349.56/291.49 WORST_CASE(Omega(n^1), O(n^2)) 349.56/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 349.56/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 349.56/291.51 349.56/291.51 349.56/291.51 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 349.56/291.51 349.56/291.51 (0) CpxRelTRS 349.56/291.51 (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 ms] 349.56/291.51 (2) CpxRelTRS 349.56/291.51 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (4) CpxWeightedTrs 349.56/291.51 (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (6) CpxWeightedTrs 349.56/291.51 (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 1 ms] 349.56/291.51 (8) CpxTypedWeightedTrs 349.56/291.51 (9) CompletionProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (10) CpxTypedWeightedCompleteTrs 349.56/291.51 (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (12) CpxTypedWeightedCompleteTrs 349.56/291.51 (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (14) CpxRNTS 349.56/291.51 (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (16) CpxRNTS 349.56/291.51 (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (18) CpxRNTS 349.56/291.51 (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (20) CpxRNTS 349.56/291.51 (21) IntTrsBoundProof [UPPER BOUND(ID), 486 ms] 349.56/291.51 (22) CpxRNTS 349.56/291.51 (23) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] 349.56/291.51 (24) CpxRNTS 349.56/291.51 (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (26) CpxRNTS 349.56/291.51 (27) IntTrsBoundProof [UPPER BOUND(ID), 1409 ms] 349.56/291.51 (28) CpxRNTS 349.56/291.51 (29) IntTrsBoundProof [UPPER BOUND(ID), 364 ms] 349.56/291.51 (30) CpxRNTS 349.56/291.51 (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (32) CpxRNTS 349.56/291.51 (33) IntTrsBoundProof [UPPER BOUND(ID), 511 ms] 349.56/291.51 (34) CpxRNTS 349.56/291.51 (35) IntTrsBoundProof [UPPER BOUND(ID), 286 ms] 349.56/291.51 (36) CpxRNTS 349.56/291.51 (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (38) CpxRNTS 349.56/291.51 (39) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] 349.56/291.51 (40) CpxRNTS 349.56/291.51 (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 349.56/291.51 (42) CpxRNTS 349.56/291.51 (43) FinalProof [FINISHED, 0 ms] 349.56/291.51 (44) BOUNDS(1, n^2) 349.56/291.51 (45) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (46) CpxRelTRS 349.56/291.51 (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 349.56/291.51 (48) typed CpxTrs 349.56/291.51 (49) OrderProof [LOWER BOUND(ID), 0 ms] 349.56/291.51 (50) typed CpxTrs 349.56/291.51 (51) RewriteLemmaProof [LOWER BOUND(ID), 302 ms] 349.56/291.51 (52) typed CpxTrs 349.56/291.51 (53) RewriteLemmaProof [LOWER BOUND(ID), 9662 ms] 349.56/291.51 (54) BEST 349.56/291.51 (55) proven lower bound 349.56/291.51 (56) LowerBoundPropagationProof [FINISHED, 0 ms] 349.56/291.51 (57) BOUNDS(n^1, INF) 349.56/291.51 (58) typed CpxTrs 349.56/291.51 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (0) 349.56/291.51 Obligation: 349.56/291.51 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 349.56/291.51 349.56/291.51 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 349.56/291.51 The (relative) TRS S consists of the following rules: 349.56/291.51 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0, S(y)) -> True 349.56/291.51 <(x, 0) -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) 349.56/291.51 proved termination of relative rules 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (2) 349.56/291.51 Obligation: 349.56/291.51 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 349.56/291.51 349.56/291.51 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 349.56/291.51 The (relative) TRS S consists of the following rules: 349.56/291.51 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0, S(y)) -> True 349.56/291.51 <(x, 0) -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Transformed relative TRS to weighted TRS 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (4) 349.56/291.51 Obligation: 349.56/291.51 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 349.56/291.51 349.56/291.51 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) [1] 349.56/291.51 isort(Nil, r) -> r [1] 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) [1] 349.56/291.51 inssort(xs) -> isort(xs, Nil) [1] 349.56/291.51 <(S(x), S(y)) -> <(x, y) [0] 349.56/291.51 <(0, S(y)) -> True [0] 349.56/291.51 <(x, 0) -> False [0] 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Renamed defined symbols to avoid conflicts with arithmetic symbols: 349.56/291.51 349.56/291.51 < => lt 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (6) 349.56/291.51 Obligation: 349.56/291.51 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 349.56/291.51 349.56/291.51 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] 349.56/291.51 isort(Nil, r) -> r [1] 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) [1] 349.56/291.51 inssort(xs) -> isort(xs, Nil) [1] 349.56/291.51 lt(S(x), S(y)) -> lt(x, y) [0] 349.56/291.51 lt(0, S(y)) -> True [0] 349.56/291.51 lt(x, 0) -> False [0] 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Infered types. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (8) 349.56/291.51 Obligation: 349.56/291.51 Runtime Complexity Weighted TRS with Types. 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] 349.56/291.51 isort(Nil, r) -> r [1] 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) [1] 349.56/291.51 inssort(xs) -> isort(xs, Nil) [1] 349.56/291.51 lt(S(x), S(y)) -> lt(x, y) [0] 349.56/291.51 lt(0, S(y)) -> True [0] 349.56/291.51 lt(x, 0) -> False [0] 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] 349.56/291.51 349.56/291.51 The TRS has the following type information: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 lt :: S:0 -> S:0 -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0 -> S:0 349.56/291.51 0 :: S:0 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (9) CompletionProof (UPPER BOUND(ID)) 349.56/291.51 The transformation into a RNTS is sound, since: 349.56/291.51 349.56/291.51 (a) The obligation is a constructor system where every type has a constant constructor, 349.56/291.51 349.56/291.51 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 349.56/291.51 349.56/291.51 isort_2 349.56/291.51 inssort_1 349.56/291.51 349.56/291.51 (c) The following functions are completely defined: 349.56/291.51 349.56/291.51 insert_2 349.56/291.51 lt_2 349.56/291.51 insert[Ite][False][Ite]_3 349.56/291.51 349.56/291.51 Due to the following rules being added: 349.56/291.51 349.56/291.51 lt(v0, v1) -> null_lt [0] 349.56/291.51 insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] 349.56/291.51 349.56/291.51 And the following fresh constants: null_lt 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (10) 349.56/291.51 Obligation: 349.56/291.51 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 349.56/291.51 349.56/291.51 Runtime Complexity Weighted TRS with Types. 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1] 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] 349.56/291.51 isort(Nil, r) -> r [1] 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) [1] 349.56/291.51 inssort(xs) -> isort(xs, Nil) [1] 349.56/291.51 lt(S(x), S(y)) -> lt(x, y) [0] 349.56/291.51 lt(0, S(y)) -> True [0] 349.56/291.51 lt(x, 0) -> False [0] 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] 349.56/291.51 lt(v0, v1) -> null_lt [0] 349.56/291.51 insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] 349.56/291.51 349.56/291.51 The TRS has the following type information: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 lt :: S:0 -> S:0 -> True:False:null_lt 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0 -> S:0 349.56/291.51 0 :: S:0 349.56/291.51 True :: True:False:null_lt 349.56/291.51 False :: True:False:null_lt 349.56/291.51 null_lt :: True:False:null_lt 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (11) NarrowingProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (12) 349.56/291.51 Obligation: 349.56/291.51 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 349.56/291.51 349.56/291.51 Runtime Complexity Weighted TRS with Types. 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), Cons(x1, xs')) -> isort(xs, insert[Ite][False][Ite](lt(x, x1), x, Cons(x1, xs'))) [2] 349.56/291.51 isort(Cons(x, xs), Nil) -> isort(xs, Cons(x, Nil)) [2] 349.56/291.51 insert(S(x''), Cons(S(y'), xs)) -> insert[Ite][False][Ite](lt(x'', y'), S(x''), Cons(S(y'), xs)) [1] 349.56/291.51 insert(0, Cons(S(y''), xs)) -> insert[Ite][False][Ite](True, 0, Cons(S(y''), xs)) [1] 349.56/291.51 insert(x', Cons(0, xs)) -> insert[Ite][False][Ite](False, x', Cons(0, xs)) [1] 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](null_lt, x', Cons(x, xs)) [1] 349.56/291.51 isort(Nil, r) -> r [1] 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) [1] 349.56/291.51 inssort(xs) -> isort(xs, Nil) [1] 349.56/291.51 lt(S(x), S(y)) -> lt(x, y) [0] 349.56/291.51 lt(0, S(y)) -> True [0] 349.56/291.51 lt(x, 0) -> False [0] 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0] 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) [0] 349.56/291.51 lt(v0, v1) -> null_lt [0] 349.56/291.51 insert[Ite][False][Ite](v0, v1, v2) -> Nil [0] 349.56/291.51 349.56/291.51 The TRS has the following type information: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil 349.56/291.51 lt :: S:0 -> S:0 -> True:False:null_lt 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0 -> S:0 349.56/291.51 0 :: S:0 349.56/291.51 True :: True:False:null_lt 349.56/291.51 False :: True:False:null_lt 349.56/291.51 null_lt :: True:False:null_lt 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 349.56/291.51 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 349.56/291.51 The constant constructors are abstracted as follows: 349.56/291.51 349.56/291.51 Nil => 0 349.56/291.51 0 => 0 349.56/291.51 True => 2 349.56/291.51 False => 1 349.56/291.51 null_lt => 0 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (14) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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'' >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x' 349.56/291.51 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' 349.56/291.51 insert(z, z') -{ 1 }-> 1 + x + 0 :|: x >= 0, z = x, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0 349.56/291.51 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 349.56/291.51 inssort(z) -{ 1 }-> isort(xs, 0) :|: xs >= 0, z = xs 349.56/291.51 isort(z, z') -{ 1 }-> r :|: r >= 0, z = 0, z' = r 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 349.56/291.51 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (15) SimplificationProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (16) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Found the following analysis order by SCC decomposition: 349.56/291.51 349.56/291.51 { lt } 349.56/291.51 { insert, insert[Ite][False][Ite] } 349.56/291.51 { isort } 349.56/291.51 { inssort } 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (18) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (19) ResultPropagationProof (UPPER BOUND(ID)) 349.56/291.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (20) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (21) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed SIZE bound using CoFloCo for: lt 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(1) with polynomial bound: 2 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (22) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {lt}, {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: ?, size: O(1) [2] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (23) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed RUNTIME bound using CoFloCo for: lt 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(1) with polynomial bound: 0 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (24) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (25) ResultPropagationProof (UPPER BOUND(ID)) 349.56/291.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (26) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (27) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed SIZE bound using CoFloCo for: insert 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: 1 + z + z' 349.56/291.51 349.56/291.51 Computed SIZE bound using CoFloCo for: insert[Ite][False][Ite] 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: 1 + z' + z'' 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (28) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {insert,insert[Ite][False][Ite]}, {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: ?, size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: ?, size: O(n^1) [1 + z' + z''] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (29) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed RUNTIME bound using CoFloCo for: insert 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: 2 + z' 349.56/291.51 349.56/291.51 Computed RUNTIME bound using CoFloCo for: insert[Ite][False][Ite] 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: 1 + z'' 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (30) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 1 }-> insert[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (31) ResultPropagationProof (UPPER BOUND(ID)) 349.56/291.