1142.10/291.50 WORST_CASE(Omega(n^1), O(n^2)) 1142.10/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1142.10/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1142.10/291.51 1142.10/291.51 1142.10/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1142.10/291.51 1142.10/291.51 (0) CpxTRS 1142.10/291.51 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1142.10/291.51 (2) CpxWeightedTrs 1142.10/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1142.10/291.51 (4) CpxTypedWeightedTrs 1142.10/291.51 (5) CompletionProof [UPPER BOUND(ID), 49 ms] 1142.10/291.51 (6) CpxTypedWeightedCompleteTrs 1142.10/291.51 (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 1142.10/291.51 (8) CpxRNTS 1142.10/291.51 (9) CompleteCoflocoProof [FINISHED, 1305 ms] 1142.10/291.51 (10) BOUNDS(1, n^2) 1142.10/291.51 (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1142.10/291.51 (12) CpxTRS 1142.10/291.51 (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1142.10/291.51 (14) typed CpxTrs 1142.10/291.51 (15) OrderProof [LOWER BOUND(ID), 0 ms] 1142.10/291.51 (16) typed CpxTrs 1142.10/291.51 (17) RewriteLemmaProof [LOWER BOUND(ID), 268 ms] 1142.10/291.51 (18) BEST 1142.10/291.51 (19) proven lower bound 1142.10/291.51 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 1142.10/291.51 (21) BOUNDS(n^1, INF) 1142.10/291.51 (22) typed CpxTrs 1142.10/291.51 1142.10/291.51 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (0) 1142.10/291.51 Obligation: 1142.10/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 1142.10/291.51 1142.10/291.51 1142.10/291.51 The TRS R consists of the following rules: 1142.10/291.51 1142.10/291.51 le(0, y) -> true 1142.10/291.51 le(s(x), 0) -> false 1142.10/291.51 le(s(x), s(y)) -> le(x, y) 1142.10/291.51 quot(x, 0) -> quotZeroErro 1142.10/291.51 quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) 1142.10/291.51 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.51 if(true, x, y, z, u, v) -> v 1142.10/291.51 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.51 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.51 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) 1142.10/291.51 1142.10/291.51 S is empty. 1142.10/291.51 Rewrite Strategy: INNERMOST 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1142.10/291.51 Transformed relative TRS to weighted TRS 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (2) 1142.10/291.51 Obligation: 1142.10/291.51 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 1142.10/291.51 1142.10/291.51 1142.10/291.51 The TRS R consists of the following rules: 1142.10/291.51 1142.10/291.51 le(0, y) -> true [1] 1142.10/291.51 le(s(x), 0) -> false [1] 1142.10/291.51 le(s(x), s(y)) -> le(x, y) [1] 1142.10/291.51 quot(x, 0) -> quotZeroErro [1] 1142.10/291.51 quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1] 1142.10/291.51 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1] 1142.10/291.51 if(true, x, y, z, u, v) -> v [1] 1142.10/291.51 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1] 1142.10/291.51 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1] 1142.10/291.51 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1] 1142.10/291.51 1142.10/291.51 Rewrite Strategy: INNERMOST 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1142.10/291.51 Infered types. 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (4) 1142.10/291.51 Obligation: 1142.10/291.51 Runtime Complexity Weighted TRS with Types. 1142.10/291.51 The TRS R consists of the following rules: 1142.10/291.51 1142.10/291.51 le(0, y) -> true [1] 1142.10/291.51 le(s(x), 0) -> false [1] 1142.10/291.51 le(s(x), s(y)) -> le(x, y) [1] 1142.10/291.51 quot(x, 0) -> quotZeroErro [1] 1142.10/291.51 quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1] 1142.10/291.51 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1] 1142.10/291.51 if(true, x, y, z, u, v) -> v [1] 1142.10/291.51 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1] 1142.10/291.51 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1] 1142.10/291.51 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1] 1142.10/291.51 1142.10/291.51 The TRS has the following type information: 1142.10/291.51 le :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> true:false 1142.10/291.51 0 :: 0:s:quotZeroErro 1142.10/291.51 true :: true:false 1142.10/291.51 s :: 0:s:quotZeroErro -> 0:s:quotZeroErro 1142.10/291.51 false :: true:false 1142.10/291.51 quot :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro 1142.10/291.51 quotZeroErro :: 0:s:quotZeroErro 1142.10/291.51 quotIter :: 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro 1142.10/291.51 if :: true:false -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro 1142.10/291.51 if2 :: true:false -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro -> 0:s:quotZeroErro 1142.10/291.51 1142.10/291.51 Rewrite Strategy: INNERMOST 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (5) CompletionProof (UPPER BOUND(ID)) 1142.10/291.51 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 1142.10/291.51 1142.10/291.51 le(v0, v1) -> null_le [0] 1142.10/291.51 quot(v0, v1) -> null_quot [0] 1142.10/291.51 quotIter(v0, v1, v2, v3, v4) -> null_quotIter [0] 1142.10/291.51 if(v0, v1, v2, v3, v4, v5) -> null_if [0] 1142.10/291.51 if2(v0, v1, v2, v3, v4, v5) -> null_if2 [0] 1142.10/291.51 1142.10/291.51 And the following fresh constants: null_le, null_quot, null_quotIter, null_if, null_if2 1142.10/291.51 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (6) 1142.10/291.51 Obligation: 1142.10/291.51 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 1142.10/291.51 1142.10/291.51 Runtime Complexity Weighted TRS with Types. 