WORST_CASE(Omega(n^1),?)
proof of input_Vts1bnq7cI.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 9 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 305 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 34.9 s]
        (18) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs))
   nestinc(Nil) -> number17(Nil)
   nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs)))
   inc(Nil) -> Cons(Nil, Nil)
   number17(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
   goal(x) -> nestinc(x)

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
   inc(Nil) -> Cons(Nil, Nil)
   nestinc(Nil) -> number17(Nil)
   nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
   number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
   goal(z0) -> nestinc(z0)
Tuples:
   INC(Cons(z0, z1)) -> c(INC(z1))
   INC(Nil) -> c1
   NESTINC(Nil) -> c2(NUMBER17(Nil))
   NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTINC(z0))
S tuples:
   INC(Cons(z0, z1)) -> c(INC(z1))
   INC(Nil) -> c1
   NESTINC(Nil) -> c2(NUMBER17(Nil))
   NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTINC(z0))
K tuples:none
Defined Rule Symbols:   inc_1, nestinc_1, number17_1, goal_1

Defined Pair Symbols:   INC_1, NESTINC_1, NUMBER17_1, GOAL_1

Compound Symbols:   c_1, c1, c2_1, c3_2, c4, c5_1


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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   INC(Cons(z0, z1)) -> c(INC(z1))
   INC(Nil) -> c1
   NESTINC(Nil) -> c2(NUMBER17(Nil))
   NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTINC(z0))

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

   inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
   inc(Nil) -> Cons(Nil, Nil)
   nestinc(Nil) -> number17(Nil)
   nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
   number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
   goal(z0) -> nestinc(z0)

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

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

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


The TRS R consists of the following rules:

   INC(Cons(z0, z1)) -> c(INC(z1))
   INC(Nil) -> c1
   NESTINC(Nil) -> c2(NUMBER17(Nil))
   NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTINC(z0))

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

   inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
   inc(Nil) -> Cons(Nil, Nil)
   nestinc(Nil) -> number17(Nil)
   nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
   number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
   goal(z0) -> nestinc(z0)

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

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

(8)
Obligation:
Innermost TRS:
Rules:
INC(Cons(z0, z1)) -> c(INC(z1))
INC(Nil) -> c1
NESTINC(Nil) -> c2(NUMBER17(Nil))
NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
NUMBER17(z0) -> c4
GOAL(z0) -> c5(NESTINC(z0))
inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
inc(Nil) -> Cons(Nil, Nil)
nestinc(Nil) -> number17(Nil)
nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(z0) -> nestinc(z0)

Types:
INC :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
NESTINC :: Cons:Nil -> c2:c3
c2 :: c4 -> c2:c3
NUMBER17 :: Cons:Nil -> c4
c3 :: c2:c3 -> c:c1 -> c2:c3
inc :: Cons:Nil -> Cons:Nil
c4 :: c4
GOAL :: Cons:Nil -> c5
c5 :: c2:c3 -> c5
nestinc :: Cons:Nil -> Cons:Nil
number17 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_6 :: c:c1
hole_Cons:Nil2_6 :: Cons:Nil
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_c55_6 :: c5
gen_c:c16_6 :: Nat -> c:c1
gen_Cons:Nil7_6 :: Nat -> Cons:Nil
gen_c2:c38_6 :: Nat -> c2:c3

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
INC, NESTINC, inc, nestinc

They will be analysed ascendingly in the following order:
INC < NESTINC
inc < NESTINC
inc < nestinc

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

(10)
Obligation:
Innermost TRS:
Rules:
INC(Cons(z0, z1)) -> c(INC(z1))
INC(Nil) -> c1
NESTINC(Nil) -> c2(NUMBER17(Nil))
NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
NUMBER17(z0) -> c4
GOAL(z0) -> c5(NESTINC(z0))
inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
inc(Nil) -> Cons(Nil, Nil)
nestinc(Nil) -> number17(Nil)
nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(z0) -> nestinc(z0)

