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

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(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)

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed 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) transformation)

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) transformation)

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

(c) The following functions are completely defined:

sort#2
leq#2
insert#3
cond_insert_ord_x_ys_1

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) transformation)

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) transformation)

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) transformation)

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) transformation)

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) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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}
Previous analysis results are:

leq#2: runtime: ?, size: O(1) [1]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: leq#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(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: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
Previous analysis results are:

leq#2: runtime: O(n1) [1 + z'], size: O(1) [1]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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') -{ 3 + 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') -{ 1 + 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(n1) [1 + z'], size: O(1) [1]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: cond_insert_ord_x_ys_1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) 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(n¹) with polynomial bound: 1 + z + z'

(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') -{ 3 + 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') -{ 1 + 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: ?, size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: ?, size: O(n1) [1 + z + z']

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: cond_insert_ord_x_ys_1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + 10·z1

Computed RUNTIME bound using CoFloCo for: insert#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 13 + 10·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') -{ 3 + 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') -{ 1 + 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z']

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z']

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: sort#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z'] sort#2: runtime: ?, size: O(n1) [z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: sort#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 17 + 22·z + 20·z²

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z'] sort#2: runtime: O(n2) [17 + 22·z + 20·z2], size: O(n1) [z]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 18 + 22·z + 20·z² }→ s8 :|: s8 >= 0, s8 <= 1 * z, z >= 0
sort#2(z) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * 0 + 1, z - 1 >= 0
sort#2(z) -{ 45 + 10·s5 + 10·s6 + 22·x2' + 20·x2'² }→ s7 :|: s5 >= 0, s5 <= 1 * x2', s6 >= 0, s6 <= 1 * x4' + 1 * s5 + 1, s7 >= 0, s7 <= 1 * x4 + 1 * 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z'] sort#2: runtime: O(n2) [17 + 22·z + 20·z2], size: O(n1) [z]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 18 + 22·z + 20·z² }→ s8 :|: s8 >= 0, s8 <= 1 * z, z >= 0
sort#2(z) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * 0 + 1, z - 1 >= 0
sort#2(z) -{ 45 + 10·s5 + 10·s6 + 22·x2' + 20·x2'² }→ s7 :|: s5 >= 0, s5 <= 1 * x2', s6 >= 0, s6 <= 1 * x4' + 1 * s5 + 1, s7 >= 0, s7 <= 1 * x4 + 1 * 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z'] sort#2: runtime: O(n2) [17 + 22·z + 20·z2], size: O(n1) [z] main: runtime: ?, size: O(n1) [z]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 18 + 22·z + 20·z²

(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) -{ 14 + 10·z1 }→ 1 + z'' + s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
insert#3(z, z') -{ 6 + 10·x2 }→ s2 :|: s2 >= 0, s2 <= 1 * 0 + 1 * x4 + 1 * x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
insert#3(z, z') -{ -4 + 10·z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)) + 1 * 0 + 1 * (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
insert#3(z, z') -{ 7 + 10·x2 + x2'' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * (1 + x2'') + 1 * 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') -{ 1 + 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) -{ 18 + 22·z + 20·z² }→ s8 :|: s8 >= 0, s8 <= 1 * z, z >= 0
sort#2(z) -{ 15 }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * 0 + 1, z - 1 >= 0
sort#2(z) -{ 45 + 10·s5 + 10·s6 + 22·x2' + 20·x2'² }→ s7 :|: s5 >= 0, s5 <= 1 * x2', s6 >= 0, s6 <= 1 * x4' + 1 * s5 + 1, s7 >= 0, s7 <= 1 * x4 + 1 * 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(n1) [1 + z'], size: O(1) [1] cond_insert_ord_x_ys_1: runtime: O(n1) [4 + 10·z1], size: O(n1) [2 + z' + z'' + z1] insert#3: runtime: O(n1) [13 + 10·z'], size: O(n1) [1 + z + z'] sort#2: runtime: O(n2) [17 + 22·z + 20·z2], size: O(n1) [z] main: runtime: O(n2) [18 + 22·z + 20·z2], size: O(n1) [z]

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(38) BOUNDS(1, n^2)