WORST_CASE(Omega(n^3),O(n^3))
proof of input_5EZ5faZO0h.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 182 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 284 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 447 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 399 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^3)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 19 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 310 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 133 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 542 ms]
            (72) proven lower bound
            (73) LowerBoundPropagationProof [FINISHED, 0 ms]
            (74) BOUNDS(n^3, INF)


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

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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

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

(4)
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
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   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

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

(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:

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

(9) 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

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

(10)
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


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

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

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

   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


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

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

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

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

(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

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

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

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

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

   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) 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'

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

(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}
Previous analysis results are:
  first: runtime: ?, size: O(n^1) [z']

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

(19) 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

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

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

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

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

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

   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) 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

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

(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']
  dbl: runtime: ?, size: O(n^1) [2*z]

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

(25) 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

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

(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: {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]

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

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

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

   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]

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

(29) 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'

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

(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]
  add: runtime: ?, size: O(n^1) [z + z']

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

(31) 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

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

(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: {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']

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

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

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

   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']

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

(35) 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

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

(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']
  sqr: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2]

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

(37) 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

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

(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: {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]

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

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

(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) -{ -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]

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

(41) 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

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

(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]
  terms: runtime: ?, size: O(n^2) [3 + 4*z + 4*z^2]

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

(43) 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

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

(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: 
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]

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

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

(46)
BOUNDS(1, n^3)

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

(47) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   terms(z0) -> cons(recip(sqr(z0)))
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1)) -> cons(z1)
Tuples:
   TERMS(z0) -> c(SQR(z0))
   SQR(0) -> c1
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c4
   DBL(s(z0)) -> c5(DBL(z0))
   ADD(0, z0) -> c6
   ADD(s(z0), z1) -> c7(ADD(z0, z1))
   FIRST(0, z0) -> c8
   FIRST(s(z0), cons(z1)) -> c9
S tuples:
   TERMS(z0) -> c(SQR(z0))
   SQR(0) -> c1
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c4
   DBL(s(z0)) -> c5(DBL(z0))
   ADD(0, z0) -> c6
   ADD(s(z0), z1) -> c7(ADD(z0, z1))
   FIRST(0, z0) -> c8
   FIRST(s(z0), cons(z1)) -> c9
K tuples:none
Defined Rule Symbols:   terms_1, sqr_1, dbl_1, add_2, first_2

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1, c2_2, c3_2, c4, c5_1, c6, c7_1, c8, c9


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

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

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


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0))
   SQR(0) -> c1
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c4
   DBL(s(z0)) -> c5(DBL(z0))
   ADD(0, z0) -> c6
   ADD(s(z0), z1) -> c7(ADD(z0, z1))
   FIRST(0, z0) -> c8
   FIRST(s(z0), cons(z1)) -> c9

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

   terms(z0) -> cons(recip(sqr(z0)))
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1)) -> cons(z1)

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

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0))
   SQR(0') -> c1
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0') -> c4
   DBL(s(z0)) -> c5(DBL(z0))
   ADD(0', z0) -> c6
   ADD(s(z0), z1) -> c7(ADD(z0, z1))
   FIRST(0', z0) -> c8
   FIRST(s(z0), cons(z1)) -> c9

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

   terms(z0) -> cons(recip(sqr(z0)))
   sqr(0') -> 0'
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0') -> 0'
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0', z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0', z0) -> nil
   first(s(z0), cons(z1)) -> cons(z1)

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

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

(54)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SQR, ADD, sqr, dbl, DBL, add

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

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

(56)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)

Induction Base:
ADD(gen_0':s9_10(0), gen_0':s9_10(b)) ->_R^Omega(1)
c6

Induction Step:
ADD(gen_0':s9_10(+(n14_10, 1)), gen_0':s9_10(b)) ->_R^Omega(1)
c7(ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b))) ->_IH
c7(gen_c6:c711_10(c15_10))

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

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)

Induction Base:
dbl(gen_0':s9_10(0)) ->_R^Omega(0)
0'

Induction Step:
dbl(gen_0':s9_10(+(n602_10, 1))) ->_R^Omega(0)
s(s(dbl(gen_0':s9_10(n602_10)))) ->_IH
s(s(gen_0':s9_10(*(2, c603_10))))

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

(64)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DBL(gen_0':s9_10(n934_10)) -> gen_c4:c512_10(n934_10), rt in Omega(1 + n934_10)

Induction Base:
DBL(gen_0':s9_10(0)) ->_R^Omega(1)
c4

Induction Step:
DBL(gen_0':s9_10(+(n934_10, 1))) ->_R^Omega(1)
c5(DBL(gen_0':s9_10(n934_10))) ->_IH
c5(gen_c4:c512_10(c935_10))

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

(66)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)
DBL(gen_0':s9_10(n934_10)) -> gen_c4:c512_10(n934_10), rt in Omega(1 + n934_10)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s9_10(n1310_10), gen_0':s9_10(b)) -> gen_0':s9_10(+(n1310_10, b)), rt in Omega(0)

Induction Base:
add(gen_0':s9_10(0), gen_0':s9_10(b)) ->_R^Omega(0)
gen_0':s9_10(b)

Induction Step:
add(gen_0':s9_10(+(n1310_10, 1)), gen_0':s9_10(b)) ->_R^Omega(0)
s(add(gen_0':s9_10(n1310_10), gen_0':s9_10(b))) ->_IH
s(gen_0':s9_10(+(b, c1311_10)))