1142.10/291.51 The TRS R consists of the following rules: 1142.10/291.51 1142.10/291.51 le(0, y) -> true [1] 1142.10/291.51 le(s(x), 0) -> false [1] 1142.10/291.51 le(s(x), s(y)) -> le(x, y) [1] 1142.10/291.51 quot(x, 0) -> quotZeroErro [1] 1142.10/291.51 quot(x, s(y)) -> quotIter(x, s(y), 0, 0, 0) [1] 1142.10/291.51 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) [1] 1142.10/291.51 if(true, x, y, z, u, v) -> v [1] 1142.10/291.51 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) [1] 1142.10/291.51 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) [1] 1142.10/291.51 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0, s(v)) [1] 1142.10/291.51 le(v0, v1) -> null_le [0] 1142.10/291.51 quot(v0, v1) -> null_quot [0] 1142.10/291.51 quotIter(v0, v1, v2, v3, v4) -> null_quotIter [0] 1142.10/291.51 if(v0, v1, v2, v3, v4, v5) -> null_if [0] 1142.10/291.51 if2(v0, v1, v2, v3, v4, v5) -> null_if2 [0] 1142.10/291.51 1142.10/291.51 The TRS has the following type information: 1142.10/291.51 le :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> true:false:null_le 1142.10/291.51 0 :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 true :: true:false:null_le 1142.10/291.51 s :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 false :: true:false:null_le 1142.10/291.51 quot :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 quotZeroErro :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 quotIter :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 if :: true:false:null_le -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 if2 :: true:false:null_le -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 -> 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 null_le :: true:false:null_le 1142.10/291.51 null_quot :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 null_quotIter :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 null_if :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 null_if2 :: 0:s:quotZeroErro:null_quot:null_quotIter:null_if:null_if2 1142.10/291.51 1142.10/291.51 Rewrite Strategy: INNERMOST 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1142.10/291.51 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1142.10/291.51 The constant constructors are abstracted as follows: 1142.10/291.51 1142.10/291.51 0 => 0 1142.10/291.51 true => 2 1142.10/291.51 false => 1 1142.10/291.51 quotZeroErro => 1 1142.10/291.51 null_le => 0 1142.10/291.51 null_quot => 0 1142.10/291.51 null_quotIter => 0 1142.10/291.51 null_if => 0 1142.10/291.51 null_if2 => 0 1142.10/291.51 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (8) 1142.10/291.51 Obligation: 1142.10/291.51 Complexity RNTS consisting of the following rules: 1142.10/291.51 1142.10/291.51 if(z', z'', z1, z2, z3, z4) -{ 1 }-> v :|: z1 = y, z4 = v, z >= 0, v >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0 1142.10/291.51 if(z', z'', z1, z2, z3, z4) -{ 1 }-> if2(le(y, 1 + u), x, y, 1 + z, 1 + u, v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0 1142.10/291.51 if(z', z'', z1, z2, z3, z4) -{ 0 }-> 0 :|: z2 = v3, z4 = v5, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, v5 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0 1142.10/291.51 if2(z', z'', z1, z2, z3, z4) -{ 1 }-> quotIter(x, y, z, u, v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0 1142.10/291.51 if2(z', z'', z1, z2, z3, z4) -{ 1 }-> quotIter(x, y, z, 0, 1 + v) :|: z1 = y, z4 = v, z >= 0, v >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0 1142.10/291.51 if2(z', z'', z1, z2, z3, z4) -{ 0 }-> 0 :|: z2 = v3, z4 = v5, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, v5 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0 1142.10/291.51 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y 1142.10/291.51 le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 1142.10/291.51 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 1142.10/291.51 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 1142.10/291.51 quot(z', z'') -{ 1 }-> quotIter(x, 1 + y, 0, 0, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y 1142.10/291.51 quot(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0 1142.10/291.51 quot(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 1142.10/291.51 quotIter(z', z'', z1, z2, z3) -{ 1 }-> if(le(x, z), x, 1 + y, z, u, v) :|: z1 = z, z2 = u, z >= 0, v >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, z3 = v, u >= 0 1142.10/291.51 quotIter(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0 1142.10/291.51 1142.10/291.51 Only complete derivations are relevant for the runtime complexity. 1142.10/291.51 1142.10/291.51 ---------------------------------------- 1142.10/291.51 1142.10/291.51 (9) CompleteCoflocoProof (FINISHED) 1142.10/291.51 Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: 1142.10/291.51 1142.10/291.51 eq(start(V, V1, V16, V14, V15, V20),0,[le(V, V1, Out)],[V >= 0,V1 >= 0]). 1142.10/291.51 eq(start(V, V1, V16, V14, V15, V20),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]). 1142.10/291.51 eq(start(V, V1, V16, V14, V15, V20),0,[quotIter(V, V1, V16, V14, V15, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0]). 1142.10/291.51 eq(start(V, V1, V16, V14, V15, V20),0,[if(V, V1, V16, V14, V15, V20, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0,V20 >= 0]). 1142.10/291.51 eq(start(V, V1, V16, V14, V15, V20),0,[if2(V, V1, V16, V14, V15, V20, Out)],[V >= 0,V1 >= 0,V16 >= 0,V14 >= 0,V15 >= 0,V20 >= 0]). 1142.10/291.51 eq(le(V, V1, Out),1,[],[Out = 2,V1 = V2,V2 >= 0,V = 0]). 1142.10/291.51 eq(le(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V3,V3 >= 0]). 1142.10/291.51 eq(le(V, V1, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]). 1142.10/291.51 eq(quot(V, V1, Out),1,[],[Out = 1,V1 = 0,V = V6,V6 >= 0]). 1142.10/291.51 eq(quot(V, V1, Out),1,[quotIter(V7, 1 + V8, 0, 0, 0, Ret1)],[Out = Ret1,V = V7,V7 >= 0,V8 >= 0,V1 = 1 + V8]). 1142.10/291.51 eq(quotIter(V, V1, V16, V14, V15, Out),1,[le(V10, V11, Ret0),if(Ret0, V10, 1 + V12, V11, V9, V13, Ret2)],[Out = Ret2,V16 = V11,V14 = V9,V11 >= 0,V13 >= 0,V = V10,V10 >= 0,V12 >= 0,V1 = 1 + V12,V15 = V13,V9 >= 0]). 1142.10/291.51 eq(if(V, V1, V16, V14, V15, V20, Out),1,[],[Out = V21,V16 = V17,V20 = V21,V18 >= 0,V21 >= 0,V = 2,V14 = V18,V22 >= 0,V17 >= 0,V1 = V22,V15 = V19,V19 >= 0]). 1142.10/291.51 eq(if(V, V1, V16, V14, V15, V20, Out),1,[le(V23, 1 + V25, Ret01),if2(Ret01, V26, V23, 1 + V24, 1 + V25, V27, Ret3)],[Out = Ret3,V16 = V23,V20 = V27,V24 >= 0,V27 >= 0,V14 = V24,V26 >= 0,V23 >= 0,V1 = V26,V = 1,V15 = V25,V25 >= 0]). 1142.10/291.51 eq(if2(V, V1, V16, V14, V15, V20, Out),1,[quotIter(V28, V29, V30, V31, V32, Ret4)],[Out = Ret4,V16 = V29,V20 = V32,V30 >= 0,V32 >= 0,V14 = V30,V28 >= 0,V29 >= 0,V1 = V28,V = 1,V15 = V31,V31 >= 0]). 1142.10/291.51 eq(if2(V, V1, V16, V14, V15, V20, Out),1,[quotIter(V34, V35, V36, 0, 1 + V37, Ret5)],[Out = Ret5,V16 = V35,V20 = V37,V36 >= 0,V37 >= 0,V = 2,V14 = V36,V34 >= 0,V35 >= 0,V1 = V34,V15 = V33,V33 >= 0]). 1142.10/291.51 eq(le(V, V1, Out),0,[],[Out = 0,V39 >= 0,V38 >= 0,V1 = V38,V = V39]). 