/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 173 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 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), 450 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 1029 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 271 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 451 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 255 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 120 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) FinalProof [FINISHED, 0 ms] (44) BOUNDS(1, n^2) (45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (46) TRS for Loop Detection (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (48) BEST (49) proven lower bound (50) LowerBoundPropagationProof [FINISHED, 0 ms] (51) BOUNDS(n^1, INF) (52) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) isort(Nil, r) -> r insert(x, Nil) -> Cons(x, Nil) inssort(xs) -> isort(xs, Nil) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) insert[Ite][False][Ite](True, x, r) -> Cons(x, r) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) isort(Nil, r) -> r insert(x, Nil) -> Cons(x, Nil) inssort(xs) -> isort(xs, Nil) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) insert[Ite][False][Ite](True, x, r) -> Cons(x, r) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) [1] isort(Nil, r) -> r [1] insert(x, Nil) -> Cons(x, Nil) [1] inssort(xs) -> isort(xs, Nil) [1] <(S(x), S(y)) -> <(x, y) [0] <(0, S(y)) -> True [0] <(x, 0) -> False [0] insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: < => lt ---------------------------------------- (6) 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: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] isort(Nil, r) -> r [1] insert(x, Nil) -> Cons(x, Nil) [1] inssort(xs) -> isort(xs, Nil) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] isort(Nil, r) -> r [1] insert(x, Nil) -> Cons(x, Nil) [1] inssort(xs) -> isort(xs, Nil) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] The TRS has the following type information: isort :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil insert :: S:0 -> Cons:Nil -> Cons:Nil insert[Ite][False][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False Nil :: Cons:Nil inssort :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 True :: True:False False :: True:False Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: isort_2 inssort_1 (c) The following functions are completely defined: insert_2 lt_2 insert[Ite][False][Ite]_3 Due to the following rules being added: lt(v0, v1) -> null_lt [0] insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: null_lt ---------------------------------------- (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: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] isort(Nil, r) -> r [1] insert(x, Nil) -> Cons(x, Nil) [1] inssort(xs) -> isort(xs, Nil) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] lt(v0, v1) -> null_lt [0] insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: isort :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil insert :: S:0 -> Cons:Nil -> Cons:Nil insert[Ite][False][Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False:null_lt Nil :: Cons:Nil inssort :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 True :: True:False:null_lt False :: True:False:null_lt null_lt :: True:False:null_lt Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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: isort(Cons(x, xs), Cons(x1, xs')) -> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, Cons(x1, xs'))) [2] isort(Cons(x, xs), Nil) -> isort(xs, Cons(x, Nil)) [2] insert(S(x''), Cons(S(y'), xs)) -> insert[Ite][False][Ite](lt(x'', y'), S(x''), Cons(S(y'), xs)) [1] insert(0, Cons(S(y''), xs)) -> insert[Ite][False][Ite](True, 0, Cons(S(y''), xs)) [1] insert(x', Cons(0, xs)) -> insert[Ite][False][Ite](False, x', Cons(0, xs)) [1] insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](null_lt, x', Cons(x, xs)) [1] isort(Nil, r) -> r [1] insert(x, Nil) -> Cons(x, Nil) [1] inssort(xs) -> isort(xs, Nil) [1] lt(S(x), S(y)) -> lt(x, y) [0] lt(0, S(y)) -> True [0] lt(x, 0) -> False [0] insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] lt(v0, v1) -> null_lt [0] insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: isort :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S:0 -> Cons:Nil -> Cons:Nil insert :: S:0 -> Cons:Nil -> Cons:Nil insert[Ite][False][Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil lt :: S:0 -> S:0 -> True:False:null_lt Nil :: Cons:Nil inssort :: Cons:Nil -> Cons:Nil S :: S:0 -> S:0 0 :: S:0 True :: True:False:null_lt False :: True:False:null_lt null_lt :: True:False:null_lt Rewrite Strategy: INNERMOST ---------------------------------------- (13) 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 0 => 0 True => 2 False => 1 null_lt => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(x'', y'), 1 + x'', 1 + (1 + y') + xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, x'' >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x' insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' insert(z, z') -{ 1 }-> 1 + x + 0 :|: x >= 0, z = x, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs inssort(z) -{ 1 }-> isort(xs, 0) :|: xs >= 0, z = xs isort(z, z') -{ 1 }-> r :|: r >= 0, z = 0, z' = r isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (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: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt } { insert, insert[Ite][False][Ite] } { isort } { inssort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} ---------------------------------------- (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: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt 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: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} Previous analysis results are: lt: runtime: ?, size: O(1) [2] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lt 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: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} Previous analysis results are: lt: 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: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' Computed SIZE bound using CoFloCo for: insert[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: ?, size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: ?, size: O(n^1) [1 + z' + z''] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' Computed RUNTIME bound using CoFloCo for: insert[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 1 }-> insert[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 2 }-> isort(xs, insert[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isort}, {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] ---------------------------------------- (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: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 4 + x1 + xs' }-> isort(xs, s1) :|: s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isort}, {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: isort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 4 + x1 + xs' }-> isort(xs, s1) :|: s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isort}, {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] isort: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: isort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z + z*z' + z^2 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 isort(z, z') -{ 4 + x1 + xs' }-> isort(xs, s1) :|: s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] ---------------------------------------- (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: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 isort(z, z') -{ 5 + s1*xs + x1 + 4*xs + xs^2 + xs' }-> s7 :|: s7 >= 0, s7 <= xs + s1, s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 3 + x*xs + 5*xs + xs^2 }-> s8 :|: s8 >= 0, s8 <= xs + (1 + x + 0), z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inssort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 isort(z, z') -{ 5 + s1*xs + x1 + 4*xs + xs^2 + xs' }-> s7 :|: s7 >= 0, s7 <= xs + s1, s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 3 + x*xs + 5*xs + xs^2 }-> s8 :|: s8 >= 0, s8 <= xs + (1 + x + 0), z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inssort} Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] inssort: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: inssort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 4*z + z^2 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: insert(z, z') -{ 4 + xs + y' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0 insert(z, z') -{ 4 + xs + y'' }-> s3 :|: s3 >= 0, s3 <= 0 + (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert[Ite][False][Ite](z, z', z'') -{ 2 + xs }-> 1 + x + s6 :|: s6 >= 0, s6 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 isort(z, z') -{ 5 + s1*xs + x1 + 4*xs + xs^2 + xs' }-> s7 :|: s7 >= 0, s7 <= xs + s1, s1 >= 0, s1 <= x + (1 + x1 + xs') + 1, s >= 0, s <= 2, z = 1 + x + xs, xs >= 0, x1 >= 0, z' = 1 + x1 + xs', x >= 0, xs' >= 0 isort(z, z') -{ 3 + x*xs + 5*xs + xs^2 }-> s8 :|: s8 >= 0, s8 <= xs + (1 + x + 0), z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: lt: runtime: O(1) [0], size: O(1) [2] insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] inssort: runtime: O(n^2) [2 + 4*z + z^2], size: O(n^1) [z] ---------------------------------------- (43) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (44) BOUNDS(1, n^2) ---------------------------------------- (45) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (46) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) isort(Nil, r) -> r insert(x, Nil) -> Cons(x, Nil) inssort(xs) -> isort(xs, Nil) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) insert[Ite][False][Ite](True, x, r) -> Cons(x, r) Rewrite Strategy: INNERMOST ---------------------------------------- (47) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence isort(Cons(x, xs), r) ->^+ isort(xs, insert(x, r)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [xs / Cons(x, xs)]. The result substitution is [r / insert(x, r)]. ---------------------------------------- (48) Complex Obligation (BEST) ---------------------------------------- (49) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) isort(Nil, r) -> r insert(x, Nil) -> Cons(x, Nil) inssort(xs) -> isort(xs, Nil) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) insert[Ite][False][Ite](True, x, r) -> Cons(x, r) Rewrite Strategy: INNERMOST ---------------------------------------- (50) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (51) BOUNDS(n^1, INF) ---------------------------------------- (52) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) isort(Nil, r) -> r insert(x, Nil) -> Cons(x, Nil) inssort(xs) -> isort(xs, Nil) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) insert[Ite][False][Ite](True, x, r) -> Cons(x, r) Rewrite Strategy: INNERMOST