =< aux(42) 1099.37/291.50 aux(36) =< aux(52) 1099.37/291.50 it(32) =< aux(52) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(36)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(110) =< aux(36) 1099.37/291.50 s(120) =< it(32)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(52) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=5,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],41]: 12*it(32)+9*it(34)+14*s(10)+12*s(12)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+5 1099.37/291.50 Such that:aux(6) =< 1 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(53) =< V1 1099.37/291.50 aux(54) =< 2*V1+4*V 1099.37/291.50 aux(55) =< 3*V1 1099.37/291.50 s(10) =< aux(6) 1099.37/291.50 s(12) =< aux(54) 1099.37/291.50 it(32) =< aux(53) 1099.37/291.50 it(34) =< aux(53) 1099.37/291.50 it(34) =< aux(55) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(53)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(55) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],39]: 12*it(32)+9*it(34)+6*s(68)+6*s(70)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+5 1099.37/291.50 Such that:aux(20) =< 1 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(56) =< V1 1099.37/291.50 aux(57) =< 2*V1+4*V 1099.37/291.50 aux(58) =< 3*V1 1099.37/291.50 s(68) =< aux(20) 1099.37/291.50 s(70) =< aux(57) 1099.37/291.50 it(32) =< aux(56) 1099.37/291.50 it(34) =< aux(56) 1099.37/291.50 it(34) =< aux(58) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(56)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(58) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],38]: 12*it(32)+9*it(34)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+4*s(118)+1*s(122)+6 1099.37/291.50 Such that:s(122) =< 1 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(59) =< V1 1099.37/291.50 aux(60) =< 2*V1+4*V 1099.37/291.50 aux(61) =< 3*V1 1099.37/291.50 s(118) =< aux(60) 1099.37/291.50 it(32) =< aux(59) 1099.37/291.50 it(34) =< aux(59) 1099.37/291.50 it(34) =< aux(61) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(59)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(61) 1099.37/291.50 1099.37/291.50 with precondition: [V1>=2,Out>=0,V>=Out+1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],37,41]: 12*it(32)+9*it(34)+15*s(10)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+3*s(118)+12 1099.37/291.50 Such that:aux(24) =< 1 1099.37/291.50 s(119) =< 2*V1+4*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(62) =< V1 1099.37/291.50 aux(63) =< 3*V1 1099.37/291.50 s(10) =< aux(24) 1099.37/291.50 it(32) =< aux(62) 1099.37/291.50 it(34) =< aux(62) 1099.37/291.50 it(34) =< aux(63) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(62)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(118) =< s(119) 1099.37/291.50 s(112) =< aux(63) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],37,40]: 12*it(32)+9*it(34)+3*s(79)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+3*s(118)+12 1099.37/291.50 Such that:aux(29) =< 1 1099.37/291.50 s(119) =< 2*V1+4*V 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(64) =< V1 1099.37/291.50 aux(65) =< 3*V1 1099.37/291.50 