1142.10/291.51 eq(quot(V, V1, Out),0,[],[Out = 0,V41 >= 0,V40 >= 0,V1 = V40,V = V41]). 1142.10/291.51 eq(quotIter(V, V1, V16, V14, V15, Out),0,[],[Out = 0,V14 = V45,V43 >= 0,V44 >= 0,V16 = V46,V42 >= 0,V1 = V42,V15 = V44,V46 >= 0,V45 >= 0,V = V43]). 1142.10/291.51 eq(if(V, V1, V16, V14, V15, V20, Out),0,[],[Out = 0,V14 = V52,V20 = V50,V47 >= 0,V51 >= 0,V16 = V48,V49 >= 0,V50 >= 0,V1 = V49,V15 = V51,V48 >= 0,V52 >= 0,V = V47]). 1142.10/291.51 eq(if2(V, V1, V16, V14, V15, V20, Out),0,[],[Out = 0,V14 = V55,V20 = V57,V54 >= 0,V58 >= 0,V16 = V56,V53 >= 0,V57 >= 0,V1 = V53,V15 = V58,V56 >= 0,V55 >= 0,V = V54]). 1142.10/291.51 input_output_vars(le(V,V1,Out),[V,V1],[Out]). 1142.10/291.51 input_output_vars(quot(V,V1,Out),[V,V1],[Out]). 1142.10/291.51 input_output_vars(quotIter(V,V1,V16,V14,V15,Out),[V,V1,V16,V14,V15],[Out]). 1142.10/291.51 input_output_vars(if(V,V1,V16,V14,V15,V20,Out),[V,V1,V16,V14,V15,V20],[Out]). 1142.10/291.51 input_output_vars(if2(V,V1,V16,V14,V15,V20,Out),[V,V1,V16,V14,V15,V20],[Out]). 1142.10/291.51 1142.10/291.51 1142.10/291.51 CoFloCo proof output: 1142.10/291.51 Preprocessing Cost Relations 1142.10/291.51 ===================================== 1142.10/291.51 1142.10/291.51 #### Computed strongly connected components 1142.10/291.51 0. recursive : [le/3] 1142.10/291.51 1. recursive : [if/7,if2/7,quotIter/6] 1142.10/291.51 2. non_recursive : [quot/3] 1142.10/291.51 3. non_recursive : [start/6] 1142.10/291.51 1142.10/291.51 #### Obtained direct recursion through partial evaluation 1142.10/291.51 0. SCC is partially evaluated into le/3 1142.10/291.51 1. SCC is partially evaluated into quotIter/6 1142.10/291.51 2. SCC is partially evaluated into quot/3 1142.10/291.51 3. SCC is partially evaluated into start/6 1142.10/291.51 1142.10/291.51 Control-Flow Refinement of Cost Relations 1142.10/291.51 ===================================== 1142.10/291.51 1142.10/291.51 ### Specialization of cost equations le/3 1142.10/291.51 * CE 14 is refined into CE [24] 1142.10/291.51 * CE 12 is refined into CE [25] 1142.10/291.51 * CE 11 is refined into CE [26] 1142.10/291.51 * CE 13 is refined into CE [27] 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Cost equations --> "Loop" of le/3 1142.10/291.51 * CEs [27] --> Loop 14 1142.10/291.51 * CEs [24] --> Loop 15 1142.10/291.51 * CEs [25] --> Loop 16 1142.10/291.51 * CEs [26] --> Loop 17 1142.10/291.51 1142.10/291.51 ### Ranking functions of CR le(V,V1,Out) 1142.10/291.51 * RF of phase [14]: [V,V1] 1142.10/291.51 1142.10/291.51 #### Partial ranking functions of CR le(V,V1,Out) 1142.10/291.51 * Partial RF of phase [14]: 1142.10/291.51 - RF of loop [14:1]: 1142.10/291.51 V 1142.10/291.51 V1 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Specialization of cost equations quotIter/6 1142.10/291.51 * CE 19 is refined into CE [28,29] 1142.10/291.51 * CE 15 is refined into CE [30,31,32,33,34,35] 1142.10/291.51 * CE 18 is refined into CE [36,37,38,39,40] 1142.10/291.51 * CE 20 is refined into CE [41] 1142.10/291.51 * CE 17 is refined into CE [42,43] 1142.10/291.51 * CE 16 is refined into CE [44,45] 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Cost equations --> "Loop" of quotIter/6 1142.10/291.51 * CEs [43] --> Loop 18 1142.10/291.51 * CEs [45] --> Loop 19 1142.10/291.51 * CEs [42] --> Loop 20 1142.10/291.51 * CEs [44] --> Loop 21 1142.10/291.51 * CEs [29] --> Loop 22 1142.10/291.51 * CEs [30,31,32,37] --> Loop 23 1142.10/291.51 * CEs [28] --> Loop 24 1142.10/291.51 * CEs [33,34,35,36,38,39,40,41] --> Loop 25 1142.10/291.51 1142.10/291.51 ### Ranking functions of CR quotIter(V,V1,V16,V14,V15,Out) 1142.10/291.51 * RF of phase [18,19]: [V-V16] 1142.10/291.51 1142.10/291.51 #### Partial ranking functions of CR quotIter(V,V1,V16,V14,V15,Out) 1142.10/291.51 * Partial RF of phase [18,19]: 1142.10/291.51 - RF of loop [18:1]: 1142.10/291.51 V1-V14-1 depends on loops [19:1] 1142.10/291.51 - RF of loop [18:1,19:1]: 1142.10/291.51 V-V16 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Specialization of cost equations quot/3 1142.10/291.51 * CE 22 is refined into CE [46,47,48,49,50,51] 1142.10/291.51 * CE 23 is refined into CE [52] 1142.10/291.51 * CE 21 is refined into CE [53] 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Cost equations --> "Loop" of quot/3 1142.10/291.51 * CEs [51] --> Loop 26 1142.10/291.51 * CEs [50] --> Loop 27 1142.10/291.51 * CEs [53] --> Loop 28 1142.10/291.51 * CEs [48] --> Loop 29 1142.10/291.51 * CEs [47] --> Loop 30 1142.10/291.51 * CEs [46,49,52] --> Loop 31 1142.10/291.51 1142.10/291.51 ### Ranking functions of CR quot(V,V1,Out) 1142.10/291.51 1142.10/291.51 #### Partial ranking functions of CR quot(V,V1,Out) 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Specialization of cost equations start/6 1142.10/291.51 * CE 4 is refined into CE [54,55,56,57,58,59,60,61] 1142.10/291.51 * CE 7 is refined into CE [62] 1142.10/291.51 * CE 1 is refined into CE [63,64,65,66] 1142.10/291.51 * CE 2 is refined into CE [67] 1142.10/291.51 * CE 3 is refined into CE [68,69,70,71,72] 1142.10/291.51 * CE 5 is refined into CE [73,74,75,76] 1142.10/291.51 * CE 6 is refined into CE [77,78,79,80,81,82,83,84] 1142.10/291.51 * CE 8 is refined into CE [85,86,87,88,89] 1142.10/291.51 * CE 9 is refined into CE [90,91,92,93,94] 1142.10/291.51 * CE 10 is refined into CE [95,96,97,98,99,100,101,102] 1142.10/291.51 1142.10/291.51 1142.10/291.51 ### Cost equations --> "Loop" of start/6 1142.10/291.51 * CEs [100] --> Loop 32 1142.10/291.51 * CEs [99] --> Loop 33 1142.10/291.51 * CEs [86,92] --> Loop 34 1142.10/291.51 * CEs [59] --> Loop 35 1142.10/291.51 * CEs [58] --> Loop 36 1142.10/291.51 * CEs [56] --> Loop 37 1142.10/291.51 * CEs [55,93] --> Loop 38 1142.10/291.51 * CEs [54,57,60,61,62] --> Loop 39 1142.10/291.51 * CEs [82] --> Loop 40 1142.10/291.51 * CEs [81] --> Loop 41 1142.10/291.51 * CEs [97] --> Loop 42 1142.10/291.51 * CEs [96] --> Loop 43 1142.10/291.51 * CEs [63,68] --> Loop 44 1142.10/291.51 * CEs [78,79,90] --> Loop 45 1142.10/291.51 * CEs [64,65,66,69,70,71,72,73,74,75,76,77,80,83,84] --> Loop 46 1142.10/291.51 * CEs [67,85,87,88,89,91,94,95,98,101,102] --> Loop 47 1142.10/291.51 1142.10/291.51 ### Ranking functions of CR start(V,V1,V16,V14,V15,V20) 1142.10/291.51 1142.10/291.51 #### Partial ranking functions of CR start(V,V1,V16,V14,V15,V20) 1142.10/291.51 1142.10/291.51 1142.10/291.51 