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (32) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (33) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed SIZE bound using CoFloCo for: isort 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: z + z' 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (34) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {isort}, {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 isort: runtime: ?, size: O(n^1) [z + z'] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (35) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed RUNTIME bound using CoFloCo for: isort 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^2) with polynomial bound: 1 + 4*z + z*z' + z^2 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (36) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 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' >= 0 349.56/291.51 isort(z, z') -{ 2 }-> isort(xs, 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (37) ResultPropagationProof (UPPER BOUND(ID)) 349.56/291.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (38) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 349.56/291.51 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' >= 0 349.56/291.51 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' = 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (39) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed SIZE bound using CoFloCo for: inssort 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^1) with polynomial bound: z 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (40) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 349.56/291.51 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' >= 0 349.56/291.51 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' = 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: {inssort} 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] 349.56/291.51 inssort: runtime: ?, size: O(n^1) [z] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (41) IntTrsBoundProof (UPPER BOUND(ID)) 349.56/291.51 349.56/291.51 Computed RUNTIME bound using KoAT for: inssort 349.56/291.51 after applying outer abstraction to obtain an ITS, 349.56/291.51 resulting in: O(n^2) with polynomial bound: 2 + 4*z + z^2 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (42) 349.56/291.51 Obligation: 349.56/291.51 Complexity RNTS consisting of the following rules: 349.56/291.51 349.56/291.51 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 >= 0 349.56/291.51 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 = 0 349.56/291.51 insert(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= z + (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0 349.56/291.51 insert(z, z') -{ 3 + x + xs }-> s5 :|: s5 >= 0, s5 <= z + (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 349.56/291.51 insert(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 349.56/291.51 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 349.56/291.51 insert[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0 349.56/291.51 inssort(z) -{ 2 + 4*z + z^2 }-> s9 :|: s9 >= 0, s9 <= z + 0, z >= 0 349.56/291.51 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' >= 0 349.56/291.51 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' = 0 349.56/291.51 isort(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 349.56/291.51 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 349.56/291.51 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 349.56/291.51 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 349.56/291.51 349.56/291.51 Function symbols to be analyzed: 349.56/291.51 Previous analysis results are: 349.56/291.51 lt: runtime: O(1) [0], size: O(1) [2] 349.56/291.51 insert: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z + z'] 349.56/291.51 insert[Ite][False][Ite]: runtime: O(n^1) [1 + z''], size: O(n^1) [1 + z' + z''] 349.56/291.51 isort: runtime: O(n^2) [1 + 4*z + z*z' + z^2], size: O(n^1) [z + z'] 349.56/291.51 inssort: runtime: O(n^2) [2 + 4*z + z^2], size: O(n^1) [z] 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (43) FinalProof (FINISHED) 349.56/291.51 Computed overall runtime complexity 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (44) 349.56/291.51 BOUNDS(1, n^2) 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (45) RenamingProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Renamed function symbols to avoid clashes with predefined symbol. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (46) 349.56/291.51 Obligation: 349.56/291.51 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 349.56/291.51 349.56/291.51 349.56/291.51 The TRS R consists of the following rules: 349.56/291.51 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 349.56/291.51 The (relative) TRS S consists of the following rules: 349.56/291.51 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Rewrite Strategy: INNERMOST 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (47) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 349.56/291.51 Infered types. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (48) 349.56/291.51 Obligation: 349.56/291.51 Innermost TRS: 349.56/291.51 Rules: 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Types: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 < :: S:0' -> S:0' -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0' -> S:0' 349.56/291.51 0' :: S:0' 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 hole_Cons:Nil1_0 :: Cons:Nil 349.56/291.51 hole_S:0'2_0 :: S:0' 349.56/291.51 hole_True:False3_0 :: True:False 349.56/291.51 gen_Cons:Nil4_0 :: Nat -> Cons:Nil 349.56/291.51 gen_S:0'5_0 :: Nat -> S:0' 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (49) OrderProof (LOWER BOUND(ID)) 349.56/291.51 Heuristically decided to analyse the following defined symbols: 349.56/291.51 isort, insert, < 349.56/291.51 349.56/291.51 They will be analysed ascendingly in the following order: 349.56/291.51 insert < isort 349.56/291.51 < < insert 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (50) 349.56/291.51 Obligation: 349.56/291.51 Innermost TRS: 349.56/291.51 Rules: 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Types: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 < :: S:0' -> S:0' -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0' -> S:0' 349.56/291.51 0' :: S:0' 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 hole_Cons:Nil1_0 :: Cons:Nil 349.56/291.51 hole_S:0'2_0 :: S:0' 349.56/291.51 hole_True:False3_0 :: True:False 349.56/291.51 gen_Cons:Nil4_0 :: Nat -> Cons:Nil 349.56/291.51 gen_S:0'5_0 :: Nat -> S:0' 349.56/291.51 349.56/291.51 349.56/291.51 Generator Equations: 349.56/291.51 gen_Cons:Nil4_0(0) <=> Nil 349.56/291.51 gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x)) 349.56/291.51 gen_S:0'5_0(0) <=> 0' 349.56/291.51 gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x)) 349.56/291.51 349.56/291.51 349.56/291.51 The following defined symbols remain to be analysed: 349.56/291.51 <, isort, insert 349.56/291.51 349.56/291.51 They will be analysed ascendingly in the following order: 349.56/291.51 insert < isort 349.56/291.51 < < insert 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (51) RewriteLemmaProof (LOWER BOUND(ID)) 349.56/291.51 Proved the following rewrite lemma: 349.56/291.51 <(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0) 349.56/291.51 349.56/291.51 Induction Base: 349.56/291.51 <(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0) 349.56/291.51 True 349.56/291.51 349.56/291.51 Induction Step: 349.56/291.51 <(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0) 349.56/291.51 <(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH 349.56/291.51 True 349.56/291.51 349.56/291.51 We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (52) 349.56/291.51 Obligation: 349.56/291.51 Innermost TRS: 349.56/291.51 Rules: 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Types: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 < :: S:0' -> S:0' -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0' -> S:0' 349.56/291.51 0' :: S:0' 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 hole_Cons:Nil1_0 :: Cons:Nil 349.56/291.51 hole_S:0'2_0 :: S:0' 349.56/291.51 hole_True:False3_0 :: True:False 349.56/291.51 gen_Cons:Nil4_0 :: Nat -> Cons:Nil 349.56/291.51 gen_S:0'5_0 :: Nat -> S:0' 349.56/291.51 349.56/291.51 349.56/291.51 Lemmas: 349.56/291.51 <(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0) 349.56/291.51 349.56/291.51 349.56/291.51 Generator Equations: 349.56/291.51 gen_Cons:Nil4_0(0) <=> Nil 349.56/291.51 gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x)) 349.56/291.51 gen_S:0'5_0(0) <=> 0' 349.56/291.51 gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x)) 349.56/291.51 349.56/291.51 349.56/291.51 The following defined symbols remain to be analysed: 349.56/291.51 insert, isort 349.56/291.51 349.56/291.51 They will be analysed ascendingly in the following order: 349.56/291.51 insert < isort 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (53) RewriteLemmaProof (LOWER BOUND(ID)) 349.56/291.51 Proved the