s(79) =< aux(29) 1099.37/291.50 it(32) =< aux(64) 1099.37/291.50 it(34) =< aux(64) 1099.37/291.50 it(34) =< aux(65) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(64)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(118) =< s(119) 1099.37/291.50 s(112) =< aux(65) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],36,41]: 12*it(32)+9*it(34)+15*s(10)+13*s(12)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+12 1099.37/291.50 Such that:aux(66) =< 1 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(68) =< V1 1099.37/291.50 aux(69) =< 2*V1+4*V 1099.37/291.50 aux(70) =< 3*V1 1099.37/291.50 s(10) =< aux(66) 1099.37/291.50 s(12) =< aux(69) 1099.37/291.50 it(32) =< aux(68) 1099.37/291.50 it(34) =< aux(68) 1099.37/291.50 it(34) =< aux(70) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(68)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(70) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [[32,34],36,40]: 12*it(32)+9*it(34)+6*s(80)+3*s(81)+1*s(111)+3*s(112)+2*s(113)+2*s(116)+12 1099.37/291.50 Such that:aux(71) =< 1 1099.37/291.50 aux(38) =< V 1099.37/291.50 aux(73) =< V1 1099.37/291.50 aux(74) =< 2*V1+4*V 1099.37/291.50 aux(75) =< 3*V1 1099.37/291.50 s(81) =< aux(71) 1099.37/291.50 s(80) =< aux(74) 1099.37/291.50 it(32) =< aux(73) 1099.37/291.50 it(34) =< aux(73) 1099.37/291.50 it(34) =< aux(75) 1099.37/291.50 aux(41) =< aux(38) 1099.37/291.50 s(113) =< aux(73)*2 1099.37/291.50 s(111) =< it(32)*aux(38) 1099.37/291.50 s(120) =< it(34)*aux(41) 1099.37/291.50 s(116) =< s(120) 1099.37/291.50 s(112) =< aux(75) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=3,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [41]: 14*s(10)+9*s(12)+5 1099.37/291.50 Such that:aux(6) =< 1 1099.37/291.50 aux(7) =< V 1099.37/291.50 s(10) =< aux(6) 1099.37/291.50 s(12) =< aux(7) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=0,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [40]: 2*s(80)+2*s(81)+5 1099.37/291.50 Such that:aux(27) =< 1 1099.37/291.50 aux(28) =< V 1099.37/291.50 s(81) =< aux(27) 1099.37/291.50 s(80) =< aux(28) 1099.37/291.50 1099.37/291.50 with precondition: [V1=0,Out=1,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [39]: 6*s(68)+3*s(70)+5 1099.37/291.50 Such that:aux(20) =< 1 1099.37/291.50 aux(21) =< V 1099.37/291.50 s(68) =< aux(20) 1099.37/291.50 s(70) =< aux(21) 1099.37/291.50 1099.37/291.50 with precondition: [V1=1,Out=0,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [38]: 1*s(122)+1*s(123)+6 1099.37/291.50 Such that:s(122) =< 1 1099.37/291.50 s(123) =< V 1099.37/291.50 1099.37/291.50 with precondition: [V1=1,Out>=0,V>=Out+1] 1099.37/291.50 1099.37/291.50 * Chain [37,41]: 15*s(10)+12 1099.37/291.50 Such that:aux(24) =< 1 1099.37/291.50 s(10) =< aux(24) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [37,40]: 3*s(79)+12 1099.37/291.50 Such that:aux(29) =< 1 1099.37/291.50 s(79) =< aux(29) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=2,V>=0] 1099.37/291.50 1099.37/291.50 * Chain [36,41]: 15*s(10)+10*s(12)+12 1099.37/291.50 Such that:aux(66) =< 1 1099.37/291.50 aux(67) =< V 1099.37/291.50 s(10) =< aux(66) 