Types:
INC :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
NESTINC :: Cons:Nil -> c2:c3
c2 :: c4 -> c2:c3
NUMBER17 :: Cons:Nil -> c4
c3 :: c2:c3 -> c:c1 -> c2:c3
inc :: Cons:Nil -> Cons:Nil
c4 :: c4
GOAL :: Cons:Nil -> c5
c5 :: c2:c3 -> c5
nestinc :: Cons:Nil -> Cons:Nil
number17 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_6 :: c:c1
hole_Cons:Nil2_6 :: Cons:Nil
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_c55_6 :: c5
gen_c:c16_6 :: Nat -> c:c1
gen_Cons:Nil7_6 :: Nat -> Cons:Nil
gen_c2:c38_6 :: Nat -> c2:c3


Generator Equations:
gen_c:c16_6(0) <=> c1
gen_c:c16_6(+(x, 1)) <=> c(gen_c:c16_6(x))
gen_Cons:Nil7_6(0) <=> Nil
gen_Cons:Nil7_6(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_6(x))
gen_c2:c38_6(0) <=> c2(c4)
gen_c2:c38_6(+(x, 1)) <=> c3(gen_c2:c38_6(x), c1)


The following defined symbols remain to be analysed:
INC, NESTINC, inc, nestinc

They will be analysed ascendingly in the following order:
INC < NESTINC
inc < NESTINC
inc < nestinc

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_Cons:Nil7_6(n10_6)) -> gen_c:c16_6(n10_6), rt in Omega(1 + n10_6)

Induction Base:
INC(gen_Cons:Nil7_6(0)) ->_R^Omega(1)
c1

Induction Step:
INC(gen_Cons:Nil7_6(+(n10_6, 1))) ->_R^Omega(1)
c(INC(gen_Cons:Nil7_6(n10_6))) ->_IH
c(gen_c:c16_6(c11_6))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
INC(Cons(z0, z1)) -> c(INC(z1))
INC(Nil) -> c1
NESTINC(Nil) -> c2(NUMBER17(Nil))
NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
NUMBER17(z0) -> c4
GOAL(z0) -> c5(NESTINC(z0))
inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
inc(Nil) -> Cons(Nil, Nil)
nestinc(Nil) -> number17(Nil)
nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(z0) -> nestinc(z0)

Types:
INC :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
NESTINC :: Cons:Nil -> c2:c3
c2 :: c4 -> c2:c3
NUMBER17 :: Cons:Nil -> c4
c3 :: c2:c3 -> c:c1 -> c2:c3
inc :: Cons:Nil -> Cons:Nil
c4 :: c4
GOAL :: Cons:Nil -> c5
c5 :: c2:c3 -> c5
nestinc :: Cons:Nil -> Cons:Nil
number17 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_6 :: c:c1
hole_Cons:Nil2_6 :: Cons:Nil
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_c55_6 :: c5
gen_c:c16_6 :: Nat -> c:c1
gen_Cons:Nil7_6 :: Nat -> Cons:Nil
gen_c2:c38_6 :: Nat -> c2:c3


Generator Equations:
gen_c:c16_6(0) <=> c1
gen_c:c16_6(+(x, 1)) <=> c(gen_c:c16_6(x))
gen_Cons:Nil7_6(0) <=> Nil
gen_Cons:Nil7_6(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_6(x))
gen_c2:c38_6(0) <=> c2(c4)
gen_c2:c38_6(+(x, 1)) <=> c3(gen_c2:c38_6(x), c1)


The following defined symbols remain to be analysed:
INC, NESTINC, inc, nestinc

They will be analysed ascendingly in the following order:
INC < NESTINC
inc < NESTINC
inc < nestinc

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

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
INC(Cons(z0, z1)) -> c(INC(z1))
INC(Nil) -> c1
NESTINC(Nil) -> c2(NUMBER17(Nil))
NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
NUMBER17(z0) -> c4
GOAL(z0) -> c5(NESTINC(z0))
inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
inc(Nil) -> Cons(Nil, Nil)
nestinc(Nil) -> number17(Nil)
nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(z0) -> nestinc(z0)

