KILLED
proof of input_orLz2LWRhR.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 5 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 317 ms]
        (16) typed CpxTrs
    (17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (18) TRS for Loop Detection
(19) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(20) CdtProblem
    (21) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (22) CdtProblem
    (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
        (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxTRS
        (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxWeightedTrs
            (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedTrs
            (33) CompletionProof [UPPER BOUND(ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
                (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (36) CpxTypedWeightedCompleteTrs
                (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (38) CpxRNTS
                (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxRNTS
            (41) CompletionProof [UPPER BOUND(ID), 0 ms]
            (42) CpxTypedWeightedCompleteTrs
                (43) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (44) CpxRNTS
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
        (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CdtProblem
        (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
        (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CdtProblem
        (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 704 ms]
        (54) CdtProblem
        (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1250 ms]
        (56) CdtProblem
(57) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(58) CpxWeightedTrs
    (59) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxTypedWeightedTrs
    (61) CompletionProof [UPPER BOUND(ID), 0 ms]
    (62) CpxTypedWeightedCompleteTrs
        (63) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CpxRNTS
    (69) CompletionProof [UPPER BOUND(ID), 0 ms]
    (70) CpxTypedWeightedCompleteTrs
        (71) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (72) 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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

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:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
K tuples:none
Defined Rule Symbols:   badd_2, goal_2

Defined Pair Symbols:   BADD_2, GOAL_2

Compound Symbols:   c_2, c1, 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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))

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

   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)

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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))

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

   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
BADD(z0, Nil) -> c1
GOAL(z0, z1) -> c2(BADD(z0, z1))
badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
badd(z0, Nil) -> z0
goal(z0, z1) -> badd(z0, z1)

Types:
BADD :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1 -> c:c1
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c:c1
GOAL :: Cons:Nil -> Cons:Nil -> c2
c2 :: c:c1 -> c2
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_3 :: c:c1
hole_Cons:Nil2_3 :: Cons:Nil
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_Cons:Nil5_3 :: Nat -> Cons:Nil

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
BADD, badd

They will be analysed ascendingly in the following order:
badd < BADD

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

(14)
Obligation:
Innermost TRS:
Rules:
BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
BADD(z0, Nil) -> c1
GOAL(z0, z1) -> c2(BADD(z0, z1))
badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
badd(z0, Nil) -> z0
goal(z0, z1) -> badd(z0, z1)

Types:
BADD :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1 -> c:c1
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c:c1
GOAL :: Cons:Nil -> Cons:Nil -> c2
c2 :: c:c1 -> c2
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_3 :: c:c1
hole_Cons:Nil2_3 :: Cons:Nil
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_Cons:Nil5_3 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(c1, gen_c:c14_3(x))
gen_Cons:Nil5_3(0) <=> Nil
gen_Cons:Nil5_3(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil5_3(x))


The following defined symbols remain to be analysed:
badd, BADD

They will be analysed ascendingly in the following order:
badd < BADD

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
badd(gen_Cons:Nil5_3(a), gen_Cons:Nil5_3(+(1, n7_3))) -> *6_3, rt in Omega(0)

Induction Base:
badd(gen_Cons:Nil5_3(a), gen_Cons:Nil5_3(+(1, 0)))

Induction Step:
badd(gen_Cons:Nil5_3(a), gen_Cons:Nil5_3(+(1, +(n7_3, 1)))) ->_R^Omega(0)
badd(Cons(Nil, Nil), badd(gen_Cons:Nil5_3(a), gen_Cons:Nil5_3(+(1, n7_3)))) ->_IH
badd(Cons(Nil, Nil), *6_3)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
BADD(z0, Nil) -> c1
GOAL(z0, z1) -> c2(BADD(z0, z1))
badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
badd(z0, Nil) -> z0
goal(z0, z1) -> badd(z0, z1)

Types:
BADD :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1 -> c:c1
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c:c1
GOAL :: Cons:Nil -> Cons:Nil -> c2
c2 :: c:c1 -> c2
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_3 :: c:c1
hole_Cons:Nil2_3 :: Cons:Nil
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_Cons:Nil5_3 :: Nat -> Cons:Nil


Lemmas:
badd(gen_Cons:Nil5_3(a), gen_Cons:Nil5_3(+(1, n7_3))) -> *6_3, rt in Omega(0)


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(c1, gen_c:c14_3(x))
gen_Cons:Nil5_3(0) <=> Nil
gen_Cons:Nil5_3(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil5_3(x))


The following defined symbols remain to be analysed:
BADD
----------------------------------------

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

(18)
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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))

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

   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)

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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
K tuples:none
Defined Rule Symbols:   badd_2, goal_2

Defined Pair Symbols:   BADD_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1


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

(21) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c2(BADD(z0, z1))
Removed 1 trailing nodes:
   BADD(z0, Nil) -> c1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
K tuples:none
Defined Rule Symbols:   badd_2, goal_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


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

(23) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> badd(z0, z1)

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


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

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

(26)
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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))

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

   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0

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

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

(28)
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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0

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

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

(30)
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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) [1]
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2)) [0]
   badd(z0, Nil) -> z0 [0]

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

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

(32)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) [1]
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2)) [0]
   badd(z0, Nil) -> z0 [0]

The TRS has the following type information:
BADD :: Cons:Nil -> Cons:Nil -> c
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c -> c -> c
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

(33) 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:

   BADD_2

(c) The following functions are completely defined:

   badd_2

Due to the following rules being added:

   badd(v0, v1) -> Nil [0]

And the following fresh constants:    const

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

(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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) [1]
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2)) [0]
   badd(z0, Nil) -> z0 [0]
   badd(v0, v1) -> Nil [0]

