/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^2), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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) 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), 65 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 46 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 299 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 513 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 178 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^2)
(45) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(46) CpxTRS
    (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) typed CpxTrs
    (49) OrderProof [LOWER BOUND(ID), 0 ms]
    (50) typed CpxTrs
    (51) RewriteLemmaProof [LOWER BOUND(ID), 265 ms]
    (52) BEST
        (53) proven lower bound
            (54) LowerBoundPropagationProof [FINISHED, 0 ms]
            (55) BOUNDS(n^1, INF)
        (56) typed CpxTrs
            (57) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
            (58) proven lower bound
            (59) LowerBoundPropagationProof [FINISHED, 0 ms]
            (60) 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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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

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

The duplicating contexts are:
x([], s(M))


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.
----------------------------------------

(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:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> 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^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> 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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 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:

   and_2
   activate_1

(c) The following functions are completely defined:

   x_2
   plus_2

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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s
const :: and:activate

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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(0)) -> plus(0, N) [2]
   x(N, s(s(M'))) -> plus(plus(x(N, M'), N), N) [2]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s
const :: and:activate

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:

   tt => 0
   0 => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
   x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0


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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   and(z, z') -{ 2 }-> X' :|: z' = X, X >= 0, z = 0, X' >= 0, X = X'
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
   x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 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:

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


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

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

   { activate }
   { and }
   { plus }
   { x }

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

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

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

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}

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

(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:

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

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: activate
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:

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
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:

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

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:
  activate: 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:

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

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

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

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [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:

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

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [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:

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

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [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:

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

Function symbols to be analyzed: {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [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:

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

Function symbols to be analyzed: {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [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: x
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z + 2*z*z'

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

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

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

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

(44)
BOUNDS(1, n^2)

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

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

(46)
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:

   and(tt, X) -> activate(X)
   plus(N, 0') -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0') -> 0'
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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

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

(48)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

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

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

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

(50)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)

Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) ->_IH
s(gen_0':s4_0(+(a, c7_0)))

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

(52)
Complex Obligation (BEST)

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

(53)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

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

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

(55)
BOUNDS(n^1, INF)

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

(56)
Obligation:
TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) -> gen_0':s4_0(*(n441_0, a)), rt in Omega(1 + a*n441_0 + n441_0)

Induction Base:
x(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
x(gen_0':s4_0(a), gen_0':s4_0(+(n441_0, 1))) ->_R^Omega(1)
plus(x(gen_0':s4_0(a), gen_0':s4_0(n441_0)), gen_0':s4_0(a)) ->_IH
plus(gen_0':s4_0(*(c442_0, a)), gen_0':s4_0(a)) ->_L^Omega(1 + a)
gen_0':s4_0(+(a, *(n441_0, a)))

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

(58)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
and(tt, X) -> activate(X)
plus(N, 0') -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0') -> 0'
x(N, s(M)) -> plus(x(N, M), N)
activate(X) -> X

Types:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
x :: 0':s -> 0':s -> 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

(60)
BOUNDS(n^2, INF)