WORST_CASE(?,O(n^1)) proof of input_wRrsFwXnXs.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 209 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 4 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 285 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 133 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 135 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 111 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 1599 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 719 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1551 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 676 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 148 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (52) CpxRNTS (53) FinalProof [FINISHED, 0 ms] (54) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) The (relative) TRS S consists of the following rules: g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) The (relative) TRS S consists of the following rules: g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True [1] lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True [1] lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) [0] The TRS has the following type information: lt0 :: Nil:Cons -> Nil:Cons -> True:False Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons True :: True:False g :: Nil:Cons -> Nil:Cons -> Nil:Cons f :: Nil:Cons -> Nil:Cons -> Nil:Cons notEmpty :: Nil:Cons -> True:False False :: True:False g[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons f[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons number4 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons -> Nil:Cons Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: g_2 f_2 notEmpty_1 number4_1 goal_2 (c) The following functions are completely defined: lt0_2 g[Ite][False][Ite]_3 f[Ite][False][Ite]_3 Due to the following rules being added: g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: const ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True [1] lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) [0] g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: lt0 :: Nil:Cons -> Nil:Cons -> True:False Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons True :: True:False g :: Nil:Cons -> Nil:Cons -> Nil:Cons f :: Nil:Cons -> Nil:Cons -> Nil:Cons notEmpty :: Nil:Cons -> True:False False :: True:False g[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons f[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons number4 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons -> Nil:Cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt0(Nil, Cons(x', xs)) -> True [1] lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lt0(x, Nil) -> False [1] g(Nil, Cons(x', xs)) -> g[Ite][False][Ite](True, Nil, Cons(x', xs)) [2] g(Cons(x'', xs''), Cons(x', xs)) -> g[Ite][False][Ite](lt0(xs'', Nil), Cons(x'', xs''), Cons(x', xs)) [2] f(Nil, Cons(x', xs)) -> f[Ite][False][Ite](True, Nil, Cons(x', xs)) [2] f(Cons(x''', xs'''), Cons(x', xs)) -> f[Ite][False][Ite](lt0(xs''', Nil), Cons(x''', xs'''), Cons(x', xs)) [2] number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs) [0] g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: lt0 :: Nil:Cons -> Nil:Cons -> True:False Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons True :: True:False g :: Nil:Cons -> Nil:Cons -> Nil:Cons f :: Nil:Cons -> Nil:Cons -> Nil:Cons notEmpty :: Nil:Cons -> True:False False :: True:False g[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons f[Ite][False][Ite] :: True:False -> Nil:Cons -> Nil:Cons -> Nil:Cons number4 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons -> Nil:Cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 True => 1 False => 0 const => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: x >= 0, z = x, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: x >= 0, z = x, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> 1 + f(x, y) + (1 + g(x, y) + 0) :|: x >= 0, y >= 0, z = x, z' = y lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: n >= 0, z = n ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt0 } { notEmpty } { number4 } { g[Ite][False][Ite], g } { f[Ite][False][Ite], f } { goal } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {lt0}, {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {lt0}, {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {lt0}, {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: ?, size: O(1) [1] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lt0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 2 }-> f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: notEmpty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: ?, size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: notEmpty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: number4 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: ?, size: O(1) [4] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: number4 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: ?, size: O(1) [4] g: runtime: ?, size: O(1) [4] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 19 + 12*z' + 4*z'' Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 23 + 12*z + 4*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 4 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] f[Ite][False][Ite]: runtime: ?, size: O(1) [4] f: runtime: ?, size: O(1) [4] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 19 + 12*z' + 4*z'' Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 23 + 12*z + 4*z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*xs''' }-> s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: goal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 10 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*xs''' }-> s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] goal: runtime: ?, size: O(1) [10] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 47 + 24*z + 8*z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*xs''' }-> s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 g(z, z') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*xs'' }-> s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 31 + 12*xs + 4*z'' }-> s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0 g[Ite][False][Ite](z, z', z'') -{ 23 + 4*xs + 12*z' }-> s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0 lt0(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0 lt0(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 0 :|: z = 0 number4(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0 Function symbols to be analyzed: Previous analysis results are: lt0: runtime: O(n^1) [2 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4] f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4] goal: runtime: O(n^1) [47 + 24*z + 8*z'], size: O(1) [10] ---------------------------------------- (53) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (54) BOUNDS(1, n^1)