1099.37/291.50 s(12) =< aux(67) 1099.37/291.50 1099.37/291.50 with precondition: [Out=0,V1>=2,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [36,40]: 3*s(80)+3*s(81)+12 1099.37/291.50 Such that:aux(71) =< 1 1099.37/291.50 aux(72) =< V 1099.37/291.50 s(81) =< aux(71) 1099.37/291.50 s(80) =< aux(72) 1099.37/291.50 1099.37/291.50 with precondition: [Out=1,V1>=2,V>=1] 1099.37/291.50 1099.37/291.50 1099.37/291.50 #### Cost of chains of start(V1,V,V17,V21): 1099.37/291.50 * Chain [48]: 33*s(356)+256*s(358)+4*s(363)+381*s(372)+139*s(383)+333*s(385)+42*s(387)+87*s(391)+36*s(392)+8*s(393)+12*s(394)+16*s(396)+24*s(398)+119*s(439)+7*s(444)+22*s(446)+36*s(448)+8*s(449)+12*s(450)+4*s(453)+13 1099.37/291.50 Such that:aux(90) =< 1 1099.37/291.50 aux(91) =< V1 1099.37/291.50 aux(92) =< V1-V+1 1099.37/291.50 aux(93) =< 2*V1 1099.37/291.50 aux(94) =< 2*V1+4*V 1099.37/291.50 aux(95) =< 3*V1 1099.37/291.50 aux(96) =< 6*V1 1099.37/291.50 aux(97) =< 6*V1+12*V 1099.37/291.50 aux(98) =< V 1099.37/291.50 s(372) =< aux(90) 1099.37/291.50 s(358) =< aux(91) 1099.37/291.50 s(363) =< aux(92) 1099.37/291.50 s(356) =< aux(98) 1099.37/291.50 s(383) =< aux(93) 1099.37/291.50 s(385) =< aux(91) 1099.37/291.50 s(385) =< aux(95) 1099.37/291.50 s(387) =< aux(91)*2 1099.37/291.50 s(391) =< aux(95) 1099.37/291.50 s(392) =< aux(93) 1099.37/291.50 s(392) =< aux(96) 1099.37/291.50 s(393) =< aux(93)*2 1099.37/291.50 s(394) =< aux(96) 1099.37/291.50 s(395) =< aux(91) 1099.37/291.50 s(395) =< aux(95) 1099.37/291.50 s(396) =< s(395)*2 1099.37/291.50 s(398) =< s(395) 1099.37/291.50 s(439) =< aux(94) 1099.37/291.50 s(442) =< aux(98) 1099.37/291.50 s(444) =< s(358)*aux(98) 1099.37/291.50 s(445) =< s(385)*s(442) 1099.37/291.50 s(446) =< s(445) 1099.37/291.50 s(448) =< aux(94) 1099.37/291.50 s(448) =< aux(97) 1099.37/291.50 s(449) =< aux(94)*2 1099.37/291.50 s(450) =< aux(97) 1099.37/291.50 s(453) =< s(385)*aux(98) 1099.37/291.50 s(363) =< aux(91) 1099.37/291.50 1099.37/291.50 with precondition: [V1>=0] 1099.37/291.50 1099.37/291.50 * Chain [47]: 1612*s(498)+98*s(499)+88*s(509)+27*s(518)+6*s(519)+9*s(520)+28*s(526)+88*s(536)+234*s(538)+28*s(540)+4*s(541)+14*s(543)+60*s(544)+27*s(545)+6*s(546)+9*s(547)+12*s(549)+3*s(550)+18*s(551)+131*s(556)+119*s(568)+132*s(569)+171*s(570)+22*s(572)+7*s(573)+22*s(575)+45*s(576)+36*s(577)+8*s(578)+12*s(579)+8*s(581)+4*s(582)+12*s(583)+119*s(649)+132*s(650)+171*s(651)+22*s(653)+7*s(654)+22*s(656)+45*s(657)+36*s(658)+8*s(659)+12*s(660)+8*s(662)+4*s(663)+12*s(664)+12*s(665)+88*s(727)+27*s(736)+6*s(737)+9*s(738)+88*s(754)+4*s(759)+14*s(761)+27*s(763)+6*s(764)+9*s(765)+3*s(768)+33*s(774)+119*s(786)+132*s(787)+171*s(788)+22*s(790)+7*s(791)+22*s(793)+45*s(794)+36*s(795)+8*s(796)+12*s(797)+8*s(799)+4*s(800)+12*s(801)+119*s(867)+132*s(868)+171*s(869)+22*s(871)+7*s(872)+22*s(874)+45*s(875)+36*s(876)+8*s(877)+12*s(878)+8*s(880)+4*s(881)+12*s(882)+12*s(883)+17 1099.37/291.50 Such that:s(721) =< 4*V17 1099.37/291.50 s(748) =< 4*V17+2 1099.37/291.50 s(723) =< 12*V17 1099.37/291.50 s(750) =< 12*V17+6 1099.37/291.50 s(503) =< 4*V21 1099.37/291.50 s(530) =< 4*V21+2 1099.37/291.50 s(505) =< 