WORST_CASE(Omega(n^1),O(n^2))
proof of input_I350Pn093p.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 245 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 50 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 1481 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 618 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 445 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 155 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^2)
(41) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CdtProblem
    (43) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CpxRelTRS
    (45) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) typed CpxTrs
    (49) OrderProof [LOWER BOUND(ID), 7 ms]
    (50) typed CpxTrs
    (51) RewriteLemmaProof [LOWER BOUND(ID), 310 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 88 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 383 ms]
            (60) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fold#3(insert_ord(z0), Nil) -> Nil
   fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
   cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
   cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
   insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
   insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
   leq#2(0, z0) -> True
   leq#2(S(z0), 0) -> False
   leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
   main(z0) -> fold#3(insert_ord(leq), z0)
Tuples:
   FOLD#3(insert_ord(z0), Nil) -> c
   FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
   COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
   COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
   INSERT_ORD#2(leq, z0, Nil) -> c4
   INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
   LEQ#2(0, z0) -> c6
   LEQ#2(S(z0), 0) -> c7
   LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
   MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
S tuples:
   FOLD#3(insert_ord(z0), Nil) -> c
   FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
   COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
   COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
   INSERT_ORD#2(leq, z0, Nil) -> c4
   INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
   LEQ#2(0, z0) -> c6
   LEQ#2(S(z0), 0) -> c7
   LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
   MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
K tuples:none
Defined Rule Symbols:   fold#3_2, cond_insert_ord_x_ys_1_4, insert_ord#2_3, leq#2_2, main_1

Defined Pair Symbols:   FOLD#3_2, COND_INSERT_ORD_X_YS_1_4, INSERT_ORD#2_3, LEQ#2_2, MAIN_1

Compound Symbols:   c, c1_2, c2, c3_1, c4, c5_2, c6, c7, c8_1, c9_1


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

(43) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(44)
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:

   FOLD#3(insert_ord(z0), Nil) -> c
   FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
   COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
   COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
   INSERT_ORD#2(leq, z0, Nil) -> c4
   INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
   LEQ#2(0, z0) -> c6
   LEQ#2(S(z0), 0) -> c7
   LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
   MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))

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

   fold#3(insert_ord(z0), Nil) -> Nil
   fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
   cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
   cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
   insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
   insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
   leq#2(0, z0) -> True
   leq#2(S(z0), 0) -> False
   leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
   main(z0) -> fold#3(insert_ord(leq), z0)

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

(45) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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:

   FOLD#3(insert_ord(z0), Nil) -> c
   FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
   COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
   COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
   INSERT_ORD#2(leq, z0, Nil) -> c4
   INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
   LEQ#2(0', z0) -> c6
   LEQ#2(S(z0), 0') -> c7
   LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
   MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))

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

   fold#3(insert_ord(z0), Nil) -> Nil
   fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
   cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
   cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
   insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
   insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
   leq#2(0', z0) -> True
   leq#2(S(z0), 0') -> False
   leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
   main(z0) -> fold#3(insert_ord(leq), z0)

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

(47) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(48)
Obligation:
Innermost TRS:
Rules:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8

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

(49) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FOLD#3, INSERT_ORD#2, fold#3, leq#2, LEQ#2, insert_ord#2

They will be analysed ascendingly in the following order:
INSERT_ORD#2 < FOLD#3
fold#3 < FOLD#3
leq#2 < INSERT_ORD#2
LEQ#2 < INSERT_ORD#2
insert_ord#2 < fold#3
leq#2 < insert_ord#2

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

(50)
Obligation:
Innermost TRS:
Rules:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8


Generator Equations:
gen_c:c111_10(0) <=> c
gen_c:c111_10(+(x, 1)) <=> c1(c4, gen_c:c111_10(x))
gen_Nil:Cons12_10(0) <=> Nil
gen_Nil:Cons12_10(+(x, 1)) <=> Cons(0', gen_Nil:Cons12_10(x))
gen_0':S13_10(0) <=> 0'
gen_0':S13_10(+(x, 1)) <=> S(gen_0':S13_10(x))
gen_c6:c7:c814_10(0) <=> c6
gen_c6:c7:c814_10(+(x, 1)) <=> c8(gen_c6:c7:c814_10(x))


The following defined symbols remain to be analysed:
leq#2, FOLD#3, INSERT_ORD#2, fold#3, LEQ#2, insert_ord#2

They will be analysed ascendingly in the following order:
INSERT_ORD#2 < FOLD#3
fold#3 < FOLD#3
leq#2 < INSERT_ORD#2
LEQ#2 < INSERT_ORD#2
insert_ord#2 < fold#3
leq#2 < insert_ord#2

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) -> True, rt in Omega(0)

Induction Base:
leq#2(gen_0':S13_10(0), gen_0':S13_10(0)) ->_R^Omega(0)
True

Induction Step:
leq#2(gen_0':S13_10(+(n16_10, 1)), gen_0':S13_10(+(n16_10, 1))) ->_R^Omega(0)
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) ->_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:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8