Computing Bounds 1142.10/291.51 ===================================== 1142.10/291.51 1142.10/291.51 #### Cost of chains of le(V,V1,Out): 1142.10/291.51 * Chain [[14],17]: 1*it(14)+1 1142.10/291.51 Such that:it(14) =< V 1142.10/291.51 1142.10/291.51 with precondition: [Out=2,V>=1,V1>=V] 1142.10/291.51 1142.10/291.51 * Chain [[14],16]: 1*it(14)+1 1142.10/291.51 Such that:it(14) =< V1 1142.10/291.51 1142.10/291.51 with precondition: [Out=1,V1>=1,V>=V1+1] 1142.10/291.51 1142.10/291.51 * Chain [[14],15]: 1*it(14)+0 1142.10/291.51 Such that:it(14) =< V1 1142.10/291.51 1142.10/291.51 with precondition: [Out=0,V>=1,V1>=1] 1142.10/291.51 1142.10/291.51 * Chain [17]: 1 1142.10/291.51 with precondition: [V=0,Out=2,V1>=0] 1142.10/291.51 1142.10/291.51 * Chain [16]: 1 1142.10/291.51 with precondition: [V1=0,Out=1,V>=1] 1142.10/291.51 1142.10/291.51 * Chain [15]: 0 1142.10/291.51 with precondition: [Out=0,V>=0,V1>=0] 1142.10/291.51 1142.10/291.51 1142.10/291.51 #### Cost of chains of quotIter(V,V1,V16,V14,V15,Out): 1142.10/291.51 * Chain [[18,19],25]: 10*it(18)+6*s(2)+3*s(3)+1*s(7)+1*s(19)+1*s(20)+1*s(21)+4 1142.10/291.51 Such that:s(7) =< V1 1142.10/291.51 aux(13) =< V1+V14 1142.10/291.51 aux(14) =< V 1142.10/291.51 aux(15) =< V-V16 1142.10/291.51 aux(16) =< V-V16+V14+1 1142.10/291.51 s(2) =< aux(14) 1142.10/291.51 s(3) =< aux(16) 1142.10/291.51 it(18) =< aux(15) 1142.10/291.51 aux(7) =< aux(14) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(13) 1142.10/291.51 s(19) =< it(18)*aux(14) 1142.10/291.51 1142.10/291.51 with precondition: [Out=0,V1>=1,V16>=1,V14>=0,V15>=0,V>=V16+1] 1142.10/291.51 1142.10/291.51 * Chain [[18,19],22]: 10*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+1*s(23)+3 1142.10/291.51 Such that:s(22) =< V-V16+V14 1142.10/291.51 aux(13) =< V1+V14 1142.10/291.51 aux(17) =< V 1142.10/291.51 aux(18) =< V-V16 1142.10/291.51 s(23) =< aux(17) 1142.10/291.51 it(18) =< aux(18) 1142.10/291.51 aux(7) =< aux(17) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(13) 1142.10/291.51 s(19) =< it(18)*aux(17) 1142.10/291.51 1142.10/291.51 with precondition: [V1>=1,V16>=1,V14>=0,V15>=0,V>=V16+1,Out>=V15,Out+V1>=V15+2,V+V15>=Out+V16] 1142.10/291.51 1142.10/291.51 * Chain [25]: 5*s(2)+2*s(3)+1*s(7)+1*s(10)+4 1142.10/291.51 Such that:s(10) =< V 1142.10/291.51 s(7) =< V1 1142.10/291.51 aux(1) =< V16 1142.10/291.51 aux(2) =< V14+1 1142.10/291.51 s(2) =< aux(1) 1142.10/291.51 s(3) =< aux(2) 1142.10/291.51 1142.10/291.51 with precondition: [Out=0,V>=0,V1>=0,V16>=0,V14>=0,V15>=0] 1142.10/291.51 1142.10/291.51 * Chain [24]: 3 1142.10/291.51 with precondition: [V=0,V15=Out,V1>=1,V16>=0,V14>=0,V15>=0] 1142.10/291.51 1142.10/291.51 * Chain [23]: 2*s(24)+1*s(26)+4 1142.10/291.51 Such that:s(26) =< V1 1142.10/291.51 aux(19) =< V14+1 1142.10/291.51 s(24) =< aux(19) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,Out=0,V>=1,V1>=1,V14>=0,V15>=0] 1142.10/291.51 1142.10/291.51 * Chain [22]: 1*s(23)+3 1142.10/291.51 Such that:s(23) =< V 1142.10/291.51 1142.10/291.51 with precondition: [V15=Out,V>=1,V1>=1,V14>=0,V15>=0,V16>=V] 1142.10/291.51 1142.10/291.51 * Chain [21,[18,19],25]: 19*it(18)+2*s(7)+1*s(19)+1*s(20)+1*s(21)+9 1142.10/291.51 Such that:aux(20) =< V 1142.10/291.51 aux(21) =< V1 1142.10/291.51 s(7) =< aux(21) 1142.10/291.51 it(18) =< aux(20) 1142.10/291.51 aux(7) =< aux(20) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(21) 1142.10/291.51 s(19) =< it(18)*aux(20) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,Out=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.51 1142.10/291.51 * Chain [21,[18,19],22]: 12*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(27)+8 1142.10/291.51 Such that:aux(22) =< V 1142.10/291.51 aux(23) =< V1 1142.10/291.51 it(18) =< aux(22) 1142.10/291.51 s(27) =< aux(23) 1142.10/291.51 aux(7) =< aux(22) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(23) 1142.10/291.51 s(19) =< it(18)*aux(22) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1,Out>=V15+1,Out+V1>=V15+3,V+V15>=Out] 1142.10/291.51 1142.10/291.51 * Chain [21,25]: 7*s(2)+2*s(7)+1*s(10)+9 1142.10/291.51 Such that:s(10) =< V 1142.10/291.51 aux(24) =< 1 1142.10/291.51 aux(25) =< V1 1142.10/291.51 s(7) =< aux(25) 1142.10/291.51 s(2) =< aux(24) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,Out=0,V>=1,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.51 1142.10/291.51 * Chain [21,22]: 1*s(23)+1*s(27)+8 1142.10/291.51 Such that:s(23) =< 1 1142.10/291.51 s(27) =< V1 1142.10/291.51 1142.10/291.51 with precondition: [V=1,V16=0,Out=V15+1,V1>=1,Out>=1,V14+1>=V1] 1142.10/291.51 1142.10/291.51 * Chain [20,[18,19],25]: 16*it(18)+3*s(3)+1*s(7)+1*s(19)+1*s(20)+1*s(21)+1*s(28)+9 1142.10/291.51 Such that:aux(16) =< V+V14+1 1142.10/291.51 s(7) =< V1 1142.10/291.51 aux(13) =< V1+V14+1 1142.10/291.51 s(28) =< V14+1 1142.10/291.51 aux(26) =< V 1142.10/291.51 it(18) =< aux(26) 1142.10/291.51 s(3) =< aux(16) 1142.10/291.51 aux(7) =< aux(26) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(13) 1142.10/291.51 s(19) =< it(18)*aux(26) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,Out=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.51 1142.10/291.51 * Chain [20,[18,19],22]: 11*it(18)+1*s(19)+1*s(20)+1*s(21)+1*s(22)+1*s(28)+8 1142.10/291.51 Such that:s(22) =< V+V14 1142.10/291.51 aux(13) =< V1+V14+1 1142.10/291.51 s(28) =< V14+1 1142.10/291.51 aux(27) =< V 1142.10/291.51 it(18) =< aux(27) 1142.10/291.51 aux(7) =< aux(27) 1142.10/291.51 s(21) =< it(18)*aux(7) 1142.10/291.51 s(20) =< it(18)*aux(13) 1142.10/291.51 s(19) =< it(18)*aux(27) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2,Out>=V15,V+V15>=Out+1] 1142.10/291.51 1142.10/291.51 * Chain [20,25]: 5*s(2)+2*s(3)+1*s(7)+1*s(10)+1*s(28)+9 1142.10/291.51 Such that:aux(1) =< 1 1142.10/291.51 s(10) =< V 1142.10/291.51 s(7) =< V1 1142.10/291.51 s(28) =< V14+1 1142.10/291.51 aux(2) =< V14+2 1142.10/291.51 s(2) =< aux(1) 1142.10/291.51 s(3) =< aux(2) 1142.10/291.51 1142.10/291.51 with precondition: [V16=0,Out=0,V>=1,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.51 1142.10/291.51 * Chain [20,22]: 1*s(23)+1*s(28)+8 1142.10/291.51 Such that:s(23) =< 1 1142.10/291.51 s(28) =< V14+1 1142.10/291.51 1142.10/291.51 with precondition: [V=1,V16=0,V15=Out,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.51 1142.10/291.51 1142.10/291.51 #### Cost of chains of quot(V,V1,Out): 1142.10/291.51 * Chain [31]: 