Convolution.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 0 contain sorry, 0 declared as axioms

Convolution of Functions on ℕ

This file defines the sum (∗) convolution of functions ℕ → R.

Main Definitions

AdditiveCombinatorics.sumConv: The sum convolution f ∗ g.

Notation

f ∗ g for sumConv f g.

TODO

f ∘ g for diffConv f g.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def AdditiveCombinatorics.sumConv The sum convolution of two functions f, g : ℕ → R, also known as the Cauchy product line 50 source
Source documentation
The sum convolution of two functions f, g : ℕ → R, also known as the Cauchy product. (f ∗ g) n = ∑_{a+b=n} f(a)g(b).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, Nat
Proof uses: Finset.sum, Prod, Finset.HasAntidiagonal.antidiagonal, HMul.hMul, Prod.fst, Prod.snd
6 instances, coercions and auxiliary constants left out.
```
def sumConv (f g : ℕ → R) (n : ℕ) : R := ∑ p ∈ antidiagonal n, f p.1 * g p.2
…
```
def AdditiveCombinatorics.sumRep The number of sum representations is the sum convolution of A's indicator function with itself: $1_A\ast 1_A(n)$ line 56 source
Source documentation
The number of sum representations is the sum convolution of A's indicator function with itself: $1_A\ast 1_A(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: AdditiveCombinatorics.sumConv, Set.indicatorOne
4 instances, coercions and auxiliary constants left out.
```
noncomputable def sumRep (A : Set ℕ) : ℕ → ℕ := (𝟙_A ∗ 𝟙_A)
…
```
lemma AdditiveCombinatorics.sumRep_def line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Eq, AdditiveCombinatorics.sumRep, Finset.card, Prod, Finset.filter, And, Membership.mem, Prod.fst, Prod.snd, Classical.propDecidable, Finset.HasAntidiagonal.antidiagonal
Proof uses: Eq.mpr, Finset.sum, ite, OfNat.ofNat, id, congrFun', congrArg, Finset.sum_congr, HMul.hMul, Set.indicatorOne, Eq.refl, Finset, Eq.trans, congr, ite_congr, Classical.decPred, Not, mul_ite, mul_one, MulZeroClass.mul_zero and 1 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem AdditiveCombinatorics.sumRep_eq_powerSeries_coeff line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Eq, AdditiveCombinatorics.sumRep, LinearMap, RingHom.id, PowerSeries, Unit, PowerSeries.coeff, HMul.hMul, PowerSeries.mk, Set.indicatorOne
Proof uses: of_eq_true, Eq.trans, Finset.sum, Prod, Finset.HasAntidiagonal.antidiagonal, ite, Membership.mem, Prod.snd, Classical.decPred, Prod.fst, OfNat.ofNat, True, congr, congrArg, funext, mul_ite, ite.congr_simp, Eq.refl, mul_one, MulZeroClass.mul_zero and 6 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.