Lemmas:
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) -> True, rt in Omega(0)


Generator Equations:
gen_c:c111_10(0) <=> c
gen_c:c111_10(+(x, 1)) <=> c1(c4, gen_c:c111_10(x))
gen_Nil:Cons12_10(0) <=> Nil
gen_Nil:Cons12_10(+(x, 1)) <=> Cons(0', gen_Nil:Cons12_10(x))
gen_0':S13_10(0) <=> 0'
gen_0':S13_10(+(x, 1)) <=> S(gen_0':S13_10(x))
gen_c6:c7:c814_10(0) <=> c6
gen_c6:c7:c814_10(+(x, 1)) <=> c8(gen_c6:c7:c814_10(x))


The following defined symbols remain to be analysed:
LEQ#2, FOLD#3, INSERT_ORD#2, fold#3, insert_ord#2

They will be analysed ascendingly in the following order:
INSERT_ORD#2 < FOLD#3
fold#3 < FOLD#3
LEQ#2 < INSERT_ORD#2
insert_ord#2 < fold#3

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LEQ#2(gen_0':S13_10(n335_10), gen_0':S13_10(n335_10)) -> gen_c6:c7:c814_10(n335_10), rt in Omega(1 + n335_10)

Induction Base:
LEQ#2(gen_0':S13_10(0), gen_0':S13_10(0)) ->_R^Omega(1)
c6

Induction Step:
LEQ#2(gen_0':S13_10(+(n335_10, 1)), gen_0':S13_10(+(n335_10, 1))) ->_R^Omega(1)
c8(LEQ#2(gen_0':S13_10(n335_10), gen_0':S13_10(n335_10))) ->_IH
c8(gen_c6:c7:c814_10(c336_10))

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

(54)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8


Lemmas:
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) -> True, rt in Omega(0)


Generator Equations:
gen_c:c111_10(0) <=> c
gen_c:c111_10(+(x, 1)) <=> c1(c4, gen_c:c111_10(x))
gen_Nil:Cons12_10(0) <=> Nil
gen_Nil:Cons12_10(+(x, 1)) <=> Cons(0', gen_Nil:Cons12_10(x))
gen_0':S13_10(0) <=> 0'
gen_0':S13_10(+(x, 1)) <=> S(gen_0':S13_10(x))
gen_c6:c7:c814_10(0) <=> c6
gen_c6:c7:c814_10(+(x, 1)) <=> c8(gen_c6:c7:c814_10(x))


The following defined symbols remain to be analysed:
LEQ#2, FOLD#3, INSERT_ORD#2, fold#3, insert_ord#2

They will be analysed ascendingly in the following order:
INSERT_ORD#2 < FOLD#3
fold#3 < FOLD#3
LEQ#2 < INSERT_ORD#2
insert_ord#2 < fold#3

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

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
Rules:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8


Lemmas:
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) -> True, rt in Omega(0)
LEQ#2(gen_0':S13_10(n335_10), gen_0':S13_10(n335_10)) -> gen_c6:c7:c814_10(n335_10), rt in Omega(1 + n335_10)


Generator Equations:
gen_c:c111_10(0) <=> c
gen_c:c111_10(+(x, 1)) <=> c1(c4, gen_c:c111_10(x))
gen_Nil:Cons12_10(0) <=> Nil
gen_Nil:Cons12_10(+(x, 1)) <=> Cons(0', gen_Nil:Cons12_10(x))
gen_0':S13_10(0) <=> 0'
gen_0':S13_10(+(x, 1)) <=> S(gen_0':S13_10(x))
gen_c6:c7:c814_10(0) <=> c6
gen_c6:c7:c814_10(+(x, 1)) <=> c8(gen_c6:c7:c814_10(x))


The following defined symbols remain to be analysed:
INSERT_ORD#2, FOLD#3, fold#3, insert_ord#2

They will be analysed ascendingly in the following order:
INSERT_ORD#2 < FOLD#3
fold#3 < FOLD#3
insert_ord#2 < fold#3

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fold#3(insert_ord(leq), gen_Nil:Cons12_10(n1768_10)) -> *15_10, rt in Omega(0)

Induction Base:
fold#3(insert_ord(leq), gen_Nil:Cons12_10(0))

Induction Step:
fold#3(insert_ord(leq), gen_Nil:Cons12_10(+(n1768_10, 1))) ->_R^Omega(0)
insert_ord#2(leq, 0', fold#3(insert_ord(leq), gen_Nil:Cons12_10(n1768_10))) ->_IH
insert_ord#2(leq, 0', *15_10)

