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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 47 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 273 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 82 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 982 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 271 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^2)
(37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CpxTRS
    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) typed CpxTrs
    (41) OrderProof [LOWER BOUND(ID), 0 ms]
    (42) typed CpxTrs
    (43) RewriteLemmaProof [LOWER BOUND(ID), 294 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) typed CpxTrs
            (49) RewriteLemmaProof [LOWER BOUND(ID), 89 ms]
            (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 47 ms]
            (52) proven lower bound
            (53) LowerBoundPropagationProof [FINISHED, 0 ms]
            (54) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   sqr(0) -> 0
   sqr(s(x)) -> +(sqr(x), s(double(x)))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   sqr(s(x)) -> s(+(sqr(x), double(x)))

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

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

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


The TRS R consists of the following rules:

   sqr(0) -> 0 [1]
   sqr(s(x)) -> +(sqr(x), s(double(x))) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   sqr(s(x)) -> s(+(sqr(x), double(x))) [1]

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

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

+ => plus

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

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


The TRS R consists of the following rules:

   sqr(0) -> 0 [1]
   sqr(s(x)) -> plus(sqr(x), s(double(x))) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   sqr(s(x)) -> s(plus(sqr(x), double(x))) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   sqr(0) -> 0 [1]
   sqr(s(x)) -> plus(sqr(x), s(double(x))) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   sqr(s(x)) -> s(plus(sqr(x), double(x))) [1]

The TRS has the following type information:
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
double :: 0:s -> 0:s

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

(7) 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:
none

(c) The following functions are completely defined:

   sqr_1
   double_1
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   sqr(0) -> 0 [1]
   sqr(s(x)) -> plus(sqr(x), s(double(x))) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   sqr(s(x)) -> s(plus(sqr(x), double(x))) [1]

The TRS has the following type information:
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
double :: 0:s -> 0:s

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

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

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

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

   sqr(0) -> 0 [1]
   sqr(s(0)) -> plus(0, s(0)) [3]
   sqr(s(s(x'))) -> plus(plus(sqr(x'), s(double(x'))), s(s(s(double(x'))))) [3]
   sqr(s(s(x''))) -> plus(s(plus(sqr(x''), double(x''))), s(s(s(double(x''))))) [3]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   sqr(s(0)) -> s(plus(0, 0)) [3]
   sqr(s(s(x1))) -> s(plus(plus(sqr(x1), s(double(x1))), s(s(double(x1))))) [3]
   sqr(s(s(x2))) -> s(plus(s(plus(sqr(x2), double(x2))), s(s(double(x2))))) [3]

The TRS has the following type information:
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
double :: 0:s -> 0:s

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

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

   0 => 0

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   sqr(z) -{ 3 }-> plus(plus(sqr(x'), 1 + double(x')), 1 + (1 + (1 + double(x')))) :|: x' >= 0, z = 1 + (1 + x')
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(x''), double(x'')), 1 + (1 + (1 + double(x'')))) :|: x'' >= 0, z = 1 + (1 + x'')
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(x1), 1 + double(x1)), 1 + (1 + double(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(x2), double(x2)), 1 + (1 + double(x2))) :|: z = 1 + (1 + x2), x2 >= 0


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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 3 }-> plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0


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

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

   { double }
   { plus }
   { sqr }

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 3 }-> plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {double}, {plus}, {sqr}

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 3 }-> plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {double}, {plus}, {sqr}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: double
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:

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 3 }-> plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {double}, {plus}, {sqr}
Previous analysis results are:
  double: runtime: ?, size: O(n^1) [2*z]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 3 }-> plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  plus: runtime: ?, size: O(n^1) [z + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sqr(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0 + (1 + 0), z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 8*z + 4*z^2

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sqr(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0 + (1 + 0), z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {sqr}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  sqr: runtime: ?, size: O(n^2) [5 + 8*z + 4*z^2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 10 + 24*z^2

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sqr(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0 + (1 + 0), z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  sqr: runtime: O(n^2) [10 + 24*z^2], size: O(n^2) [5 + 8*z + 4*z^2]

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

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

(36)
BOUNDS(1, n^2)

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

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

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   sqr(0') -> 0'
   sqr(s(x)) -> +'(sqr(x), s(double(x)))
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   sqr(s(x)) -> s(+'(sqr(x), double(x)))

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

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

(40)
Obligation:
Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(41) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sqr, +', double

They will be analysed ascendingly in the following order:
+' < sqr
double < sqr

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

(42)
Obligation:
Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
+', sqr, double

They will be analysed ascendingly in the following order:
+' < sqr
double < sqr

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_0(+(a, c5_0)))

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

(44)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
+', sqr, double

They will be analysed ascendingly in the following order:
+' < sqr
double < sqr

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

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

(47)
BOUNDS(n^1, INF)

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

(48)
Obligation:
Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
double, sqr

They will be analysed ascendingly in the following order:
double < sqr

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_0)

Induction Base:
double(gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s2_0(+(n513_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s2_0(n513_0)))) ->_IH
s(s(gen_0':s2_0(*(2, c514_0))))

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

(50)
Obligation:
Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
sqr
----------------------------------------

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s2_0(n777_0)) -> gen_0':s2_0(*(n777_0, n777_0)), rt in Omega(1 + n777_0 + n777_0^2)

Induction Base:
sqr(gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
sqr(gen_0':s2_0(+(n777_0, 1))) ->_R^Omega(1)
+'(sqr(gen_0':s2_0(n777_0)), s(double(gen_0':s2_0(n777_0)))) ->_IH
+'(gen_0':s2_0(*(c778_0, c778_0)), s(double(gen_0':s2_0(n777_0)))) ->_L^Omega(1 + n777_0)
+'(gen_0':s2_0(*(n777_0, n777_0)), s(gen_0':s2_0(*(2, n777_0)))) ->_L^Omega(2 + 2*n777_0)
gen_0':s2_0(+(+(*(2, n777_0), 1), *(n777_0, n777_0)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(52)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
sqr(0') -> 0'
sqr(s(x)) -> +'(sqr(x), s(double(x)))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
sqr(s(x)) -> s(+'(sqr(x), double(x)))

Types:
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
double(gen_0':s2_0(n513_0)) -> gen_0':s2_0(*(2, n513_0)), rt in Omega(1 + n513_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
sqr
----------------------------------------

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

(54)
BOUNDS(n^2, INF)