WORST_CASE(Omega(n^1),?)
proof of input_InxTT0uwCp.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 14 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 228 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), 80 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 342 ms]
        (20) typed CpxTrs


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

(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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
   binom(Nil, k) -> Cons(Nil, Nil)
   goal(x, y) -> binom(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   goal(z0, z1) -> binom(z0, z1)
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))
K tuples:none
Defined Rule Symbols:   @_2, binom_2, goal_2

Defined Pair Symbols:   @'_2, BINOM_2, GOAL_2

Compound Symbols:   c_1, c1, c2_2, c3_2, c4, c5, c6_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:

   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))

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

   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   goal(z0, z1) -> binom(z0, z1)

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:

   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))

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

   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   goal(z0, z1) -> binom(z0, z1)

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

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

(8)
Obligation:
Innermost TRS:
Rules:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
@', BINOM, binom, @

They will be analysed ascendingly in the following order:
@' < BINOM
binom < BINOM
@ < binom

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

(10)
Obligation:
Innermost TRS:
Rules:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5


Generator Equations:
gen_c:c15_7(0) <=> c1
gen_c:c15_7(+(x, 1)) <=> c(gen_c:c15_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c2:c3:c4:c57_7(0) <=> c4
gen_c2:c3:c4:c57_7(+(x, 1)) <=> c2(c1, gen_c2:c3:c4:c57_7(x))


The following defined symbols remain to be analysed:
@', BINOM, binom, @

They will be analysed ascendingly in the following order:
@' < BINOM
binom < BINOM
@ < binom

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
@'(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c:c15_7(n9_7), rt in Omega(1 + n9_7)

Induction Base:
@'(gen_Cons:Nil6_7(0), gen_Cons:Nil6_7(b)) ->_R^Omega(1)
c1

Induction Step:
@'(gen_Cons:Nil6_7(+(n9_7, 1)), gen_Cons:Nil6_7(b)) ->_R^Omega(1)
c(@'(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b))) ->_IH
c(gen_c:c15_7(c10_7))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5


Generator Equations:
gen_c:c15_7(0) <=> c1
gen_c:c15_7(+(x, 1)) <=> c(gen_c:c15_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c2:c3:c4:c57_7(0) <=> c4
gen_c2:c3:c4:c57_7(+(x, 1)) <=> c2(c1, gen_c2:c3:c4:c57_7(x))


The following defined symbols remain to be analysed:
@', BINOM, binom, @

They will be analysed ascendingly in the following order:
@' < BINOM
binom < BINOM
@ < binom

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5


Lemmas:
@'(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c:c15_7(n9_7), rt in Omega(1 + n9_7)


Generator Equations:
gen_c:c15_7(0) <=> c1
gen_c:c15_7(+(x, 1)) <=> c(gen_c:c15_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c2:c3:c4:c57_7(0) <=> c4
gen_c2:c3:c4:c57_7(+(x, 1)) <=> c2(c1, gen_c2:c3:c4:c57_7(x))


The following defined symbols remain to be analysed:
@, BINOM, binom

They will be analysed ascendingly in the following order:
binom < BINOM
@ < binom

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
@(gen_Cons:Nil6_7(n435_7), gen_Cons:Nil6_7(b)) -> gen_Cons:Nil6_7(+(n435_7, b)), rt in Omega(0)

Induction Base:
@(gen_Cons:Nil6_7(0), gen_Cons:Nil6_7(b)) ->_R^Omega(0)
gen_Cons:Nil6_7(b)

Induction Step:
@(gen_Cons:Nil6_7(+(n435_7, 1)), gen_Cons:Nil6_7(b)) ->_R^Omega(0)
Cons(Nil, @(gen_Cons:Nil6_7(n435_7), gen_Cons:Nil6_7(b))) ->_IH
Cons(Nil, gen_Cons:Nil6_7(+(b, c436_7)))

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:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5


