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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 1 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 361 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 1596 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 539 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 310 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^2)
(41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CpxTRS
    (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) typed CpxTrs
    (45) OrderProof [LOWER BOUND(ID), 0 ms]
    (46) typed CpxTrs
    (47) RewriteLemmaProof [LOWER BOUND(ID), 236 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) typed CpxTrs
            (53) RewriteLemmaProof [LOWER BOUND(ID), 322 ms]
            (54) BOUNDS(1, INF)


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

(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:

   sort#2(Nil) -> Nil
   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
   insert#3(x2, Nil) -> Cons(x2, Nil)
   insert#3(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(x1) -> sort#2(x1)

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:

   sort#2(Nil) -> Nil [1]
   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(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, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
   insert#3(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(x1) -> sort#2(x1) [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:

   sort#2(Nil) -> Nil [1]
   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(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, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
   insert#3(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(x1) -> sort#2(x1) [1]

The TRS has the following type information:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#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:

   sort#2_1
   leq#2_2
   insert#3_2
   cond_insert_ord_x_ys_1_4

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   sort#2(Nil) -> Nil [1]
   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(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, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
   insert#3(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(x1) -> sort#2(x1) [1]

The TRS has the following type information:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons

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:

   sort#2(Nil) -> Nil [1]
   sort#2(Cons(x4, Nil)) -> insert#3(x4, Nil) [2]
   sort#2(Cons(x4, Cons(x4', x2'))) -> insert#3(x4, insert#3(x4', sort#2(x2'))) [2]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
   insert#3(0, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(True, 0, x4, x2) [2]
   insert#3(S(x12'), Cons(0, x2)) -> cond_insert_ord_x_ys_1(False, S(x12'), 0, x2) [2]
   insert#3(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(x1) -> sort#2(x1) [1]

The TRS has the following type information:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons

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
   0 => 0

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x2 + insert#3(x3, x1) :|: x1 >= 0, z' = x3, z1 = x1, z = 0, z'' = x2, x3 >= 0, x2 >= 0
   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
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(x4'', x2''), 1 + x4'', 1 + x2'', x2) :|: x4'' >= 0, z = 1 + x4'', z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + x12', 0, x2) :|: z = 1 + x12', z' = 1 + 0 + x2, x12' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + x2 + 0 :|: z = x2, x2 >= 0, z' = 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 }-> sort#2(x1) :|: x1 >= 0, z = x1
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, 0) :|: z = 1 + x4 + 0, x4 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: 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#3 }
   { sort#2 }
   { 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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3
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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {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#3: 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#3
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#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(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, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {sort#2}, {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#3: 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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {sort#2}, {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#3: 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: sort#2
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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {sort#2}, {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#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
  sort#2: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: sort#2
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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> sort#2(z) :|: z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: 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#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
  sort#2: 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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: 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#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
  sort#2: 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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: 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#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
  sort#2: 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'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
   insert#3(z, z') -{ 1 }-> 1 + 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 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
   sort#2(z) -{ 1 }-> 0 :|: 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#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
  sort#2: 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) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(42)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   sort#2(Nil) -> Nil
   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
   insert#3(x2, Nil) -> Cons(x2, Nil)
   insert#3(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(x1) -> sort#2(x1)

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

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

(44)
Obligation:
Innermost TRS:
Rules:
sort#2(Nil) -> Nil
sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
insert#3(x2, Nil) -> Cons(x2, Nil)
insert#3(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(x1) -> sort#2(x1)

Types:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0':S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0':S -> 0':S -> True:False
0' :: 0':S
S :: 0':S -> 0':S
main :: Nil:Cons -> Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_0':S2_0 :: 0':S
hole_True:False3_0 :: True:False
gen_Nil:Cons4_0 :: Nat -> Nil:Cons
gen_0':S5_0 :: Nat -> 0':S

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

(45) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sort#2, insert#3, leq#2

They will be analysed ascendingly in the following order:
insert#3 < sort#2
leq#2 < insert#3

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

(46)
Obligation:
Innermost TRS:
Rules:
sort#2(Nil) -> Nil
sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
insert#3(x2, Nil) -> Cons(x2, Nil)
insert#3(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(x1) -> sort#2(x1)

