fact stringlengths 28 3.53k | type stringclasses 8
values | library stringclasses 3
values | imports listlengths 1 7 | filename stringlengths 33 87 | symbolic_name stringlengths 1 87 | docstring stringlengths 17 494 ⌀ |
|---|---|---|---|---|---|---|
Prime.squarefree {p : ℕ} (hp : p.Prime) : Squarefree p := Irreducible.squarefree hp | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | Prime.squarefree | null |
squarefree_squarefreePart (n : ℕ) : Squarefree n.squarefreePart := by refine Nat.squarefree_iff_factorization_le_one n.squarefreePart_ne_zero |>.2 fun p ↦ ?_ by_cases hp : p.Prime · linarith [n.squarefreePart_factorization hp, Nat.mod_lt (n.factorization p) two_pos] · linarith [factorization_eq_zero_of_non_prime n.squa... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarefree_squarefreePart | null |
squarefreePart_dvd (n : ℕ) : squarefreePart n ∣ n := by rcases eq_or_ne n 0 with (rfl | h₀); simp exact Nat.factorization_prime_le_iff_dvd n.squarefreePart_ne_zero h₀ |>.1 fun p hp ↦ squarefreePart_factorization _ hp ▸ Nat.mod_le _ _ /-- The square part is the value of `b ^ 2` in the squarefree decomposition of `n = a₀... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarefreePart_dvd | null |
squarePart (n : ℕ) : ℕ := n / n.squarefreePart | def | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarePart | /-- The square part is the value of `b ^ 2` in the squarefree decomposition of `n = a₀ * b ^ 2`.-/ |
squarePart_zero : squarePart 0 = 0 := by simp [squarePart] /-- The squarefree decomposition of a natural number. -/ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarePart_zero | /-- The square part is the value of `b ^ 2` in the squarefree decomposition of `n = a₀ * b ^ 2`.-/ |
squarefreePart_mul_squarePart (n : ℕ) : n.squarefreePart * n.squarePart = n := Nat.mul_div_eq_iff_dvd.2 n.squarefreePart_dvd | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarefreePart_mul_squarePart | /-- The squarefree decomposition of a natural number. -/ |
squarefree_infinite : Set.Infinite { n : ℕ | Squarefree n } := Set.Infinite.mono (fun _ hp ↦ hp.squarefree) Nat.infinite_setOf_prime | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Squarefree",
"import FormalConjecturesForMathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Squarefree.lean | squarefree_infinite | /-- The squarefree decomposition of a natural number. -/ |
gompertzConstant : ℝ := -exp 1 * ∫ (t:ℝ) in Set.Ioi 1, exp (-t) / t /-- **Catalan's constant** $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ -/ | def | FormalConjecturesForMathlib | [
"import Mathlib.MeasureTheory.Integral.Bochner.Basic",
"import Mathlib.MeasureTheory.Measure.Haar.OfBasis"
] | FormalConjecturesForMathlib/Data/Real/Constants.lean | gompertzConstant | /--
**Gompertz constant**
$$\delta = -e * \int_1^∞ e^{-t}/t dt \approx 0.59634$$
-/ |
catalanConstant : ℝ := ∑' n : ℕ, (-1)^n / (2*n + 1)^2 | def | FormalConjecturesForMathlib | [
"import Mathlib.MeasureTheory.Integral.Bochner.Basic",
"import Mathlib.MeasureTheory.Measure.Haar.OfBasis"
] | FormalConjecturesForMathlib/Data/Real/Constants.lean | catalanConstant | /--
**Catalan's constant**
$$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$
-/ |
Set.interIio (S : Set β) (b : β) : Set β := S ∩ Set.Iio b /-- Given a set `S` and elements `a` and `b` in an order `β`, `Set.interIcc S a b` is the intersection `S ∩ Icc a b`. -/ @[inline] | abbrev | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIio | /--
Given a set `S` and an element `b` in an order `β`, `Set.interIio S b A` is the intersection
`S ∩ Iio b`.