54*s(92)+9*s(93)+18*s(94)+2*s(96)+3*s(98)+2*s(99)+3*s(100)+6*s(102)+1*s(103)+10 1142.10/291.51 Such that:aux(32) =< 1 1142.10/291.51 s(87) =< 2 1142.10/291.51 aux(33) =< V 1142.10/291.51 aux(34) =< V+1 1142.10/291.51 aux(35) =< V1 1142.10/291.51 s(85) =< V1+1 1142.10/291.51 s(92) =< aux(33) 1142.10/291.51 s(93) =< aux(35) 1142.10/291.51 s(94) =< aux(32) 1142.10/291.51 s(96) =< s(87) 1142.10/291.51 s(97) =< aux(33) 1142.10/291.51 s(98) =< s(92)*s(97) 1142.10/291.51 s(99) =< s(92)*aux(35) 1142.10/291.51 s(100) =< s(92)*aux(33) 1142.10/291.51 s(102) =< aux(34) 1142.10/291.51 s(103) =< s(92)*s(85) 1142.10/291.51 1142.10/291.51 with precondition: [Out=0,V>=0,V1>=0] 1142.10/291.51 1142.10/291.51 * Chain [30]: 2*s(109)+9 1142.10/291.51 Such that:aux(36) =< 1 1142.10/291.51 s(109) =< aux(36) 1142.10/291.51 1142.10/291.51 with precondition: [V=1,V1=1,Out=1] 1142.10/291.51 1142.10/291.51 * Chain [29]: 2*s(111)+9 1142.10/291.51 Such that:aux(37) =< 1 1142.10/291.51 s(111) =< aux(37) 1142.10/291.51 1142.10/291.51 with precondition: [V=1,Out=0,V1>=2] 1142.10/291.51 1142.10/291.51 * Chain [28]: 1 1142.10/291.51 with precondition: [V1=0,Out=1,V>=0] 1142.10/291.51 1142.10/291.51 * Chain [27]: 12*s(115)+1*s(116)+1*s(118)+1*s(119)+1*s(120)+9 1142.10/291.51 Such that:s(114) =< 1 1142.10/291.51 s(113) =< V 1142.10/291.51 s(115) =< s(113) 1142.10/291.51 s(116) =< s(114) 1142.10/291.51 s(117) =< s(113) 1142.10/291.51 s(118) =< s(115)*s(117) 1142.10/291.51 s(119) =< s(115)*s(114) 1142.10/291.51 s(120) =< s(115)*s(113) 1142.10/291.51 1142.10/291.51 with precondition: [V1=1,Out>=2,V>=Out] 1142.10/291.51 1142.10/291.51 * Chain [26]: 12*s(121)+1*s(123)+1*s(127)+1*s(128)+1*s(129)+9 1142.10/291.51 Such that:s(123) =< 1 1142.10/291.51 s(122) =< V1+1 1142.10/291.51 aux(38) =< V 1142.10/291.51 s(121) =< aux(38) 1142.10/291.51 s(126) =< aux(38) 1142.10/291.51 s(127) =< s(121)*s(126) 1142.10/291.51 s(128) =< s(121)*s(122) 1142.10/291.51 s(129) =< s(121)*aux(38) 1142.10/291.51 1142.10/291.51 with precondition: [V>=2,V1>=2,Out>=0,V>=Out+1] 1142.10/291.51 1142.10/291.51 1142.10/291.51 #### Cost of chains of start(V,V1,V16,V14,V15,V20): 1142.10/291.51 * Chain [47]: 20*s(148)+113*s(150)+33*s(157)+2*s(160)+6*s(162)+3*s(163)+6*s(164)+6*s(165)+2*s(166)+6*s(188)+2*s(190)+5*s(195)+3*s(196)+1*s(197)+3*s(198)+20*s(199)+2*s(200)+2*s(201)+2*s(202)+1*s(204)+10 1142.10/291.51 Such that:s(151) =< 2 1142.10/291.51 s(153) =< V+1 1142.10/291.51 s(204) =< V-V16+V14 1142.10/291.51 s(176) =< V-V16+V14+1 1142.10/291.51 s(177) =< V+V14+1 1142.10/291.51 s(179) =< V1+V14+1 1142.10/291.51 s(180) =< V16 1142.10/291.51 s(185) =< V14+1 1142.10/291.51 s(181) =< V14+2 1142.10/291.51 aux(40) =< 1 1142.10/291.51 aux(41) =< V 1142.10/291.51 aux(42) =< V-V16 1142.10/291.51 aux(43) =< V1 1142.10/291.51 aux(44) =< V1+1 1142.10/291.51 aux(45) =< V1+V14 1142.10/291.51 s(157) =< aux(40) 1142.10/291.51 s(150) =< aux(41) 1142.10/291.51 s(148) =< aux(43) 1142.10/291.51 s(160) =< s(151) 1142.10/291.51 s(161) =< aux(41) 1142.10/291.51 s(162) =< s(150)*s(161) 1142.10/291.51 s(163) =< s(150)*aux(43) 1142.10/291.51 s(164) =< s(150)*aux(41) 1142.10/291.51 s(165) =< s(153) 1142.10/291.51 s(166) =< s(150)*aux(44) 1142.10/291.51 s(188) =< s(185) 1142.10/291.51 s(190) =< s(181) 1142.10/291.51 s(195) =< s(180) 1142.10/291.51 s(196) =< s(177) 1142.10/291.51 s(197) =< s(150)*s(179) 1142.10/291.51 s(198) =< s(176) 1142.10/291.51 s(199) =< aux(42) 1142.10/291.51 s(200) =< s(199)*s(161) 1142.10/291.51 s(201) =< s(199)*aux(45) 1142.10/291.51 s(202) =< s(199)*aux(41) 1142.10/291.51 1142.10/291.51 with precondition: [V>=0,V1>=0] 1142.10/291.51 1142.10/291.51 * Chain [46]: 12*s(214)+32*s(216)+138*s(230)+42*s(232)+2*s(234)+6*s(236)+3*s(237)+6*s(238)+10*s(239)+3*s(240)+1*s(241)+64*s(242)+6*s(244)+2*s(245)+6*s(246)+8*s(275)+2*s(277)+3*s(283)+1*s(284)+6*s(285)+2*s(288)+2*s(293)+5*s(323)+3*s(324)+1*s(325)+2*s(329)+12 1142.10/291.51 Such that:s(225) =< 2 1142.10/291.51 s(221) =< V1+1 1142.10/291.51 s(305) =< V1+V15+1 1142.10/291.51 s(264) =< V1+V15+2 1142.10/291.51 s(223) =< V16+1 1142.10/291.51 s(266) =< V16+V15+2 1142.10/291.51 s(308) =< V14 1142.10/291.51 s(268) =< V15+3 1142.10/291.51 aux(51) =< 1 1142.10/291.51 aux(52) =< V1 1142.10/291.51 aux(53) =< V1-V14 1142.10/291.51 aux(54) =< V1-V14+V15 1142.10/291.51 aux(55) =< V1-V14+V15+1 1142.10/291.51 aux(56) =< V16 1142.10/291.51 aux(57) =< V16+V15 1142.10/291.51 aux(58) =< V16+V15+1 1142.10/291.51 aux(59) =< V14+1 1142.10/291.51 aux(60) =< V15+1 1142.10/291.51 aux(61) =< V15+2 1142.10/291.51 s(230) =< aux(52) 1142.10/291.51 s(293) =< aux(54) 1142.10/291.51 s(216) =< aux(56) 1142.10/291.51 s(214) =< aux(60) 1142.10/291.51 s(275) =< aux(61) 1142.10/291.51 s(232) =< aux(51) 1142.10/291.51 s(277) =< s(268) 1142.10/291.51 s(235) =< aux(52) 1142.10/291.51 s(236) =< s(230)*s(235) 1142.10/291.51 s(237) =< s(230)*aux(56) 1142.10/291.51 s(238) =< s(230)*aux(52) 1142.10/291.51 s(239) =< aux(59) 1142.10/291.51 s(283) =< s(264) 1142.10/291.51 s(284) =< s(230)*s(266) 1142.10/291.52 s(285) =< aux(55) 1142.10/291.52 s(242) =< aux(53) 1142.10/291.52 s(244) =< s(242)*s(235) 1142.10/291.52 s(288) =< s(242)*aux(58) 1142.10/291.52 s(246) =< s(242)*aux(52) 1142.10/291.52 s(323) =< s(308) 1142.10/291.52 s(324) =< s(305) 1142.10/291.52 s(325) =< s(230)*aux(58) 1142.10/291.52 s(329) =< s(242)*aux(57) 1142.10/291.52 s(234) =< s(225) 1142.10/291.52 s(240) =< s(221) 1142.10/291.52 s(241) =< s(230)*s(223) 1142.10/291.52 s(245) =< s(242)*aux(56) 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [45]: 4*s(342)+1*s(343)+1*s(345)+9 1142.10/291.52 Such that:s(343) =< V16 1142.10/291.52 s(345) =< V15+1 1142.10/291.52 aux(62) =< 1 1142.10/291.52 s(342) =< aux(62) 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V1=1] 1142.10/291.52 1142.10/291.52 * Chain [44]: 44*s(359)+18*s(361)+2*s(363)+2*s(365)+2*s(367)+5*s(368)+3*s(369)+1*s(370)+13*s(371)+1*s(373)+1*s(375)+12 1142.10/291.52 Such that:aux(64) =< 1 1142.10/291.52 s(354) =< 2 1142.10/291.52 s(356) =< V1 1142.10/291.52 s(350) =< V1+1 1142.10/291.52 aux(65) =< V1-V14 1142.10/291.52 s(353) =< V14+1 1142.10/291.52 s(359) =< s(356) 1142.10/291.52 s(361) =< aux(64) 1142.10/291.52 s(363) =< s(354) 1142.10/291.52 s(364) =< s(356) 1142.10/291.52 s(365) =< s(359)*s(364) 1142.10/291.52 s(367) =< s(359)*s(356) 1142.10/291.52 s(368) =< s(353) 1142.10/291.52 s(369) =< s(350) 1142.10/291.52 