MAYBE
proof of input_W5sNeyp3FW.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 9 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxTRS
    (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTRS
    (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxWeightedTrs
        (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxTypedWeightedTrs
        (31) CompletionProof [UPPER BOUND(ID), 0 ms]
        (32) CpxTypedWeightedCompleteTrs
            (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
            (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (36) CpxRNTS
            (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (38) CpxRNTS
            (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (40) CpxRNTS
            (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (42) CpxRNTS
            (43) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
            (46) CpxRNTS
        (47) CompletionProof [UPPER BOUND(ID), 0 ms]
        (48) CpxTypedWeightedCompleteTrs
            (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (50) CpxRNTS
(51) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(52) CpxWeightedTrs
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxTypedWeightedTrs
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxRNTS
        (63) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxRNTS
        (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 164 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
        (70) CpxRNTS
    (71) CompletionProof [UPPER BOUND(ID), 0 ms]
    (72) CpxTypedWeightedCompleteTrs
        (73) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS


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

(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, zeros)
   tail(cons(X, XS)) -> XS

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', zeros)
   tail(cons(X, XS)) -> XS

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1
S tuples:
   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1
K tuples:none
Defined Rule Symbols:   zeros, tail_1

Defined Pair Symbols:   ZEROS, TAIL_1

Compound Symbols:   c_1, c1


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

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

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


The TRS R consists of the following rules:

   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1

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

   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1

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

   zeros -> cons(0', zeros)
   tail(cons(z0, z1)) -> z1

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

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

(10)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c(ZEROS)
TAIL(cons(z0, z1)) -> c1
zeros -> cons(0', zeros)
tail(cons(z0, z1)) -> z1

Types:
ZEROS :: c
c :: c -> c
TAIL :: cons -> c1
cons :: 0' -> cons -> cons
c1 :: c1
zeros :: cons
0' :: 0'
tail :: cons -> cons
hole_c1_2 :: c
hole_c12_2 :: c1
hole_cons3_2 :: cons
hole_0'4_2 :: 0'
gen_c5_2 :: Nat -> c
gen_cons6_2 :: Nat -> cons

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ZEROS, zeros
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c(ZEROS)
TAIL(cons(z0, z1)) -> c1
zeros -> cons(0', zeros)
tail(cons(z0, z1)) -> z1

Types:
ZEROS :: c
c :: c -> c
TAIL :: cons -> c1
cons :: 0' -> cons -> cons
c1 :: c1
zeros :: cons
0' :: 0'
tail :: cons -> cons
hole_c1_2 :: c
hole_c12_2 :: c1
hole_cons3_2 :: cons
hole_0'4_2 :: 0'
gen_c5_2 :: Nat -> c
gen_cons6_2 :: Nat -> cons


Generator Equations:
gen_c5_2(0) <=> hole_c1_2
gen_c5_2(+(x, 1)) <=> c(gen_c5_2(x))
gen_cons6_2(0) <=> hole_cons3_2
gen_cons6_2(+(x, 1)) <=> cons(0', gen_cons6_2(x))


The following defined symbols remain to be analysed:
ZEROS, zeros
----------------------------------------

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1

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

   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1

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

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

(16)
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, zeros)
   tail(cons(X, XS)) -> XS

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1
S tuples:
   ZEROS -> c(ZEROS)
   TAIL(cons(z0, z1)) -> c1
K tuples:none
Defined Rule Symbols:   zeros, tail_1

Defined Pair Symbols:   ZEROS, TAIL_1

Compound Symbols:   c_1, c1


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TAIL(cons(z0, z1)) -> c1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c(ZEROS)
S tuples:
   ZEROS -> c(ZEROS)
K tuples:none
Defined Rule Symbols:   zeros, tail_1

Defined Pair Symbols:   ZEROS

Compound Symbols:   c_1


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

(21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   zeros -> cons(0, zeros)
   tail(cons(z0, z1)) -> z1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ZEROS -> c(ZEROS)
S tuples:
   ZEROS -> c(ZEROS)
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   ZEROS

Compound Symbols:   c_1


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

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

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   ZEROS -> c(ZEROS)

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

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

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   ZEROS -> c(ZEROS)

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

(27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
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 -> c(ZEROS) [1]

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

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

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   ZEROS -> c(ZEROS) [1]

The TRS has the following type information:
ZEROS :: c
c :: c -> c

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

(31) 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:

   ZEROS

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(32)
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 -> c(ZEROS) [1]

The TRS has the following type information:
ZEROS :: c
c :: c -> c
const :: c

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

(33) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(34)
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 -> c(ZEROS) [1]

The TRS has the following type information:
ZEROS :: c
c :: c -> c
const :: c

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

(35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   const => 0

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

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

   ZEROS -{ 1 }-> 1 + ZEROS :|: 


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

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

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

   ZEROS -{ 1 }-> 1 + ZEROS :|: 


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

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

   { ZEROS }

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

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

   ZEROS -{ 1 }-> 1 + ZEROS :|: 

Function symbols to be analyzed: {ZEROS}

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

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

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

   ZEROS -{ 1 }-> 1 + ZEROS :|: 

Function symbols to be analyzed: {ZEROS}

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

(43) 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: 0

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   ZEROS -{ 1 }-> 1 + ZEROS :|: 

Function symbols to be analyzed: {ZEROS}
Previous analysis results are:
  ZEROS: runtime: ?, size: O(1) [0]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   ZEROS -{ 1 }-> 1 + ZEROS :|: 

Function symbols to be analyzed: {ZEROS}
Previous analysis results are:
  ZEROS: runtime: INF, size: O(1) [0]

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

(47) 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:
none

And the following fresh constants:    const

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

(48)
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 -> c(ZEROS) [1]

The TRS has the following type information:
ZEROS :: c
c :: c -> c
const :: c

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

(49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   const => 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   ZEROS -{ 1 }-> 1 + ZEROS :|: 

Only complete derivations are relevant for the runtime complexity.

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

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

(52)
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, zeros) [1]
   tail(cons(X, XS)) -> XS [1]

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

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

(54)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   zeros -> cons(0, zeros) [1]
   tail(cons(X, XS)) -> XS [1]

The TRS has the following type information:
zeros :: cons
cons :: 0 -> cons -> cons
0 :: 0
tail :: cons -> cons

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

(55) 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:

   zeros
   tail_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(56)
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, zeros) [1]
   tail(cons(X, XS)) -> XS [1]

The TRS has the following type information:
zeros :: cons
cons :: 0 -> cons -> cons
0 :: 0
tail :: cons -> cons
const :: cons

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

(57) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(58)
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, zeros) [1]
   tail(cons(X, XS)) -> XS [1]

The TRS has the following type information:
zeros :: cons
cons :: 0 -> cons -> cons
0 :: 0
tail :: cons -> cons
const :: cons

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

(59) 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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

   { zeros }
   { tail }

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {zeros}, {tail}

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

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {zeros}, {tail}

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

(67) 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: 0

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {zeros}, {tail}
Previous analysis results are:
  zeros: runtime: ?, size: O(1) [0]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {zeros}, {tail}
Previous analysis results are:
  zeros: runtime: INF, size: O(1) [0]

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

(71) 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:

   tail(v0) -> null_tail [0]

And the following fresh constants:    null_tail

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

(72)
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, zeros) [1]
   tail(cons(X, XS)) -> XS [1]
   tail(v0) -> null_tail [0]

The TRS has the following type information:
zeros :: cons:null_tail
cons :: 0 -> cons:null_tail -> cons:null_tail
0 :: 0
tail :: cons:null_tail -> cons:null_tail
null_tail :: cons:null_tail

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

(73) 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_tail => 0

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   tail(z) -{ 1 }-> XS :|: z = 1 + X + XS, X >= 0, XS >= 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Only complete derivations are relevant for the runtime complexity.