12*V21 1099.37/291.50 s(532) =< 12*V21+6 1099.37/291.50 aux(119) =< 1 1099.37/291.50 aux(120) =< 2 1099.37/291.50 aux(121) =< 3 1099.37/291.50 aux(122) =< V 1099.37/291.50 aux(123) =< V+1 1099.37/291.50 aux(124) =< V+4*V17 1099.37/291.50 aux(125) =< V+4*V17+1 1099.37/291.50 aux(126) =< 3*V+12*V17 1099.37/291.50 aux(127) =< 3*V+12*V17+3 1099.37/291.50 aux(128) =< V/2 1099.37/291.50 aux(129) =< V/2+1/2 1099.37/291.50 aux(130) =< 3/2*V 1099.37/291.50 aux(131) =< 3/2*V+3/2 1099.37/291.50 aux(132) =< V17 1099.37/291.50 aux(133) =< V17+1 1099.37/291.50 aux(134) =< V17+4*V21 1099.37/291.50 aux(135) =< V17+4*V21+1 1099.37/291.50 aux(136) =< 3*V17+12*V21 1099.37/291.50 aux(137) =< 3*V17+12*V21+3 1099.37/291.50 aux(138) =< V17/2 1099.37/291.50 aux(139) =< V17/2+1/2 1099.37/291.50 aux(140) =< 3/2*V17 1099.37/291.50 aux(141) =< 3/2*V17+3/2 1099.37/291.50 aux(142) =< V21 1099.37/291.50 s(498) =< aux(119) 1099.37/291.50 s(526) =< aux(120) 1099.37/291.50 s(556) =< aux(132) 1099.37/291.50 s(499) =< aux(142) 1099.37/291.50 s(509) =< s(503) 1099.37/291.50 s(518) =< s(503) 1099.37/291.50 s(518) =< s(505) 1099.37/291.50 s(519) =< s(503)*2 1099.37/291.50 s(520) =< s(505) 1099.37/291.50 s(568) =< aux(134) 1099.37/291.50 s(569) =< aux(138) 1099.37/291.50 s(570) =< aux(138) 1099.37/291.50 s(570) =< aux(140) 1099.37/291.50 s(539) =< aux(142) 1099.37/291.50 s(572) =< aux(138)*2 1099.37/291.50 s(573) =< s(569)*aux(142) 1099.37/291.50 s(574) =< s(570)*s(539) 1099.37/291.50 s(575) =< s(574) 1099.37/291.50 s(576) =< aux(140) 1099.37/291.50 s(577) =< aux(134) 1099.37/291.50 s(577) =< aux(136) 1099.37/291.50 s(578) =< aux(134)*2 1099.37/291.50 s(579) =< aux(136) 1099.37/291.50 s(580) =< aux(138) 1099.37/291.50 s(580) =< aux(140) 1099.37/291.50 s(581) =< s(580)*2 1099.37/291.50 s(582) =< s(570)*aux(142) 1099.37/291.50 s(583) =< s(580) 1099.37/291.50 s(649) =< aux(135) 1099.37/291.50 s(650) =< aux(139) 1099.37/291.50 s(651) =< aux(139) 1099.37/291.50 s(651) =< aux(141) 1099.37/291.50 s(653) =< aux(139)*2 1099.37/291.50 s(654) =< s(650)*aux(142) 1099.37/291.50 s(655) =< s(651)*s(539) 1099.37/291.50 s(656) =< s(655) 1099.37/291.50 s(657) =< aux(141) 1099.37/291.50 s(658) =< aux(135) 1099.37/291.50 s(658) =< aux(137) 1099.37/291.50 s(659) =< aux(135)*2 1099.37/291.50 s(660) =< aux(137) 1099.37/291.50 s(661) =< aux(139) 1099.37/291.50 s(661) =< aux(141) 1099.37/291.50 s(662) =< s(661)*2 1099.37/291.50 s(663) =< s(651)*aux(142) 1099.37/291.50 s(664) =< s(661) 1099.37/291.50 s(665) =< aux(133) 1099.37/291.50 s(727) =< s(721) 1099.37/291.50 s(736) =< s(721) 1099.37/291.50 s(736) =< s(723) 1099.37/291.50 s(737) =< s(721)*2 1099.37/291.50 s(738) =< s(723) 1099.37/291.50 s(774) =< aux(122) 1099.37/291.50 s(786) =< aux(124) 1099.37/291.50 s(787) =< aux(128) 1099.37/291.50 s(788) =< aux(128) 1099.37/291.50 s(788) =< aux(130) 1099.37/291.50 s(757) =< aux(132) 1099.37/291.50 s(790) =< aux(128)*2 1099.37/291.50 s(791) =< s(787)*aux(132) 1099.37/291.50 s(792) =< s(788)*s(757) 1099.37/291.50 s(793) =< s(792) 1099.37/291.50 s(794) =< aux(130) 1099.37/291.50 s(795) =< aux(124) 1099.37/291.50 s(795) =< aux(126) 