/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 162 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 458 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 1420 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 370 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 280 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 205 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^2)
(45) RenamingProof [BOTH BOUNDS(ID, ID), 1 ms]
(46) CpxRelTRS
    (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) typed CpxTrs
    (49) OrderProof [LOWER BOUND(ID), 0 ms]
    (50) typed CpxTrs
    (51) RewriteLemmaProof [LOWER BOUND(ID), 281 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 465 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

Rewrite Strategy: INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   <(S(x), S(y)) -> <(x, y) [0]
   <(0, S(y)) -> True [0]
   <(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

< => lt

----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

The TRS has the following type information:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
insert :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0 -> S:0
insert[Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False
inssort :: Cons:Nil -> Cons:Nil
0 :: S:0
True :: True:False
False :: True:False

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   isort_2
   inssort_1

(c) The following functions are completely defined:

   insert_2
   lt_2
   insert[Ite]_3

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   insert[Ite](v0, v1, v2) -> Nil [0]
   insert(v0, v1) -> Nil [0]

And the following fresh constants:    null_lt

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]
   lt(v0, v1) -> null_lt [0]
   insert[Ite](v0, v1, v2) -> Nil [0]
   insert(v0, v1) -> Nil [0]

The TRS has the following type information:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
insert :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0 -> S:0
insert[Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_lt
inssort :: Cons:Nil -> Cons:Nil
0 :: S:0
True :: True:False:null_lt
False :: True:False:null_lt
null_lt :: True:False:null_lt

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   isort(Cons(S(x''), xs), r) -> isort(xs, insert[Ite](lt(S(x''), x''), S(x''), r)) [2]
   isort(Cons(x, xs), r) -> isort(xs, Nil) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(S(y')), r) -> insert[Ite](lt(S(y'), y'), S(S(y')), r) [1]
   insert(S(0), r) -> insert[Ite](False, S(0), r) [1]
   insert(S(x), r) -> insert[Ite](null_lt, S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]
   lt(v0, v1) -> null_lt [0]
   insert[Ite](v0, v1, v2) -> Nil [0]
   insert(v0, v1) -> Nil [0]

The TRS has the following type information:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
insert :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0 -> S:0
insert[Ite] :: True:False:null_lt -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_lt
inssort :: Cons:Nil -> Cons:Nil
0 :: S:0
True :: True:False:null_lt
False :: True:False:null_lt
null_lt :: True:False:null_lt

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   0 => 0
   True => 2
   False => 1
   null_lt => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + y', y'), 1 + (1 + y'), r) :|: r >= 0, z = 1 + (1 + y'), y' >= 0, z' = r
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, r) :|: r >= 0, z = 1 + 0, z' = r
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + x, r) :|: r >= 0, x >= 0, z = 1 + x, z' = r
   insert(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0
   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
   inssort(z) -{ 1 }-> isort(xs, 0) :|: xs >= 0, z = xs
   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
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, r >= 0, x >= 0, z' = r
   isort(z, z') -{ 1 }-> 0 :|: r >= 0, z = 0, z' = r
   lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


----------------------------------------

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


----------------------------------------

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { lt }
   { insert[Ite], insert }
   { isort }
   { inssort }

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort}

----------------------------------------

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort}

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:
  lt: runtime: ?, size: O(1) [2]

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: insert[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

Computed SIZE bound using KoAT for: insert
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: ?, size: O(n^1) [1 + z' + z'']
  insert: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: insert[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z''

Computed RUNTIME bound using CoFloCo for: insert
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z'

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 1 }-> insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
   insert(z, z') -{ 1 }-> insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 2 }-> isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']

----------------------------------------

(31) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 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
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']

----------------------------------------

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isort
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 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
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']
  isort: runtime: ?, size: O(1) [0]

----------------------------------------

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 3*z + z*z' + z^2

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 1 }-> isort(z, 0) :|: z >= 0
   isort(z, z') -{ 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
   isort(z, z') -{ 1 }-> isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']
  isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0]

----------------------------------------

(37) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0
   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
   isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']
  isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0]

----------------------------------------

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: inssort
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0
   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
   isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {inssort}
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']
  isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0]
  inssort: runtime: ?, size: O(1) [0]

----------------------------------------

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: inssort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 3*z + z^2

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + (z - 2)) + z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
   insert(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1 + 0 + z' + 1, z' >= 0, z = 1 + 0
   insert(z, z') -{ 3 + z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + z' + 1, z' >= 0, z - 1 >= 0
   insert(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   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
   insert[Ite](z, z', z'') -{ 0 }-> 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
   inssort(z) -{ 2 + 3*z + z^2 }-> s8 :|: s8 >= 0, s8 <= 0, z >= 0
   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
   isort(z, z') -{ 2 + 3*xs + xs^2 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   isort(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   lt(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  lt: runtime: O(1) [0], size: O(1) [2]
  insert[Ite]: runtime: O(n^1) [2 + z''], size: O(n^1) [1 + z' + z'']
  insert: runtime: O(n^1) [3 + z'], size: O(n^1) [1 + z + z']
  isort: runtime: O(n^2) [1 + 3*z + z*z' + z^2], size: O(1) [0]
  inssort: runtime: O(n^2) [2 + 3*z + z^2], size: O(1) [0]

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(43) FinalProof (FINISHED)
Computed overall runtime complexity
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(44)
BOUNDS(1, n^2)

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(45) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(46)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0', S(y)) -> True
   <(x, 0') -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

Rewrite Strategy: INNERMOST
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(47) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(48)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'

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(49) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
isort, insert, <

They will be analysed ascendingly in the following order:
insert < isort
< < insert

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(50)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
<, isort, insert

They will be analysed ascendingly in the following order:
insert < isort
< < insert

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(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)

Induction Base:
<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
True

Induction Step:
<(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
True

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(52)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
insert, isort

They will be analysed ascendingly in the following order:
insert < isort

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(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n215_0)) -> *6_0, rt in Omega(n215_0)

Induction Base:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(0))

Induction Step:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n215_0, 1))) ->_R^Omega(1)
insert[Ite](<(S(gen_S:0'5_0(0)), gen_S:0'5_0(0)), S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(n215_0, 1))) ->_R^Omega(0)
insert[Ite](False, S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(1, n215_0))) ->_R^Omega(0)
Cons(0', insert(S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(n215_0))) ->_IH
Cons(0', *6_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(54)
Complex Obligation (BEST)

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(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
insert, isort

They will be analysed ascendingly in the following order:
insert < isort

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(56) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(57)
BOUNDS(n^1, INF)

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(58)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n215_0)) -> *6_0, rt in Omega(n215_0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
isort