Types:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0':S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0':S -> 0':S -> True:False
0' :: 0':S
S :: 0':S -> 0':S
main :: Nil:Cons -> Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_0':S2_0 :: 0':S
hole_True:False3_0 :: True:False
gen_Nil:Cons4_0 :: Nat -> Nil:Cons
gen_0':S5_0 :: Nat -> 0':S


Generator Equations:
gen_Nil:Cons4_0(0) <=> Nil
gen_Nil:Cons4_0(+(x, 1)) <=> Cons(0', gen_Nil:Cons4_0(x))
gen_0':S5_0(0) <=> 0'
gen_0':S5_0(+(x, 1)) <=> S(gen_0':S5_0(x))


The following defined symbols remain to be analysed:
leq#2, sort#2, insert#3

They will be analysed ascendingly in the following order:
insert#3 < sort#2
leq#2 < insert#3

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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq#2(gen_0':S5_0(n7_0), gen_0':S5_0(n7_0)) -> True, rt in Omega(1 + n7_0)

Induction Base:
leq#2(gen_0':S5_0(0), gen_0':S5_0(0)) ->_R^Omega(1)
True

Induction Step:
leq#2(gen_0':S5_0(+(n7_0, 1)), gen_0':S5_0(+(n7_0, 1))) ->_R^Omega(1)
leq#2(gen_0':S5_0(n7_0), gen_0':S5_0(n7_0)) ->_IH
True

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

(48)
Complex Obligation (BEST)

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

(49)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
sort#2(Nil) -> Nil
sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
insert#3(x2, Nil) -> Cons(x2, Nil)
insert#3(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(x1) -> sort#2(x1)

Types:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0':S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0':S -> 0':S -> True:False
0' :: 0':S
S :: 0':S -> 0':S
main :: Nil:Cons -> Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_0':S2_0 :: 0':S
hole_True:False3_0 :: True:False
gen_Nil:Cons4_0 :: Nat -> Nil:Cons
gen_0':S5_0 :: Nat -> 0':S


Generator Equations:
gen_Nil:Cons4_0(0) <=> Nil
gen_Nil:Cons4_0(+(x, 1)) <=> Cons(0', gen_Nil:Cons4_0(x))
gen_0':S5_0(0) <=> 0'
gen_0':S5_0(+(x, 1)) <=> S(gen_0':S5_0(x))


The following defined symbols remain to be analysed:
leq#2, sort#2, insert#3

They will be analysed ascendingly in the following order:
insert#3 < sort#2
leq#2 < insert#3

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

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

(51)
BOUNDS(n^1, INF)

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

(52)
Obligation:
Innermost TRS:
Rules:
sort#2(Nil) -> Nil
sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
insert#3(x2, Nil) -> Cons(x2, Nil)
insert#3(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(x1) -> sort#2(x1)

Types:
sort#2 :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: 0':S -> Nil:Cons -> Nil:Cons
insert#3 :: 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#2 :: 0':S -> 0':S -> True:False
0' :: 0':S
S :: 0':S -> 0':S
main :: Nil:Cons -> Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_0':S2_0 :: 0':S
hole_True:False3_0 :: True:False
gen_Nil:Cons4_0 :: Nat -> Nil:Cons
gen_0':S5_0 :: Nat -> 0':S


Lemmas:
leq#2(gen_0':S5_0(n7_0), gen_0':S5_0(n7_0)) -> True, rt in Omega(1 + n7_0)


Generator Equations:
gen_Nil:Cons4_0(0) <=> Nil
gen_Nil:Cons4_0(+(x, 1)) <=> Cons(0', gen_Nil:Cons4_0(x))
gen_0':S5_0(0) <=> 0'
gen_0':S5_0(+(x, 1)) <=> S(gen_0':S5_0(x))


The following defined symbols remain to be analysed:
insert#3, sort#2

They will be analysed ascendingly in the following order:
insert#3 < sort#2

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sort#2(gen_Nil:Cons4_0(n428_0)) -> *6_0, rt in Omega(n428_0)

Induction Base:
sort#2(gen_Nil:Cons4_0(0))

Induction Step:
sort#2(gen_Nil:Cons4_0(+(n428_0, 1))) ->_R^Omega(1)
insert#3(0', sort#2(gen_Nil:Cons4_0(n428_0))) ->_IH
insert#3(0', *6_0)

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

(54)
BOUNDS(1, INF)