/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^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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), 0 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), 990 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 408 ms]
    (22) CpxRNTS
    (23) FinalProof [FINISHED, 0 ms]
    (24) BOUNDS(1, n^1)
(25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CpxTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 0 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 371 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) 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^1).


The TRS R consists of the following rules:

   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

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

Rewrite Strategy: INNERMOST
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(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:

   div_2
   quot_3

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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(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:

   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

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

Rewrite Strategy: INNERMOST
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(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:

   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

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

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

   div(z', z'') -{ 1 }-> quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z


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(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 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0


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

   { div, quot }

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {div,quot}

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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 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: {div,quot}

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(19) 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'

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

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

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']

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(23) FinalProof (FINISHED)
Computed overall runtime complexity
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(24)
BOUNDS(1, n^1)

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

(26)
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:

   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(0', s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))

S is empty.
Rewrite Strategy: FULL
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(27) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(28)
Obligation:
TRS:
Rules:
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))

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

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(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
div, quot

They will be analysed ascendingly in the following order:
div = quot

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(30)
Obligation:
TRS:
Rules:
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))

Types:
div :: 0':s -> 0':s -> 0':s
0' :: 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
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:
quot, div

They will be analysed ascendingly in the following order:
div = quot

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

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

Induction Step:
quot(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(c)) ->_R^Omega(1)
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) ->_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).
----------------------------------------

(32)
Complex Obligation (BEST)

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(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))

Types:
div :: 0':s -> 0':s -> 0':s
0' :: 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
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:
quot, div

They will be analysed ascendingly in the following order:
div = quot

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(34) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(35)
BOUNDS(n^1, INF)

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(36)
Obligation:
TRS:
Rules:
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))

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


Lemmas:
quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(c)) -> 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

They will be analysed ascendingly in the following order:
div = quot