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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms]
    (10) CpxRNTS
    (11) InliningProof [UPPER BOUND(ID), 91 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), 228 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 181 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1457 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 711 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 37 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) FinalProof [FINISHED, 0 ms]
    (54) BOUNDS(1, n^2)
(55) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(56) CdtProblem
    (57) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxRelTRS
    (59) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxRelTRS
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) typed CpxTrs
    (63) OrderProof [LOWER BOUND(ID), 9 ms]
    (64) typed CpxTrs
    (65) RewriteLemmaProof [LOWER BOUND(ID), 354 ms]
    (66) BEST
        (67) proven lower bound
            (68) LowerBoundPropagationProof [FINISHED, 0 ms]
            (69) BOUNDS(n^1, INF)
        (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (72) typed CpxTrs
            (73) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
            (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 66 ms]
            (76) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   inc(0) -> 0
   inc(s(x)) -> s(inc(x))
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(x)) -> x
   bits(x) -> bitIter(x, 0)
   bitIter(x, y) -> if(zero(x), x, inc(y))
   if(true, x, y) -> p(y)
   if(false, x, y) -> bitIter(half(x), 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^2).


The TRS R consists of the following rules:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), 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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
inc :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
p :: 0:s -> 0:s
bits :: 0:s -> 0:s
bitIter :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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:

   p_1
   bits_1
   bitIter_2
   if_3

(c) The following functions are completely defined:

   zero_1
   inc_1
   half_1

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
inc :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
p :: 0:s -> 0:s
bits :: 0:s -> 0:s
bitIter :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(0, 0) -> if(true, 0, 0) [3]
   bitIter(0, s(x'')) -> if(true, 0, s(inc(x''))) [3]
   bitIter(s(x'), 0) -> if(false, s(x'), 0) [3]
   bitIter(s(x'), s(x1)) -> if(false, s(x'), s(inc(x1))) [3]
   if(true, x, y) -> p(y) [1]
   if(false, 0, y) -> bitIter(0, y) [2]
   if(false, s(0), y) -> bitIter(0, y) [2]
   if(false, s(s(x2)), y) -> bitIter(s(half(x2)), y) [2]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
inc :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
p :: 0:s -> 0:s
bits :: 0:s -> 0:s
bitIter :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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
   true => 1
   false => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(x'')) :|: z' = 1 + x'', x'' >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + x', 1 + inc(x1)) :|: z = 1 + x', x1 >= 0, x' >= 0, z' = 1 + x1
   bits(z) -{ 1 }-> bitIter(x, 0) :|: x >= 0, z = x
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   if(z, z', z'') -{ 1 }-> p(y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, y) :|: z'' = y, y >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(x2), y) :|: z'' = y, z' = 1 + (1 + x2), y >= 0, z = 0, x2 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(x) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x


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

(11) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(x'')) :|: z' = 1 + x'', x'' >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + x', 1 + inc(x1)) :|: z = 1 + x', x1 >= 0, x' >= 0, z' = 1 + x1
   bits(z) -{ 1 }-> bitIter(x, 0) :|: x >= 0, z = x
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   if(z, z', z'') -{ 2 }-> x' :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0, x' >= 0, y = 1 + x'
   if(z, z', z'') -{ 2 }-> bitIter(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, y) :|: z'' = y, y >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(x2), y) :|: z'' = y, z' = 1 + (1 + x2), y >= 0, z = 0, x2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0, y = 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(x) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x


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

(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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0


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

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

   { half }
   { inc }
   { zero }
   { p }
   { if, bitIter }
   { bits }

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {half}, {inc}, {zero}, {p}, {if,bitIter}, {bits}

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

(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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {half}, {inc}, {zero}, {p}, {if,bitIter}, {bits}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: half
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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {half}, {inc}, {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: ?, size: O(n^1) [z]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> bitIter(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {inc}, {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], 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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {inc}, {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {inc}, {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: ?, size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: inc
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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: ?, size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]

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

(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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: ?, size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(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:

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if,bitIter}, {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: ?, size: O(n^1) [z'']
  bitIter: runtime: ?, size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 12 + 7*z' + z'*z'' + z'^2 + 2*z''

Computed RUNTIME bound using KoAT for: bitIter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 60 + 14*z + z*z' + 2*z^2 + 6*z'

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

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

   bitIter(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(1, 0, 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 3 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 3 + z' }-> if(0, 1 + (z - 1), 1 + s2) :|: s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 1 }-> bitIter(z, 0) :|: z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> bitIter(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> bitIter(1 + s', z'') :|: s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: O(n^2) [12 + 7*z' + z'*z'' + z'^2 + 2*z''], size: O(n^1) [z'']
  bitIter: runtime: O(n^2) [60 + 14*z + z*z' + 2*z^2 + 6*z'], size: O(n^1) [z']

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   bitIter(z, z') -{ 15 }-> s4 :|: s4 >= 0, s4 <= 0, z = 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s1 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + s1, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 15 + 7*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s2 + s2*z + 8*z + z^2 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + s2, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 61 + 14*z + 2*z^2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 78 + 18*s' + s'*z'' + 2*s'^2 + z' + 7*z'' }-> s10 :|: s10 >= 0, s10 <= z'', s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s8 :|: s8 >= 0, s8 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s9 :|: s9 >= 0, s9 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: O(n^2) [12 + 7*z' + z'*z'' + z'^2 + 2*z''], size: O(n^1) [z'']
  bitIter: runtime: O(n^2) [60 + 14*z + z*z' + 2*z^2 + 6*z'], size: O(n^1) [z']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: bits
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   bitIter(z, z') -{ 15 }-> s4 :|: s4 >= 0, s4 <= 0, z = 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s1 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + s1, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 15 + 7*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s2 + s2*z + 8*z + z^2 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + s2, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 61 + 14*z + 2*z^2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 78 + 18*s' + s'*z'' + 2*s'^2 + z' + 7*z'' }-> s10 :|: s10 >= 0, s10 <= z'', s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s8 :|: s8 >= 0, s8 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s9 :|: s9 >= 0, s9 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {bits}
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: O(n^2) [12 + 7*z' + z'*z'' + z'^2 + 2*z''], size: O(n^1) [z'']
  bitIter: runtime: O(n^2) [60 + 14*z + z*z' + 2*z^2 + 6*z'], size: O(n^1) [z']
  bits: runtime: ?, size: O(1) [0]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: bits
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 61 + 14*z + 2*z^2

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   bitIter(z, z') -{ 15 }-> s4 :|: s4 >= 0, s4 <= 0, z = 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s1 + z' }-> s5 :|: s5 >= 0, s5 <= 1 + s1, s1 >= 0, s1 <= z' - 1, z' - 1 >= 0, z = 0
   bitIter(z, z') -{ 15 + 7*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' = 0
   bitIter(z, z') -{ 17 + 2*s2 + s2*z + 8*z + z^2 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + s2, s2 >= 0, s2 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   bits(z) -{ 61 + 14*z + 2*z^2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 78 + 18*s' + s'*z'' + 2*s'^2 + z' + 7*z'' }-> s10 :|: s10 >= 0, s10 <= z'', s' >= 0, s' <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s8 :|: s8 >= 0, s8 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 62 + 6*z'' }-> s9 :|: s9 >= 0, s9 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> 0 :|: z = 1, z' >= 0, z'' >= 0, z'' = 0
   if(z, z', z'') -{ 2 }-> z'' - 1 :|: z = 1, z' >= 0, z'' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  zero: runtime: O(1) [1], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: O(n^2) [12 + 7*z' + z'*z'' + z'^2 + 2*z''], size: O(n^1) [z'']
  bitIter: runtime: O(n^2) [60 + 14*z + z*z' + 2*z^2 + 6*z'], size: O(n^1) [z']
  bits: runtime: O(n^2) [61 + 14*z + 2*z^2], size: O(1) [0]

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

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

(54)
BOUNDS(1, n^2)

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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(z0)) -> z0
   bits(z0) -> bitIter(z0, 0)
   bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
   if(true, z0, z1) -> p(z1)
   if(false, z0, z1) -> bitIter(half(z0), z1)
Tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(z0)) -> c8
   BITS(z0) -> c9(BITITER(z0, 0))
   BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
   BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
   IF(true, z0, z1) -> c12(P(z1))
   IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
S tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(z0)) -> c8
   BITS(z0) -> c9(BITITER(z0, 0))
   BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
   BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
   IF(true, z0, z1) -> c12(P(z1))
   IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
K tuples:none
Defined Rule Symbols:   half_1, inc_1, zero_1, p_1, bits_1, bitIter_2, if_3

Defined Pair Symbols:   HALF_1, INC_1, ZERO_1, P_1, BITS_1, BITITER_2, IF_3

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6, c7, c8, c9_1, c10_2, c11_2, c12_1, c13_2


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

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

(58)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   INC(0) -> c3
   INC(s(z0)) -> c4(INC(z0))
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(z0)) -> c8
   BITS(z0) -> c9(BITITER(z0, 0))
   BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
   BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
   IF(true, z0, z1) -> c12(P(z1))
   IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))

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

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(z0)) -> z0
   bits(z0) -> bitIter(z0, 0)
   bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
   if(true, z0, z1) -> p(z1)
   if(false, z0, z1) -> bitIter(half(z0), z1)

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

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

(60)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   HALF(0') -> c
   HALF(s(0')) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   INC(0') -> c3
   INC(s(z0)) -> c4(INC(z0))
   ZERO(0') -> c5
   ZERO(s(z0)) -> c6
   P(0') -> c7
   P(s(z0)) -> c8
   BITS(z0) -> c9(BITITER(z0, 0'))
   BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
   BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
   IF(true, z0, z1) -> c12(P(z1))
   IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))

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

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(z0))) -> s(half(z0))
   inc(0') -> 0'
   inc(s(z0)) -> s(inc(z0))
   zero(0') -> true
   zero(s(z0)) -> false
   p(0') -> 0'
   p(s(z0)) -> z0
   bits(z0) -> bitIter(z0, 0')
   bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
   if(true, z0, z1) -> p(z1)
   if(false, z0, z1) -> bitIter(half(z0), z1)

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

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

(62)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4

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

(63) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
HALF, INC, BITITER, inc, half, bitIter

They will be analysed ascendingly in the following order:
HALF < BITITER
INC < BITITER
inc < BITITER
half < BITITER
inc < bitIter
half < bitIter

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

(64)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
HALF, INC, BITITER, inc, half, bitIter

They will be analysed ascendingly in the following order:
HALF < BITITER
INC < BITITER
inc < BITITER
half < BITITER
inc < bitIter
half < bitIter

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14)

Induction Base:
HALF(gen_0':s11_14(*(2, 0))) ->_R^Omega(1)
c

Induction Step:
HALF(gen_0':s11_14(*(2, +(n14_14, 1)))) ->_R^Omega(1)
c2(HALF(gen_0':s11_14(*(2, n14_14)))) ->_IH
c2(gen_c:c1:c210_14(c15_14))

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

(66)
Complex Obligation (BEST)

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

(67)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
HALF, INC, BITITER, inc, half, bitIter

They will be analysed ascendingly in the following order:
HALF < BITITER
INC < BITITER
inc < BITITER
half < BITITER
inc < bitIter
half < bitIter

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

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

(69)
BOUNDS(n^1, INF)

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

(70)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14)


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
INC, BITITER, inc, half, bitIter

They will be analysed ascendingly in the following order:
INC < BITITER
inc < BITITER
half < BITITER
inc < bitIter
half < bitIter

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14)

Induction Base:
INC(gen_0':s11_14(0)) ->_R^Omega(1)
c3

Induction Step:
INC(gen_0':s11_14(+(n471_14, 1))) ->_R^Omega(1)
c4(INC(gen_0':s11_14(n471_14))) ->_IH
c4(gen_c3:c412_14(c472_14))

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

(72)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14)
INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14)


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
inc, BITITER, half, bitIter

They will be analysed ascendingly in the following order:
inc < BITITER
half < BITITER
inc < bitIter
half < bitIter

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0)

Induction Base:
inc(gen_0':s11_14(0)) ->_R^Omega(0)
0'

Induction Step:
inc(gen_0':s11_14(+(n848_14, 1))) ->_R^Omega(0)
s(inc(gen_0':s11_14(n848_14))) ->_IH
s(gen_0':s11_14(c849_14))

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

(74)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14)
INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14)
inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
half, BITITER, bitIter

They will be analysed ascendingly in the following order:
half < BITITER
half < bitIter

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s11_14(*(2, n1204_14))) -> gen_0':s11_14(n1204_14), rt in Omega(0)

Induction Base:
half(gen_0':s11_14(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
half(gen_0':s11_14(*(2, +(n1204_14, 1)))) ->_R^Omega(0)
s(half(gen_0':s11_14(*(2, n1204_14)))) ->_IH
s(gen_0':s11_14(c1205_14))

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

(76)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
INC(0') -> c3
INC(s(z0)) -> c4(INC(z0))
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(z0)) -> c8
BITS(z0) -> c9(BITITER(z0, 0'))
BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0))
BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1))
IF(true, z0, z1) -> c12(P(z1))
IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(z0)) -> z0
bits(z0) -> bitIter(z0, 0')
bitIter(z0, z1) -> if(zero(z0), z0, inc(z1))
if(true, z0, z1) -> p(z1)
if(false, z0, z1) -> bitIter(half(z0), z1)

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
INC :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7:c8
c7 :: c7:c8
c8 :: c7:c8
BITS :: 0':s -> c9
c9 :: c10:c11 -> c9
BITITER :: 0':s -> 0':s -> c10:c11
c10 :: c12:c13 -> c5:c6 -> c10:c11
IF :: true:false -> 0':s -> 0':s -> c12:c13
zero :: 0':s -> true:false
inc :: 0':s -> 0':s
c11 :: c12:c13 -> c3:c4 -> c10:c11
true :: true:false
c12 :: c7:c8 -> c12:c13
false :: true:false
c13 :: c10:c11 -> c:c1:c2 -> c12:c13
half :: 0':s -> 0':s
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c43_14 :: c3:c4
hole_c5:c64_14 :: c5:c6
hole_c7:c85_14 :: c7:c8
hole_c96_14 :: c9
hole_c10:c117_14 :: c10:c11
hole_c12:c138_14 :: c12:c13
hole_true:false9_14 :: true:false
gen_c:c1:c210_14 :: Nat -> c:c1:c2
gen_0':s11_14 :: Nat -> 0':s
gen_c3:c412_14 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14)
INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14)
inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0)
half(gen_0':s11_14(*(2, n1204_14))) -> gen_0':s11_14(n1204_14), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_14(0) <=> c
gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x))
gen_0':s11_14(0) <=> 0'
gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x))
gen_c3:c412_14(0) <=> c3
gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x))


The following defined symbols remain to be analysed:
BITITER, bitIter