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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 382 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 1860 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 638 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 645 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^2)
(41) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(42) TRS for Loop Detection
    (43) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(2)
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:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]

The TRS has the following type information:
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert_ord#2 :: leq -> 0:S -> Nil:Cons -> Nil:Cons
cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
True :: True:False
False :: True:False
leq :: leq
leq#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons

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

(5) 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:

   main_1

(c) The following functions are completely defined:

   leq#2_2
   fold#3_2
   insert_ord#2_3
   cond_insert_ord_x_ys_1_4

Due to the following rules being added:

   fold#3(v0, v1) -> Nil [0]

And the following fresh constants:    const

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

(6)
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:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]
   fold#3(v0, v1) -> Nil [0]

The TRS has the following type information:
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert_ord#2 :: leq -> 0:S -> Nil:Cons -> Nil:Cons
cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
True :: True:False
False :: True:False
leq :: leq
leq#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons
const :: insert_ord

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, Nil)) -> insert_ord#2(x6, x4, Nil) [2]
   fold#3(insert_ord(x6), Cons(x4, Cons(x4', x2'))) -> insert_ord#2(x6, x4, insert_ord#2(x6, x4', fold#3(insert_ord(x6), x2'))) [2]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, Nil) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, 0, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(True, 0, x4, x2) [2]
   insert_ord#2(leq, S(x12'), Cons(0, x2)) -> cond_insert_ord_x_ys_1(False, S(x12'), 0, x2) [2]
   insert_ord#2(leq, S(x4''), Cons(S(x2''), x2)) -> cond_insert_ord_x_ys_1(leq#2(x4'', x2''), S(x4''), S(x2''), x2) [2]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]
   fold#3(v0, v1) -> Nil [0]

The TRS has the following type information:
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert_ord#2 :: leq -> 0:S -> Nil:Cons -> Nil:Cons
cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
True :: True:False
False :: True:False
leq :: leq
leq#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons
const :: insert_ord

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

(9) 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
   True => 1
   False => 0
   leq => 0
   0 => 0
   const => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x3 + (1 + x2 + x1) :|: x1 >= 0, z = 1, z' = x3, z1 = x1, z'' = x2, x3 >= 0, x2 >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x5 + insert_ord#2(0, x0, x2) :|: x0 >= 0, x5 >= 0, z1 = x2, z'' = x5, z = 0, x2 >= 0, z' = x0
   fold#3(z, z') -{ 2 }-> insert_ord#2(x6, x4, insert_ord#2(x6, x4', fold#3(1 + x6, x2'))) :|: x2' >= 0, x4 >= 0, z = 1 + x6, x6 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(x6, x4, 0) :|: x4 >= 0, z = 1 + x6, x6 >= 0, z' = 1 + x4 + 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(x6, x4, 0) :|: x4 >= 0, z = 1 + x6, z' = 1 + x4 + x2, x6 >= 0, x2 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z = 1 + x2, x2 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(x4'', x2''), 1 + x4'', 1 + x2'', x2) :|: x4'' >= 0, z'' = 1 + (1 + x2'') + x2, z' = 1 + x4'', z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + x12', 0, x2) :|: z'' = 1 + 0 + x2, z' = 1 + x12', x12' >= 0, z = 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + x2 + 0 :|: z'' = 0, z' = x2, z = 0, x2 >= 0
   leq#2(z, z') -{ 1 }-> leq#2(x4, x2) :|: x4 >= 0, z' = 1 + x2, z = 1 + x4, x2 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: x8 >= 0, z = 0, z' = x8
   leq#2(z, z') -{ 1 }-> 0 :|: z = 1 + x12, x12 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, x3) :|: z = x3, x3 >= 0


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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z' - 1, x2''), 1 + (z' - 1), 1 + x2'', x2) :|: z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0


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

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

   { leq#2 }
   { cond_insert_ord_x_ys_1, insert_ord#2 }
   { fold#3 }
   { main }

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z' - 1, x2''), 1 + (z' - 1), 1 + x2'', x2) :|: z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z' - 1, x2''), 1 + (z' - 1), 1 + x2'', x2) :|: z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z' - 1, x2''), 1 + (z' - 1), 1 + x2'', x2) :|: z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z' - 1, x2''), 1 + (z' - 1), 1 + x2'', x2) :|: z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z' - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond_insert_ord_x_ys_1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z' - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert_ord#2}, {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cond_insert_ord_x_ys_1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 7*z1

