WORST_CASE(Omega(n^1),O(n^1))
proof of input_DvBwPlu1cz.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) CpxWeightedTrsRenamingProof [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), 207 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 262 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^1)
(43) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
    (45) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CpxRelTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 13 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 1917 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs


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

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


The TRS R consists of the following rules:

   +(X, 0) -> X
   +(X, s(Y)) -> s(+(X, Y))
   double(X) -> +(X, X)
   f(0, s(0), X) -> f(X, double(X), X)
   g(X, Y) -> X
   g(X, Y) -> Y

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
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:

   +(X, 0) -> X [1]
   +(X, s(Y)) -> s(+(X, Y)) [1]
   double(X) -> +(X, X) [1]
   f(0, s(0), X) -> f(X, double(X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

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

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

+ => plus

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

(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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   double(X) -> plus(X, X) [1]
   f(0, s(0), X) -> f(X, double(X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   double(X) -> plus(X, X) [1]
   f(0, s(0), X) -> f(X, double(X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
double :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> f
g :: g -> g -> g

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:

   f_3
   g_2

(c) The following functions are completely defined:

   double_1
   plus_2

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   double(X) -> plus(X, X) [1]
   f(0, s(0), X) -> f(X, double(X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
double :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> f
g :: g -> g -> g
const :: f
const1 :: g

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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   double(X) -> plus(X, X) [1]
   f(0, s(0), X) -> f(X, plus(X, X), X) [2]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
double :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> f
g :: g -> g -> g
const :: f
const1 :: g

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
   const1 => 0

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

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

   double(z) -{ 1 }-> plus(X, X) :|: X >= 0, z = X
   f(z, z', z'') -{ 2 }-> f(X, plus(X, X), X) :|: z'' = X, X >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> X :|: z' = Y, Y >= 0, X >= 0, z = X
   g(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, X >= 0, z = X
   plus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X


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

(13) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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


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

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

   { g }
   { plus }
   { f }
   { double }

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {g}, {plus}, {f}, {double}

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

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {g}, {plus}, {f}, {double}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

(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:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

(25) 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'

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

(27) 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'

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

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

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   f(z, z', z'') -{ 2 }-> f(z'', plus(z'', z''), z'') :|: z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 3 + z'' }-> f(z'', s'', z'') :|: s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 3 + z'' }-> f(z'', s'', z'') :|: s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {f}, {double}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: ?, size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z''

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 3 + z'' }-> f(z'', s'', z'') :|: s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {double}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: O(n^1) [3 + z''], size: O(1) [0]

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

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 6 + 2*z'' }-> s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {double}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: O(n^1) [3 + z''], size: O(1) [0]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 6 + 2*z'' }-> s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: {double}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: O(n^1) [3 + z''], size: O(1) [0]
  double: runtime: ?, size: O(n^1) [2*z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= z + z, z >= 0
   f(z, z', z'') -{ 6 + 2*z'' }-> s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= z'' + z'', z'' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0
   g(z, z') -{ 1 }-> 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

Function symbols to be analyzed: 
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: O(n^1) [3 + z''], size: O(1) [0]
  double: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]

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

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

(42)
BOUNDS(1, n^1)

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

(43) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   double(z0) -> +(z0, z0)
   f(0, s(0), z0) -> f(z0, double(z0), z0)
   g(z0, z1) -> z0
   g(z0, z1) -> z1
Tuples:
   +'(z0, 0) -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   DOUBLE(z0) -> c2(+'(z0, z0))
   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
   G(z0, z1) -> c4
   G(z0, z1) -> c5
S tuples:
   +'(z0, 0) -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   DOUBLE(z0) -> c2(+'(z0, z0))
   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
   G(z0, z1) -> c4
   G(z0, z1) -> c5
K tuples:none
Defined Rule Symbols:   +_2, double_1, f_3, g_2

