WORST_CASE(Omega(n^1),O(n^1))
proof of input_fllKSpXtNl.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 211 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 7 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 252 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 47 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^1)
(37) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CdtProblem
    (39) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
    (41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CpxRelTRS
    (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) typed CpxTrs
    (45) OrderProof [LOWER BOUND(ID), 18 ms]
    (46) typed CpxTrs
    (47) RewriteLemmaProof [LOWER BOUND(ID), 1871 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) 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))
   f(0, s(0), X) -> f(X, +(X, 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]
   f(0, s(0), X) -> f(X, +(X, 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]
   f(0, s(0), X) -> f(X, plus(X, 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]
   f(0, s(0), X) -> f(X, plus(X, 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
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:

   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]
   f(0, s(0), X) -> f(X, plus(X, 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
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]
   f(0, s(0), 0) -> f(0, 0, 0) [2]
   f(0, s(0), s(Y')) -> f(s(Y'), s(plus(s(Y'), Y')), s(Y')) [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
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:

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

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

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

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

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

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

(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:

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

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

(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:

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

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

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

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

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

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

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

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

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

   f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0
   f(z, z', z'') -{ 2 + z'' }-> f(1 + (z'' - 1), 1 + s', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1 + (z'' - 1) + (z'' - 1), z'' - 1 >= 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) [2 + z''], size: O(1) [0]

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

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

(36)
BOUNDS(1, n^1)

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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   +'_2, F_3, G_2

Compound Symbols:   c, c1_1, c2_2, c3, c4


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

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

(40)
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))
   F(0, s(0), z0) -> c2(F(z0, +(z0, z0), z0), +'(z0, z0))
   G(z0, z1) -> c3
   G(z0, z1) -> c4

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

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

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

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

(42)
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))
   F(0', s(0'), z0) -> c2(F(z0, +'(z0, z0), z0), +'(z0, z0))
   G(z0, z1) -> c3
   G(z0, z1) -> c4

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

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

Rewrite Strategy: INNERMOST
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(43) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(44)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
F(0', s(0'), z0) -> c2(F(z0, +'(z0, z0), z0), +'(z0, z0))
G(z0, z1) -> c3
G(z0, z1) -> c4
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
f(0', s(0'), z0) -> f(z0, +'(z0, 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
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c2
c2 :: c2 -> 0':c:s:c1 -> c2
G :: a -> b -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_5 :: 0':c:s:c1
hole_c22_5 :: c2
hole_c3:c43_5 :: c3:c4
hole_a4_5 :: a
hole_b5_5 :: b
hole_f6_5 :: f
hole_g7_5 :: g
gen_0':c:s:c18_5 :: Nat -> 0':c:s:c1
gen_c29_5 :: Nat -> c2

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

They will be analysed ascendingly in the following order:
+' < F
+' < f

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(46)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
F(0', s(0'), z0) -> c2(F(z0, +'(z0, z0), z0), +'(z0, z0))
G(z0, z1) -> c3
G(z0, z1) -> c4
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
f(0', s(0'), z0) -> f(z0, +'(z0, 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
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c2
c2 :: c2 -> 0':c:s:c1 -> c2
G :: a -> b -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_5 :: 0':c:s:c1
hole_c22_5 :: c2
hole_c3:c43_5 :: c3:c4
hole_a4_5 :: a
hole_b5_5 :: b
hole_f6_5 :: f
hole_g7_5 :: g
gen_0':c:s:c18_5 :: Nat -> 0':c:s:c1
gen_c29_5 :: Nat -> c2


Generator Equations:
gen_0':c:s:c18_5(0) <=> 0'
gen_0':c:s:c18_5(+(x, 1)) <=> s(gen_0':c:s:c18_5(x))
gen_c29_5(0) <=> hole_c22_5
gen_c29_5(+(x, 1)) <=> c2(gen_c29_5(x), 0')


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

They will be analysed ascendingly in the following order:
+' < F
+' < f

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(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':c:s:c18_5(a), gen_0':c:s:c18_5(+(1, n11_5))) -> *10_5, rt in Omega(n11_5)

Induction Base:
+'(gen_0':c:s:c18_5(a), gen_0':c:s:c18_5(+(1, 0)))

Induction Step:
+'(gen_0':c:s:c18_5(a), gen_0':c:s:c18_5(+(1, +(n11_5, 1)))) ->_R^Omega(1)
c1(+'(gen_0':c:s:c18_5(a), gen_0':c:s:c18_5(+(1, n11_5)))) ->_IH
c1(*10_5)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(48)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
F(0', s(0'), z0) -> c2(F(z0, +'(z0, z0), z0), +'(z0, z0))
G(z0, z1) -> c3
G(z0, z1) -> c4
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
f(0', s(0'), z0) -> f(z0, +'(z0, 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
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c2
c2 :: c2 -> 0':c:s:c1 -> c2
G :: a -> b -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_5 :: 0':c:s:c1
hole_c22_5 :: c2
hole_c3:c43_5 :: c3:c4
hole_a4_5 :: a
hole_b5_5 :: b
hole_f6_5 :: f
hole_g7_5 :: g
gen_0':c:s:c18_5 :: Nat -> 0':c:s:c1
gen_c29_5 :: Nat -> c2


Generator Equations:
gen_0':c:s:c18_5(0) <=> 0'
gen_0':c:s:c18_5(+(x, 1)) <=> s(gen_0':c:s:c18_5(x))
gen_c29_5(0) <=> hole_c22_5
gen_c29_5(+(x, 1)) <=> c2(gen_c29_5(x), 0')


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

They will be analysed ascendingly in the following order:
+' < F
+' < f

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(50) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(51)
BOUNDS(n^1, INF)

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(52)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(z0, s(z1)) -> c1(+'(z0, z1))
F(0', s(0'), z0) -> c2(F(z0, +'(z0, z0), z0), +'(z0, z0))
G(z0, z1) -> c3
G(z0, z1) -> c4
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))
f(0', s(0'), z0) -> f(z0, +'(z0, 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
F :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> c2
c2 :: c2 -> 0':c:s:c1 -> c2
G :: a -> b -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
f :: 0':c:s:c1 -> 0':c:s:c1 -> 0':c:s:c1 -> f
g :: g -> g -> g
hole_0':c:s:c11_5 :: 0':c:s:c1
hole_c22_5 :: c2
hole_c3:c43_5 :: c3:c4
hole_a4_5 :: a
hole_b5_5 :: b
hole_f6_5 :: f
hole_g7_5 :: g
gen_0':c:s:c18_5 :: Nat -> 0':c:s:c1
gen_c29_5 :: Nat -> c2


Lemmas:
+'(gen_0':c:s:c18_5(a), gen_0':c:s:c18_5(+(1, n11_5))) -> *10_5, rt in Omega(n11_5)


Generator Equations:
gen_0':c:s:c18_5(0) <=> 0'
gen_0':c:s:c18_5(+(x, 1)) <=> s(gen_0':c:s:c18_5(x))
gen_c29_5(0) <=> hole_c22_5
gen_c29_5(+(x, 1)) <=> c2(gen_c29_5(x), 0')


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