-/ |
Set.interIcc [OrderBot β] (S : Set β) (a b : β) : Set β := S ∩ (Set.Icc a b) variable {S b} in | abbrev | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIcc | /--
Given a set `S` and elements `a` and `b` in an order `β`, `Set.interIcc S a b` is the intersection
`S ∩ Icc a b`.
-/ |
Set.finite_interIio [LocallyFiniteOrderBot β] : (S.interIio b).Finite := Set.finite_Iio b |>.inter_of_right S | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.finite_interIio | /--
Given a set `S` and elements `a` and `b` in an order `β`, `Set.interIcc S a b` is the intersection
`S ∩ Icc a b`.
-/ |
Set.finite_interIcc [LocallyFiniteOrder β] [OrderBot β] : (S.interIcc a b).Finite := Set.finite_Icc a b |>.inter_of_right S | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.finite_interIcc | /--
Given a set `S` and elements `a` and `b` in an order `β`, `Set.interIcc S a b` is the intersection
`S ∩ Icc a b`.
-/ |
Set.interIio_univ (b : β) : Set.interIio .univ b = Set.Iio b := by rw [Set.interIio, univ_inter] @[simp] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIio_univ | null |
Set.interIio_univ' (b : β) : Set.univ.interIio b = Set.Iio b := by rw [Set.interIio_univ] @[simp] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIio_univ' | null |
Set.interIio_empty (b : β) : Set.interIio ∅ b = ∅ := by rw [Set.interIio, empty_inter] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIio_empty | null |
Set.interIio_mono {S T : Set β} (h : S ⊆ T) (b : β) : S.interIio b ⊆ T.interIio b := Set.inter_subset_inter_left _ h | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Defs"
] | FormalConjecturesForMathlib/Data/Set/Bdd.lean | Set.interIio_mono | null |
partialDensity {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) (b : β) : ℝ := (Set.interIio (S ∩ A) b).ncard / (Set.interIio A b).ncard | abbrev | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | partialDensity | /--
Given a set `S` and an element `b` in an order `β`, where all intervals bounded above are finite,
we define the partial density of `S` (relative to a set `A`) to be the proportion of elements in
`{x ∈ A | x < b}` that lie in `S ∩ A`.
This definition was inspired from https://github.com/b-mehta/unit-fractions
-/ |
partialDensity_le_one {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) (b : β) : S.partialDensity A b ≤ 1 := by apply div_le_one_of_le₀ _ (Nat.cast_nonneg _) exact mod_cast Set.ncard_le_ncard <| Set.inter_subset_inter_left _ inter_subset_right /-- Given a set `S` in an order `β`, w... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | partialDensity_le_one | null |
upperDensity {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) : ℝ := atTop.limsup fun (b : β) ↦ S.partialDensity A b /-- Given a set `S` in an order `β`, where all intervals bounded above are finite, we define the lower density of `S` (relative to a set `A`) to be the liminf of the... | def | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | upperDensity | /--
Given a set `S` in an order `β`, where all intervals bounded above are finite, we define the upper
density of `S` (relative to a set `A`) to be the limsup of the partial densities of `S`
(relative to `A`) for `b → ∞`.
-/ |
lowerDensity {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) : ℝ := atTop.liminf fun (b : β) ↦ S.partialDensity A b | def | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | lowerDensity | /--
Given a set `S` in an order `β`, where all intervals bounded above are finite, we define the lower
density of `S` (relative to a set `A`) to be the liminf of the partial densities of `S`
(relative to `A`) for `b → ∞`.
-/ |
lowerDensity_le_one {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) : S.lowerDensity A ≤ 1 := by by_cases h : atTop (α := β) = ⊥ · field_simp [h, Set.lowerDensity, Filter.liminf_eq] · have : (atTop (α := β)).NeBot := ⟨h⟩ apply Real.sSup_le (fun x hx ↦ ?_) one_pos.le simpa using hx... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | lowerDensity_le_one | /--
Given a set `S` in an order `β`, where all intervals bounded above are finite, we define the lower
density of `S` (relative to a set `A`) to be the liminf of the partial densities of `S`
(relative to `A`) for `b → ∞`.
