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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 228 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 17 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (28) CpxRNTS
    (29) FinalProof [FINISHED, 0 ms]
    (30) BOUNDS(1, n^1)
(31) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CpxTRS
    (33) SlicingProof [LOWER BOUND(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), 1939 ms]
    (40) BEST
        (41) proven lower bound
            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
            (43) BOUNDS(n^1, INF)
        (44) 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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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.
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(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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

S is empty.
Rewrite Strategy: INNERMOST
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(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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, x))) [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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1]

The TRS has the following type information:
sum :: 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
sum1 :: 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:

   sum_1
   sum1_1

(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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1]

The TRS has the following type information:
sum :: 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
sum1 :: 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.
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(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)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1]

The TRS has the following type information:
sum :: 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
sum1 :: 0:s:+ -> 0:s:+

Rewrite Strategy: INNERMOST
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(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:

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(x) + (1 + x) :|: x >= 0, z = 1 + x
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(x) + (1 + x + x)) :|: x >= 0, z = 1 + x


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(13) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(14)
Obligation:
Complexity RNTS consisting of the following rules:

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0


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

   { sum1 }
   { sum }

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum1}, {sum}

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(17) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum1}, {sum}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: sum1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z + 2*z^2

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum1}, {sum}
Previous analysis results are:
  sum1: runtime: ?, size: O(n^2) [z + 2*z^2]

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: sum1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum}
Previous analysis results are:
  sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2]

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(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:

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0

Function symbols to be analyzed: {sum}
Previous analysis results are:
  sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2]

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0

Function symbols to be analyzed: {sum}
Previous analysis results are:
  sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2]
  sum: runtime: ?, size: O(n^2) [z + z^2]

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(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2]
  sum: runtime: O(n^1) [1 + z], size: O(n^2) [z + z^2]

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(29) FinalProof (FINISHED)
Computed overall runtime complexity
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(30)
BOUNDS(1, n^1)

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(31) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(32)
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))
   sum1(0') -> 0'
   sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))

S is empty.
Rewrite Strategy: FULL
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(33) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
+'/1

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(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))
   sum1(0') -> 0'
   sum1(s(x)) -> s(+'(sum1(x)))

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

(36)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'

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(37) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum, sum1
----------------------------------------

(38)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum, sum1
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(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
sum(gen_0':s:+'2_0(+(1, 0)))

Induction Step:
sum(gen_0':s:+'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s:+'2_0(+(1, n4_0)))) ->_IH
+'(*3_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(40)
Complex Obligation (BEST)

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

TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum, sum1
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(42) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(43)
BOUNDS(n^1, INF)

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(44)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum1