/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^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [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), 1 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 301 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 334 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 103 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), 272 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), 698 ms]
            (46) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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

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

The duplicating contexts are:
sum(s([]))


The defined contexts are:
+([], s(x1))
+([], 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^2).


The TRS R consists of the following rules:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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^2).


The TRS R consists of the following rules:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sum(x), s(x)) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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

(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:

   sum(0) -> 0 [1]
   sum(s(x)) -> plus(sum(x), s(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:

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

The TRS has the following type information:
sum :: 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:

   sum_1
   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:

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

The TRS has the following type information:
sum :: 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:

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

The TRS has the following type information:
sum :: 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
   sum(z) -{ 2 }-> plus(plus(sum(x'), 1 + x'), 1 + (1 + x')) :|: x' >= 0, z = 1 + (1 + x')
   sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sum(z) -{ 1 }-> 0 :|: 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
   sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0
   sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sum(z) -{ 1 }-> 0 :|: z = 0


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

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

   { plus }
   { sum }

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

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

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

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

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

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

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

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

Function symbols to be analyzed: {plus}, {sum}
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
   sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0
   sum(z) -{ 2 }-> plus(0, 1 + 0) :|: z = 1 + 0
   sum(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {sum}
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
   sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0
   sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {sum}
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: sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 2*z^2

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

(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
   sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0
   sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   sum(z) -{ 4 }-> s :|: s >= 0, s <= 0 + (1 + 0), z = 1 + 0
   sum(z) -{ 2 }-> plus(plus(sum(z - 2), 1 + (z - 2)), 1 + (1 + (z - 2))) :|: z - 2 >= 0
   sum(z) -{ 1 }-> 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']
  sum: runtime: O(n^2) [5 + 3*z + 2*z^2], size: O(n^2) [1 + 2*z^2]

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

(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^1, INF).


The TRS R consists of the following rules:

   sum(0') -> 0'
   sum(s(x)) -> +'(sum(x), s(x))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))

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

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

(36)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(37) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum, +'

They will be analysed ascendingly in the following order:
+' < sum

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

(38)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 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:
+', sum

They will be analysed ascendingly in the following order:
+' < sum

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

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

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

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_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:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 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:
+', sum

They will be analysed ascendingly in the following order:
+' < sum

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

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

(43)
BOUNDS(n^1, INF)

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

(44)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), 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:
sum
----------------------------------------

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s2_0(+(1, n403_0))) -> *3_0, rt in Omega(n403_0)

Induction Base:
sum(gen_0':s2_0(+(1, 0)))

Induction Step:
sum(gen_0':s2_0(+(1, +(n403_0, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s2_0(+(1, n403_0))), s(gen_0':s2_0(+(1, n403_0)))) ->_IH
+'(*3_0, s(gen_0':s2_0(+(1, n403_0))))

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

(46)
BOUNDS(1, INF)