-/ |
lowerDensity_nonneg {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) : 0 ≤ S.lowerDensity A := by rw [Set.lowerDensity, Filter.liminf_eq] exact (em _).elim (le_csSup · <| .of_forall fun _ ↦ by positivity) (Real.sSup_of_not_bddAbove · |>.ge) /-- A set `S` in an order `β` where all i... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | lowerDensity_nonneg | null |
HasDensity {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (α : ℝ) (A : Set β := Set.univ) : Prop := Tendsto (fun (b : β) => S.partialDensity A b) atTop (𝓝 α) /-- A set `S` in an order `β` where all intervals bounded above are finite is said to have positive density (relative to a set `A`) if there exis... | def | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | HasDensity | /--
A set `S` in an order `β` where all intervals bounded above are finite is said to have
density `α : ℝ` (relative to a set `A`) if the proportion of `x ∈ S` such that `x < n`
in `A` tends to `α` as `n → ∞`.
When `β = ℕ` this by default defines the natural density of a set
(i.e., relative to all of `ℕ`).
-/ |
HasPosDensity {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (S : Set β) (A : Set β := Set.univ) : Prop := ∃ α > 0, S.HasDensity α A namespace HasDensity -- TODO(mercuris): generalise these to non-univ `A` /-- In a non-trivial partial order with a least element, the set of all elements has density one. -/ @[simp] | def | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | HasPosDensity | /--
A set `S` in an order `β` where all intervals bounded above are finite is said to have
positive density (relative to a set `A`) if there exists a positive `α : ℝ` such that
`S` has density `α` (relative to a set `A`).
-/ |
univ {β : Type*} [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : (@Set.univ β).HasDensity 1 := by by_cases h : atTop (α := β) = ⊥ · field_simp [h, HasDensity] · simp [HasDensity, partialDensity] let ⟨b, hb⟩ := Set.Iio_eventually_ncard_ne_zero β refine tendsto_const_nhds.congr' ?_ exact (eventually... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | univ | /-- In a non-trivial partial order with a least element, the set of all
elements has density one. -/ |
univ_nat_hasDensity_one : (@Set.univ ℕ).HasDensity 1 := univ @[simp] | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | univ_nat_hasDensity_one | null |
empty {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] (A : Set β := Set.univ) : Set.HasDensity (∅ : Set β) 0 A := by simpa [HasDensity, partialDensity] using tendsto_const_nhds | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | empty | null |
mono {β : Type*} [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] {S T : Set β} {αS αT : ℝ} [(atTop (α := β)).NeBot] (h : S ⊆ T) (hS : S.HasDensity αS) (hT : T.HasDensity αT) : αS ≤ αT := by rw [HasDensity] at hS hT apply le_of_tendsto_of_tendsto hS hT filter_upwards [eventually_ge_atTop ⊥] with b hb apply div_le_d... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | mono | null |
nonneg {β : Type*} [Preorder β] [LocallyFiniteOrderBot β] [(atTop : Filter β).NeBot] {S : Set β} {α : ℝ} (h : S.HasDensity α) : 0 ≤ α := le_of_tendsto_of_tendsto' empty h fun b => by simp [div_nonneg, partialDensity] | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | nonneg | null |
hasDensity_even : {n : ℕ | Even n}.HasDensity (1 / 2) := by simp [HasDensity, partialDensity, Set.interIio] have h {n : ℕ} (hn : 1 ≤ n) : (({n : ℕ | Even n} ∩ Iio n).ncard : ℝ) / n = if Even n then 2⁻¹ else (n + 1 : ℝ) / n * 2⁻¹ := by split_ifs with h · rw [← image_mul_two_Iio_even h, ncard_image_of_injective _ (mul_ri... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | hasDensity_even | /--
The natural density of the set of even numbers is `1 / 2`.