1099.37/291.50 s(796) =< aux(124)*2 1099.37/291.50 s(797) =< aux(126) 1099.37/291.50 s(798) =< aux(128) 1099.37/291.50 s(798) =< aux(130) 1099.37/291.50 s(799) =< s(798)*2 1099.37/291.50 s(800) =< s(788)*aux(132) 1099.37/291.50 s(801) =< s(798) 1099.37/291.50 s(867) =< aux(125) 1099.37/291.50 s(868) =< aux(129) 1099.37/291.50 s(869) =< aux(129) 1099.37/291.50 s(869) =< aux(131) 1099.37/291.50 s(871) =< aux(129)*2 1099.37/291.50 s(872) =< s(868)*aux(132) 1099.37/291.50 s(873) =< s(869)*s(757) 1099.37/291.50 s(874) =< s(873) 1099.37/291.50 s(875) =< aux(131) 1099.37/291.50 s(876) =< aux(125) 1099.37/291.50 s(876) =< aux(127) 1099.37/291.50 s(877) =< aux(125)*2 1099.37/291.50 s(878) =< aux(127) 1099.37/291.50 s(879) =< aux(129) 1099.37/291.50 s(879) =< aux(131) 1099.37/291.50 s(880) =< s(879)*2 1099.37/291.50 s(881) =< s(869)*aux(132) 1099.37/291.50 s(882) =< s(879) 1099.37/291.50 s(883) =< aux(123) 1099.37/291.50 s(536) =< s(530) 1099.37/291.50 s(538) =< aux(119) 1099.37/291.50 s(538) =< aux(121) 1099.37/291.50 s(540) =< aux(119)*2 1099.37/291.50 s(541) =< s(498)*aux(142) 1099.37/291.50 s(542) =< s(538)*s(539) 1099.37/291.50 s(543) =< s(542) 1099.37/291.50 s(544) =< aux(121) 1099.37/291.50 s(545) =< s(530) 1099.37/291.50 s(545) =< s(532) 1099.37/291.50 s(546) =< s(530)*2 1099.37/291.50 s(547) =< s(532) 1099.37/291.50 s(548) =< aux(119) 1099.37/291.50 s(548) =< aux(121) 1099.37/291.50 s(549) =< s(548)*2 1099.37/291.50 s(550) =< s(538)*aux(142) 1099.37/291.50 s(551) =< s(548) 1099.37/291.50 s(754) =< s(748) 1099.37/291.50 s(759) =< s(498)*aux(132) 1099.37/291.50 s(760) =< s(538)*s(757) 1099.37/291.50 s(761) =< s(760) 1099.37/291.50 s(763) =< s(748) 1099.37/291.50 s(763) =< s(750) 1099.37/291.50 s(764) =< s(748)*2 1099.37/291.50 s(765) =< s(750) 1099.37/291.50 s(768) =< s(538)*aux(132) 1099.37/291.50 1099.37/291.50 with precondition: [V1=1,V>=0,V17>=0] 1099.37/291.50 1099.37/291.50 * Chain [46]: 1*s(932)+1*s(933)+6 1099.37/291.50 Such that:s(932) =< 1 1099.37/291.50 s(933) =< V 1099.37/291.50 1099.37/291.50 with precondition: [V1=1,V>=1] 1099.37/291.50 1099.37/291.50 * Chain [45]: 1 1099.37/291.50 with precondition: [V1=2,V>=0,V17>=0,V21>=0] 1099.37/291.50 1099.37/291.50 * Chain [44]: 1 1099.37/291.50 with precondition: [V1=2,V>=0,V17>=1] 1099.37/291.50 1099.37/291.50 * Chain [43]: 1 1099.37/291.50 with precondition: [V=0,V1>=0] 1099.37/291.50 1099.37/291.50 * Chain [42]: 1*s(934)+4*s(936)+3 1099.37/291.50 Such that:s(934) =< 1 1099.37/291.50 s(935) =< V1 1099.37/291.50 s(936) =< s(935) 1099.37/291.50 1099.37/291.50 with precondition: [V=1,V1>=1] 1099.37/291.50 1099.37/291.50 1099.37/291.50 Closed-form bounds of start(V1,V,V17,V21): 1099.37/291.50 ------------------------------------- 1099.37/291.50 * Chain [48] with precondition: [V1>=0] 1099.37/291.50 - Upper bound: 729*V1+394+33*V1*nat(V)+nat(V)*33+382*V1+261*V1+72*V1+nat(2*V1+4*V)*171+nat(6*V1+12*V)*12+nat(V1-V+1)*4 1099.37/291.50 - Complexity: n^2 1099.37/291.50 * Chain [47] with precondition: [V1=1,V>=0,V17>=0] 1099.37/291.50 - Upper bound: 