MAYBE
proof of input_LErKy0o1jC.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
        (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) typed CpxTrs
        (13) OrderProof [LOWER BOUND(ID), 10 ms]
        (14) typed CpxTrs
    (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (16) TRS for Loop Detection
(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), 411 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 114 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), 153 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
        (70) CpxRNTS
        (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 824 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
        (76) CpxRNTS
    (77) CompletionProof [UPPER BOUND(ID), 0 ms]
    (78) CpxTypedWeightedCompleteTrs
        (79) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (80) 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:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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:

   from(X) -> cons(X)
   length -> 0'
   length -> s(length1)
   length1 -> length

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:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length
Tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:   from_1, length, length1

Defined Pair Symbols:   FROM_1, LENGTH, LENGTH1

Compound Symbols:   c, c1, c2_1, c3_1


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

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

(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:

   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)

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

   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length

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

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

(10)
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:

   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)

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

   from(z0) -> cons(z0)
   length -> 0'
   length -> s(length1)
   length1 -> length

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

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

(12)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c
LENGTH -> c1
LENGTH -> c2(LENGTH1)
LENGTH1 -> c3(LENGTH)
from(z0) -> cons(z0)
length -> 0'
length -> s(length1)
length1 -> length

Types:
FROM :: a -> c
c :: c
LENGTH :: c1:c2
c1 :: c1:c2
c2 :: c3 -> c1:c2
LENGTH1 :: c3
c3 :: c1:c2 -> c3
from :: b -> cons
cons :: b -> cons
length :: 0':s
0' :: 0':s
s :: 0':s -> 0':s
length1 :: 0':s
hole_c1_4 :: c
hole_a2_4 :: a
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_cons5_4 :: cons
hole_b6_4 :: b
hole_0':s7_4 :: 0':s
gen_0':s8_4 :: Nat -> 0':s

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LENGTH, LENGTH1, length, length1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(14)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c
LENGTH -> c1
LENGTH -> c2(LENGTH1)
LENGTH1 -> c3(LENGTH)
from(z0) -> cons(z0)
length -> 0'
length -> s(length1)
length1 -> length

Types:
FROM :: a -> c
c :: c
LENGTH :: c1:c2
c1 :: c1:c2
c2 :: c3 -> c1:c2
LENGTH1 :: c3
c3 :: c1:c2 -> c3
from :: b -> cons
cons :: b -> cons
length :: 0':s
0' :: 0':s
s :: 0':s -> 0':s
length1 :: 0':s
hole_c1_4 :: c
hole_a2_4 :: a
hole_c1:c23_4 :: c1:c2
hole_c34_4 :: c3
hole_cons5_4 :: cons
hole_b6_4 :: b
hole_0':s7_4 :: 0':s
gen_0':s8_4 :: Nat -> 0':s


Generator Equations:
gen_0':s8_4(0) <=> 0'
gen_0':s8_4(+(x, 1)) <=> s(gen_0':s8_4(x))


The following defined symbols remain to be analysed:
length1, LENGTH, LENGTH1, length

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

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

(16)
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:

   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)

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

   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length
Tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:   from_1, length, length1

Defined Pair Symbols:   FROM_1, LENGTH, LENGTH1

Compound Symbols:   c, c1, c2_1, c3_1


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   LENGTH -> c1
   FROM(z0) -> c

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length
Tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:   from_1, length, length1

Defined Pair Symbols:   LENGTH, LENGTH1

Compound Symbols:   c2_1, c3_1


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

(21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH, LENGTH1

Compound Symbols:   c2_1, c3_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:

   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)

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:

   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)

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:

   LENGTH -> c2(LENGTH1) [1]
   LENGTH1 -> c3(LENGTH) [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:

   LENGTH -> c2(LENGTH1) [1]
   LENGTH1 -> c3(LENGTH) [1]

The TRS has the following type information:
LENGTH :: c2
c2 :: c3 -> c2
LENGTH1 :: c3
c3 :: c2 -> c3

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:

   LENGTH
   LENGTH1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   LENGTH -> c2(LENGTH1) [1]
   LENGTH1 -> c3(LENGTH) [1]

The TRS has the following type information:
LENGTH :: c2
c2 :: c3 -> c2
LENGTH1 :: c3
c3 :: c2 -> c3
const :: c2
const1 :: c3

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:

   LENGTH -> c2(LENGTH1) [1]
   LENGTH1 -> c3(LENGTH) [1]

The TRS has the following type information:
LENGTH :: c2
c2 :: c3 -> c2
LENGTH1 :: c3
c3 :: c2 -> c3
const :: c2
const1 :: c3

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
   const1 => 0

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

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

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 


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

(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:

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 


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

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

   { LENGTH, LENGTH1 }

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

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

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}

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

(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:

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: LENGTH1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

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

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  LENGTH: runtime: ?, size: O(1) [0]
  LENGTH1: runtime: ?, size: O(1) [1]

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

(45) 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: ?

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

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

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  LENGTH: runtime: INF, size: O(1) [0]
  LENGTH1: runtime: ?, size: O(1) [1]

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

(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, const1

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

(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:

   LENGTH -> c2(LENGTH1) [1]
   LENGTH1 -> c3(LENGTH) [1]

The TRS has the following type information:
LENGTH :: c2
c2 :: c3 -> c2
LENGTH1 :: c3
c3 :: c2 -> c3
const :: c2
const1 :: c3

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
   const1 => 0

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

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

   LENGTH -{ 1 }-> 1 + LENGTH1 :|: 
   LENGTH1 -{ 1 }-> 1 + LENGTH :|: 

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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s

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:

   from_1
   length
   length1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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
   const1 => 0

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

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

   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 


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

(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:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 


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

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

   { from }
   { length1, length }

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}

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

(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:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}
Previous analysis results are:
  from: runtime: ?, size: O(n^1) [1 + z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]
  length1: runtime: ?, size: INF
  length: runtime: ?, size: INF

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]
  length1: runtime: INF, size: INF
  length: runtime: ?, size: INF

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

(77) 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, const1

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

(78)
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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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

(79) 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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Only complete derivations are relevant for the runtime complexity.