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


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WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3).

(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), 6 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), 4 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 363 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 762 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 290 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 433 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 278 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^3)
(41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CpxTRS
    (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) typed CpxTrs
    (45) OrderProof [LOWER BOUND(ID), 0 ms]
    (46) typed CpxTrs
    (47) RewriteLemmaProof [LOWER BOUND(ID), 258 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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^3).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
log :: 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

(c) The following functions are completely defined:

   le_2
   minus_2
   quot_2
   if_minus_3

Due to the following rules being added:

   quot(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]
   quot(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
log :: 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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), 0) -> if_minus(false, s(x), 0) [2]
   minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(0), s(y)) -> s(quot(0, s(y))) [2]
   quot(s(s(x')), s(y)) -> s(quot(if_minus(le(s(x'), y), s(x'), y), s(y))) [2]
   log(s(0)) -> 0 [1]
   log(s(s(0))) -> s(log(s(0))) [2]
   log(s(s(s(x'')))) -> s(log(s(s(quot(minus(x'', s(0)), s(s(0))))))) [2]
   log(s(s(x))) -> s(log(s(0))) [1]
   quot(v0, v1) -> 0 [0]
   if_minus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
log :: 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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
   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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: x >= 0, z = 1 + (1 + x)
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(x'', 1 + 0), 1 + (1 + 0)))) :|: x'' >= 0, z = 1 + (1 + (1 + x''))
   minus(z, z') -{ 2 }-> if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   minus(z, z') -{ 2 }-> if_minus(0, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + x', y), 1 + x', y), 1 + y) :|: z' = 1 + y, x' >= 0, y >= 0, z = 1 + (1 + x')
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0


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

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

   { le }
   { minus, if_minus }
   { quot }
   { log }

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {quot}, {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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {quot}, {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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {quot}, {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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 2 }-> 1 + quot(if_minus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {minus,if_minus}, {quot}, {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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {minus,if_minus}, {quot}, {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: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

Computed SIZE bound using CoFloCo for: if_minus
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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*z'

Computed RUNTIME bound using KoAT for: if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z''

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ 2 }-> 1 + log(1 + (1 + quot(minus(z - 3, 1 + 0), 1 + (1 + 0)))) :|: z - 3 >= 0
   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(if_minus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + 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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s5, 1 + (1 + 0)))) :|: s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: quot
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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s5, 1 + (1 + 0)))) :|: s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {quot}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  quot: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 7*z + z*z' + 3*z^2 + z^2*z'

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -3 + 5*z }-> 1 + log(1 + (1 + quot(s5, 1 + (1 + 0)))) :|: s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  quot: runtime: O(n^3) [1 + 7*z + z*z' + 3*z^2 + z^2*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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 9*s5 + 5*s5^2 + 5*z }-> 1 + log(1 + (1 + s8)) :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s6 :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 7*s4 + s4*z' + 3*s4^2 + s4^2*z' + 3*z + z*z' + z' }-> 1 + s7 :|: s7 >= 0, s7 <= s4, s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  quot: runtime: O(n^3) [1 + 7*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: log
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:

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 9*s5 + 5*s5^2 + 5*z }-> 1 + log(1 + (1 + s8)) :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s6 :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 7*s4 + s4*z' + 3*s4^2 + s4^2*z' + 3*z + z*z' + z' }-> 1 + s7 :|: s7 >= 0, s7 <= s4, s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  quot: runtime: O(n^3) [1 + 7*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: log
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 3*z + 14*z^2 + 5*z^3

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 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 }-> 0 :|: z = 1 + 0
   log(z) -{ 2 }-> 1 + log(1 + 0) :|: z = 1 + (1 + 0)
   log(z) -{ 1 }-> 1 + log(1 + 0) :|: z - 2 >= 0
   log(z) -{ -2 + 9*s5 + 5*s5^2 + 5*z }-> 1 + log(1 + (1 + s8)) :|: s8 >= 0, s8 <= s5, s5 >= 0, s5 <= z - 3, z - 3 >= 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quot(z, z') -{ 3 }-> 1 + s6 :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' - 1 >= 0
   quot(z, z') -{ 6 + 7*s4 + s4*z' + 3*s4^2 + s4^2*z' + 3*z + z*z' + z' }-> 1 + s7 :|: s7 >= 0, s7 <= s4, s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  quot: runtime: O(n^3) [1 + 7*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]
  log: runtime: O(n^3) [1 + 3*z + 14*z^2 + 5*z^3], size: O(n^1) [z]

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(39) FinalProof (FINISHED)
Computed overall runtime complexity
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(40)
BOUNDS(1, n^3)

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(41) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(42)
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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0', y) -> 0'
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0'
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0')) -> 0'
   log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

S is empty.
Rewrite Strategy: INNERMOST
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(43) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(44)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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(45) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus, quot, log

They will be analysed ascendingly in the following order:
le < minus
minus < quot
quot < log

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(46)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, quot, log

They will be analysed ascendingly in the following order:
le < minus
minus < quot
quot < log

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(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_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(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(48)
Complex Obligation (BEST)

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(49)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, quot, log

They will be analysed ascendingly in the following order:
le < minus
minus < quot
quot < log

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(50) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(51)
BOUNDS(n^1, INF)

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(52)
Obligation:
Innermost TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(0', y) -> 0'
minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) -> 0'
if_minus(false, s(x), y) -> s(minus(x, y))
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if_minus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, 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:
minus, quot, log

They will be analysed ascendingly in the following order:
minus < quot
quot < log