Types:
INC :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
NESTINC :: Cons:Nil -> c2:c3
c2 :: c4 -> c2:c3
NUMBER17 :: Cons:Nil -> c4
c3 :: c2:c3 -> c:c1 -> c2:c3
inc :: Cons:Nil -> Cons:Nil
c4 :: c4
GOAL :: Cons:Nil -> c5
c5 :: c2:c3 -> c5
nestinc :: Cons:Nil -> Cons:Nil
number17 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_6 :: c:c1
hole_Cons:Nil2_6 :: Cons:Nil
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_c55_6 :: c5
gen_c:c16_6 :: Nat -> c:c1
gen_Cons:Nil7_6 :: Nat -> Cons:Nil
gen_c2:c38_6 :: Nat -> c2:c3


Lemmas:
INC(gen_Cons:Nil7_6(n10_6)) -> gen_c:c16_6(n10_6), rt in Omega(1 + n10_6)


Generator Equations:
gen_c:c16_6(0) <=> c1
gen_c:c16_6(+(x, 1)) <=> c(gen_c:c16_6(x))
gen_Cons:Nil7_6(0) <=> Nil
gen_Cons:Nil7_6(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_6(x))
gen_c2:c38_6(0) <=> c2(c4)
gen_c2:c38_6(+(x, 1)) <=> c3(gen_c2:c38_6(x), c1)


The following defined symbols remain to be analysed:
inc, NESTINC, nestinc

They will be analysed ascendingly in the following order:
inc < NESTINC
inc < nestinc

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_Cons:Nil7_6(+(1, n240_6))) -> *9_6, rt in Omega(0)

Induction Base:
inc(gen_Cons:Nil7_6(+(1, 0)))

Induction Step:
inc(gen_Cons:Nil7_6(+(1, +(n240_6, 1)))) ->_R^Omega(0)
Cons(Cons(Nil, Nil), inc(gen_Cons:Nil7_6(+(1, n240_6)))) ->_IH
Cons(Cons(Nil, Nil), *9_6)

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

(18)
Obligation:
Innermost TRS:
Rules:
INC(Cons(z0, z1)) -> c(INC(z1))
INC(Nil) -> c1
NESTINC(Nil) -> c2(NUMBER17(Nil))
NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1)))
NUMBER17(z0) -> c4
GOAL(z0) -> c5(NESTINC(z0))
inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1))
inc(Nil) -> Cons(Nil, Nil)
nestinc(Nil) -> number17(Nil)
nestinc(Cons(z0, z1)) -> nestinc(inc(Cons(z0, z1)))
number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(z0) -> nestinc(z0)

Types:
INC :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
NESTINC :: Cons:Nil -> c2:c3
c2 :: c4 -> c2:c3
NUMBER17 :: Cons:Nil -> c4
c3 :: c2:c3 -> c:c1 -> c2:c3
inc :: Cons:Nil -> Cons:Nil
c4 :: c4
GOAL :: Cons:Nil -> c5
c5 :: c2:c3 -> c5
nestinc :: Cons:Nil -> Cons:Nil
number17 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_6 :: c:c1
hole_Cons:Nil2_6 :: Cons:Nil
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_c55_6 :: c5
gen_c:c16_6 :: Nat -> c:c1
gen_Cons:Nil7_6 :: Nat -> Cons:Nil
gen_c2:c38_6 :: Nat -> c2:c3


Lemmas:
INC(gen_Cons:Nil7_6(n10_6)) -> gen_c:c16_6(n10_6), rt in Omega(1 + n10_6)
inc(gen_Cons:Nil7_6(+(1, n240_6))) -> *9_6, rt in Omega(0)


Generator Equations:
gen_c:c16_6(0) <=> c1
gen_c:c16_6(+(x, 1)) <=> c(gen_c:c16_6(x))
gen_Cons:Nil7_6(0) <=> Nil
gen_Cons:Nil7_6(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_6(x))
gen_c2:c38_6(0) <=> c2(c4)
gen_c2:c38_6(+(x, 1)) <=> c3(gen_c2:c38_6(x), c1)


The following defined symbols remain to be analysed:
NESTINC, nestinc