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


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WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 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), 220 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 199 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 82 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 434 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 428 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) FinalProof [FINISHED, 0 ms]
    (48) BOUNDS(1, n^3)
(49) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CpxTRS
    (51) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) typed CpxTrs
    (53) OrderProof [LOWER BOUND(ID), 0 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 250 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 80 ms]
            (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
            (64) proven lower bound
            (65) LowerBoundPropagationProof [FINISHED, 0 ms]
            (66) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
sqr(s([]))


The defined contexts are:
add([], x1)
add(x0, [])


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y)) -> cons(Y) [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:

   terms(N) -> cons(recip(sqr(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y)) -> cons(Y) [1]

The TRS has the following type information:
terms :: 0:s -> cons:nil
cons :: recip -> cons:nil
recip :: 0:s -> recip
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
add :: 0:s -> 0:s -> 0:s
dbl :: 0:s -> 0:s
first :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil

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:

   terms_1
   first_2

(c) The following functions are completely defined:

   sqr_1
   dbl_1
   add_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   terms(N) -> cons(recip(sqr(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y)) -> cons(Y) [1]

The TRS has the following type information:
terms :: 0:s -> cons:nil
cons :: recip -> cons:nil
recip :: 0:s -> recip
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
add :: 0:s -> 0:s -> 0:s
dbl :: 0:s -> 0:s
first :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
const :: recip

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:

   terms(N) -> cons(recip(sqr(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(0)) -> s(add(0, 0)) [3]
   sqr(s(s(X'))) -> s(add(s(add(sqr(X'), dbl(X'))), s(s(dbl(X'))))) [3]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y)) -> cons(Y) [1]

The TRS has the following type information:
terms :: 0:s -> cons:nil
cons :: recip -> cons:nil
recip :: 0:s -> recip
sqr :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
add :: 0:s -> 0:s -> 0:s
dbl :: 0:s -> 0:s
first :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
const :: recip

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
   nil => 0
   const => 0

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

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

   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(X'), dbl(X')), 1 + (1 + dbl(X'))) :|: X' >= 0, z = 1 + (1 + X')
   terms(z) -{ 1 }-> 1 + (1 + sqr(N)) :|: z = N, N >= 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:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0


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

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

   { first }
   { dbl }
   { add }
   { sqr }
   { terms }

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}

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

(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:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: ?, size: O(n^1) [z']

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [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:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: dbl
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:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 4*z + 4*z^2

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 22*z + 8*z^3

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) :|: z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: terms
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3 + 4*z + 4*z^2

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: ?, size: O(n^2) [3 + 4*z + 4*z^2]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: terms
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 6 + 22*z + 8*z^3

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  first: runtime: O(1) [1], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: O(n^3) [6 + 22*z + 8*z^3], size: O(n^2) [3 + 4*z + 4*z^2]

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

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

(48)
BOUNDS(1, n^3)

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

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

(50)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0') -> 0'
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0') -> 0'
   dbl(s(X)) -> s(s(dbl(X)))
   add(0', X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0', X) -> nil
   first(s(X), cons(Y)) -> cons(Y)

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

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

(52)
Obligation:
TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s

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

(53) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sqr, add, dbl

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

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

(54)
Obligation:
TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
add, sqr, dbl

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

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n6_0, b)), rt in Omega(1 + n6_0)

Induction Base:
add(gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
gen_0':s4_0(b)

Induction Step:
add(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
s(add(gen_0':s4_0(n6_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c7_0)))

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

(56)
Complex Obligation (BEST)

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

(57)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
add, sqr, dbl

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

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

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

(59)
BOUNDS(n^1, INF)

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

(60)
Obligation:
TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbl(gen_0':s4_0(n495_0)) -> gen_0':s4_0(*(2, n495_0)), rt in Omega(1 + n495_0)

Induction Base:
dbl(gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
dbl(gen_0':s4_0(+(n495_0, 1))) ->_R^Omega(1)
s(s(dbl(gen_0':s4_0(n495_0)))) ->_IH
s(s(gen_0':s4_0(*(2, c496_0))))

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

(62)
Obligation:
TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n6_0, b)), rt in Omega(1 + n6_0)
dbl(gen_0':s4_0(n495_0)) -> gen_0':s4_0(*(2, n495_0)), rt in Omega(1 + n495_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s4_0(n739_0)) -> gen_0':s4_0(*(n739_0, n739_0)), rt in Omega(1 + n739_0 + n739_0^2 + n739_0^3)

Induction Base:
sqr(gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
sqr(gen_0':s4_0(+(n739_0, 1))) ->_R^Omega(1)
s(add(sqr(gen_0':s4_0(n739_0)), dbl(gen_0':s4_0(n739_0)))) ->_IH
s(add(gen_0':s4_0(*(c740_0, c740_0)), dbl(gen_0':s4_0(n739_0)))) ->_L^Omega(1 + n739_0)
s(add(gen_0':s4_0(*(n739_0, n739_0)), gen_0':s4_0(*(2, n739_0)))) ->_L^Omega(1 + n739_0^2)
s(gen_0':s4_0(+(*(n739_0, n739_0), *(2, n739_0))))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
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(64)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
terms(N) -> cons(recip(sqr(N)))
sqr(0') -> 0'
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0') -> 0'
dbl(s(X)) -> s(s(dbl(X)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0', X) -> nil
first(s(X), cons(Y)) -> cons(Y)

Types:
terms :: 0':s -> cons:nil
cons :: recip -> cons:nil
recip :: 0':s -> recip
sqr :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
add :: 0':s -> 0':s -> 0':s
dbl :: 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_recip3_0 :: recip
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
add(gen_0':s4_0(n6_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n6_0, b)), rt in Omega(1 + n6_0)
dbl(gen_0':s4_0(n495_0)) -> gen_0':s4_0(*(2, n495_0)), rt in Omega(1 + n495_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
sqr
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(65) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(66)
BOUNDS(n^3, INF)