335.04/291.51 WORST_CASE(Omega(n^1), O(n^2)) 335.08/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 335.08/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 335.08/291.52 335.08/291.52 335.08/291.52 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.52 335.08/291.52 (0) CpxRelTRS 335.08/291.52 (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 175 ms] 335.08/291.52 (2) CpxRelTRS 335.08/291.52 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 335.08/291.52 (4) CpxWeightedTrs 335.08/291.52 (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 335.08/291.52 (6) CpxWeightedTrs 335.08/291.52 (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms] 335.08/291.52 (8) CpxTypedWeightedTrs 335.08/291.52 (9) CompletionProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (10) CpxTypedWeightedCompleteTrs 335.08/291.52 (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 335.08/291.52 (12) CpxTypedWeightedCompleteTrs 335.08/291.52 (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (14) CpxRNTS 335.08/291.52 (15) SimplificationProof [BOTH BOUNDS(ID, ID), 4 ms] 335.08/291.52 (16) CpxRNTS 335.08/291.52 (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 335.08/291.52 (18) CpxRNTS 335.08/291.52 (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (20) CpxRNTS 335.08/291.52 (21) IntTrsBoundProof [UPPER BOUND(ID), 538 ms] 335.08/291.52 (22) CpxRNTS 335.08/291.52 (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] 335.08/291.52 (24) CpxRNTS 335.08/291.52 (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (26) CpxRNTS 335.08/291.52 (27) IntTrsBoundProof [UPPER BOUND(ID), 1683 ms] 335.08/291.52 (28) CpxRNTS 335.08/291.52 (29) IntTrsBoundProof [UPPER BOUND(ID), 451 ms] 335.08/291.52 (30) CpxRNTS 335.08/291.52 (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (32) CpxRNTS 335.08/291.52 (33) IntTrsBoundProof [UPPER BOUND(ID), 381 ms] 335.08/291.52 (34) CpxRNTS 335.08/291.52 (35) IntTrsBoundProof [UPPER BOUND(ID), 206 ms] 335.08/291.52 (36) CpxRNTS 335.08/291.52 (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (38) CpxRNTS 335.08/291.52 (39) IntTrsBoundProof [UPPER BOUND(ID), 167 ms] 335.08/291.52 (40) CpxRNTS 335.08/291.52 (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 335.08/291.52 (42) CpxRNTS 335.08/291.52 (43) FinalProof [FINISHED, 0 ms] 335.08/291.52 (44) BOUNDS(1, n^2) 335.08/291.52 (45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 335.08/291.52 (46) TRS for Loop Detection 335.08/291.52 (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 335.08/291.52 (48) BEST 335.08/291.52 (49) proven lower bound 335.08/291.52 (50) LowerBoundPropagationProof [FINISHED, 0 ms] 335.08/291.52 (51) BOUNDS(n^1, INF) 335.08/291.52 (52) TRS for Loop Detection 335.08/291.52 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (0) 335.08/291.52 Obligation: 335.08/291.52 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.52 335.08/291.52 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 335.08/291.52 isort(Nil, r) -> Nil 335.08/291.52 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) 335.08/291.52 inssort(xs) -> isort(xs, Nil) 335.08/291.52 335.08/291.52 The (relative) TRS S consists of the following rules: 335.08/291.52 335.08/291.52 <(S(x), S(y)) -> <(x, y) 335.08/291.52 <(0, S(y)) -> True 335.08/291.52 <(x, 0) -> False 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) 335.08/291.52 proved termination of relative rules 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (2) 335.08/291.52 Obligation: 335.08/291.52 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.52 335.08/291.52 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 335.08/291.52 isort(Nil, r) -> Nil 335.08/291.52 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) 335.08/291.52 inssort(xs) -> isort(xs, Nil) 335.08/291.52 335.08/291.52 The (relative) TRS S consists of the following rules: 335.08/291.52 335.08/291.52 <(S(x), S(y)) -> <(x, y) 335.08/291.52 <(0, S(y)) -> True 335.08/291.52 <(x, 0) -> False 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Transformed relative TRS to weighted TRS 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (4) 335.08/291.52 Obligation: 335.08/291.52 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 335.08/291.52 335.08/291.52 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 335.08/291.52 isort(Nil, r) -> Nil [1] 335.08/291.52 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) [1] 335.08/291.52 inssort(xs) -> isort(xs, Nil) [1] 335.08/291.52 <(S(x), S(y)) -> <(x, y) [0] 335.08/291.52 <(0, S(y)) -> True [0] 335.08/291.52 <(x, 0) -> False [0] 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) [0] 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Renamed defined symbols to avoid conflicts with arithmetic symbols: 335.08/291.52 335.08/291.52 < => lt 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (6) 335.08/291.52 Obligation: 335.08/291.52 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 335.08/291.52 335.08/291.52 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 335.08/291.52 isort(Nil, r) -> Nil [1] 335.08/291.52 insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1] 335.08/291.52 inssort(xs) -> isort(xs, Nil) [1] 335.08/291.52 lt(S(x), S(y)) -> lt(x, y) [0] 335.08/291.52 lt(0, S(y)) -> True [0] 335.08/291.52 lt(x, 0) -> False [0] 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) [0] 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Infered types. 