The TRS has the following type information:
BADD :: Cons:Nil -> Cons:Nil -> c
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c -> c -> c
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
const :: c

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

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

(36)
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:

   BADD(z0, Cons(z1, Cons(z1', z2'))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2'))), BADD(z0, Cons(z1', z2'))) [1]
   BADD(z0, Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil)) [1]
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), Nil), BADD(z0, z2)) [1]
   badd(z0, Cons(z1, Cons(z1'', z2''))) -> badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2''))) [0]
   badd(z0, Cons(z1, Nil)) -> badd(Cons(Nil, Nil), z0) [0]
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), Nil) [0]
   badd(z0, Nil) -> z0 [0]
   badd(v0, v1) -> Nil [0]

The TRS has the following type information:
BADD :: Cons:Nil -> Cons:Nil -> c
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c -> c -> c
Nil :: Cons:Nil
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
const :: c

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

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

   Nil => 0
   const => 0

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

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

   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, z0) + BADD(z0, 0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1 + 0
   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, badd(1 + 0 + 0, badd(z0, z2'))) + BADD(z0, 1 + z1' + z2') :|: z = z0, z1 >= 0, z1' >= 0, z2' >= 0, z' = 1 + z1 + (1 + z1' + z2'), z0 >= 0
   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, 0) + BADD(z0, z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1 + 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(z0, z2''))) :|: z = z0, z1 >= 0, z' = 1 + z1 + (1 + z1'' + z2''), z0 >= 0, z2'' >= 0, z1'' >= 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, 0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, z) + BADD(z, 0) :|: z' - 1 >= 0, z >= 0
   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, badd(1 + 0 + 0, badd(z, z2'))) + BADD(z, 1 + z1' + z2') :|: z1 >= 0, z1' >= 0, z2' >= 0, z' = 1 + z1 + (1 + z1' + z2'), z >= 0
   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, 0) + BADD(z, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, z) :|: z' - 1 >= 0, z >= 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(z, z2''))) :|: z1 >= 0, z' = 1 + z1 + (1 + z1'' + z2''), z >= 0, z2'' >= 0, z1'' >= 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

(41) 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:

   badd(v0, v1) -> null_badd [0]
   BADD(v0, v1) -> null_BADD [0]

And the following fresh constants:    null_badd, null_BADD

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

(42)
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:

   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) [1]
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2)) [0]
   badd(z0, Nil) -> z0 [0]
   badd(v0, v1) -> null_badd [0]
   BADD(v0, v1) -> null_BADD [0]

The TRS has the following type information:
BADD :: Cons:Nil:null_badd -> Cons:Nil:null_badd -> c:null_BADD
Cons :: Cons:Nil:null_badd -> Cons:Nil:null_badd -> Cons:Nil:null_badd
c :: c:null_BADD -> c:null_BADD -> c:null_BADD
Nil :: Cons:Nil:null_badd
badd :: Cons:Nil:null_badd -> Cons:Nil:null_badd -> Cons:Nil:null_badd
null_badd :: Cons:Nil:null_badd
null_BADD :: c:null_BADD

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

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

   Nil => 0
   null_badd => 0
   null_BADD => 0

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

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

   BADD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   BADD(z, z') -{ 1 }-> 1 + BADD(1 + 0 + 0, badd(z0, z2)) + BADD(z0, z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   badd(z, z') -{ 0 }-> badd(1 + 0 + 0, badd(z0, z2)) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   badd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) by 
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))
S tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
S tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2))) by 
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
S tuples:
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_1, c_2


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0)) by 
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
We considered the (Usable) Rules:none
And the Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( badd_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( Cons_2(x_1, x_2) ) =  	[[0], [4]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 4], [0, 0]] 	* 	x_2
>>>

<<<
 M( Nil ) =  	[[0], [0]]
>>>

Tuple symbols: 
<<<
 M( c_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( BADD_2(x_1, x_2) ) =  	[[0], [2]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
K tuples:
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


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

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
We considered the (Usable) Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
And the Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( badd_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[2, 0], [0, 1]] 	* 	x_1 	+ 	[[0, 0], [0, 1]] 	* 	x_2
>>>

<<<
 M( Cons_2(x_1, x_2) ) =  	[[0], [2]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 4], [0, 1]] 	* 	x_2
>>>

<<<
 M( Nil ) =  	[[0], [0]]
>>>

Tuple symbols: 
<<<
 M( c_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( BADD_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
K tuples:
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


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

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

(58)
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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

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

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

(60)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

(61) 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:

   goal_2

(c) The following functions are completely defined:

   badd_2

Due to the following rules being added:
none

And the following fresh constants: none

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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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
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(63) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(64)
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:

   badd(x', Cons(x, Cons(x1, xs'))) -> badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(x', xs'))) [2]
   badd(x', Cons(x, Nil)) -> badd(Cons(Nil, Nil), x') [2]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

   Nil => 0

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

   badd(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, x') :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x'
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(x', xs'))) :|: x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z = x', z' = 1 + x + (1 + x1 + xs')
   goal(z, z') -{ 1 }-> badd(x, y) :|: x >= 0, y >= 0, z = x, z' = y


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(67) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(68)
Obligation:
Complexity RNTS consisting of the following rules:

   badd(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, z) :|: z >= 0, z' - 1 >= 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   goal(z, z') -{ 1 }-> badd(z, z') :|: z >= 0, z' >= 0


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(69) 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: none

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(70)
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:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

   Nil => 0

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

   badd(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   badd(z, z') -{ 1 }-> badd(1 + 0 + 0, badd(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   goal(z, z') -{ 1 }-> badd(x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.