/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^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 488 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 219 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 899 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 379 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 372 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^3)
(37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CpxTRS
    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) typed CpxTrs
    (41) OrderProof [LOWER BOUND(ID), 0 ms]
    (42) typed CpxTrs
    (43) RewriteLemmaProof [LOWER BOUND(ID), 278 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) typed CpxTrs


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


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
div(s(x), s([]))
f(x, s(y), [])


The defined contexts are:
div([], x1)
f(x0, [], x2)
div([], s(x1))
minus([], x1)
minus(s([]), s(0))


[] just represents basic- or constructor-terms in the following defined contexts:
f(x0, [], x2)
div([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(4)
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:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> 0:s

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

(7) 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:
none

(c) The following functions are completely defined:

   f_3
   minus_2
   div_2

Due to the following rules being added:

   div(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(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:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
   div(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> 0:s

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

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

(10)
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:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(x)) -> s(div(0, s(x))) [2]
   div(s(s(x')), s(s(y'))) -> s(div(minus(x', y'), s(s(y')))) [2]
   div(s(0), s(y)) -> s(div(0, s(y))) [2]
   div(s(x), s(0)) -> s(div(x, s(0))) [2]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(0), b) -> div(f(x, 0, b), b) [2]
   f(x, s(y), b) -> div(f(x, minus(y, 0), b), b) [2]
   div(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> 0:s

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

(11) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   div(z, z') -{ 2 }-> 1 + div(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   div(z, z') -{ 2 }-> 1 + div(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   div(z, z') -{ 2 }-> 1 + div(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 1 + x
   div(z, z') -{ 2 }-> 1 + div(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z', z'') -{ 1 }-> x :|: b >= 0, z'' = b, x >= 0, z = x, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(x, minus(y, 0), b), b) :|: z' = 1 + y, b >= 0, z'' = b, x >= 0, y >= 0, z = x
   f(z, z', z'') -{ 2 }-> div(f(x, 0, b), b) :|: b >= 0, z'' = b, x >= 0, z' = 1 + 0, z = x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = x
   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0


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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


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

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

   { minus }
   { div }
   { f }

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {div}, {f}

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {div}, {f}

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

(19) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {div}, {f}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {div}, {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {div}, {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {div}, {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
  div: runtime: ?, size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 9*z + z*z'

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
  f: runtime: ?, size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 18*z*z' + 2*z*z'*z'' + 9*z'

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 18*z*z' + 2*z*z'*z'' + 9*z'], size: O(n^1) [z]

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

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

(36)
BOUNDS(1, n^3)

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

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

(38)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus(x, x) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   minus(0', x) -> 0'
   minus(x, 0') -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0', s(y)) -> 0'
   f(x, 0', b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

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

(40)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
minus(0', x) -> 0'
minus(x, 0') -> x
div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
div(0', s(y)) -> 0'
f(x, 0', b) -> x
f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(41) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, div, f

They will be analysed ascendingly in the following order:
minus < div
minus < f
div < f

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

(42)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
minus(0', x) -> 0'
minus(x, 0') -> x
div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
div(0', s(y)) -> 0'
f(x, 0', b) -> x
f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div, f

They will be analysed ascendingly in the following order:
minus < div
minus < f
div < f

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
gen_0':s2_0(0)

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

(44)
Complex Obligation (BEST)

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

(45)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
minus(x, x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
minus(0', x) -> 0'
minus(x, 0') -> x
div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
div(0', s(y)) -> 0'
f(x, 0', b) -> x
f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div, f

They will be analysed ascendingly in the following order:
minus < div
minus < f
div < f

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

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

(47)
BOUNDS(n^1, INF)

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

(48)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(s(x), s(y)) -> minus(x, y)
minus(0', x) -> 0'
minus(x, 0') -> x
div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
div(0', s(y)) -> 0'
f(x, 0', b) -> x
f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
div, f

They will be analysed ascendingly in the following order:
div < f