MAYBE
proof of input_vpQNFtUlLF.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(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), 0 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), 212 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 337 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 3177 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 727 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 72 ms]
    (48) CdtProblem
    (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem


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

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


The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

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

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

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0') -> s(0')
   g(s(X)) -> s(s(g(X)))
   sel(0', cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [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:

   f(X) -> cons(X, f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]

The TRS has the following type information:
f :: 0:s -> cons
cons :: 0:s -> cons -> cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> cons -> 0:s

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:

   f_1
   sel_2

(c) The following functions are completely defined:

   g_1

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   f(X) -> cons(X, f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]

The TRS has the following type information:
f :: 0:s -> cons
cons :: 0:s -> cons -> cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> cons -> 0:s
const :: 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:

   f(0) -> cons(0, f(s(0))) [2]
   f(s(X')) -> cons(s(X'), f(s(s(g(X'))))) [2]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]

The TRS has the following type information:
f :: 0:s -> cons
cons :: 0:s -> cons -> cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> cons -> 0:s
const :: 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:

   0 => 0
   const => 0

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

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

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


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

(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:

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


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

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

   { g }
   { sel }
   { f }

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

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

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

Function symbols to be analyzed: {g}, {sel}, {f}

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(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:

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

Function symbols to be analyzed: {g}, {sel}, {f}

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

(21) 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: 1 + 2*z

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sel
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:

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sel
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:

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

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

(33) 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

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

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

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   f(z) -{ 2 }-> 1 + 0 + f(1 + 0) :|: z = 0
   f(z) -{ 2 + z }-> 1 + (1 + (z - 1)) + f(1 + (1 + s)) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

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

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

(37) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   sel(v0, v1) -> null_sel [0]
   g(v0) -> null_g [0]

And the following fresh constants:    null_sel, null_g, const

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

(38)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   sel(v0, v1) -> null_sel [0]
   g(v0) -> null_g [0]

The TRS has the following type information:
f :: 0:s:null_sel:null_g -> cons
cons :: 0:s:null_sel:null_g -> cons -> cons
g :: 0:s:null_sel:null_g -> 0:s:null_sel:null_g
0 :: 0:s:null_sel:null_g
s :: 0:s:null_sel:null_g -> 0:s:null_sel:null_g
sel :: 0:s:null_sel:null_g -> cons -> 0:s:null_sel:null_g
null_sel :: 0:s:null_sel:null_g
null_g :: 0:s:null_sel:null_g
const :: cons

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

(39) 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
   null_sel => 0
   null_g => 0
   const => 0

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

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

   f(z) -{ 1 }-> 1 + X + f(g(X)) :|: X >= 0, z = X
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + g(X)) :|: z = 1 + X, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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(41) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> cons(z0, f(g(z0)))
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(0) -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(0) -> c1
   G(s(z0)) -> c2(G(z0))
   SEL(0, cons(z0, z1)) -> c3
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   f_1, g_1, sel_2

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   G(0) -> c1
   SEL(0, cons(z0, z1)) -> c3

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> cons(z0, f(g(z0)))
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   f_1, g_1, sel_2

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c2_1, c4_1


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0) -> cons(z0, f(g(z0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c2_1, c4_1


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

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(F(x_1)) = 0
   POL(G(x_1)) = 0
   POL(SEL(x_1, x_2)) = x_2^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(g(x_1)) = [1]
   POL(s(x_1)) = 0

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(z0)) -> c2(G(z0))
K tuples:
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c2_1, c4_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(s(z0)) -> c2(G(z0)) by 
   G(s(s(y0))) -> c2(G(s(y0)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
   G(s(s(y0))) -> c2(G(s(y0)))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(y0))) -> c2(G(s(y0)))
K tuples:
   SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2))
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, SEL_2, G_1

Compound Symbols:   c_2, c4_1, c2_1


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) by 
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(y0))) -> c2(G(s(y0)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2)))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(y0))) -> c2(G(s(y0)))
K tuples:
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c2_1, c4_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(s(s(y0))) -> c2(G(s(y0))) by 
   G(s(s(s(y0)))) -> c2(G(s(s(y0))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2)))
   G(s(s(s(y0)))) -> c2(G(s(s(y0))))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(s(y0)))) -> c2(G(s(s(y0))))
K tuples:
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, SEL_2, G_1

Compound Symbols:   c_2, c4_1, c2_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) by 
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> s(0)
   g(s(z0)) -> s(s(g(z0)))
Tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(s(y0)))) -> c2(G(s(s(y0))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   F(z0) -> c(F(g(z0)), G(z0))
   G(s(s(s(y0)))) -> c2(G(s(s(y0))))
K tuples:
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:   g_1

Defined Pair Symbols:   F_1, G_1, SEL_2

Compound Symbols:   c_2, c2_1, c4_1