/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^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^2).

(0) CpxTRS
(1) DependencyGraphProof [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), 2 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), 149 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 60 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 453 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 298 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 145 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^2)
(45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(46) TRS for Loop Detection
    (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) TRS for Loop Detection


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

(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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
   pred(cons(s(x), nil)) -> cons(x, nil)

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

(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))

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

(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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   pred(cons(s(x), nil)) -> cons(x, nil)

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

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

As the TRS is a non-duplicating overlay system, 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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   pred(cons(s(x), nil)) -> cons(x, nil)

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:

   app(nil, k) -> k [1]
   app(l, nil) -> l [1]
   app(cons(x, l), k) -> cons(x, app(l, k)) [1]
   sum(cons(x, nil)) -> cons(x, nil) [1]
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   pred(cons(s(x), nil)) -> cons(x, nil) [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:

   app(nil, k) -> k [1]
   app(l, nil) -> l [1]
   app(cons(x, l), k) -> cons(x, app(l, k)) [1]
   sum(cons(x, nil)) -> cons(x, nil) [1]
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   pred(cons(s(x), nil)) -> cons(x, nil) [1]

The TRS has the following type information:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
pred :: nil:cons -> nil:cons

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:

   app_2
   sum_1
   pred_1

(c) The following functions are completely defined:

   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:

   app(nil, k) -> k [1]
   app(l, nil) -> l [1]
   app(cons(x, l), k) -> cons(x, app(l, k)) [1]
   sum(cons(x, nil)) -> cons(x, nil) [1]
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   pred(cons(s(x), nil)) -> cons(x, nil) [1]

The TRS has the following type information:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
pred :: nil:cons -> nil:cons

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:

   app(nil, k) -> k [1]
   app(l, nil) -> l [1]
   app(cons(x, l), k) -> cons(x, app(l, k)) [1]
   sum(cons(x, nil)) -> cons(x, nil) [1]
   sum(cons(0, cons(y, l))) -> sum(cons(y, l)) [2]
   sum(cons(s(x'), cons(y, l))) -> sum(cons(s(plus(x', y)), l)) [2]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   pred(cons(s(x), nil)) -> cons(x, nil) [1]

The TRS has the following type information:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
pred :: nil:cons -> nil:cons

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:

   nil => 0
   0 => 0

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

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

   app(z, z') -{ 1 }-> k :|: k >= 0, z' = k, z = 0
   app(z, z') -{ 1 }-> l :|: z = l, l >= 0, z' = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, k) :|: x >= 0, l >= 0, z = 1 + x + l, k >= 0, z' = k
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   pred(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + (1 + x) + 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 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:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0


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

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

   { pred }
   { app }
   { plus }
   { sum }

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

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

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {pred}, {app}, {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:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {plus}, {sum}
Previous analysis results are:
  pred: runtime: O(1) [1], size: O(n^1) [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:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(33) 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'

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(35) 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

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 2 }-> sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 3 + x' }-> sum(1 + (1 + s') + l) :|: s' >= 0, s' <= x' + y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 3 + x' }-> sum(1 + (1 + s') + l) :|: s' >= 0, s' <= x' + y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + l }-> 1 + x + s :|: s >= 0, s <= l + z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   pred(z) -{ 1 }-> 1 + (z - 2) + 0 :|: z - 2 >= 0
   sum(z) -{ 2 }-> sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
   sum(z) -{ 3 + x' }-> sum(1 + (1 + s') + l) :|: s' >= 0, s' <= x' + y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
   sum(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  pred: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sum: runtime: O(n^2) [3 + 2*z + z^2], size: O(n^1) [z]

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

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

(44)
BOUNDS(1, n^2)

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(45) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(46)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
   pred(cons(s(x), nil)) -> cons(x, nil)

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

(47) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

app(cons(x, l), k) ->^+ cons(x, app(l, k))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [l / cons(x, l)].

The result substitution is [ ].




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

(48)
Complex Obligation (BEST)

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

(49)
Obligation:
Proved the lower bound n^1 for the following 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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
   pred(cons(s(x), nil)) -> cons(x, nil)

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

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

(51)
BOUNDS(n^1, INF)

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

(52)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
   pred(cons(s(x), nil)) -> cons(x, nil)

S is empty.
Rewrite Strategy: FULL