-/ |
hasDensity_zero_of_finite {S : Set ℕ} (h : S.Finite) : S.HasDensity 0 := by simp [HasDensity, partialDensity, Set.interIio] have (n : ℕ) : ((S ∩ Set.Iio n).ncard : ℝ) / n ≤ S.ncard / n := by by_cases h₀ : n = 0; simp [h₀] exact div_le_div₀ (by simp) (by simpa using Set.ncard_inter_le_ncard_left _ _ h) (by simpa using n... | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | hasDensity_zero_of_finite | /-- A finite set has natural density zero. -/ |
infinite_of_hasDensity_pos {S : Set ℕ} {α : ℝ} (h : S.HasDensity α) (hα : 0 < α) : S.Infinite := mt hasDensity_zero_of_finite fun h' => (_root_.ne_of_lt hα).symm (tendsto_nhds_unique h h') | theorem | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | infinite_of_hasDensity_pos | /-- A set of positive natural density is infinite. -/ |
Set.HasLogDensity (A : Set ℕ) (d : ℝ) : Prop := Tendsto (fun n : ℕ => (∑ k ≤ n with k ∈ A, (k : ℝ)⁻¹ / .log n : ℝ)) atTop (𝓝 d) /-- If a set has natural density `d`, then it also has logarithmic density `d`. -/ proof_wanted Set.HasDensity.hasLogDensity {A : Set ℕ} {d : ℝ} (h : A.HasDensity d) : A.HasLogDensity d | def | FormalConjecturesForMathlib | [
"import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval",
"import FormalConjecturesForMathlib.Data.Set.Bdd",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Basic",
"import FormalConjecturesForMathlib.Order.Interval.Finset.Nat",
"import Batteries.Util.ProofWanted"
] | FormalConjecturesForMathlib/Data/Set/Density.lean | Set.HasLogDensity | /--
A set `A` of natural numbers has logarithmic density `d` if the sequence
$(1 / \log n) \cdot \sum_{k \in A, k \le n} (1/k)$ converges to `d`.
Logarithmic density is a weaker notion than natural density: if a set has natural density `d`,
then it also has logarithmic density `d` (see `Set.HasDensity.hasLogDensity`), ... |
Triplewise {α : Type*} (r : α → α → α → Prop) : Prop := ∀ ⦃i j k ⦄, i ≠ j → j ≠ k → i ≠ k → r i j k namespace Set variable {α : Type*} {r : α → α → α → Prop} {s t : Set α} {x y z : α} /-- The ternary relation `r` holds triplewise on the set `s` if `r x y z` for all *distinct* `x y z ∈ s`. -/ | def | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise | null |
Triplewise (s : Set α) (r : α → α → α → Prop) : Prop := ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ∀ ⦃z⦄, z ∈ s → x ≠ y → y ≠ z → x ≠ z → r x y z | def | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise | /--
The ternary relation `r` holds triplewise on the set `s`
if `r x y z` for all *distinct* `x y z ∈ s`.
-/ |
Triplewise.eq (hs : s.Triplewise r) (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ s) (h : ¬r x y z) : x = y ∨ y = z ∨ x = z := by have := (h <| hs hx hy hz · · ·) tauto | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise.eq | /--
The ternary relation `r` holds triplewise on the set `s`
if `r x y z` for all *distinct* `x y z ∈ s`.
-/ |
Triplewise.mono (h : t ⊆ s) (hs : s.Triplewise r) : t.Triplewise r := fun _ hx _ hy _ hz hxy hyz hxz ↦ hs (h hx) (h hy) (h hz) hxy hyz hxz @[simp] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise.mono | /--
The ternary relation `r` holds triplewise on the set `s`
if `r x y z` for all *distinct* `x y z ∈ s`.
-/ |
Triplewise_univ_iff : Set.univ.Triplewise r ↔ Triplewise r := by simp [Set.Triplewise, Triplewise] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise_univ_iff | null |
triplewise_of_encard_lt (r : α → α → α → Prop) (h : s.encard < 3) : s.Triplewise r := by unfold Set.Triplewise contrapose! h obtain ⟨x, hx, y, hy, z, hz, hxy, hyz, hxz, _⟩ := h trans encard {x, y, z} · norm_num [encard_insert_of_notMem, *] · exact encard_le_encard_of_injOn (by simp [MapsTo, hx, hy, hz]) (injOn_id _) @[... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_of_encard_lt | null |
triplewise_empty (r : α → α → α → Prop) : Set.Triplewise ∅ r := (notMem_empty · · |>.elim) @[simp] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_empty | null |
triplewise_singleton (r : α → α → α → Prop) : Set.Triplewise {x} r := by apply triplewise_of_encard_lt simp @[simp] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_singleton | null |
triplewise_pair (r : α → α → α → Prop) : Set.Triplewise {x, y} r := by apply triplewise_of_encard_lt apply (encard_insert_le {y} x).trans_lt norm_num | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_pair | null |
triplewise_insert : (insert x s).Triplewise r ↔ s.Triplewise r ∧ ∀ y ∈ s, ∀ z ∈ s, x ≠ y → x ≠ z → y ≠ z → r x y z ∧ r y x z ∧ r y z x := by refine ⟨fun hxs ↦ ?_, fun ⟨hs, hxs⟩ ↦ ?_⟩ · refine ⟨hxs.mono (subset_insert x s), fun y hy z hz hxy hxz hyz ↦ ⟨?_, ?_, ?_⟩⟩ · exact hxs (mem_insert x s) (mem_insert_of_mem x hy) (... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_insert | null |
triplewise_insert_of_not_mem (hx : x ∉ s) : (insert x s).Triplewise r ↔ s.Triplewise r ∧ ∀ y ∈ s, ∀ z ∈ s, y ≠ z → r x y z ∧ r y x z ∧ r y z x := by refine Set.triplewise_insert.trans <| and_congr_right' <| forall₄_congr ?_ intro y hy z hz simp [(ne_of_mem_of_not_mem hy hx).symm, (ne_of_mem_of_not_mem hz hx).symm] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_insert_of_not_mem | null |
Triplewise.insert (hs : s.Triplewise r) (h : ∀ y ∈ s, ∀ z ∈ s, x ≠ y → x ≠ z → y ≠ z → r x y z ∧ r y x z ∧ r y z x) : (insert x s).Triplewise r := Set.triplewise_insert.mpr ⟨hs, h⟩ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise.insert | null |
Triplewise.insert_of_not_mem (hx : x ∉ s) (hs : s.Triplewise r) (h : ∀ y ∈ s, ∀ z ∈ s, y ≠ z → r x y z ∧ r y x z ∧ r y z x) : (insert x s).Triplewise r := (Set.triplewise_insert_of_not_mem hx).mpr ⟨hs, h⟩ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | Triplewise.insert_of_not_mem | null |
triplewise_set_pred_iff {S : Set α} {P : Set α → Prop} : S.Triplewise (P {·, ·, ·}) ↔ ∀ T ⊆ S, T.ncard = 3 → P T := by constructor · intro h T hT T_ncard obtain ⟨x, y, z, hxy, hxz, hyz, rfl⟩ := ncard_eq_three.mp T_ncard simp_rw [insert_subset_iff, singleton_subset_iff] at hT exact h hT.1 hT.2.1 hT.2.2 hxy hyz hxz · int... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.ENat.Lattice",
"import Mathlib.Data.Set.Card",
"import Mathlib.Order.CompletePartialOrder",
"import Mathlib.Tactic.NormNum.Ineq"
] | FormalConjecturesForMathlib/Data/Set/Triplewise.lean | triplewise_set_pred_iff | null |
IsPerfectDifferenceSet (D : Set ℕ) (n : ℕ) : Prop := D.offDiag.BijOn (fun (a, b) => (a - b : ZMod n)) {x : ZMod n | x ≠ 0} | def | FormalConjecturesForMathlib | [
"import Mathlib.Data.ZMod.Defs"
] | FormalConjecturesForMathlib/Data/ZMod/PerfectDifferenceSet.lean | IsPerfectDifferenceSet | /--
A perfect difference set modulo `n` is a set `D` such that the map `(a, b) ↦ a - b` from
`D.offDiag` to `{x : ZMod n | x ≠ 0}` is a bijection.