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (8) 335.08/291.52 Obligation: 335.08/291.52 Runtime Complexity Weighted TRS with Types. 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 335.08/291.52 isort(Nil, r) -> Nil [1] 335.08/291.52 insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1] 335.08/291.52 inssort(xs) -> isort(xs, Nil) [1] 335.08/291.52 lt(S(x), S(y)) -> lt(x, y) [0] 335.08/291.52 lt(0, S(y)) -> True [0] 335.08/291.52 lt(x, 0) -> False [0] 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) [0] 335.08/291.52 335.08/291.52 The TRS has the following type information: 335.08/291.52 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 335.08/291.52 Cons :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 insert :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 Nil :: Cons:Nil 335.08/291.52 S :: S:0 -> S:0 335.08/291.52 insert[Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 lt :: S:0 -> S:0 -> True:False 335.08/291.52 inssort :: Cons:Nil -> Cons:Nil 335.08/291.52 0 :: S:0 335.08/291.52 True :: True:False 335.08/291.52 False :: True:False 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (9) CompletionProof (UPPER BOUND(ID)) 335.08/291.52 The transformation into a RNTS is sound, since: 335.08/291.52 335.08/291.52 (a) The obligation is a constructor system where every type has a constant constructor, 335.08/291.52 335.08/291.52 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 335.08/291.52 335.08/291.52 isort_2 335.08/291.52 inssort_1 335.08/291.52 335.08/291.52 (c) The following functions are completely defined: 335.08/291.52 335.08/291.52 insert_2 335.08/291.52 lt_2 335.08/291.52 insert[Ite]_3 335.08/291.52 335.08/291.52 Due to the following rules being added: 335.08/291.52 335.08/291.52 lt(v0, v1) -> null_lt [0] 335.08/291.52 insert[Ite](v0, v1, v2) -> Nil [0] 335.08/291.52 insert(v0, v1) -> Nil [0] 335.08/291.52 335.08/291.52 And the following fresh constants: null_lt 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (10) 335.08/291.52 Obligation: 335.08/291.52 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 335.08/291.52 335.08/291.52 Runtime Complexity Weighted TRS with Types. 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 335.08/291.52 isort(Nil, r) -> Nil [1] 335.08/291.52 insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1] 335.08/291.52 inssort(xs) -> isort(xs, Nil) [1] 335.08/291.52 lt(S(x), S(y)) -> lt(x, y) [0] 335.08/291.52 lt(0, S(y)) -> True [0] 335.08/291.52 lt(x, 0) -> False [0] 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) [0] 335.08/291.52 lt(v0, v1) -> null_lt [0] 335.08/291.52 insert[Ite](v0, v1, v2) -> Nil [0] 335.08/291.52 insert(v0, v1) -> Nil [0] 335.08/291.52 335.08/291.52 The TRS has the following type information: 335.08/291.52 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 335.08/291.52 Cons :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 insert :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 Nil :: Cons:Nil 335.08/291.52 S :: S:0 -> S:0 335.08/291.52 insert[Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 lt :: S:0 -> S:0 -> True:False:null_lt 335.08/291.52 inssort :: Cons:Nil -> Cons:Nil 335.08/291.52 0 :: S:0 335.08/291.52 True :: True:False:null_lt 335.08/291.52 False :: True:False:null_lt 335.08/291.52 null_lt :: True:False:null_lt 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (11) NarrowingProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (12) 335.08/291.52 Obligation: 335.08/291.52 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 335.08/291.52 335.08/291.52 Runtime Complexity Weighted TRS with Types. 