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

(60)
Obligation:
Innermost TRS:
Rules:
FOLD#3(insert_ord(z0), Nil) -> c
FOLD#3(insert_ord(z0), Cons(z1, z2)) -> c1(INSERT_ORD#2(z0, z1, fold#3(insert_ord(z0), z2)), FOLD#3(insert_ord(z0), z2))
COND_INSERT_ORD_X_YS_1(True, z0, z1, z2) -> c2
COND_INSERT_ORD_X_YS_1(False, z0, z1, z2) -> c3(INSERT_ORD#2(leq, z0, z2))
INSERT_ORD#2(leq, z0, Nil) -> c4
INSERT_ORD#2(leq, z0, Cons(z1, z2)) -> c5(COND_INSERT_ORD_X_YS_1(leq#2(z0, z1), z0, z1, z2), LEQ#2(z0, z1))
LEQ#2(0', z0) -> c6
LEQ#2(S(z0), 0') -> c7
LEQ#2(S(z0), S(z1)) -> c8(LEQ#2(z0, z1))
MAIN(z0) -> c9(FOLD#3(insert_ord(leq), z0))
fold#3(insert_ord(z0), Nil) -> Nil
fold#3(insert_ord(z0), Cons(z1, z2)) -> insert_ord#2(z0, z1, fold#3(insert_ord(z0), z2))
cond_insert_ord_x_ys_1(True, z0, z1, z2) -> Cons(z0, Cons(z1, z2))
cond_insert_ord_x_ys_1(False, z0, z1, z2) -> Cons(z1, insert_ord#2(leq, z0, z2))
insert_ord#2(leq, z0, Nil) -> Cons(z0, Nil)
insert_ord#2(leq, z0, Cons(z1, z2)) -> cond_insert_ord_x_ys_1(leq#2(z0, z1), z0, z1, z2)
leq#2(0', z0) -> True
leq#2(S(z0), 0') -> False
leq#2(S(z0), S(z1)) -> leq#2(z0, z1)
main(z0) -> fold#3(insert_ord(leq), z0)

Types:
FOLD#3 :: insert_ord -> Nil:Cons -> c:c1
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
c :: c:c1
Cons :: 0':S -> Nil:Cons -> Nil:Cons
c1 :: c4:c5 -> c:c1 -> c:c1
INSERT_ORD#2 :: leq -> 0':S -> Nil:Cons -> c4:c5
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
COND_INSERT_ORD_X_YS_1 :: True:False -> 0':S -> 0':S -> Nil:Cons -> c2:c3
True :: True:False
c2 :: c2:c3
False :: True:False
c3 :: c4:c5 -> c2:c3
leq :: leq
c4 :: c4:c5
c5 :: c2:c3 -> c6:c7:c8 -> c4:c5
leq#2 :: 0':S -> 0':S -> True:False
LEQ#2 :: 0':S -> 0':S -> c6:c7:c8
0' :: 0':S
c6 :: c6:c7:c8
S :: 0':S -> 0':S
c7 :: c6:c7:c8
c8 :: c6:c7:c8 -> c6:c7:c8
MAIN :: Nil:Cons -> c9
c9 :: c:c1 -> c9
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
main :: Nil:Cons -> Nil:Cons
hole_c:c11_10 :: c:c1
hole_insert_ord2_10 :: insert_ord
hole_Nil:Cons3_10 :: Nil:Cons
hole_leq4_10 :: leq
hole_0':S5_10 :: 0':S
hole_c4:c56_10 :: c4:c5
hole_c2:c37_10 :: c2:c3
hole_True:False8_10 :: True:False
hole_c6:c7:c89_10 :: c6:c7:c8
hole_c910_10 :: c9
gen_c:c111_10 :: Nat -> c:c1
gen_Nil:Cons12_10 :: Nat -> Nil:Cons
gen_0':S13_10 :: Nat -> 0':S
gen_c6:c7:c814_10 :: Nat -> c6:c7:c8


Lemmas:
leq#2(gen_0':S13_10(n16_10), gen_0':S13_10(n16_10)) -> True, rt in Omega(0)
LEQ#2(gen_0':S13_10(n335_10), gen_0':S13_10(n335_10)) -> gen_c6:c7:c814_10(n335_10), rt in Omega(1 + n335_10)
fold#3(insert_ord(leq), gen_Nil:Cons12_10(n1768_10)) -> *15_10, rt in Omega(0)


Generator Equations:
gen_c:c111_10(0) <=> c
gen_c:c111_10(+(x, 1)) <=> c1(c4, gen_c:c111_10(x))
gen_Nil:Cons12_10(0) <=> Nil
gen_Nil:Cons12_10(+(x, 1)) <=> Cons(0', gen_Nil:Cons12_10(x))
gen_0':S13_10(0) <=> 0'
gen_0':S13_10(+(x, 1)) <=> S(gen_0':S13_10(x))
gen_c6:c7:c814_10(0) <=> c6
gen_c6:c7:c814_10(+(x, 1)) <=> c8(gen_c6:c7:c814_10(x))


The following defined symbols remain to be analysed:
FOLD#3