Defined Pair Symbols:   +'_2, DOUBLE_1, F_3, G_2

Compound Symbols:   c, c1_1, c2_1, c3_2, c4, c5


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

(45) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(46)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   +'(z0, 0) -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   DOUBLE(z0) -> c2(+'(z0, z0))
   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
   G(z0, z1) -> c4
   G(z0, z1) -> c5

The (relative) TRS S consists of the following rules:

   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   double(z0) -> +(z0, z0)
   f(0, s(0), z0) -> f(z0, double(z0), z0)
   g(z0, z1) -> z0
   g(z0, z1) -> z1

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

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   +'(z0, 0') -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   DOUBLE(z0) -> c2(+'(z0, z0))
   F(0', s(0'), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
   G(z0, z1) -> c4
   G(z0, z1) -> c5

The (relative) TRS S consists of the following rules:

   +'(z0, 0') -> z0
   +'(z0, s(z1)) -> s(+'(z0, z1))
   double(z0) -> +'(z0, z0)
   f(0', s(0'), z0) -> f(z0, double(z0), z0)
   g(z0, z1) -> z0
   g(z0, z1) -> z1

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

(49) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(50)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
DOUBLE(z0) -> c2(+'(z0, z0))
F(0', s(0'), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
G(z0, z1) -> c4
G(z0, z1) -> c5
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
double(z0) -> +'(z0, z0)
f(0', s(0'), z0) -> f(z0, double(z0), z0)
g(z0, z1) -> z0
g(z0, z1) -> z1

Types:
+' :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1
0' :: 0':c:s:c1
c :: 0':c:s:c1
s :: 0':c:s:c1 -> 0':c:s:c1
c1 :: 0':c:s:c1 -> 0':c:s:c1
DOUBLE :: 0':c:s:c1 -> c2
c2 :: 0':c:s:c1 -> c2
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c3
c3 :: c3 -> c2 -> c3
double :: 0':c:s:c1 -> 0':c:s:c1
G :: a -> b -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_6 :: 0':c:s:c1
hole_c22_6 :: c2
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_a5_6 :: a
hole_b6_6 :: b
hole_f7_6 :: f
hole_g8_6 :: g
gen_0':c:s:c19_6 :: Nat -> 0':c:s:c1
gen_c310_6 :: Nat -> c3

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', F, f
----------------------------------------

(52)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
DOUBLE(z0) -> c2(+'(z0, z0))
F(0', s(0'), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
G(z0, z1) -> c4
G(z0, z1) -> c5
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
double(z0) -> +'(z0, z0)
f(0', s(0'), z0) -> f(z0, double(z0), z0)
g(z0, z1) -> z0
g(z0, z1) -> z1

Types:
+' :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1
0' :: 0':c:s:c1
c :: 0':c:s:c1
s :: 0':c:s:c1 -> 0':c:s:c1
c1 :: 0':c:s:c1 -> 0':c:s:c1
DOUBLE :: 0':c:s:c1 -> c2
c2 :: 0':c:s:c1 -> c2
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c3
c3 :: c3 -> c2 -> c3
double :: 0':c:s:c1 -> 0':c:s:c1
G :: a -> b -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_6 :: 0':c:s:c1
hole_c22_6 :: c2
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_a5_6 :: a
hole_b6_6 :: b
hole_f7_6 :: f
hole_g8_6 :: g
gen_0':c:s:c19_6 :: Nat -> 0':c:s:c1
gen_c310_6 :: Nat -> c3


Generator Equations:
gen_0':c:s:c19_6(0) <=> 0'
gen_0':c:s:c19_6(+(x, 1)) <=> s(gen_0':c:s:c19_6(x))
gen_c310_6(0) <=> hole_c33_6
gen_c310_6(+(x, 1)) <=> c3(gen_c310_6(x), c2(0'))


The following defined symbols remain to be analysed:
+', F, f
----------------------------------------

