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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 291 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 259 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 162 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1326 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 576 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CpxTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 0 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 251 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 60 ms]
            (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (62) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (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: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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: INNERMOST
----------------------------------------

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

(50)
Obligation:
Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, le, inc, log2

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

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

(52)
Obligation:
Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n5_0)))) ->_IH
s(gen_0':s3_0(c6_0))

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

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n343_0), gen_0':s3_0(n343_0)) -> true, rt in Omega(1 + n343_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n343_0, 1)), gen_0':s3_0(+(n343_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n343_0), gen_0':s3_0(n343_0)) ->_IH
true

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

(60)
Obligation:
Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
le(gen_0':s3_0(n343_0), gen_0':s3_0(n343_0)) -> true, rt in Omega(1 + n343_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s3_0(n662_0)) -> gen_0':s3_0(n662_0), rt in Omega(1 + n662_0)

Induction Base:
inc(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
inc(gen_0':s3_0(+(n662_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s3_0(n662_0))) ->_IH
s(gen_0':s3_0(c663_0))

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

(62)
Obligation:
Innermost TRS:
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)

Types:
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
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
le(gen_0':s3_0(n343_0), gen_0':s3_0(n343_0)) -> true, rt in Omega(1 + n343_0)
inc(gen_0':s3_0(n662_0)) -> gen_0':s3_0(n662_0), rt in Omega(1 + n662_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
log2