MAYBE
proof of input_KflHDtd6fD.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 183 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 227 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 718 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
        (38) CpxRNTS
    (39) CompletionProof [UPPER BOUND(ID), 0 ms]
    (40) CpxTypedWeightedCompleteTrs
        (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
(43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(44) CdtProblem
    (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) 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:

   zeros -> cons(0, n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0
   length(cons(N, L)) -> s(length(activate(L)))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

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:

   zeros -> cons(0', n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0'
   length(cons(N, L)) -> s(length(activate(L)))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

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:

   zeros -> cons(0, n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0
   length(cons(N, L)) -> s(length(activate(L)))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
and :: tt -> n__zeros:cons:nil -> n__zeros:cons:nil
tt :: tt
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil
s :: 0:s -> 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:

   and_2
   length_1

(c) The following functions are completely defined:

   activate_1
   zeros

Due to the following rules being added:
none

And the following fresh constants: none

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
and :: tt -> n__zeros:cons:nil -> n__zeros:cons:nil
tt :: tt
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil
s :: 0:s -> 0:s

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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, n__zeros)) -> s(length(zeros)) [2]
   length(cons(N, L)) -> s(length(L)) [2]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
and :: tt -> n__zeros:cons:nil -> n__zeros:cons:nil
tt :: tt
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil
s :: 0:s -> 0:s

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
   n__zeros => 0
   tt => 0
   nil => 1

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(zeros) :|: z = 1 + N + 0, N >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

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

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

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> X' :|: z' = X, X >= 0, z = 0, X' >= 0, X = X'
   and(z, z') -{ 2 }-> zeros :|: z' = X, X >= 0, z = 0, X = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 1 + N + 0, N >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

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

   { activate }
   { zeros }
   { length }
   { and }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {length}, {and}

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {length}, {and}

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

(23) 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: 1 + z

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(25) 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: 2

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   and(z, z') -{ 2 }-> zeros :|: z' >= 0, z = 0, z' = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 3 }-> s :|: s >= 0, s <= 1, z' >= 0, z = 0, z' = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 3 }-> s :|: s >= 0, s <= 1, z' >= 0, z = 0, z' = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 3 }-> s :|: s >= 0, s <= 1, z' >= 0, z = 0, z' = 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(39) 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:

   length(v0) -> null_length [0]

And the following fresh constants:    null_length

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

(40)
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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]
   length(v0) -> null_length [0]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s:null_length -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s:null_length
n__zeros :: n__zeros:cons:nil
and :: tt -> n__zeros:cons:nil -> n__zeros:cons:nil
tt :: tt
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
length :: n__zeros:cons:nil -> 0:s:null_length
nil :: n__zeros:cons:nil
s :: 0:s:null_length -> 0:s:null_length
null_length :: 0:s:null_length

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

(41) 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
   n__zeros => 0
   tt => 0
   nil => 1
   null_length => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length(activate(L)) :|: z = 1 + N + L, L >= 0, N >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Only complete derivations are relevant for the runtime complexity.

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   ZEROS -> c
   ZEROS -> c1
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c5(ZEROS)
   ACTIVATE(z0) -> c6
S tuples:
   ZEROS -> c
   ZEROS -> c1
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c5(ZEROS)
   ACTIVATE(z0) -> c6
K tuples:none
Defined Rule Symbols:   zeros, and_2, length_1, activate_1

Defined Pair Symbols:   ZEROS, AND_2, LENGTH_1, ACTIVATE_1

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5_1, c6


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

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   ACTIVATE(n__zeros) -> c5(ZEROS)
   LENGTH(nil) -> c3
   ZEROS -> c1
   ZEROS -> c
   ACTIVATE(z0) -> c6
   AND(tt, z0) -> c2(ACTIVATE(z0))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
S tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   zeros, and_2, length_1, activate_1

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_2


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

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)))
S tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)))
K tuples:none
Defined Rule Symbols:   zeros, and_2, length_1, activate_1

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(49) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)))
S tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1))) by 
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   activate(n__zeros) -> zeros
   activate(z0) -> z0

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
We considered the (Usable) Rules:
   zeros -> n__zeros
   zeros -> cons(0, n__zeros)
And the Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(LENGTH(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(n__zeros) = [1]
   POL(zeros) = [2]

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
Defined Rule Symbols:   zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) by 
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
Defined Rule Symbols:   zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
Defined Rule Symbols:   zeros

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(x0, z0)) -> c4(LENGTH(z0)) by 
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1


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

(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) by 
   LENGTH(cons(z0, cons(z1, n__zeros))) -> c4(LENGTH(cons(z1, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1

Compound Symbols:   c4_1