33*V+152*V17+2197+(V/2+1/2)*(33*V17)+V/2*(33*V17)+nat(V21)*119+(V17/2+1/2)*(nat(V21)*33)+V17/2*(nat(V21)*33)+508*V17+nat(4*V21)*127+108*V17+nat(12*V21)*9+135/2*V+135/2*V17+(12*V+12)+(171*V+684*V17)+(12*V17+12)+nat(V17+4*V21)*171+(36*V+144*V17)+nat(3*V17+12*V21)*12+(508*V17+254)+nat(4*V21+2)*127+(108*V17+54)+nat(12*V21+6)*9+(135/2*V+135/2)+(135/2*V17+135/2)+(171*V+684*V17+171)+nat(V17+4*V21+1)*171+(36*V+144*V17+36)+nat(3*V17+12*V21+3)*12+(375/2*V+375/2)+(375/2*V17+375/2)+375/2*V+375/2*V17 1099.37/291.50 - Complexity: n^2 1099.37/291.50 * Chain [46] with precondition: [V1=1,V>=1] 1099.37/291.50 - Upper bound: V+7 1099.37/291.50 - Complexity: n 1099.37/291.50 * Chain [45] with precondition: [V1=2,V>=0,V17>=0,V21>=0] 1099.37/291.50 - Upper bound: 1 1099.37/291.50 - Complexity: constant 1099.37/291.50 * Chain [44] with precondition: [V1=2,V>=0,V17>=1] 1099.37/291.50 - Upper bound: 1 1099.37/291.50 - Complexity: constant 1099.37/291.50 * Chain [43] with precondition: [V=0,V1>=0] 1099.37/291.50 - Upper bound: 1 1099.37/291.50 - Complexity: constant 1099.37/291.50 * Chain [42] with precondition: [V=1,V1>=1] 1099.37/291.50 - Upper bound: 4*V1+4 1099.37/291.50 - Complexity: n 1099.37/291.50 1099.37/291.50 ### Maximum cost of start(V1,V,V17,V21): max([4*V1+3,nat(V)*32+387+max([33*V1*nat(V)+729*V1+382*V1+261*V1+72*V1+nat(2*V1+4*V)*171+nat(6*V1+12*V)*12+nat(V1-V+1)*4,nat(V17)*152+1803+nat(V17)*33*nat(V/2+1/2)+nat(V17)*33*nat(V/2)+nat(V21)*119+nat(V21)*33*nat(V17/2+1/2)+nat(V21)*33*nat(V17/2)+nat(4*V17)*127+nat(4*V21)*127+nat(12*V17)*9+nat(12*V21)*9+nat(3/2*V)*45+nat(3/2*V17)*45+nat(V+1)*12+nat(V+4*V17)*171+nat(V17+1)*12+nat(V17+4*V21)*171+nat(3*V+12*V17)*12+nat(3*V17+12*V21)*12+nat(4*V17+2)*127+nat(4*V21+2)*127+nat(12*V17+6)*9+nat(12*V21+6)*9+nat(3/2*V+3/2)*45+nat(3/2*V17+3/2)*45+nat(V+4*V17+1)*171+nat(V17+4*V21+1)*171+nat(3*V+12*V17+3)*12+nat(3*V17+12*V21+3)*12+nat(V/2+1/2)*375+nat(V17/2+1/2)*375+nat(V/2)*375+nat(V17/2)*375])+(nat(V)+6)])+1 1099.37/291.50 Asymptotic class: n^2 1099.37/291.50 * Total analysis performed in 1968 ms. 1099.37/291.50 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (10) 1099.37/291.50 BOUNDS(1, n^2) 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1099.37/291.50 Transformed a relative TRS into a decreasing-loop problem. 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (12) 1099.37/291.50 Obligation: 1099.37/291.50 Analyzing the following TRS for decreasing loops: 1099.37/291.50 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1099.37/291.50 1099.37/291.50 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true 1099.37/291.50 le(s(x), 0) -> false 1099.37/291.50 le(s(x), s(y)) -> le(x, y) 1099.37/291.50 inc(0) -> 0 1099.37/291.50 inc(s(x)) -> s(inc(x)) 1099.37/291.50 minus(0, y) -> 0 1099.37/291.50 minus(x, 0) -> x 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) 1099.37/291.50 quot(0, s(y)) -> 0 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 1099.37/291.50 log(x) -> log2(x, 0) 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) 1099.37/291.50 if(true, b, x, y) -> log_undefined 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) 1099.37/291.50 if2(true, x, s(y)) -> y 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) 1099.37/291.50 1099.37/291.50 S is empty. 