s(370) =< s(359)*aux(64) 1142.10/291.52 s(371) =< aux(65) 1142.10/291.52 s(373) =< s(371)*s(364) 1142.10/291.52 s(375) =< s(371)*s(356) 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V16=0,V1>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [43]: 1*s(376)+1*s(377)+8 1142.10/291.52 Such that:s(376) =< 1 1142.10/291.52 s(377) =< V1 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V16=0,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.52 1142.10/291.52 * Chain [42]: 1*s(378)+1*s(379)+8 1142.10/291.52 Such that:s(378) =< 1 1142.10/291.52 s(379) =< V14+1 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V16=0,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.52 1142.10/291.52 * Chain [41]: 12*s(382)+1*s(383)+1*s(385)+1*s(386)+1*s(387)+9 1142.10/291.52 Such that:s(380) =< V1 1142.10/291.52 s(381) =< V16 1142.10/291.52 s(382) =< s(380) 1142.10/291.52 s(383) =< s(381) 1142.10/291.52 s(384) =< s(380) 1142.10/291.52 s(385) =< s(382)*s(384) 1142.10/291.52 s(386) =< s(382)*s(381) 1142.10/291.52 s(387) =< s(382)*s(380) 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V14=0,V1>=2,V16>=1,V20>=0,V15+1>=V16] 1142.10/291.52 1142.10/291.52 * Chain [40]: 1*s(388)+1*s(390)+11*s(392)+1*s(394)+1*s(395)+1*s(396)+9 1142.10/291.52 Such that:s(391) =< V1 1142.10/291.52 s(388) =< V1+V15 1142.10/291.52 s(389) =< V16+V15+1 1142.10/291.52 s(390) =< V15+1 1142.10/291.52 s(392) =< s(391) 1142.10/291.52 s(393) =< s(391) 1142.10/291.52 s(394) =< s(392)*s(393) 1142.10/291.52 s(395) =< s(392)*s(389) 1142.10/291.52 s(396) =< s(392)*s(391) 1142.10/291.52 1142.10/291.52 with precondition: [V=1,V14=0,V1>=2,V15>=0,V20>=0,V16>=V15+2] 1142.10/291.52 1142.10/291.52 * Chain [39]: 46*s(408)+9*s(409)+18*s(410)+2*s(412)+2*s(414)+1*s(415)+2*s(416)+5*s(417)+3*s(418)+1*s(419)+3*s(420)+21*s(421)+2*s(422)+2*s(423)+2*s(424)+10 1142.10/291.52 Such that:aux(66) =< 1 1142.10/291.52 s(403) =< 2 1142.10/291.52 s(399) =< V1+1 1142.10/291.52 s(398) =< V1-V14+1 1142.10/291.52 s(401) =< V16+1 1142.10/291.52 s(402) =< V14 1142.10/291.52 aux(69) =< V1 1142.10/291.52 aux(70) =< V1-V14 1142.10/291.52 aux(71) =< V16 1142.10/291.52 s(408) =< aux(69) 1142.10/291.52 s(409) =< aux(71) 1142.10/291.52 s(410) =< aux(66) 1142.10/291.52 s(412) =< s(403) 1142.10/291.52 s(413) =< aux(69) 1142.10/291.52 s(414) =< s(408)*s(413) 1142.10/291.52 s(415) =< s(408)*aux(71) 1142.10/291.52 s(416) =< s(408)*aux(69) 1142.10/291.52 s(417) =< s(402) 1142.10/291.52 s(418) =< s(399) 1142.10/291.52 s(419) =< s(408)*s(401) 1142.10/291.52 s(420) =< s(398) 1142.10/291.52 s(421) =< aux(70) 1142.10/291.52 s(422) =< s(421)*s(413) 1142.10/291.52 s(423) =< s(421)*aux(71) 1142.10/291.52 s(424) =< s(421)*aux(69) 1142.10/291.52 1142.10/291.52 with precondition: [V=2,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [38]: 3*s(436)+12*s(440)+1*s(443)+1*s(444)+1*s(445)+9 1142.10/291.52 Such that:s(439) =< V 1142.10/291.52 aux(73) =< 1 1142.10/291.52 s(436) =< aux(73) 1142.10/291.52 s(440) =< s(439) 1142.10/291.52 s(442) =< s(439) 1142.10/291.52 s(443) =< s(440)*s(442) 1142.10/291.52 s(444) =< s(440)*aux(73) 1142.10/291.52 s(445) =< s(440)*s(439) 1142.10/291.52 1142.10/291.52 with precondition: [V1=1,V>=2] 1142.10/291.52 1142.10/291.52 * Chain [37]: 2*s(446)+9 1142.10/291.52 Such that:aux(74) =< 1 1142.10/291.52 s(446) =< aux(74) 1142.10/291.52 1142.10/291.52 with precondition: [V=2,V1=1,V14=0,V16>=2,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [36]: 12*s(450)+1*s(451)+1*s(453)+1*s(454)+1*s(455)+9 1142.10/291.52 Such that:s(449) =< 1 1142.10/291.52 s(448) =< V1 1142.10/291.52 s(450) =< s(448) 1142.10/291.52 s(451) =< s(449) 1142.10/291.52 s(452) =< s(448) 1142.10/291.52 s(453) =< s(450)*s(452) 1142.10/291.52 s(454) =< s(450)*s(449) 1142.10/291.52 s(455) =< s(450)*s(448) 1142.10/291.52 1142.10/291.52 with precondition: [V=2,V16=1,V14=0,V1>=2,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [35]: 12*s(456)+1*s(458)+1*s(462)+1*s(463)+1*s(464)+9 1142.10/291.52 Such that:s(458) =< 1 1142.10/291.52 s(457) =< V16+1 1142.10/291.52 aux(75) =< V1 1142.10/291.52 s(456) =< aux(75) 1142.10/291.52 s(461) =< aux(75) 1142.10/291.52 s(462) =< s(456)*s(461) 1142.10/291.52 s(463) =< s(456)*s(457) 1142.10/291.52 s(464) =< s(456)*aux(75) 1142.10/291.52 1142.10/291.52 with precondition: [V=2,V14=0,V1>=2,V16>=2,V15>=0,V20>=0] 1142.10/291.52 1142.10/291.52 * Chain [34]: 1 1142.10/291.52 with precondition: [V1=0,V>=0] 1142.10/291.52 1142.10/291.52 * Chain [33]: 12*s(467)+1*s(468)+1*s(470)+1*s(471)+1*s(472)+8 1142.10/291.52 Such that:s(465) =< V 1142.10/291.52 s(466) =< V1 1142.10/291.52 s(467) =< s(465) 1142.10/291.52 s(468) =< s(466) 1142.10/291.52 s(469) =< s(465) 1142.10/291.52 s(470) =< s(467)*s(469) 1142.10/291.52 s(471) =< s(467)*s(466) 1142.10/291.52 s(472) =< s(467)*s(465) 1142.10/291.52 1142.10/291.52 with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.52 1142.10/291.52 * Chain [32]: 1*s(473)+1*s(475)+11*s(477)+1*s(479)+1*s(480)+1*s(481)+8 1142.10/291.52 Such that:s(476) =< V 1142.10/291.52 s(473) =< V+V14 1142.10/291.52 s(474) =< V1+V14+1 1142.10/291.52 s(475) =< V14+1 1142.10/291.52 s(477) =< s(476) 1142.10/291.52 s(478) =< s(476) 1142.10/291.52 s(479) =< s(477)*s(478) 1142.10/291.52 s(480) =< s(477)*s(474) 1142.10/291.52 s(481) =< s(477)*s(476) 1142.10/291.52 1142.10/291.52 with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.52 1142.10/291.52 1142.10/291.52 Closed-form bounds of start(V,V1,V16,V14,V15,V20): 1142.10/291.52 ------------------------------------- 1142.10/291.52 * Chain [47] with precondition: [V>=0,V1>=0] 1142.10/291.52 - Upper bound: 113*V+47+12*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V14+1)*V+4*V*nat(V-V16)+20*V1+nat(V16)*5+(6*V+6)+nat(V1+V14)*2*nat(V-V16)+nat(V14+1)*6+nat(V14+2)*2+nat(V+V14+1)*3+nat(V-V16+V14+1)*3+nat(V-V16+V14)+nat(V-V16)*20 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [46] with precondition: [V=1,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 138*V1+58+12*V1*V1+3*V1*V16+(V16+1)*V1+(V16+V15+1)*V1+(V16+V15+2)*V1+12*V1*nat(V1-V14)+32*V16+2*V16*nat(V1-V14)+5*V14+(3*V1+3)+(2*V16+2*V15)*nat(V1-V14)+(10*V14+10)+(12*V15+12)+(8*V15+16)+(2*V15+6)+(3*V1+3*V15+3)+(3*V1+3*V15+6)+(2*V16+2*V15+2)*nat(V1-V14)+nat(V1-V14+V15+1)*6+nat(V1-V14+V15)*2+nat(V1-V14)*64 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [45] with precondition: [V=1,V1=1] 1142.10/291.52 - Upper