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

(68)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)
DBL(gen_0':s9_10(n934_10)) -> gen_c4:c512_10(n934_10), rt in Omega(1 + n934_10)
add(gen_0':s9_10(n1310_10), gen_0':s9_10(b)) -> gen_0':s9_10(+(n1310_10, b)), rt in Omega(0)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s9_10(n2387_10)) -> gen_0':s9_10(*(n2387_10, n2387_10)), rt in Omega(0)

Induction Base:
sqr(gen_0':s9_10(0)) ->_R^Omega(0)
0'

Induction Step:
sqr(gen_0':s9_10(+(n2387_10, 1))) ->_R^Omega(0)
s(add(sqr(gen_0':s9_10(n2387_10)), dbl(gen_0':s9_10(n2387_10)))) ->_IH
s(add(gen_0':s9_10(*(c2388_10, c2388_10)), dbl(gen_0':s9_10(n2387_10)))) ->_L^Omega(0)
s(add(gen_0':s9_10(*(n2387_10, n2387_10)), gen_0':s9_10(*(2, n2387_10)))) ->_L^Omega(0)
s(gen_0':s9_10(+(*(n2387_10, n2387_10), *(2, n2387_10))))

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

(70)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)
DBL(gen_0':s9_10(n934_10)) -> gen_c4:c512_10(n934_10), rt in Omega(1 + n934_10)
add(gen_0':s9_10(n1310_10), gen_0':s9_10(b)) -> gen_0':s9_10(+(n1310_10, b)), rt in Omega(0)
sqr(gen_0':s9_10(n2387_10)) -> gen_0':s9_10(*(n2387_10, n2387_10)), rt in Omega(0)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SQR(gen_0':s9_10(n2856_10)) -> *13_10, rt in Omega(n2856_10 + n2856_10^3)

Induction Base:
SQR(gen_0':s9_10(0))

Induction Step:
SQR(gen_0':s9_10(+(n2856_10, 1))) ->_R^Omega(1)
c2(ADD(sqr(gen_0':s9_10(n2856_10)), dbl(gen_0':s9_10(n2856_10))), SQR(gen_0':s9_10(n2856_10))) ->_L^Omega(0)
c2(ADD(gen_0':s9_10(*(n2856_10, n2856_10)), dbl(gen_0':s9_10(n2856_10))), SQR(gen_0':s9_10(n2856_10))) ->_L^Omega(0)
c2(ADD(gen_0':s9_10(*(n2856_10, n2856_10)), gen_0':s9_10(*(2, n2856_10))), SQR(gen_0':s9_10(n2856_10))) ->_L^Omega(1 + n2856_10^2)
c2(gen_c6:c711_10(*(n2856_10, n2856_10)), SQR(gen_0':s9_10(n2856_10))) ->_IH
c2(gen_c6:c711_10(*(n2856_10, n2856_10)), *13_10)

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(72)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
SQR(0') -> c1
SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c4
DBL(s(z0)) -> c5(DBL(z0))
ADD(0', z0) -> c6
ADD(s(z0), z1) -> c7(ADD(z0, z1))
FIRST(0', z0) -> c8
FIRST(s(z0), cons(z1)) -> c9
terms(z0) -> cons(recip(sqr(z0)))
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1)) -> cons(z1)

Types:
TERMS :: 0':s -> c
c :: c1:c2:c3 -> c
SQR :: 0':s -> c1:c2:c3
0' :: 0':s
c1 :: c1:c2:c3
s :: 0':s -> 0':s
c2 :: c6:c7 -> c1:c2:c3 -> c1:c2:c3
ADD :: 0':s -> 0':s -> c6:c7
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c3 :: c6:c7 -> c4:c5 -> c1:c2:c3
DBL :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
FIRST :: 0':s -> cons:nil -> c8:c9
c8 :: c8:c9
cons :: recip -> cons:nil
c9 :: c8:c9
terms :: 0':s -> cons:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
hole_c1_10 :: c
hole_0':s2_10 :: 0':s
hole_c1:c2:c33_10 :: c1:c2:c3
hole_c6:c74_10 :: c6:c7
hole_c4:c55_10 :: c4:c5
hole_c8:c96_10 :: c8:c9
hole_cons:nil7_10 :: cons:nil
hole_recip8_10 :: recip
gen_0':s9_10 :: Nat -> 0':s
gen_c1:c2:c310_10 :: Nat -> c1:c2:c3
gen_c6:c711_10 :: Nat -> c6:c7
gen_c4:c512_10 :: Nat -> c4:c5


Lemmas:
ADD(gen_0':s9_10(n14_10), gen_0':s9_10(b)) -> gen_c6:c711_10(n14_10), rt in Omega(1 + n14_10)
dbl(gen_0':s9_10(n602_10)) -> gen_0':s9_10(*(2, n602_10)), rt in Omega(0)
DBL(gen_0':s9_10(n934_10)) -> gen_c4:c512_10(n934_10), rt in Omega(1 + n934_10)
add(gen_0':s9_10(n1310_10), gen_0':s9_10(b)) -> gen_0':s9_10(+(n1310_10, b)), rt in Omega(0)
sqr(gen_0':s9_10(n2387_10)) -> gen_0':s9_10(*(n2387_10, n2387_10)), rt in Omega(0)


Generator Equations:
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c1:c2:c310_10(0) <=> c1
gen_c1:c2:c310_10(+(x, 1)) <=> c2(c6, gen_c1:c2:c310_10(x))
gen_c6:c711_10(0) <=> c6
gen_c6:c711_10(+(x, 1)) <=> c7(gen_c6:c711_10(x))
gen_c4:c512_10(0) <=> c4
gen_c4:c512_10(+(x, 1)) <=> c5(gen_c4:c512_10(x))


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

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

(74)
BOUNDS(n^3, INF)