335.08/291.52 The TRS R consists of the following rules: 335.08/291.52 335.08/291.52 isort(Cons(S(x''), xs), r) -> isort(xs, insert[Ite](lt(S(x''), x''), S(x''), r)) [2] 335.08/291.52 isort(Cons(x, xs), r) -> isort(xs, Nil) [1] 335.08/291.52 isort(Nil, r) -> Nil [1] 335.08/291.52 insert(S(S(y')), r) -> insert[Ite](lt(S(y'), y'), S(S(y')), r) [1] 335.08/291.52 insert(S(0), r) -> insert[Ite](False, S(0), r) [1] 335.08/291.52 insert(S(x), r) -> insert[Ite](null_lt, S(x), r) [1] 335.08/291.52 inssort(xs) -> isort(xs, Nil) [1] 335.08/291.52 lt(S(x), S(y)) -> lt(x, y) [0] 335.08/291.52 lt(0, S(y)) -> True [0] 335.08/291.52 lt(x, 0) -> False [0] 335.08/291.52 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 335.08/291.52 insert[Ite](True, x, r) -> Cons(x, r) [0] 335.08/291.52 lt(v0, v1) -> null_lt [0] 335.08/291.52 insert[Ite](v0, v1, v2) -> Nil [0] 335.08/291.52 insert(v0, v1) -> Nil [0] 335.08/291.52 335.08/291.52 The TRS has the following type information: 335.08/291.52 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 335.08/291.52 Cons :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 insert :: S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 Nil :: Cons:Nil 335.08/291.52 S :: S:0 -> S:0 335.08/291.52 insert[Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil 335.08/291.52 lt :: S:0 -> S:0 -> True:False:null_lt 335.08/291.52 inssort :: Cons:Nil -> Cons:Nil 335.08/291.52 0 :: S:0 335.08/291.52 True :: True:False:null_lt 335.08/291.52 False :: True:False:null_lt 335.08/291.52 null_lt :: True:False:null_lt 335.08/291.52 335.08/291.52 Rewrite Strategy: INNERMOST 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 335.08/291.52 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 335.08/291.52 The constant constructors are abstracted as follows: 335.08/291.52 335.08/291.52 Nil => 0 335.08/291.52 0 => 0 335.08/291.52 True => 2 335.08/291.52 False => 1 335.08/291.52 null_lt => 0 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (14) 335.08/291.52 Obligation: 335.08/291.52 Complexity RNTS consisting of the following rules: 335.08/291.52 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + y', y'), 1 + (1 + y'), r) :|: r >= 0, z = 1 + (1 + y'), y' >= 0, z' = r 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, r) :|: r >= 0, z = 1 + 0, z' = r 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + x, r) :|: r >= 0, x >= 0, z = 1 + x, z' = r 335.08/291.52 insert(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.52 inssort(z) -{ 1 }-> isort(xs, 0) :|: xs >= 0, z = xs 335.08/291.52 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', r)) :|: xs >= 0, r >= 0, z = 1 + (1 + x'') + xs, x'' >= 0, z' = r 335.08/291.52 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, r >= 0, x >= 0, z' = r 335.08/291.52 isort(z, z') -{ 1 }-> 0 :|: r >= 0, z = 0, z' = r 335.08/291.52 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 335.08/291.52 lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 335.08/291.52 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 335.08/291.52 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 335.08/291.52 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (15) SimplificationProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (16) 335.08/291.52 Obligation: 335.08/291.52 Complexity RNTS consisting of the following rules: 335.08/291.52 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.52 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.52 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.52 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.52 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.52 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.52 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 335.08/291.52 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.52 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.52 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.52 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 335.08/291.52 Found the following analysis order by SCC decomposition: 335.08/291.52 335.08/291.52 { lt } 335.08/291.52 { insert[Ite], insert } 335.08/291.52 { isort } 335.08/291.52 { inssort } 335.08/291.52 335.08/291.52 ---------------------------------------- 335.08/291.52 335.08/291.52 (18) 335.08/291.52 Obligation: 335.08/291.52 Complexity RNTS consisting of the following rules: 335.08/291.52 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.52 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.52 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.52 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (19) ResultPropagationProof (UPPER BOUND(ID)) 335.08/291.53 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (20) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (21) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed SIZE bound using CoFloCo for: lt 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(1) with polynomial bound: 2 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (22) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: ?, size: O(1) [2] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (23) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed RUNTIME bound using CoFloCo for: lt 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(1) with polynomial bound: 0 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (24) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (25) ResultPropagationProof (UPPER BOUND(ID)) 335.08/291.53 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (26) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (27) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed SIZE bound using CoFloCo for: insert[Ite] 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^1) with polynomial bound: 1 + z' + z'' 335.08/291.53 335.08/291.53 Computed SIZE bound using KoAT for: insert 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^1) with polynomial bound: 1 + z + z' 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (28) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: ?, size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: ?, size: O(n^1) [1 + z + z'] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (29) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed RUNTIME bound using CoFloCo for: insert[Ite] 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^1) with polynomial bound: 2 + z'' 335.08/291.53 335.08/291.53 Computed RUNTIME bound using CoFloCo for: insert 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^1) with polynomial bound: 3 + z' 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (30) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (31) ResultPropagationProof (UPPER BOUND(ID)) 335.08/291.53 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (32) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 4 + z' }-> isort(xs, s1) :|: s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (33) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed SIZE bound using CoFloCo for: isort 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(1) with polynomial bound: 0 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (34) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 4 + z' }-> isort(xs, s1) :|: s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {isort}, {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 isort: runtime: ?, size: O(1) [0] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (35) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed RUNTIME bound using CoFloCo for: isort 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^2) with polynomial bound: 1 + 3*z + z*z' + z^2 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (36) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 335.08/291.53 isort(z, z') -{ 4 + z' }-> isort(xs, s1) :|: s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (37) ResultPropagationProof (UPPER BOUND(ID)) 335.08/291.53 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (38) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0 335.08/291.53 isort(z, z') -{ 5 + s1*xs + 3*xs + xs^2 + z' }-> s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (39) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed SIZE bound using CoFloCo for: inssort 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(1) with polynomial bound: 0 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (40) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0 335.08/291.53 isort(z, z') -{ 5 + s1*xs + 3*xs + xs^2 + z' }-> s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: {inssort} 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0] 335.08/291.53 inssort: runtime: ?, size: O(1) [0] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (41) IntTrsBoundProof (UPPER BOUND(ID)) 335.08/291.53 335.08/291.53 Computed RUNTIME bound using KoAT for: inssort 335.08/291.53 after applying outer abstraction to obtain an ITS, 335.08/291.53 resulting in: O(n^2) with polynomial bound: 2 + 3*z + z^2 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (42) 335.08/291.53 Obligation: 335.08/291.53 Complexity RNTS consisting of the following rules: 335.08/291.53 335.08/291.53 insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0 335.08/291.53 insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0 335.08/291.53 insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 335.08/291.53 insert[Ite](z, z', z'') -{ 3 + xs }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs 335.08/291.53 insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 335.08/291.53 inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0 335.08/291.53 isort(z, z') -{ 5 + s1*xs + 3*xs + xs^2 + z' }-> s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 + x'' + z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0 335.08/291.53 isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 335.08/291.53 isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 