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (13) DecreasingLoopProof (LOWER BOUND(ID)) 1099.37/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1099.37/291.50 1099.37/291.50 The rewrite sequence 1099.37/291.50 1099.37/291.50 le(s(x), s(y)) ->^+ le(x, y) 1099.37/291.50 1099.37/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1099.37/291.50 1099.37/291.50 The pumping substitution is [x / s(x), y / s(y)]. 1099.37/291.50 1099.37/291.50 The result substitution is [ ]. 1099.37/291.50 1099.37/291.50 1099.37/291.50 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (14) 1099.37/291.50 Complex Obligation (BEST) 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (15) 1099.37/291.50 Obligation: 1099.37/291.50 Proved the lower bound n^1 for the following obligation: 1099.37/291.50 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1099.37/291.50 1099.37/291.50 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true 1099.37/291.50 le(s(x), 0) -> false 1099.37/291.50 le(s(x), s(y)) -> le(x, y) 1099.37/291.50 inc(0) -> 0 1099.37/291.50 inc(s(x)) -> s(inc(x)) 1099.37/291.50 minus(0, y) -> 0 1099.37/291.50 minus(x, 0) -> x 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) 1099.37/291.50 quot(0, s(y)) -> 0 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 1099.37/291.50 log(x) -> log2(x, 0) 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) 1099.37/291.50 if(true, b, x, y) -> log_undefined 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) 1099.37/291.50 if2(true, x, s(y)) -> y 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) 1099.37/291.50 1099.37/291.50 S is empty. 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (16) LowerBoundPropagationProof (FINISHED) 1099.37/291.50 Propagated lower bound. 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (17) 1099.37/291.50 BOUNDS(n^1, INF) 1099.37/291.50 1099.37/291.50 ---------------------------------------- 1099.37/291.50 1099.37/291.50 (18) 1099.37/291.50 Obligation: 1099.37/291.50 Analyzing the following TRS for decreasing loops: 1099.37/291.50 1099.37/291.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1099.37/291.50 1099.37/291.50 1099.37/291.50 The TRS R consists of the following rules: 1099.37/291.50 1099.37/291.50 le(0, y) -> true 1099.37/291.50 le(s(x), 0) -> false 1099.37/291.50 le(s(x), s(y)) -> le(x, y) 1099.37/291.50 inc(0) -> 0 1099.37/291.50 inc(s(x)) -> s(inc(x)) 1099.37/291.50 minus(0, y) -> 0 1099.37/291.50 minus(x, 0) -> x 1099.37/291.50 minus(s(x), s(y)) -> minus(x, y) 1099.37/291.50 quot(0, s(y)) -> 0 1099.37/291.50 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 1099.37/291.50 log(x) -> log2(x, 0) 1099.37/291.50 log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y)) 1099.37/291.50 if(true, b, x, y) -> log_undefined 1099.37/291.50 if(false, b, x, y) -> if2(b, x, y) 1099.37/291.50 if2(true, x, s(y)) -> y 1099.37/291.50 if2(false, x, y) -> log2(quot(x, s(s(0))), y) 1099.37/291.50 1099.37/291.50 S is empty. 1099.37/291.50 Rewrite Strategy: INNERMOST 1099.60/291.59 EOF