WORST_CASE(?,O(n^2)) proof of input_hT65wS5427.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 115 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), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 12 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), 133 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 9 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 459 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 78 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 1032 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 369 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 995 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 396 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] (46) CpxRNTS (47) FinalProof [FINISHED, 0 ms] (48) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) overlap(Nil, ys) -> False member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) member(x, Nil) -> False notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs, ys) -> overlap(xs, ys) The (relative) TRS S consists of the following rules: !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0, S(y)) -> False !EQ(S(x), 0) -> False !EQ(0, 0) -> True overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) overlap[Ite][True][Ite](True, xs, ys) -> True member[Ite][True][Ite](True, x, xs) -> 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^2). The TRS R consists of the following rules: overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) overlap(Nil, ys) -> False member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) member(x, Nil) -> False notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs, ys) -> overlap(xs, ys) The (relative) TRS S consists of the following rules: !EQ(S(x), S(y)) -> !EQ(x, y) !EQ(0, S(y)) -> False !EQ(S(x), 0) -> False !EQ(0, 0) -> True overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) overlap[Ite][True][Ite](True, xs, ys) -> True member[Ite][True][Ite](True, x, xs) -> 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^2). The TRS R consists of the following rules: overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1] overlap(Nil, ys) -> False [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs, ys) -> overlap(xs, ys) [1] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) -> True [0] member[Ite][True][Ite](True, x, xs) -> 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: overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1] overlap(Nil, ys) -> False [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs, ys) -> overlap(xs, ys) [1] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) -> True [0] member[Ite][True][Ite](True, x, xs) -> True [0] The TRS has the following type information: overlap :: Cons:Nil -> Cons:Nil -> False:True Cons :: S:0 -> Cons:Nil -> Cons:Nil overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True member :: S:0 -> Cons:Nil -> False:True Nil :: Cons:Nil False :: False:True member[Ite][True][Ite] :: False:True -> S:0 -> Cons:Nil -> False:True !EQ :: S:0 -> S:0 -> False:True notEmpty :: Cons:Nil -> False:True True :: False:True goal :: Cons:Nil -> Cons:Nil -> False:True S :: S:0 -> S:0 0 :: S:0 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: overlap_2 notEmpty_1 goal_2 (c) The following functions are completely defined: member_2 !EQ_2 overlap[Ite][True][Ite]_3 member[Ite][True][Ite]_3 Due to the following rules being added: !EQ(v0, v1) -> null_!EQ [0] overlap[Ite][True][Ite](v0, v1, v2) -> null_overlap[Ite][True][Ite] [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] And the following fresh constants: null_!EQ, null_overlap[Ite][True][Ite], null_member[Ite][True][Ite] ---------------------------------------- (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: overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1] overlap(Nil, ys) -> False [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1] member(x, Nil) -> False [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs, ys) -> overlap(xs, ys) [1] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) -> True [0] member[Ite][True][Ite](True, x, xs) -> True [0] !EQ(v0, v1) -> null_!EQ [0] overlap[Ite][True][Ite](v0, v1, v2) -> null_overlap[Ite][True][Ite] [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] The TRS has the following type information: overlap :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Cons :: S:0 -> Cons:Nil -> Cons:Nil overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member :: S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Nil :: Cons:Nil False :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] !EQ :: S:0 -> S:0 -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] notEmpty :: Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] True :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] goal :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] S :: S:0 -> S:0 0 :: S:0 null_!EQ :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] 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: overlap(Cons(x, xs), Cons(x1, xs')) -> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, Cons(x1, xs')), Cons(x, xs), Cons(x1, xs')) [2] overlap(Cons(x, xs), Nil) -> overlap[Ite][True][Ite](False, Cons(x, xs), Nil) [2] overlap(Nil, ys) -> False [1] member(S(y'), Cons(S(x''), xs)) -> member[Ite][True][Ite](!EQ(x'', y'), S(y'), Cons(S(x''), xs)) [1] member(S(y''), Cons(0, xs)) -> member[Ite][True][Ite](False, S(y''), Cons(0, xs)) [1] member(0, Cons(S(x2), xs)) -> member[Ite][True][Ite](False, 0, Cons(S(x2), xs)) [1] member(0, Cons(0, xs)) -> member[Ite][True][Ite](True, 0, Cons(0, xs)) [1] member(x', Cons(x, xs)) -> member[Ite][True][Ite](null_!EQ, x', Cons(x, xs)) [1] member(x, Nil) -> False [1] notEmpty(Cons(x, xs)) -> True [1] notEmpty(Nil) -> False [1] goal(xs, ys) -> overlap(xs, ys) [1] !EQ(S(x), S(y)) -> !EQ(x, y) [0] !EQ(0, S(y)) -> False [0] !EQ(S(x), 0) -> False [0] !EQ(0, 0) -> True [0] overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0] member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0] overlap[Ite][True][Ite](True, xs, ys) -> True [0] member[Ite][True][Ite](True, x, xs) -> True [0] !EQ(v0, v1) -> null_!EQ [0] overlap[Ite][True][Ite](v0, v1, v2) -> null_overlap[Ite][True][Ite] [0] member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0] The TRS has the following type information: overlap :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Cons :: S:0 -> Cons:Nil -> Cons:Nil overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member :: S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] Nil :: Cons:Nil False :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] !EQ :: S:0 -> S:0 -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] notEmpty :: Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] True :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] goal :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] S :: S:0 -> S:0 0 :: S:0 null_!EQ :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] null_member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] 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 0 => 0 null_!EQ => 0 null_overlap[Ite][True][Ite] => 0 null_member[Ite][True][Ite] => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x goal(z, z') -{ 1 }-> overlap(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + y'', 1 + 0 + xs) :|: xs >= 0, z = 1 + y'', z' = 1 + 0 + xs, y'' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', y'), 1 + y', 1 + (1 + x'') + xs) :|: xs >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' = ys, ys >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, ys) :|: xs >= 0, z' = 1 + x + xs, z = 1, ys >= 0, x >= 0, z'' = ys overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, ys >= 0, z'' = ys, z' = xs overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ---------------------------------------- (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: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { notEmpty } { !EQ } { member, member[Ite][True][Ite] } { overlap[Ite][True][Ite], overlap } { goal } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {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: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} ---------------------------------------- (19) 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 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {notEmpty}, {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: ?, size: O(1) [2] ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (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: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: !EQ after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {!EQ}, {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: ?, size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: !EQ after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](!EQ(x1, x), x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (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: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: member after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 Computed SIZE bound using CoFloCo for: member[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {member,member[Ite][True][Ite]}, {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: ?, size: O(1) [2] member[Ite][True][Ite]: runtime: ?, size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: member after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' Computed RUNTIME bound using CoFloCo for: member[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](s', 1 + (z - 1), 1 + (1 + x'') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 1 }-> member[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](member[Ite][True][Ite](s, x, 1 + x1 + xs'), 1 + x + xs, 1 + x1 + xs') :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] ---------------------------------------- (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: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 4 + x1 + xs' }-> overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: overlap[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 Computed SIZE bound using CoFloCo for: overlap after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 4 + x1 + xs' }-> overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {overlap[Ite][True][Ite],overlap}, {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] overlap[Ite][True][Ite]: runtime: ?, size: O(1) [2] overlap: runtime: ?, size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: overlap[Ite][True][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 3*z' + z'*z'' + z'' Computed RUNTIME bound using KoAT for: overlap after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 23 + 14*z + 4*z*z' + 6*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 1 }-> overlap(z, z') :|: z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 4 + x1 + xs' }-> overlap[Ite][True][Ite](s1, 1 + x + xs, 1 + x1 + xs') :|: s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 2 }-> overlap[Ite][True][Ite](1, 1 + x + xs, 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2] overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2] ---------------------------------------- (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: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*xs + xs*xs' + 3*xs' }-> s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 9 + 3*x + 3*xs }-> s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 23 + 14*xs + 4*xs*z'' + 6*z'' }-> s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2] overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2] ---------------------------------------- (43) 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 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*xs + xs*xs' + 3*xs' }-> s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 9 + 3*x + 3*xs }-> s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 23 + 14*xs + 4*xs*z'' + 6*z'' }-> s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2] overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2] goal: runtime: ?, size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 24 + 14*z + 4*z*z' + 6*z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 goal(z, z') -{ 24 + 14*z + 4*z*z' + 6*z' }-> s10 :|: s10 >= 0, s10 <= 2, z >= 0, z' >= 0 member(z, z') -{ 4 + x'' + xs }-> s2 :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0 member(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 2, z' - 1 >= 0, z - 1 >= 0 member(z, z') -{ 4 + x2 + xs }-> s4 :|: s4 >= 0, s4 <= 2, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0 member(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 2, z' - 1 >= 0, z = 0 member(z, z') -{ 3 + x + xs }-> s6 :|: s6 >= 0, s6 <= 2, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 member(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 member[Ite][True][Ite](z, z', z'') -{ 2 + xs }-> s7 :|: s7 >= 0, s7 <= 2, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z'' >= 0, z' >= 0 member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 notEmpty(z) -{ 1 }-> 1 :|: z = 0 overlap(z, z') -{ 13 + 4*x + x*x1 + x*xs' + 3*x1 + x1*xs + 4*xs + xs*xs' + 3*xs' }-> s8 :|: s8 >= 0, s8 <= 2, s1 >= 0, s1 <= 2, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 overlap(z, z') -{ 9 + 3*x + 3*xs }-> s9 :|: s9 >= 0, s9 <= 2, z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 overlap(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 overlap[Ite][True][Ite](z, z', z'') -{ 23 + 14*xs + 4*xs*z'' + 6*z'' }-> s11 :|: s11 >= 0, s11 <= 2, xs >= 0, z' = 1 + x + xs, z = 1, z'' >= 0, x >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, z' >= 0, z'' >= 0 overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: Previous analysis results are: notEmpty: runtime: O(1) [1], size: O(1) [2] !EQ: runtime: O(1) [0], size: O(1) [2] member: runtime: O(n^1) [2 + z'], size: O(1) [2] member[Ite][True][Ite]: runtime: O(n^1) [1 + z''], size: O(1) [2] overlap[Ite][True][Ite]: runtime: O(n^2) [4 + 3*z' + z'*z'' + z''], size: O(1) [2] overlap: runtime: O(n^2) [23 + 14*z + 4*z*z' + 6*z'], size: O(1) [2] goal: runtime: O(n^2) [24 + 14*z + 4*z*z' + 6*z'], size: O(1) [2] ---------------------------------------- (47) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (48) BOUNDS(1, n^2)