-/ |
prod_primeFactors_factorization_apply (n : ℕ) {p : ℕ} (hp : p.Prime) {f : ℕ → ℕ → ℕ} (hf : ∀ q n, ¬ q ∣ n → f q n = 0) (hf₀ : ∀ q, f q 0 = 0): (∏ q ∈ n.primeFactors, q ^ f q n).factorization p = f p n := by rw [factorization_prod fun _ _ _ ↦ by simp_all, Finset.sum_congr rfl fun x hx ↦ by rw [factorization_pow, (mem_pr... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Factorization.Basic"
] | FormalConjecturesForMathlib/Data/Nat/Factorization/Basic.lean | prod_primeFactors_factorization_apply | null |
Nat.Composite (n : ℕ) : Prop := 1 < n ∧ ¬n.Prime | abbrev | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Prime.Defs"
] | FormalConjecturesForMathlib/Data/Nat/Prime/Composite.lean | Nat.Composite | null |
Nat.exists_prime_not_dvd (n : ℕ) (hn : n ≠ 0) : ∃ p, p.Prime ∧ ¬p ∣ n := by let ⟨p, hp, h_lt, _⟩ := exists_prime_lt_and_le_two_mul n hn exact ⟨p, hp, not_dvd_of_pos_of_lt (Nat.pos_of_ne_zero hn) h_lt⟩ | lemma | FormalConjecturesForMathlib | [
"import Mathlib.Data.Nat.Prime.Defs",
"import Mathlib.NumberTheory.Bertrand"
] | FormalConjecturesForMathlib/Data/Nat/Prime/Defs.lean | Nat.exists_prime_not_dvd | null |
Finset.Coprime (S : Finset ℕ) : Prop := S.gcd id = 1 | def | FormalConjecturesForMathlib | [
"import Batteries.Data.Nat.Gcd",
"import Mathlib.Algebra.GCDMonoid.Finset",
"import Mathlib.Algebra.GCDMonoid.Nat"
] | FormalConjecturesForMathlib/Data/Nat/Prime/Finset.lean | Finset.Coprime | /-- A Finset of numbers is coprime, or relatively prime, if its `gcd` is 1. -/ |
Finset.Coprime_pair_iff (S : Finset ℕ) (a b : ℕ) (h : S = {a, b}) : S.Coprime ↔ Nat.Coprime a b := by simp only [h, Finset.Coprime, Nat.Coprime, gcd_insert, gcd_singleton, normalize_eq] rfl | theorem | FormalConjecturesForMathlib | [
"import Batteries.Data.Nat.Gcd",
"import Mathlib.Algebra.GCDMonoid.Finset",
"import Mathlib.Algebra.GCDMonoid.Nat"
] | FormalConjecturesForMathlib/Data/Nat/Prime/Finset.lean | Finset.Coprime_pair_iff | /-- A Finset of numbers is coprime, or relatively prime, if its `gcd` is 1. -/ |
lt_finRotate_of_ne_last {i : Fin (n + 1)} (hi : i ≠ Fin.last n) : i < finRotate _ i := by rw [Fin.lt_iff_val_lt_val, coe_finRotate_of_ne_last hi, Nat.lt_add_one_iff] | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Logic.Equiv.Fin.Rotate"
] | FormalConjecturesForMathlib/Logic/Equiv/Fin/Rotate.lean | lt_finRotate_of_ne_last | null |
cofinite_hasBasis_Ioi {α : Type*} [LinearOrder α] [LocallyFiniteOrder α] [OrderBot α] : cofinite.HasBasis (fun (_ : α) ↦ True) (Set.Ioi ·) := by constructor intro A simp only [mem_cofinite, true_and] constructor · intro hA obtain ⟨a, ha⟩ := Set.Finite.bddAbove hA use a rw [← Set.compl_subset_compl, compl_Ioi] exact ha ... | lemma | FormalConjecturesForMathlib | [
"import Mathlib.Order.Filter.Cofinite",
"import Mathlib.Order.Interval.Finset.Defs",
"import Mathlib.Order.SuccPred.Basic"
] | FormalConjecturesForMathlib/Order/Filter/Cofinite.lean | cofinite_hasBasis_Ioi | null |
