WORST_CASE(?,O(n^1)) proof of input_gmZ9TGmqqB.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).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), 136 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), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 276 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 264 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 151 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 396 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 119 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 251 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 447 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 164 ms] (48) CpxRNTS (49) FinalProof [FINISHED, 0 ms] (50) 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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) lte(Cons(x, xs), Nil) -> False even(Cons(x, Nil)) -> False even(Cons(x', Cons(x, xs))) -> even(xs) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False lte(Nil, y) -> True even(Nil) -> True goal(x, y) -> and(lte(x, y), even(x)) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) lte(Cons(x, xs), Nil) -> False even(Cons(x, Nil)) -> False even(Cons(x', Cons(x, xs))) -> even(xs) notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False lte(Nil, y) -> True even(Nil) -> True goal(x, y) -> and(lte(x, y), even(x)) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1] lte(Cons(x, xs), Nil) -> False [1] even(Cons(x, Nil)) -> False [1] even(Cons(x', Cons(x, xs))) -> even(xs) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lte(Nil, y) -> True [1] even(Nil) -> True [1] goal(x, y) -> and(lte(x, y), even(x)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1] lte(Cons(x, xs), Nil) -> False [1] even(Cons(x, Nil)) -> False [1] even(Cons(x', Cons(x, xs))) -> even(xs) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lte(Nil, y) -> True [1] even(Nil) -> True [1] goal(x, y) -> and(lte(x, y), even(x)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] The TRS has the following type information: lte :: Cons:Nil -> Cons:Nil -> False:True Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil False :: False:True even :: Cons:Nil -> False:True notEmpty :: Cons:Nil -> False:True True :: False:True goal :: Cons:Nil -> Cons:Nil -> False:True and :: False:True -> False:True -> False:True 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: notEmpty_1 goal_2 (c) The following functions are completely defined: lte_2 even_1 and_2 Due to the following rules being added: and(v0, v1) -> null_and [0] And the following fresh constants: null_and, 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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1] lte(Cons(x, xs), Nil) -> False [1] even(Cons(x, Nil)) -> False [1] even(Cons(x', Cons(x, xs))) -> even(xs) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lte(Nil, y) -> True [1] even(Nil) -> True [1] goal(x, y) -> and(lte(x, y), even(x)) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] and(v0, v1) -> null_and [0] The TRS has the following type information: lte :: Cons:Nil -> Cons:Nil -> False:True:null_and Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil False :: False:True:null_and even :: Cons:Nil -> False:True:null_and notEmpty :: Cons:Nil -> False:True:null_and True :: False:True:null_and goal :: Cons:Nil -> Cons:Nil -> False:True:null_and and :: False:True:null_and -> False:True:null_and -> False:True:null_and null_and :: False:True:null_and 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: lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1] lte(Cons(x, xs), Nil) -> False [1] even(Cons(x, Nil)) -> False [1] even(Cons(x', Cons(x, xs))) -> even(xs) [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] lte(Nil, y) -> True [1] even(Nil) -> True [1] goal(Cons(x'', Nil), Cons(x1, xs1)) -> and(lte(Nil, xs1), False) [3] goal(Cons(x'', Cons(x3, xs3)), Cons(x1, xs1)) -> and(lte(Cons(x3, xs3), xs1), even(xs3)) [3] goal(Cons(x2, Nil), Nil) -> and(False, False) [3] goal(Cons(x2, Cons(x4, xs4)), Nil) -> and(False, even(xs4)) [3] goal(Nil, y) -> and(True, True) [3] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] and(v0, v1) -> null_and [0] The TRS has the following type information: lte :: Cons:Nil -> Cons:Nil -> False:True:null_and Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil False :: False:True:null_and even :: Cons:Nil -> False:True:null_and notEmpty :: Cons:Nil -> False:True:null_and True :: False:True:null_and goal :: Cons:Nil -> Cons:Nil -> False:True:null_and and :: False:True:null_and -> False:True:null_and -> False:True:null_and null_and :: False:True:null_and 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 False => 1 True => 2 null_and => 0 const => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x + 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, x'' >= 0, z = 1 + x'' + 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(2, 2) :|: y >= 0, z = 0, z' = y goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> and(1, 1) :|: z = 1 + x2 + 0, x2 >= 0, z' = 0 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x + 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, x'' >= 0, z = 1 + x'' + 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: y >= 0, z = 0, z' = y, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z = 1 + x2 + 0, x2 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z = 1 + x2 + 0, x2 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: y >= 0, z = 0, z' = y, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { and } { notEmpty } { lte } { even } { goal } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {and}, {notEmpty}, {lte}, {even}, {goal} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {and}, {notEmpty}, {lte}, {even}, {goal} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {and}, {notEmpty}, {lte}, {even}, {goal} Previous analysis results are: and: runtime: ?, size: O(1) [2] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (27) 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: 2 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {notEmpty}, {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: ?, size: O(1) [2] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lte after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {lte}, {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: ?, size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lte after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 3 }-> and(lte(0, xs1), 1) :|: x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 3 }-> and(lte(1 + x3 + xs3, xs1), even(xs3)) :|: z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: even after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {even}, {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] even: runtime: ?, size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: even after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 4 + xs1 }-> and(s1, even(xs3)) :|: s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 3 }-> and(1, even(xs4)) :|: x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] even: runtime: O(n^1) [2 + z], size: O(1) [2] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] even: runtime: O(n^1) [2 + z], size: O(1) [2] ---------------------------------------- (45) 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: 2 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] even: runtime: O(n^1) [2 + z], size: O(1) [2] goal: runtime: ?, size: O(1) [2] ---------------------------------------- (47) 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: 3 + z + z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 even(z) -{ 3 + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z - 1 >= 0 goal(z, z') -{ 4 + xs1 }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, x1 >= 0, xs1 >= 0, z - 1 >= 0, z' = 1 + x1 + xs1 goal(z, z') -{ 6 + xs1 + xs3 }-> s4 :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 2, z = 1 + x'' + (1 + x3 + xs3), x1 >= 0, xs1 >= 0, x'' >= 0, x3 >= 0, z' = 1 + x1 + xs1, xs3 >= 0 goal(z, z') -{ 5 + xs4 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, x4 >= 0, xs4 >= 0, z = 1 + x2 + (1 + x4 + xs4), x2 >= 0, z' = 0 goal(z, z') -{ 3 }-> 2 :|: z' >= 0, z = 0, 2 = 2 goal(z, z') -{ 3 }-> 1 :|: z - 1 >= 0, z' = 0, 1 = 1 goal(z, z') -{ 3 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 goal(z, z') -{ 3 }-> 0 :|: z' >= 0, z = 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 lte(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lte(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 Function symbols to be analyzed: Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] notEmpty: runtime: O(1) [1], size: O(1) [2] lte: runtime: O(n^1) [1 + z'], size: O(1) [2] even: runtime: O(n^1) [2 + z], size: O(1) [2] goal: runtime: O(n^1) [3 + z + z'], size: O(1) [2] ---------------------------------------- (49) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (50) BOUNDS(1, n^1)