Lemmas:
@'(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c:c15_7(n9_7), rt in Omega(1 + n9_7)
@(gen_Cons:Nil6_7(n435_7), gen_Cons:Nil6_7(b)) -> gen_Cons:Nil6_7(+(n435_7, b)), rt in Omega(0)


Generator Equations:
gen_c:c15_7(0) <=> c1
gen_c:c15_7(+(x, 1)) <=> c(gen_c:c15_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c2:c3:c4:c57_7(0) <=> c4
gen_c2:c3:c4:c57_7(+(x, 1)) <=> c2(c1, gen_c2:c3:c4:c57_7(x))


The following defined symbols remain to be analysed:
binom, BINOM

They will be analysed ascendingly in the following order:
binom < BINOM

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
binom(gen_Cons:Nil6_7(+(1, n1366_7)), gen_Cons:Nil6_7(+(1, n1366_7))) -> *8_7, rt in Omega(0)

Induction Base:
binom(gen_Cons:Nil6_7(+(1, 0)), gen_Cons:Nil6_7(+(1, 0)))

Induction Step:
binom(gen_Cons:Nil6_7(+(1, +(n1366_7, 1))), gen_Cons:Nil6_7(+(1, +(n1366_7, 1)))) ->_R^Omega(0)
@(binom(gen_Cons:Nil6_7(+(1, n1366_7)), gen_Cons:Nil6_7(+(1, n1366_7))), binom(gen_Cons:Nil6_7(+(1, n1366_7)), Cons(Nil, gen_Cons:Nil6_7(+(1, n1366_7))))) ->_IH
@(*8_7, binom(gen_Cons:Nil6_7(+(1, n1366_7)), Cons(Nil, gen_Cons:Nil6_7(+(1, n1366_7))))) ->_R^Omega(0)
@(*8_7, @(binom(gen_Cons:Nil6_7(n1366_7), gen_Cons:Nil6_7(+(1, n1366_7))), binom(gen_Cons:Nil6_7(n1366_7), Cons(Nil, gen_Cons:Nil6_7(+(1, n1366_7))))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(20)
Obligation:
Innermost TRS:
Rules:
@'(Cons(z0, z1), z2) -> c(@'(z1, z2))
@'(Nil, z0) -> c1
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
BINOM(Cons(z0, z1), Nil) -> c4
BINOM(Nil, z0) -> c5
GOAL(z0, z1) -> c6(BINOM(z0, z1))
@(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
@(Nil, z0) -> z0
binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
binom(Nil, z0) -> Cons(Nil, Nil)
goal(z0, z1) -> binom(z0, z1)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
BINOM :: Cons:Nil -> Cons:Nil -> c2:c3:c4:c5
c2 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
c3 :: c:c1 -> c2:c3:c4:c5 -> c2:c3:c4:c5
c4 :: c2:c3:c4:c5
c5 :: c2:c3:c4:c5
GOAL :: Cons:Nil -> Cons:Nil -> c6
c6 :: c2:c3:c4:c5 -> c6
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_c2:c3:c4:c53_7 :: c2:c3:c4:c5
hole_c64_7 :: c6
gen_c:c15_7 :: Nat -> c:c1
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c2:c3:c4:c57_7 :: Nat -> c2:c3:c4:c5


Lemmas:
@'(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c:c15_7(n9_7), rt in Omega(1 + n9_7)
@(gen_Cons:Nil6_7(n435_7), gen_Cons:Nil6_7(b)) -> gen_Cons:Nil6_7(+(n435_7, b)), rt in Omega(0)
binom(gen_Cons:Nil6_7(+(1, n1366_7)), gen_Cons:Nil6_7(+(1, n1366_7))) -> *8_7, rt in Omega(0)


Generator Equations:
gen_c:c15_7(0) <=> c1
gen_c:c15_7(+(x, 1)) <=> c(gen_c:c15_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c2:c3:c4:c57_7(0) <=> c4
gen_c2:c3:c4:c57_7(+(x, 1)) <=> c2(c1, gen_c2:c3:c4:c57_7(x))


The following defined symbols remain to be analysed:
BINOM