bound: nat(V16)+13+nat(V15+1) 1142.10/291.52 - Complexity: n 1142.10/291.52 * Chain [44] with precondition: [V=1,V16=0,V1>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 45*V1+34+4*V1*V1+2*V1*nat(V1-V14)+(3*V1+3)+(5*V14+5)+nat(V1-V14)*13 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [43] with precondition: [V=1,V16=0,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.52 - Upper bound: V1+9 1142.10/291.52 - Complexity: n 1142.10/291.52 * Chain [42] with precondition: [V=1,V16=0,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.52 - Upper bound: V14+10 1142.10/291.52 - Complexity: n 1142.10/291.52 * Chain [41] with precondition: [V=1,V14=0,V1>=2,V16>=1,V20>=0,V15+1>=V16] 1142.10/291.52 - Upper bound: 12*V1+9+2*V1*V1+V16*V1+V16 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [40] with precondition: [V=1,V14=0,V1>=2,V15>=0,V20>=0,V16>=V15+2] 1142.10/291.52 - Upper bound: 11*V1+9+2*V1*V1+(V16+V15+1)*V1+(V1+V15)+(V15+1) 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [39] with precondition: [V=2,V1>=0,V16>=0,V14>=0,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 46*V1+32+4*V1*V1+V16*V1+(V16+1)*V1+4*V1*nat(V1-V14)+9*V16+2*V16*nat(V1-V14)+5*V14+(3*V1+3)+nat(V1-V14+1)*3+nat(V1-V14)*21 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [38] with precondition: [V1=1,V>=2] 1142.10/291.52 - Upper bound: 13*V+12+2*V*V 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [37] with precondition: [V=2,V1=1,V14=0,V16>=2,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 11 1142.10/291.52 - Complexity: constant 1142.10/291.52 * Chain [36] with precondition: [V=2,V16=1,V14=0,V1>=2,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 13*V1+10+2*V1*V1 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [35] with precondition: [V=2,V14=0,V1>=2,V16>=2,V15>=0,V20>=0] 1142.10/291.52 - Upper bound: 12*V1+10+2*V1*V1+(V16+1)*V1 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [34] with precondition: [V1=0,V>=0] 1142.10/291.52 - Upper bound: 1 1142.10/291.52 - Complexity: constant 1142.10/291.52 * Chain [33] with precondition: [V16=0,V>=2,V1>=1,V15>=0,V14+1>=V1] 1142.10/291.52 - Upper bound: 12*V+8+2*V*V+V1*V+V1 1142.10/291.52 - Complexity: n^2 1142.10/291.52 * Chain [32] with precondition: [V16=0,V>=2,V14>=0,V15>=0,V1>=V14+2] 1142.10/291.52 - Upper bound: 11*V+8+2*V*V+(V1+V14+1)*V+(V+V14)+(V14+1) 1142.10/291.52 - Complexity: n^2 1142.10/291.52 1142.10/291.52 ### Maximum cost of start(V,V1,V16,V14,V15,V20): max([max([max([10,nat(V14+1)+8,nat(V16)+12+nat(V15+1)]),11*V+7+2*V*V+max([2*V+4,nat(V1+V14+1)*V+nat(V+V14)+nat(V14+1)])]),V1+7+max([max([1,2*V*V+12*V+V1*V]),10*V1+1+max([max([2*V1*V1+max([max([nat(V16)*V1+nat(V16),nat(V16+1)*V1+1]),32*V1+22+2*V1*V1+2*V1*nat(V1-V14)+(3*V1+3)+nat(V1-V14)*13+max([nat(V14+1)*5+2,nat(V16)*V1+V1+nat(V16+1)*V1+2*V1*nat(V1-V14)+nat(V16)*9+nat(V16)*2*nat(V1-V14)+nat(V14)*5+nat(V1-V14)*8+max([nat(V1-V14+1)*3,92*V1+26+8*V1*V1+2*V1*nat(V16)+nat(V16+V15+1)*V1+nat(V16+V15+2)*V1+8*V1*nat(V1-V14)+nat(V16)*23+nat(V16+V15)*2*nat(V1-V14)+nat(V14+1)*10+nat(V15+1)*12+nat(V15+2)*8+nat(V15+3)*2+nat(V1+V15+1)*3+nat(V1+V15+2)*3+nat(V16+V15+1)*2*nat(V1-V14)+nat(V1-V14+V15+1)*6+nat(V1-V14+V15)*2+nat(V1-V14)*43])])+(V1+1)]),113*V+38+12*V*V+3*V*V1+(V1+1)*(2*V)+nat(V1+V14+1)*V+4*V*nat(V-V16)+8*V1+nat(V16)*5+(6*V+6)+nat(V1+V14)*2*nat(V-V16)+nat(V14+1)*6+nat(V14+2)*2+nat(V+V14+1)*3+nat(V-V16+V14+1)*3+nat(V-V16+V14)+nat(V-V16)*20])+V1,2*V1*V1+nat(V16+V15+1)*V1+nat(V1+V15)+nat(V15+1)])])])+1 1142.10/291.52 Asymptotic class: n^2 1142.10/291.52 * Total analysis performed in 1171 ms. 1142.10/291.52 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (10) 1142.10/291.52 BOUNDS(1, n^2) 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (11) RenamingProof (BOTH BOUNDS(ID, ID)) 1142.10/291.52 Renamed function symbols to avoid clashes with predefined symbol. 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (12) 1142.10/291.52 Obligation: 1142.10/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1142.10/291.52 1142.10/291.52 1142.10/291.52 The TRS R consists of the following rules: 1142.10/291.52 1142.10/291.52 le(0', y) -> true 1142.10/291.52 le(s(x), 0') -> false 1142.10/291.52 le(s(x), s(y)) -> le(x, y) 1142.10/291.52 quot(x, 0') -> quotZeroErro 1142.10/291.52 quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0') 1142.10/291.52 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.52 if(true, x, y, z, u, v) -> v 1142.10/291.52 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.52 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.52 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v)) 1142.10/291.52 1142.10/291.52 S is empty. 1142.10/291.52 Rewrite Strategy: INNERMOST 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1142.10/291.52 Infered types. 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (14) 1142.10/291.52 Obligation: 1142.10/291.52 Innermost TRS: 1142.10/291.52 Rules: 1142.10/291.52 le(0', y) -> true 1142.10/291.52 le(s(x), 0') -> false 1142.10/291.52 le(s(x), s(y)) -> le(x, y) 1142.10/291.52 quot(x, 0') -> quotZeroErro 1142.10/291.52 quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0') 1142.10/291.52 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.52 if(true, x, y, z, u, v) -> v 1142.10/291.52 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.52 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.52 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v)) 1142.10/291.52 1142.10/291.52 Types: 1142.10/291.52 le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false 1142.10/291.52 0' :: 0':s:quotZeroErro 1142.10/291.52 true :: true:false 1142.10/291.52 s :: 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 false :: true:false 1142.10/291.52 quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 quotZeroErro :: 0':s:quotZeroErro 1142.10/291.52 quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 hole_true:false1_0 :: true:false 1142.10/291.52 hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro 1142.10/291.52 gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (15) OrderProof (LOWER BOUND(ID)) 1142.10/291.52 Heuristically decided to analyse the following defined symbols: 1142.10/291.52 le, quotIter 1142.10/291.52 1142.10/291.52 They will be analysed ascendingly in the following order: 1142.10/291.52 le < quotIter 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (16) 