/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) InliningProof [UPPER BOUND(ID), 28 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), 369 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 178 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CpxTRS
    (35) SlicingProof [LOWER BOUND(ID), 0 ms]
    (36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 729 ms]
    (42) proven lower bound
    (43) LowerBoundPropagationProof [FINISHED, 0 ms]
    (44) BOUNDS(n^1, INF)


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

(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)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(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.
----------------------------------------

(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)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

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^1).


The TRS R consists of the following rules:

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

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

   sum_1
   sqr_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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)) -> +(sqr(s(x)), sum(x)) [1]
   sqr(x) -> *(x, x) [1]
   sum(s(x)) -> +(*(s(x), s(x)), sum(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:+
sqr :: 0:s:+ -> *
* :: 0:s:+ -> 0:s:+ -> *
const :: *

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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sqr(s(x)), sum(x)) [1]
   sqr(x) -> *(x, x) [1]
   sum(s(x)) -> +(*(s(x), s(x)), sum(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:+
sqr :: 0:s:+ -> *
* :: 0:s:+ -> 0:s:+ -> *
const :: *

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
   const => 0

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

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

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


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

(13) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

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

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

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

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


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

(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:

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


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

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

   { sum }
   { sqr }

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

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

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

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

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

(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:

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

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

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

(21) 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: 2*z + 2*z^2

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

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

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

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

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

(23) 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 + 2*z

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

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

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

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

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

(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:

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

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z

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

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

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

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

(32)
BOUNDS(1, n^1)

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

(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)) -> +'(sqr(s(x)), sum(x))
   sqr(x) -> *'(x, x)
   sum(s(x)) -> +'(*'(s(x), s(x)), sum(x))

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

(35) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
sqr/0
*'/0
*'/1

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

(36)
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)) -> +'(sqr, sum(x))
   sqr -> *'
   sum(s(x)) -> +'(*', sum(x))

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

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

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

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'

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

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

(40)
Obligation:
TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sqr, sum(x))
sqr -> *'
sum(s(x)) -> +'(*', sum(x))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'3_0(0) <=> 0'
gen_0':s:+'3_0(+(x, 1)) <=> s(gen_0':s:+'3_0(x))


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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:+'3_0(n5_0)) -> *4_0, rt in Omega(n5_0)

Induction Base:
sum(gen_0':s:+'3_0(0))

Induction Step:
sum(gen_0':s:+'3_0(+(n5_0, 1))) ->_R^Omega(1)
+'(*', sum(gen_0':s:+'3_0(n5_0))) ->_IH
+'(*', *4_0)

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

(42)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sqr, sum(x))
sqr -> *'
sum(s(x)) -> +'(*', sum(x))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'3_0(0) <=> 0'
gen_0':s:+'3_0(+(x, 1)) <=> s(gen_0':s:+'3_0(x))


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

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

(44)
BOUNDS(n^1, INF)