MAYBE
proof of input_oAzeo8StMy.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 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), 381 ms]
                (44) CpxRNTS
                (45) IntTrsBoundProof [UPPER BOUND(ID), 184 ms]
                (46) CpxRNTS
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
    (51) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
        (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
(55) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(56) CpxWeightedTrs
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxTypedWeightedTrs
    (59) CompletionProof [UPPER BOUND(ID), 0 ms]
    (60) CpxTypedWeightedCompleteTrs
        (61) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxTypedWeightedCompleteTrs
        (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
        (65) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CpxRNTS
        (67) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CpxRNTS
        (69) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 874 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (74) CpxRNTS
    (75) CompletionProof [UPPER BOUND(ID), 0 ms]
    (76) CpxTypedWeightedCompleteTrs
        (77) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (78) 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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

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:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1

Defined Pair Symbols:   2ND_1, FROM_1

Compound Symbols:   c, c1_1, c2_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:

   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))

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

   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))

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:

   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))

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

   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
2ND(cons1(z0, cons(z1, z2))) -> c
2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
FROM(z0) -> c2(FROM(s(z0)))
2nd(cons1(z0, cons(z1, z2))) -> z1
2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
from(z0) -> cons(z0, from(s(z0)))

Types:
2ND :: cons:cons1 -> c:c1
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
c :: c:c1
c1 :: c:c1 -> c:c1
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
2nd :: cons:cons1 -> s
from :: s -> cons:cons1
hole_c:c11_3 :: c:c1
hole_cons:cons12_3 :: cons:cons1
hole_s3_3 :: s
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_cons:cons16_3 :: Nat -> cons:cons1
gen_s7_3 :: Nat -> s
gen_c28_3 :: Nat -> c2

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
2ND, FROM, 2nd, from
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
2ND(cons1(z0, cons(z1, z2))) -> c
2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
FROM(z0) -> c2(FROM(s(z0)))
2nd(cons1(z0, cons(z1, z2))) -> z1
2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
from(z0) -> cons(z0, from(s(z0)))

Types:
2ND :: cons:cons1 -> c:c1
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
c :: c:c1
c1 :: c:c1 -> c:c1
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
2nd :: cons:cons1 -> s
from :: s -> cons:cons1
hole_c:c11_3 :: c:c1
hole_cons:cons12_3 :: cons:cons1
hole_s3_3 :: s
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_cons:cons16_3 :: Nat -> cons:cons1
gen_s7_3 :: Nat -> s
gen_c28_3 :: Nat -> c2


Generator Equations:
gen_c:c15_3(0) <=> c
gen_c:c15_3(+(x, 1)) <=> c1(gen_c:c15_3(x))
gen_cons:cons16_3(0) <=> hole_cons:cons12_3
gen_cons:cons16_3(+(x, 1)) <=> cons(hole_s3_3, gen_cons:cons16_3(x))
gen_s7_3(0) <=> hole_s3_3
gen_s7_3(+(x, 1)) <=> s(gen_s7_3(x))
gen_c28_3(0) <=> hole_c24_3
gen_c28_3(+(x, 1)) <=> c2(gen_c28_3(x))


The following defined symbols remain to be analysed:
2ND, FROM, 2nd, from
----------------------------------------

(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:

   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))

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

   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))

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:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1

Defined Pair Symbols:   2ND_1, FROM_1

Compound Symbols:   c, c1_1, c2_1


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   2ND(cons1(z0, cons(z1, z2))) -> c

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

(21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_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:

   FROM(z0) -> c2(FROM(s(z0)))

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:

   FROM(z0) -> c2(FROM(s(z0)))

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:

   FROM(z0) -> c2(FROM(s(z0))) [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:

   FROM(z0) -> c2(FROM(s(z0))) [1]

The TRS has the following type information:
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s

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:

   FROM_1

(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:

   FROM(z0) -> c2(FROM(s(z0))) [1]

The TRS has the following type information:
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
const :: c2
const1 :: s

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:

   FROM(z0) -> c2(FROM(s(z0))) [1]

The TRS has the following type information:
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
const :: c2
const1 :: s

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:

   FROM(z) -{ 1 }-> 1 + FROM(1 + z0) :|: z = z0, z0 >= 0


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

(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:

   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0


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

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

   { FROM }

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

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

   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0

Function symbols to be analyzed: {FROM}

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

(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:

   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0

Function symbols to be analyzed: {FROM}

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: FROM
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:

   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0

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

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0

Function symbols to be analyzed: {FROM}
Previous analysis results are:
  FROM: 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, 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:

   FROM(z0) -> c2(FROM(s(z0))) [1]

The TRS has the following type information:
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
const :: c2
const1 :: s

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:

   FROM(z) -{ 1 }-> 1 + FROM(1 + z0) :|: z = z0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(51) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by 
   FROM(s(x0)) -> c2(FROM(s(s(x0))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
S tuples:
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

(53) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
S tuples:
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

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

(56)
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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

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

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

(58)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s

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

(59) 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:

   2nd_1
   from_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(60)
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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s
const :: s
const1 :: cons:cons1

Rewrite Strategy: INNERMOST
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(61) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(62)
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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s
const :: s
const1 :: cons:cons1

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

(63) 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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X


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(65) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(66)
Obligation:
Complexity RNTS consisting of the following rules:

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0


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

   { from }
   { 2nd }

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}

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(69) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(70)
Obligation:
Complexity RNTS consisting of the following rules:

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}

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(71) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}
Previous analysis results are:
  from: runtime: ?, size: O(1) [0]

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(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}
Previous analysis results are:
  from: runtime: INF, size: O(1) [0]

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(75) 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:

   2nd(v0) -> null_2nd [0]

And the following fresh constants:    null_2nd, const

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(76)
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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   2nd(v0) -> null_2nd [0]

The TRS has the following type information:
2nd :: cons:cons1 -> s:null_2nd
cons1 :: s:null_2nd -> cons:cons1 -> cons:cons1
cons :: s:null_2nd -> cons:cons1 -> cons:cons1
from :: s:null_2nd -> cons:cons1
s :: s:null_2nd -> s:null_2nd
null_2nd :: s:null_2nd
const :: cons:cons1

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

   null_2nd => 0
   const => 0

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(78)
Obligation:
Complexity RNTS consisting of the following rules:

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   2nd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X

Only complete derivations are relevant for the runtime complexity.