Computed RUNTIME bound using CoFloCo for: insert_ord#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + 7*z''

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert_ord#2(0, z', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> insert_ord#2(z - 1, x4, 0) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z' - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z' - 1), 0, z'' - 1) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {fold#3}, {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']
  fold#3: runtime: ?, size: O(n^1) [z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: fold#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 15 + 21*z' + 14*z'^2

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 2 }-> insert_ord#2(z - 1, x4, insert_ord#2(z - 1, x4', fold#3(1 + (z - 1), x2'))) :|: x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, z) :|: z >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']
  fold#3: runtime: O(n^2) [15 + 21*z' + 14*z'^2], size: O(n^1) [z']

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 29 + 7*s6 + 7*s7 + 21*x2' + 14*x2'^2 }-> s8 :|: s6 >= 0, s6 <= x2', s7 >= 0, s7 <= x4' + s6 + 1, s8 >= 0, s8 <= x4 + s7 + 1, x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 16 + 21*z + 14*z^2 }-> s9 :|: s9 >= 0, s9 <= z, z >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']
  fold#3: runtime: O(n^2) [15 + 21*z' + 14*z'^2], size: O(n^1) [z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 29 + 7*s6 + 7*s7 + 21*x2' + 14*x2'^2 }-> s8 :|: s6 >= 0, s6 <= x2', s7 >= 0, s7 <= x4' + s6 + 1, s8 >= 0, s8 <= x4 + s7 + 1, x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 16 + 21*z + 14*z^2 }-> s9 :|: s9 >= 0, s9 <= z, z >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']
  fold#3: runtime: O(n^2) [15 + 21*z' + 14*z'^2], size: O(n^1) [z']
  main: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 16 + 21*z + 14*z^2

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s2 :|: s2 >= 0, s2 <= z' + z1 + 1, z' >= 0, z'' >= 0, z = 0, z1 >= 0
   fold#3(z, z') -{ 8 }-> s'' :|: s'' >= 0, s'' <= z' - 1 + 0 + 1, z' - 1 >= 0, z - 1 >= 0
   fold#3(z, z') -{ 7 }-> s1 :|: s1 >= 0, s1 <= x4 + 0 + 1, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   fold#3(z, z') -{ 29 + 7*s6 + 7*s7 + 21*x2' + 14*x2'^2 }-> s8 :|: s6 >= 0, s6 <= x2', s7 >= 0, s7 <= x4' + s6 + 1, s8 >= 0, s8 <= x4 + s7 + 1, x2' >= 0, x4 >= 0, z - 1 >= 0, z' = 1 + x4 + (1 + x4' + x2'), x4' >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   insert_ord#2(z, z', z'') -{ 6 + 7*x2 }-> s3 :|: s3 >= 0, s3 <= 0 + x4 + x2 + 2, x4 >= 0, z'' = 1 + x4 + x2, z = 0, x2 >= 0, z' = 0
   insert_ord#2(z, z', z'') -{ -1 + 7*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z' - 1) + 0 + (z'' - 1) + 2, z' - 1 >= 0, z = 0, z'' - 1 >= 0
   insert_ord#2(z, z', z'') -{ 8 + 7*x2 + x2'' }-> s5 :|: s5 >= 0, s5 <= 1 + (z' - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z' - 1 >= 0, z'' = 1 + (1 + x2'') + x2, z = 0, x2'' >= 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + z' + 0 :|: z'' = 0, z = 0, z' >= 0
   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   main(z) -{ 16 + 21*z + 14*z^2 }-> s9 :|: s9 >= 0, s9 <= z, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
  insert_ord#2: runtime: O(n^1) [6 + 7*z''], size: O(n^1) [1 + z' + z'']
  fold#3: runtime: O(n^2) [15 + 21*z' + 14*z'^2], size: O(n^1) [z']
  main: runtime: O(n^2) [16 + 21*z + 14*z^2], size: O(n^1) [z]

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

(39) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(40)
BOUNDS(1, n^2)

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

(41) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(42)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(43) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

leq#2(S(x4), S(x2)) ->^+ leq#2(x4, x2)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x4 / S(x4), x2 / S(x2)].

The result substitution is [ ].




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

(44)
Complex Obligation (BEST)

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

(45)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(46) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(47)
BOUNDS(n^1, INF)

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

(48)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST