WORST_CASE(Omega(n^1),O(n^2))
proof of input_Mn8gwdUE1a.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 3 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), 339 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1333 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 597 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(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), 9 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), 172 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 66 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 47 ms]
            (72) 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))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   inc(0) -> 0
   inc(s(x)) -> s(inc(x))
   log(x) -> log2(x, 0)
   log2(x, y) -> if(le(x, s(0)), x, inc(y))
   if(true, x, s(y)) -> y
   if(false, x, y) -> log2(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]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   log(x) -> log2(x, 0) [1]
   log2(x, y) -> if(le(x, s(0)), x, inc(y)) [1]
   if(true, x, s(y)) -> y [1]
   if(false, x, y) -> log2(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]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   log(x) -> log2(x, 0) [1]
   log2(x, y) -> if(le(x, s(0)), x, inc(y)) [1]
   if(true, x, s(y)) -> y [1]
   if(false, x, y) -> log2(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
inc :: 0:s -> 0:s
log :: 0:s -> 0:s
log2 :: 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:

   log_1
   log2_2
   if_3

(c) The following functions are completely defined:

   half_1
   le_2
   inc_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]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   log(x) -> log2(x, 0) [1]
   log2(x, y) -> if(le(x, s(0)), x, inc(y)) [1]
   if(true, x, s(y)) -> y [1]
   if(false, x, y) -> log2(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
inc :: 0:s -> 0:s
log :: 0:s -> 0:s
log2 :: 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]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   log(x) -> log2(x, 0) [1]
   log2(0, 0) -> if(true, 0, 0) [3]
   log2(0, s(x'')) -> if(true, 0, s(inc(x''))) [3]
   log2(s(x'), 0) -> if(le(x', 0), s(x'), 0) [3]
   log2(s(x'), s(x1)) -> if(le(x', 0), s(x'), s(inc(x1))) [3]
   if(true, x, s(y)) -> y [1]
   if(false, 0, y) -> log2(0, y) [2]
   if(false, s(0), y) -> log2(0, y) [2]
   if(false, s(s(x2)), y) -> log2(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
le :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
inc :: 0:s -> 0:s
log :: 0:s -> 0:s
log2 :: 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:

   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 }-> y :|: z' = x, z = 1, x >= 0, y >= 0, z'' = 1 + y
   if(z, z', z'') -{ 2 }-> log2(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, y) :|: z'' = y, y >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(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
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   log(z) -{ 1 }-> log2(x, 0) :|: x >= 0, z = x
   log2(z, z') -{ 3 }-> if(le(x', 0), 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(x', 0), 1 + x', 1 + inc(x1)) :|: z = 1 + x', x1 >= 0, x' >= 0, z' = 1 + x1
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(x'')) :|: z' = 1 + x'', x'' >= 0, z = 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:

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0


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

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

   { le }
   { half }
   { inc }
   { log2, if }
   { log }

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

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

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {le}, {half}, {inc}, {log2,if}, {log}

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

(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:

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {le}, {half}, {inc}, {log2,if}, {log}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {le}, {half}, {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   log2(z, z') -{ 3 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(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:

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(23) 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

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

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

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: ?, size: O(n^1) [z]

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

(25) 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

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

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

   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 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [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:

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(29) 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

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: ?, size: O(n^1) [z]

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

(31) 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

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(z - 1) :|: z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 }-> if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 }-> if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [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:

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 + z' }-> if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 + z' }-> if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: log2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 + z' }-> if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 + z' }-> if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  log2: runtime: ?, size: O(n^1) [z']
  if: runtime: ?, size: O(n^1) [z'']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: log2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 18 + 8*z + z*z' + z^2 + 3*z'

Computed RUNTIME bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 54 + 7*z' + z'*z'' + z'^2 + 8*z''

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 2 + z' }-> log2(1 + s2, z'') :|: s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 1 }-> log2(z, 0) :|: z >= 0
   log2(z, z') -{ 5 }-> if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
   log2(z, z') -{ 5 + z' }-> if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 3 }-> if(1, 0, 0) :|: z = 0, z' = 0
   log2(z, z') -{ 3 + z' }-> if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  log2: runtime: O(n^2) [18 + 8*z + z*z' + z^2 + 3*z'], size: O(n^1) [z']
  if: runtime: O(n^2) [54 + 7*z' + z'*z'' + z'^2 + 8*z''], size: O(n^1) [z'']

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

(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:

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s11 :|: s11 >= 0, s11 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s12 :|: s12 >= 0, s12 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 29 + 10*s2 + s2*z'' + s2^2 + z' + 4*z'' }-> s13 :|: s13 >= 0, s13 <= z'', s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 19 + 8*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z >= 0
   log2(z, z') -{ 67 + 8*s5 + s5*z + 8*z + z^2 + z' }-> s10 :|: s10 >= 0, s10 <= 1 + s5, s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 57 }-> s7 :|: s7 >= 0, s7 <= 0, z = 0, z' = 0
   log2(z, z') -{ 65 + 8*s4 + z' }-> s8 :|: s8 >= 0, s8 <= 1 + s4, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0
   log2(z, z') -{ 59 + 7*z + z^2 }-> s9 :|: s9 >= 0, s9 <= 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  log2: runtime: O(n^2) [18 + 8*z + z*z' + z^2 + 3*z'], size: O(n^1) [z']
  if: runtime: O(n^2) [54 + 7*z' + z'*z'' + z'^2 + 8*z''], size: O(n^1) [z'']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s11 :|: s11 >= 0, s11 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s12 :|: s12 >= 0, s12 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 29 + 10*s2 + s2*z'' + s2^2 + z' + 4*z'' }-> s13 :|: s13 >= 0, s13 <= z'', s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 19 + 8*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z >= 0
   log2(z, z') -{ 67 + 8*s5 + s5*z + 8*z + z^2 + z' }-> s10 :|: s10 >= 0, s10 <= 1 + s5, s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 57 }-> s7 :|: s7 >= 0, s7 <= 0, z = 0, z' = 0
   log2(z, z') -{ 65 + 8*s4 + z' }-> s8 :|: s8 >= 0, s8 <= 1 + s4, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0
   log2(z, z') -{ 59 + 7*z + z^2 }-> s9 :|: s9 >= 0, s9 <= 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  log2: runtime: O(n^2) [18 + 8*z + z*z' + z^2 + 3*z'], size: O(n^1) [z']
  if: runtime: O(n^2) [54 + 7*z' + z'*z'' + z'^2 + 8*z''], size: O(n^1) [z'']
  log: runtime: ?, size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: log
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 19 + 8*z + z^2

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

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

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 2, z - 2 >= 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s11 :|: s11 >= 0, s11 <= z'', z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 20 + 3*z'' }-> s12 :|: s12 >= 0, s12 <= z'', z'' >= 0, z' = 1 + 0, z = 0
   if(z, z', z'') -{ 29 + 10*s2 + s2*z'' + s2^2 + z' + 4*z'' }-> s13 :|: s13 >= 0, s13 <= z'', s2 >= 0, s2 <= z' - 2, z'' >= 0, z = 0, z' - 2 >= 0
   if(z, z', z'') -{ 1 }-> z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   log(z) -{ 19 + 8*z + z^2 }-> s6 :|: s6 >= 0, s6 <= 0, z >= 0
   log2(z, z') -{ 67 + 8*s5 + s5*z + 8*z + z^2 + z' }-> s10 :|: s10 >= 0, s10 <= 1 + s5, s5 >= 0, s5 <= z' - 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   log2(z, z') -{ 57 }-> s7 :|: s7 >= 0, s7 <= 0, z = 0, z' = 0
   log2(z, z') -{ 65 + 8*s4 + z' }-> s8 :|: s8 >= 0, s8 <= 1 + s4, s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z = 0
   log2(z, z') -{ 59 + 7*z + z^2 }-> s9 :|: s9 >= 0, s9 <= 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  half: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  inc: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  log2: runtime: O(n^2) [18 + 8*z + z*z' + z^2 + 3*z'], size: O(n^1) [z']
  if: runtime: O(n^2) [54 + 7*z' + z'*z'' + z'^2 + 8*z''], size: O(n^1) [z'']
  log: runtime: O(n^2) [19 + 8*z + z^2], size: O(1) [0]

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

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

(46)
BOUNDS(1, n^2)

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1))
   if(true, z0, s(z1)) -> z1
   if(false, z0, z1) -> log2(half(z0), z1)
Tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   LOG(z0) -> c8(LOG2(z0, 0))
   LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, s(z1)) -> c11
   IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
S tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   LOG(z0) -> c8(LOG2(z0, 0))
   LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, s(z1)) -> c11
   IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
K tuples:none
Defined Rule Symbols:   half_1, le_2, inc_1, log_1, log2_2, if_3

Defined Pair Symbols:   HALF_1, LE_2, INC_1, LOG_1, LOG2_2, IF_3

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8_1, c9_2, c10_2, c11, c12_2


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

(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^1, INF).


The TRS R consists of the following rules:

   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   LOG(z0) -> c8(LOG2(z0, 0))
   LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0)))
   LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1))
   IF(true, z0, s(z1)) -> c11
   IF(false, z0, z1) -> c12(LOG2(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))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0) -> 0
   inc(s(z0)) -> s(inc(z0))
   log(z0) -> log2(z0, 0)
   log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1))
   if(true, z0, s(z1)) -> z1
   if(false, z0, z1) -> log2(half(z0), 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^1, INF).


The TRS R consists of the following rules:

   HALF(0') -> c
   HALF(s(0')) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LE(0', z0) -> c3
   LE(s(z0), 0') -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   INC(0') -> c6
   INC(s(z0)) -> c7(INC(z0))
   LOG(z0) -> c8(LOG2(z0, 0'))
   LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
   LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
   IF(true, z0, s(z1)) -> c11
   IF(false, z0, z1) -> c12(LOG2(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))
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   inc(0') -> 0'
   inc(s(z0)) -> s(inc(z0))
   log(z0) -> log2(z0, 0')
   log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
   if(true, z0, s(z1)) -> z1
   if(false, z0, z1) -> log2(half(z0), z1)

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

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

(54)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
HALF, LE, INC, LOG2, le, inc, half, log2

They will be analysed ascendingly in the following order:
HALF < LOG2
LE < LOG2
INC < LOG2
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

(56)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
HALF, LE, INC, LOG2, le, inc, half, log2

They will be analysed ascendingly in the following order:
HALF < LOG2
LE < LOG2
INC < LOG2
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)

Induction Base:
HALF(gen_0':s10_13(*(2, 0))) ->_R^Omega(1)
c

Induction Step:
HALF(gen_0':s10_13(*(2, +(n14_13, 1)))) ->_R^Omega(1)
c2(HALF(gen_0':s10_13(*(2, n14_13)))) ->_IH
c2(gen_c:c1:c29_13(c15_13))

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:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
HALF, LE, INC, LOG2, le, inc, half, log2

They will be analysed ascendingly in the following order:
HALF < LOG2
LE < LOG2
INC < LOG2
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
LE, INC, LOG2, le, inc, half, log2

They will be analysed ascendingly in the following order:
LE < LOG2
INC < LOG2
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)

Induction Base:
LE(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(1)
c3

Induction Step:
LE(gen_0':s10_13(+(n465_13, 1)), gen_0':s10_13(+(n465_13, 1))) ->_R^Omega(1)
c5(LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13))) ->_IH
c5(gen_c3:c4:c511_13(c466_13))

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

(64)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
INC, LOG2, le, inc, half, log2

They will be analysed ascendingly in the following order:
INC < LOG2
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_0':s10_13(n1175_13)) -> gen_c6:c712_13(n1175_13), rt in Omega(1 + n1175_13)

Induction Base:
INC(gen_0':s10_13(0)) ->_R^Omega(1)
c6

Induction Step:
INC(gen_0':s10_13(+(n1175_13, 1))) ->_R^Omega(1)
c7(INC(gen_0':s10_13(n1175_13))) ->_IH
c7(gen_c6:c712_13(c1176_13))

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:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)
INC(gen_0':s10_13(n1175_13)) -> gen_c6:c712_13(n1175_13), rt in Omega(1 + n1175_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
le, LOG2, inc, half, log2

They will be analysed ascendingly in the following order:
le < LOG2
inc < LOG2
half < LOG2
le < log2
inc < log2
half < log2

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s10_13(n1584_13), gen_0':s10_13(n1584_13)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s10_13(+(n1584_13, 1)), gen_0':s10_13(+(n1584_13, 1))) ->_R^Omega(0)
le(gen_0':s10_13(n1584_13), gen_0':s10_13(n1584_13)) ->_IH
true

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:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)
INC(gen_0':s10_13(n1175_13)) -> gen_c6:c712_13(n1175_13), rt in Omega(1 + n1175_13)
le(gen_0':s10_13(n1584_13), gen_0':s10_13(n1584_13)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
inc, LOG2, half, log2

They will be analysed ascendingly in the following order:
inc < LOG2
half < LOG2
inc < log2
half < log2

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s10_13(n2005_13)) -> gen_0':s10_13(n2005_13), rt in Omega(0)

Induction Base:
inc(gen_0':s10_13(0)) ->_R^Omega(0)
0'

Induction Step:
inc(gen_0':s10_13(+(n2005_13, 1))) ->_R^Omega(0)
s(inc(gen_0':s10_13(n2005_13))) ->_IH
s(gen_0':s10_13(c2006_13))

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:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)
INC(gen_0':s10_13(n1175_13)) -> gen_c6:c712_13(n1175_13), rt in Omega(1 + n1175_13)
le(gen_0':s10_13(n1584_13), gen_0':s10_13(n1584_13)) -> true, rt in Omega(0)
inc(gen_0':s10_13(n2005_13)) -> gen_0':s10_13(n2005_13), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
half, LOG2, log2

They will be analysed ascendingly in the following order:
half < LOG2
half < log2

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s10_13(*(2, n2361_13))) -> gen_0':s10_13(n2361_13), rt in Omega(0)

Induction Base:
half(gen_0':s10_13(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
half(gen_0':s10_13(*(2, +(n2361_13, 1)))) ->_R^Omega(0)
s(half(gen_0':s10_13(*(2, n2361_13)))) ->_IH
s(gen_0':s10_13(c2362_13))

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

(72)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LE(0', z0) -> c3
LE(s(z0), 0') -> c4
LE(s(z0), s(z1)) -> c5(LE(z0, z1))
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
LOG(z0) -> c8(LOG2(z0, 0'))
LOG2(z0, z1) -> c9(IF(le(z0, s(0')), z0, inc(z1)), LE(z0, s(0')))
LOG2(z0, z1) -> c10(IF(le(z0, s(0')), z0, inc(z1)), INC(z1))
IF(true, z0, s(z1)) -> c11
IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
inc(0') -> 0'
inc(s(z0)) -> s(inc(z0))
log(z0) -> log2(z0, 0')
log2(z0, z1) -> if(le(z0, s(0')), z0, inc(z1))
if(true, z0, s(z1)) -> z1
if(false, z0, z1) -> log2(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
LE :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
INC :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
LOG :: 0':s -> c8
c8 :: c9:c10 -> c8
LOG2 :: 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c3:c4:c5 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
c10 :: c11:c12 -> c6:c7 -> c9:c10
true :: true:false
c11 :: c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
half :: 0':s -> 0':s
log :: 0':s -> 0':s
log2 :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c9:c106_13 :: c9:c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
HALF(gen_0':s10_13(*(2, n14_13))) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
LE(gen_0':s10_13(n465_13), gen_0':s10_13(n465_13)) -> gen_c3:c4:c511_13(n465_13), rt in Omega(1 + n465_13)
INC(gen_0':s10_13(n1175_13)) -> gen_c6:c712_13(n1175_13), rt in Omega(1 + n1175_13)
le(gen_0':s10_13(n1584_13), gen_0':s10_13(n1584_13)) -> true, rt in Omega(0)
inc(gen_0':s10_13(n2005_13)) -> gen_0':s10_13(n2005_13), rt in Omega(0)
half(gen_0':s10_13(*(2, n2361_13))) -> gen_0':s10_13(n2361_13), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
LOG2, log2