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':c:s:c19_6(a), gen_0':c:s:c19_6(+(1, n12_6))) -> *11_6, rt in Omega(n12_6)

Induction Base:
+'(gen_0':c:s:c19_6(a), gen_0':c:s:c19_6(+(1, 0)))

Induction Step:
+'(gen_0':c:s:c19_6(a), gen_0':c:s:c19_6(+(1, +(n12_6, 1)))) ->_R^Omega(1)
c1(+'(gen_0':c:s:c19_6(a), gen_0':c:s:c19_6(+(1, n12_6)))) ->_IH
c1(*11_6)

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

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
DOUBLE(z0) -> c2(+'(z0, z0))
F(0', s(0'), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
G(z0, z1) -> c4
G(z0, z1) -> c5
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
double(z0) -> +'(z0, z0)
f(0', s(0'), z0) -> f(z0, double(z0), z0)
g(z0, z1) -> z0
g(z0, z1) -> z1

Types:
+' :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1
0' :: 0':c:s:c1
c :: 0':c:s:c1
s :: 0':c:s:c1 -> 0':c:s:c1
c1 :: 0':c:s:c1 -> 0':c:s:c1
DOUBLE :: 0':c:s:c1 -> c2
c2 :: 0':c:s:c1 -> c2
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c3
c3 :: c3 -> c2 -> c3
double :: 0':c:s:c1 -> 0':c:s:c1
G :: a -> b -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_6 :: 0':c:s:c1
hole_c22_6 :: c2
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_a5_6 :: a
hole_b6_6 :: b
hole_f7_6 :: f
hole_g8_6 :: g
gen_0':c:s:c19_6 :: Nat -> 0':c:s:c1
gen_c310_6 :: Nat -> c3


Generator Equations:
gen_0':c:s:c19_6(0) <=> 0'
gen_0':c:s:c19_6(+(x, 1)) <=> s(gen_0':c:s:c19_6(x))
gen_c310_6(0) <=> hole_c33_6
gen_c310_6(+(x, 1)) <=> c3(gen_c310_6(x), c2(0'))


The following defined symbols remain to be analysed:
+', F, f
----------------------------------------

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

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
DOUBLE(z0) -> c2(+'(z0, z0))
F(0', s(0'), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
G(z0, z1) -> c4
G(z0, z1) -> c5
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
double(z0) -> +'(z0, z0)
f(0', s(0'), z0) -> f(z0, double(z0), z0)
g(z0, z1) -> z0
g(z0, z1) -> z1

Types:
+' :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1
0' :: 0':c:s:c1
c :: 0':c:s:c1
s :: 0':c:s:c1 -> 0':c:s:c1
c1 :: 0':c:s:c1 -> 0':c:s:c1
DOUBLE :: 0':c:s:c1 -> c2
c2 :: 0':c:s:c1 -> c2
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c3
c3 :: c3 -> c2 -> c3
double :: 0':c:s:c1 -> 0':c:s:c1
G :: a -> b -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_6 :: 0':c:s:c1
hole_c22_6 :: c2
hole_c33_6 :: c3
hole_c4:c54_6 :: c4:c5
hole_a5_6 :: a
hole_b6_6 :: b
hole_f7_6 :: f
hole_g8_6 :: g
gen_0':c:s:c19_6 :: Nat -> 0':c:s:c1
gen_c310_6 :: Nat -> c3


Lemmas:
+'(gen_0':c:s:c19_6(a), gen_0':c:s:c19_6(+(1, n12_6))) -> *11_6, rt in Omega(n12_6)


Generator Equations:
gen_0':c:s:c19_6(0) <=> 0'
gen_0':c:s:c19_6(+(x, 1)) <=> s(gen_0':c:s:c19_6(x))
gen_c310_6(0) <=> hole_c33_6
gen_c310_6(+(x, 1)) <=> c3(gen_c310_6(x), c2(0'))


The following defined symbols remain to be analysed:
F, f