1142.10/291.52 Obligation: 1142.10/291.52 Innermost TRS: 1142.10/291.52 Rules: 1142.10/291.52 le(0', y) -> true 1142.10/291.52 le(s(x), 0') -> false 1142.10/291.52 le(s(x), s(y)) -> le(x, y) 1142.10/291.52 quot(x, 0') -> quotZeroErro 1142.10/291.52 quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0') 1142.10/291.52 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.52 if(true, x, y, z, u, v) -> v 1142.10/291.52 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.52 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.52 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v)) 1142.10/291.52 1142.10/291.52 Types: 1142.10/291.52 le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false 1142.10/291.52 0' :: 0':s:quotZeroErro 1142.10/291.52 true :: true:false 1142.10/291.52 s :: 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 false :: true:false 1142.10/291.52 quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 quotZeroErro :: 0':s:quotZeroErro 1142.10/291.52 quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 hole_true:false1_0 :: true:false 1142.10/291.52 hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro 1142.10/291.52 gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro 1142.10/291.52 1142.10/291.52 1142.10/291.52 Generator Equations: 1142.10/291.52 gen_0':s:quotZeroErro3_0(0) <=> 0' 1142.10/291.52 gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x)) 1142.10/291.52 1142.10/291.52 1142.10/291.52 The following defined symbols remain to be analysed: 1142.10/291.52 le, quotIter 1142.10/291.52 1142.10/291.52 They will be analysed ascendingly in the following order: 1142.10/291.52 le < quotIter 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1142.10/291.52 Proved the following rewrite lemma: 1142.10/291.52 le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1142.10/291.52 1142.10/291.52 Induction Base: 1142.10/291.52 le(gen_0':s:quotZeroErro3_0(0), gen_0':s:quotZeroErro3_0(0)) ->_R^Omega(1) 1142.10/291.52 true 1142.10/291.52 1142.10/291.52 Induction Step: 1142.10/291.52 le(gen_0':s:quotZeroErro3_0(+(n5_0, 1)), gen_0':s:quotZeroErro3_0(+(n5_0, 1))) ->_R^Omega(1) 1142.10/291.52 le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) ->_IH 1142.10/291.52 true 1142.10/291.52 1142.10/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (18) 1142.10/291.52 Complex Obligation (BEST) 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (19) 1142.10/291.52 Obligation: 1142.10/291.52 Proved the lower bound n^1 for the following obligation: 1142.10/291.52 1142.10/291.52 Innermost TRS: 1142.10/291.52 Rules: 1142.10/291.52 le(0', y) -> true 1142.10/291.52 le(s(x), 0') -> false 1142.10/291.52 le(s(x), s(y)) -> le(x, y) 1142.10/291.52 quot(x, 0') -> quotZeroErro 1142.10/291.52 quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0') 1142.10/291.52 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.52 if(true, x, y, z, u, v) -> v 1142.10/291.52 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.52 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.52 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v)) 1142.10/291.52 1142.10/291.52 Types: 1142.10/291.52 le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false 1142.10/291.52 0' :: 0':s:quotZeroErro 1142.10/291.52 true :: true:false 1142.10/291.52 s :: 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 false :: true:false 1142.10/291.52 quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 quotZeroErro :: 0':s:quotZeroErro 1142.10/291.52 quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 hole_true:false1_0 :: true:false 1142.10/291.52 hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro 1142.10/291.52 gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro 1142.10/291.52 1142.10/291.52 1142.10/291.52 Generator Equations: 1142.10/291.52 gen_0':s:quotZeroErro3_0(0) <=> 0' 1142.10/291.52 gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x)) 1142.10/291.52 1142.10/291.52 1142.10/291.52 The following defined symbols remain to be analysed: 1142.10/291.52 le, quotIter 1142.10/291.52 1142.10/291.52 They will be analysed ascendingly in the following order: 1142.10/291.52 le < quotIter 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (20) LowerBoundPropagationProof (FINISHED) 1142.10/291.52 Propagated lower bound. 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (21) 1142.10/291.52 BOUNDS(n^1, INF) 1142.10/291.52 1142.10/291.52 ---------------------------------------- 1142.10/291.52 1142.10/291.52 (22) 1142.10/291.52 Obligation: 1142.10/291.52 Innermost TRS: 1142.10/291.52 Rules: 1142.10/291.52 le(0', y) -> true 1142.10/291.52 le(s(x), 0') -> false 1142.10/291.52 le(s(x), s(y)) -> le(x, y) 1142.10/291.52 quot(x, 0') -> quotZeroErro 1142.10/291.52 quot(x, s(y)) -> quotIter(x, s(y), 0', 0', 0') 1142.10/291.52 quotIter(x, s(y), z, u, v) -> if(le(x, z), x, s(y), z, u, v) 1142.10/291.52 if(true, x, y, z, u, v) -> v 1142.10/291.52 if(false, x, y, z, u, v) -> if2(le(y, s(u)), x, y, s(z), s(u), v) 1142.10/291.52 if2(false, x, y, z, u, v) -> quotIter(x, y, z, u, v) 1142.10/291.52 if2(true, x, y, z, u, v) -> quotIter(x, y, z, 0', s(v)) 1142.10/291.52 1142.10/291.52 Types: 1142.10/291.52 le :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> true:false 1142.10/291.52 0' :: 0':s:quotZeroErro 1142.10/291.52 true :: true:false 1142.10/291.52 s :: 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 false :: true:false 1142.10/291.52 quot :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 quotZeroErro :: 0':s:quotZeroErro 1142.10/291.52 quotIter :: 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 if2 :: true:false -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro -> 0':s:quotZeroErro 1142.10/291.52 hole_true:false1_0 :: true:false 1142.10/291.52 hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro 1142.10/291.52 gen_0':s:quotZeroErro3_0 :: Nat -> 0':s:quotZeroErro 1142.10/291.52 1142.10/291.52 1142.10/291.52 Lemmas: 1142.10/291.52 le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1142.10/291.52 1142.10/291.52 1142.10/291.52 Generator Equations: 1142.10/291.52 gen_0':s:quotZeroErro3_0(0) <=> 0' 1142.10/291.52 gen_0':s:quotZeroErro3_0(+(x, 1)) <=> s(gen_0':s:quotZeroErro3_0(x)) 1142.10/291.52 1142.10/291.52 1142.10/291.52 The following defined symbols remain to be analysed: 1142.10/291.52 quotIter 1142.41/291.59 EOF