/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^2), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, n^2).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 297 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 465 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^2)
(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CpxTRS
    (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) typed CpxTrs
    (37) OrderProof [LOWER BOUND(ID), 0 ms]
    (38) typed CpxTrs
    (39) RewriteLemmaProof [LOWER BOUND(ID), 287 ms]
    (40) BEST
        (41) proven lower bound
            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
            (43) BOUNDS(n^1, INF)
        (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (46) proven lower bound
            (47) LowerBoundPropagationProof [FINISHED, 0 ms]
            (48) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of plus: plus, times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

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

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


The TRS R consists of the following rules:

   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))

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

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

The duplicating contexts are:
times([], s(y))


The defined contexts are:
plus([], x1)


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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))

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

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

(6)
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:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(9) 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:

   times_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

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

(12)
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:

   times(x, 0) -> 0 [1]
   times(x, s(0)) -> plus(0, x) [2]
   times(x, s(s(y'))) -> plus(plus(times(x, y'), x), x) [2]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(13) 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

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

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

   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   times(z, z') -{ 2 }-> plus(plus(times(x, y'), x), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
   times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0


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

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

   { plus }
   { times }

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z']

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

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

(32)
BOUNDS(1, n^2)

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

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

(34)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
   times(x, 0') -> 0'
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))

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

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

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

Types:
times :: s:0' -> s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'

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

(37) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
times, plus

They will be analysed ascendingly in the following order:
plus < times

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

(38)
Obligation:
TRS:
Rules:
times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') -> 0'
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))

Types:
times :: s:0' -> s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
plus(gen_s:0'2_0(a), gen_s:0'2_0(0)) ->_R^Omega(1)
gen_s:0'2_0(a)

Induction Step:
plus(gen_s:0'2_0(a), gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
s(plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0))) ->_IH
s(gen_s:0'2_0(+(a, c5_0)))

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

(40)
Complex Obligation (BEST)

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

(41)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') -> 0'
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))

Types:
times :: s:0' -> s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

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

(43)
BOUNDS(n^1, INF)

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

(44)
Obligation:
TRS:
Rules:
times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') -> 0'
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))

Types:
times :: s:0' -> s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
times
----------------------------------------

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_s:0'2_0(a), gen_s:0'2_0(n421_0)) -> gen_s:0'2_0(*(n421_0, a)), rt in Omega(1 + a*n421_0 + n421_0)

Induction Base:
times(gen_s:0'2_0(a), gen_s:0'2_0(0)) ->_R^Omega(1)
0'

Induction Step:
times(gen_s:0'2_0(a), gen_s:0'2_0(+(n421_0, 1))) ->_R^Omega(1)
plus(times(gen_s:0'2_0(a), gen_s:0'2_0(n421_0)), gen_s:0'2_0(a)) ->_IH
plus(gen_s:0'2_0(*(c422_0, a)), gen_s:0'2_0(a)) ->_L^Omega(1 + a)
gen_s:0'2_0(+(a, *(n421_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(46)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z)))
times(x, 0') -> 0'
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))

Types:
times :: s:0' -> s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
times
----------------------------------------

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

(48)
BOUNDS(n^2, INF)