335.08/291.53 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 335.08/291.53 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 335.08/291.53 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 335.08/291.53 335.08/291.53 Function symbols to be analyzed: 335.08/291.53 Previous analysis results are: 335.08/291.53 lt: runtime: O(1) [0], size: O(1) [2] 335.08/291.53 insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z''] 335.08/291.53 insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z'] 335.08/291.53 isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0] 335.08/291.53 inssort: runtime: O(n^2) [2 + 3*z + z^2], size: O(1) [0] 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (43) FinalProof (FINISHED) 335.08/291.53 Computed overall runtime complexity 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (44) 335.08/291.53 BOUNDS(1, n^2) 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (45) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 335.08/291.53 Transformed a relative TRS into a decreasing-loop problem. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (46) 335.08/291.53 Obligation: 335.08/291.53 Analyzing the following TRS for decreasing loops: 335.08/291.53 335.08/291.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.53 335.08/291.53 335.08/291.53 The TRS R consists of the following rules: 335.08/291.53 335.08/291.53 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 335.08/291.53 isort(Nil, r) -> Nil 335.08/291.53 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) 335.08/291.53 inssort(xs) -> isort(xs, Nil) 335.08/291.53 335.08/291.53 The (relative) TRS S consists of the following rules: 335.08/291.53 335.08/291.53 <(S(x), S(y)) -> <(x, y) 335.08/291.53 <(0, S(y)) -> True 335.08/291.53 <(x, 0) -> False 335.08/291.53 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 335.08/291.53 insert[Ite](True, x, r) -> Cons(x, r) 335.08/291.53 335.08/291.53 Rewrite Strategy: INNERMOST 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (47) DecreasingLoopProof (LOWER BOUND(ID)) 335.08/291.53 The following loop(s) give(s) rise to the lower bound Omega(n^1): 335.08/291.53 335.08/291.53 The rewrite sequence 335.08/291.53 335.08/291.53 isort(Cons(x, xs), r) ->^+ isort(xs, insert(x, r)) 335.08/291.53 335.08/291.53 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 335.08/291.53 335.08/291.53 The pumping substitution is [xs / Cons(x, xs)]. 335.08/291.53 335.08/291.53 The result substitution is [r / insert(x, r)]. 335.08/291.53 335.08/291.53 335.08/291.53 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (48) 335.08/291.53 Complex Obligation (BEST) 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (49) 335.08/291.53 Obligation: 335.08/291.53 Proved the lower bound n^1 for the following obligation: 335.08/291.53 335.08/291.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.53 335.08/291.53 335.08/291.53 The TRS R consists of the following rules: 335.08/291.53 335.08/291.53 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 335.08/291.53 isort(Nil, r) -> Nil 335.08/291.53 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) 335.08/291.53 inssort(xs) -> isort(xs, Nil) 335.08/291.53 335.08/291.53 The (relative) TRS S consists of the following rules: 335.08/291.53 335.08/291.53 <(S(x), S(y)) -> <(x, y) 335.08/291.53 <(0, S(y)) -> True 335.08/291.53 <(x, 0) -> False 335.08/291.53 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 335.08/291.53 insert[Ite](True, x, r) -> Cons(x, r) 335.08/291.53 335.08/291.53 Rewrite Strategy: INNERMOST 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (50) LowerBoundPropagationProof (FINISHED) 335.08/291.53 Propagated lower bound. 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (51) 335.08/291.53 BOUNDS(n^1, INF) 335.08/291.53 335.08/291.53 ---------------------------------------- 335.08/291.53 335.08/291.53 (52) 335.08/291.53 Obligation: 335.08/291.53 Analyzing the following TRS for decreasing loops: 335.08/291.53 335.08/291.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 335.08/291.53 335.08/291.53 335.08/291.53 The TRS R consists of the following rules: 335.08/291.53 335.08/291.53 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 335.08/291.53 isort(Nil, r) -> Nil 335.08/291.53 insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) 335.08/291.53 inssort(xs) -> isort(xs, Nil) 335.08/291.53 335.08/291.53 The (relative) TRS S consists of the following rules: 335.08/291.53 335.08/291.53 <(S(x), S(y)) -> <(x, y) 335.08/291.53 <(0, S(y)) -> True 335.08/291.53 <(x, 0) -> False 335.08/291.53 insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 335.08/291.53 insert[Ite](True, x, r) -> Cons(x, r) 335.08/291.53 335.08/291.53 Rewrite Strategy: INNERMOST 335.13/291.58 EOF