following rewrite lemma: 349.56/291.51 insert(gen_S:0'5_0(a), gen_Cons:Nil4_0(+(1, n221_0))) -> *6_0, rt in Omega(n221_0) 349.56/291.51 349.56/291.51 Induction Base: 349.56/291.51 insert(gen_S:0'5_0(a), gen_Cons:Nil4_0(+(1, 0))) 349.56/291.51 349.56/291.51 Induction Step: 349.56/291.51 insert(gen_S:0'5_0(a), gen_Cons:Nil4_0(+(1, +(n221_0, 1)))) ->_R^Omega(1) 349.56/291.51 insert[Ite][False][Ite](<(gen_S:0'5_0(a), 0'), gen_S:0'5_0(a), Cons(0', gen_Cons:Nil4_0(+(1, n221_0)))) ->_R^Omega(0) 349.56/291.51 insert[Ite][False][Ite](False, gen_S:0'5_0(a), Cons(0', gen_Cons:Nil4_0(+(1, n221_0)))) ->_R^Omega(0) 349.56/291.51 Cons(0', insert(gen_S:0'5_0(a), gen_Cons:Nil4_0(+(1, n221_0)))) ->_IH 349.56/291.51 Cons(0', *6_0) 349.56/291.51 349.56/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (54) 349.56/291.51 Complex Obligation (BEST) 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (55) 349.56/291.51 Obligation: 349.56/291.51 Proved the lower bound n^1 for the following obligation: 349.56/291.51 349.56/291.51 Innermost TRS: 349.56/291.51 Rules: 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Types: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 < :: S:0' -> S:0' -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0' -> S:0' 349.56/291.51 0' :: S:0' 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 hole_Cons:Nil1_0 :: Cons:Nil 349.56/291.51 hole_S:0'2_0 :: S:0' 349.56/291.51 hole_True:False3_0 :: True:False 349.56/291.51 gen_Cons:Nil4_0 :: Nat -> Cons:Nil 349.56/291.51 gen_S:0'5_0 :: Nat -> S:0' 349.56/291.51 349.56/291.51 349.56/291.51 Lemmas: 349.56/291.51 <(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0) 349.56/291.51 349.56/291.51 349.56/291.51 Generator Equations: 349.56/291.51 gen_Cons:Nil4_0(0) <=> Nil 349.56/291.51 gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x)) 349.56/291.51 gen_S:0'5_0(0) <=> 0' 349.56/291.51 gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x)) 349.56/291.51 349.56/291.51 349.56/291.51 The following defined symbols remain to be analysed: 349.56/291.51 insert, isort 349.56/291.51 349.56/291.51 They will be analysed ascendingly in the following order: 349.56/291.51 insert < isort 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (56) LowerBoundPropagationProof (FINISHED) 349.56/291.51 Propagated lower bound. 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (57) 349.56/291.51 BOUNDS(n^1, INF) 349.56/291.51 349.56/291.51 ---------------------------------------- 349.56/291.51 349.56/291.51 (58) 349.56/291.51 Obligation: 349.56/291.51 Innermost TRS: 349.56/291.51 Rules: 349.56/291.51 isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) 349.56/291.51 insert(x', Cons(x, xs)) -> insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 349.56/291.51 isort(Nil, r) -> r 349.56/291.51 insert(x, Nil) -> Cons(x, Nil) 349.56/291.51 inssort(xs) -> isort(xs, Nil) 349.56/291.51 <(S(x), S(y)) -> <(x, y) 349.56/291.51 <(0', S(y)) -> True 349.56/291.51 <(x, 0') -> False 349.56/291.51 insert[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) 349.56/291.51 insert[Ite][False][Ite](True, x, r) -> Cons(x, r) 349.56/291.51 349.56/291.51 Types: 349.56/291.51 isort :: Cons:Nil -> Cons:Nil -> Cons:Nil 349.56/291.51 Cons :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert :: S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 insert[Ite][False][Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil 349.56/291.51 < :: S:0' -> S:0' -> True:False 349.56/291.51 Nil :: Cons:Nil 349.56/291.51 inssort :: Cons:Nil -> Cons:Nil 349.56/291.51 S :: S:0' -> S:0' 349.56/291.51 0' :: S:0' 349.56/291.51 True :: True:False 349.56/291.51 False :: True:False 349.56/291.51 hole_Cons:Nil1_0 :: Cons:Nil 349.56/291.51 hole_S:0'2_0 :: S:0' 349.56/291.51 hole_True:False3_0 :: True:False 349.56/291.51 gen_Cons:Nil4_0 :: Nat -> Cons:Nil 349.56/291.51 gen_S:0'5_0 :: Nat -> S:0' 349.56/291.51 349.56/291.51 349.56/291.51 Lemmas: 349.56/291.51 <(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0) 349.56/291.51 insert(gen_S:0'5_0(a), gen_Cons:Nil4_0(+(1, n221_0))) -> *6_0, rt in Omega(n221_0) 349.56/291.51 349.56/291.51 349.56/291.51 Generator Equations: 349.56/291.51 gen_Cons:Nil4_0(0) <=> Nil 349.56/291.51 gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x)) 349.56/291.51 gen_S:0'5_0(0) <=> 0' 349.56/291.51 gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x)) 349.56/291.51 349.56/291.51 349.56/291.51 The following defined symbols remain to be analysed: 349.56/291.51 isort 349.72/291.54 EOF