cofinite_hasBasis_Ici {α : Type*} [LinearOrder α] [LocallyFiniteOrder α] [OrderBot α] [SuccOrder α] [NoMaxOrder α] : cofinite.HasBasis (fun (_ : α) ↦ True) (Set.Ici ·) := by constructor intro A simp only [mem_cofinite, true_and] constructor · intro hA obtain ⟨a, ha⟩ := Set.Finite.bddAbove hA use Order.succ a rw [← Set.... | lemma | FormalConjecturesForMathlib | [
"import Mathlib.Order.Filter.Cofinite",
"import Mathlib.Order.Interval.Finset.Defs",
"import Mathlib.Order.SuccPred.Basic"
] | FormalConjecturesForMathlib/Order/Filter/Cofinite.lean | cofinite_hasBasis_Ici | /--
Note(csonne): According to a TODO in `Mathlib.Order.Interval.Finset.Defs`, Assuming `SuccOrder α`
should not be necessary as an instance should follow from `[LocallyFiniteOrder α] [OrderBot α]`.
This has not been implemented in mathlib however.
-/ |
cofinite_eq_atTop {α : Type*} [LinearOrder α] [LocallyFiniteOrder α] [OrderBot α] [SuccOrder α] [NoMaxOrder α] : @cofinite α = atTop := by simp only [atTop_eq_generate_Ici, HasBasis.eq_generate cofinite_hasBasis_Ici, true_and] rfl | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Filter.Cofinite",
"import Mathlib.Order.Interval.Finset.Defs",
"import Mathlib.Order.SuccPred.Basic"
] | FormalConjecturesForMathlib/Order/Filter/Cofinite.lean | cofinite_eq_atTop | null |
Finset.tendsto_card_atTop {α : Type*} [Infinite α] : atTop.Tendsto (Finset.card (α := α)) atTop := by classical apply tendsto_atTop_atTop_of_monotone Finset.card_mono fun n ↦ ?_ obtain ⟨S, hS⟩ := (Set.infinite_univ (α := α)).exists_subset_card_eq n obtain ⟨g, hg⟩ := (Set.infinite_univ (α := α)).exists_notMem_finset S e... | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Filter.AtTopBot.Finset"
] | FormalConjecturesForMathlib/Order/Filter/atTopBot/Finset.lean | Finset.tendsto_card_atTop | null |
Finset.Iio_eventually_nonempty (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Finset.Iio n).Nonempty := let ⟨b, hb⟩ := exists_ne (⊥ : β) ⟨b, fun n hn => by simp [ne_bot_of_le_ne_bot hb hn]⟩ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Basic",
"import Mathlib.Data.Set.Card"
] | FormalConjecturesForMathlib/Order/Interval/Finset/Basic.lean | Finset.Iio_eventually_nonempty | null |
Finset.Iio_eventually_card_ne_zero (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Finset.Iio n).card ≠ 0 := let ⟨b, hb⟩ := Iio_eventually_nonempty β ⟨b, fun n hn => by simp [nonempty_iff_ne_empty.1 <| hb n hn]⟩ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Basic",
"import Mathlib.Data.Set.Card"
] | FormalConjecturesForMathlib/Order/Interval/Finset/Basic.lean | Finset.Iio_eventually_card_ne_zero | null |
Set.Iio_eventually_ncard_ne_zero (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Set.Iio n).ncard ≠ 0 := let ⟨b, hb⟩ := Finset.Iio_eventually_card_ne_zero β ⟨b, fun n hn => by rw [← Finset.coe_Iio, Set.ncard_coe_finset]; exact hb n hn⟩ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Basic",
