DifferenceBasis.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Difference bases

A difference basis for a finite set $S$ is a finite set $A$ such that every element of $S$ can be written as a difference $a - b$ of two elements $a, b \in A$; equivalently $S \subseteq A - A$. The multiplicative version is a ratio basis ($S \subseteq A / A$).

The classical case $S = \{0, 1, \ldots, N\}$ over $\mathbb{Z}$ (or $\mathbb{N}$) recovers the notion of a restricted difference basis, dual to the recreational notion of a sparse ruler: a difference basis for $\{0, \ldots, N\}$ of minimal size is an optimal ruler of length $N$.

Main definitions

Finset.IsRatioBasis A S (additive: Finset.IsDifferenceBasis A S): $S \subseteq A / A$.

Main results

Finset.isRatioBasis_iff: the elementwise characterization.
Finset.IsRatioBasis.mono_left / Finset.IsRatioBasis.mono_right: monotonicity.
Finset.IsRatioBasis.card_le_sq: a ratio basis for $S$ has at least $\sqrt{|S|}$ elements, i.e. $|S| \le |A|^2$.

References

[Wi63] Wichmann, B. "A note on restricted difference bases." Journal of the London Mathematical Society 38 (1963): 465-466.

Imports (1)

Mathlib.Algebra.Group.Pointwise.Finset.Basic

Imported by 1 file

FormalConjecturesForMathlib.lean

def Finset.IsRatioBasis line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Div, Finset
Proof uses: LE.le, HDiv.hDiv
5 instances, coercions and auxiliary constants left out.
theorem Finset.isRatioBasis_iff line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Div, Finset, Iff, Finset.IsRatioBasis, Membership.mem, Exists, And, Eq, HDiv.hDiv
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, LE.le, forall_congr, implies_congr, Eq.refl, iff_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Finset.IsRatioBasis.mono_left Shrinking the target set preserves being a difference basis line 70 source
Source documentation
Shrinking the target set preserves being a difference basis.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Div, Finset, Finset.IsRatioBasis, LE.le
Proof uses: LE.le.trans, HDiv.hDiv
8 instances, coercions and auxiliary constants left out.
```
theorem IsRatioBasis.mono_left (h : IsRatioBasis A S) (hTS : T ⊆ S) : IsRatioBasis A T :=
…
```
Mentions: Finset.IsRatioBasis
theorem Finset.IsRatioBasis.mono_right Enlarging the basis preserves being a difference basis line 75 source
Source documentation
Enlarging the basis preserves being a difference basis.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Div, Finset, Finset.IsRatioBasis, LE.le
Proof uses: LE.le.trans, HDiv.hDiv, Finset.div_subset_div
8 instances, coercions and auxiliary constants left out.
```
theorem IsRatioBasis.mono_right (h : IsRatioBasis A S) (hAB : A ⊆ B) : IsRatioBasis B S :=
…
```
Mentions: Finset.IsRatioBasis
theorem Finset.IsRatioBasis.card_le_sq A difference basis for $S$ has at least $\sqrt{|S|}$ elements: $|S| \le |A|^2$ line 80 source
Source documentation
A difference basis for $S$ has at least $\sqrt{|S|}$ elements: $|S| \le |A|^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Div, Finset, Finset.IsRatioBasis, LE.le, Nat, Finset.card, HPow.hPow, OfNat.ofNat
Proof uses: Trans.trans, Eq, HMul.hMul, HDiv.hDiv, Finset.card_le_card, Finset.card_div_le, Eq.symm, pow_two
21 instances, coercions and auxiliary constants left out.
```
theorem IsRatioBasis.card_le_sq (h : IsRatioBasis A S) : S.card ≤ A.card ^ 2 :=
…
```
Mentions: Finset.IsRatioBasis