"import Mathlib.Data.Set.Card"
] | FormalConjecturesForMathlib/Order/Interval/Finset/Basic.lean | Set.Iio_eventually_ncard_ne_zero | null |
Nat.ncard_Iio (b : ℕ) : (Set.Iio b).ncard = b := by rw [← Finset.coe_Iio, Set.ncard_coe_finset] exact Nat.card_Iio _ | theorem | FormalConjecturesForMathlib | [
"import Mathlib.Order.Interval.Finset.Nat",
"import Mathlib.Data.Set.Card"
] | FormalConjecturesForMathlib/Order/Interval/Finset/Nat.lean | Nat.ncard_Iio | null |
aleph0_le_mk_set : ℵ₀ ≤ #s ↔ s.Infinite := by rw [aleph0_le_mk_iff, Set.infinite_coe_iff] @[simp] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.SetTheory.Cardinal.Arithmetic"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/Arithmetic.lean | aleph0_le_mk_set | null |
mk_diff_eq_left' (hs : s.Infinite) (hst : #↑(s ∩ t) < #s) : #↑(s \ t) = #s := by refine (mk_le_mk_of_subset Set.diff_subset).eq_of_not_lt fun h ↦ (add_lt_of_lt (by simpa) hst h).not_ge ?_ grw [← mk_union_le .., Set.inter_union_diff] @[simp] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.SetTheory.Cardinal.Arithmetic"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/Arithmetic.lean | mk_diff_eq_left' | null |
mk_diff_eq_left (hs : s.Infinite) (hts : #t < #s) : #↑(s \ t) = #s := mk_diff_eq_left' hs <| hts.trans_le' <| mk_subtype_mono Set.inter_subset_right @[simp] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.SetTheory.Cardinal.Arithmetic"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/Arithmetic.lean | mk_diff_eq_left | null |
mk_diff_eq_left_of_finite' (hs : s.Infinite) (hst : (s ∩ t).Finite) : #↑(s \ t) = #s := mk_diff_eq_left' hs <| (aleph0_le_mk_set.2 hs).trans_lt' <| by simpa @[simp] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.SetTheory.Cardinal.Arithmetic"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/Arithmetic.lean | mk_diff_eq_left_of_finite' | null |
mk_diff_eq_left_of_finite (hs : s.Infinite) (ht : t.Finite) : #↑(s \ t) = #s := mk_diff_eq_left hs <| (aleph0_le_mk_set.2 hs).trans_lt' <| by simpa | lemma | FormalConjecturesForMathlib | [
"import Mathlib.SetTheory.Cardinal.Arithmetic"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/Arithmetic.lean | mk_diff_eq_left_of_finite | null |
OrdinalCardinalRamsey (α β : Ordinal.{u}) (c : Cardinal.{u}) : Prop := -- For any 2-coloring of `α`, ∀ red blue : SimpleGraph α.toType, IsCompl red blue → -- either there is a red `K_β` (∃ s, red.IsClique s ∧ typeLT s = β) ∨ -- or there is a blue `K_c`. ∃ s, blue.IsClique s ∧ #s = c | def | FormalConjecturesForMathlib | [
"import Mathlib.Combinatorics.SimpleGraph.Clique",
"import Mathlib.SetTheory.Ordinal.Exponential"
] | FormalConjecturesForMathlib/SetTheory/Cardinal/SimpleGraph.lean | OrdinalCardinalRamsey | /--
This proposition asserts the ordinal Ramsey property `α → (β, c)²`.
It states that for any 2-coloring of the complete graph on the ordinal `α`,
one of the following must hold:
* There is a red clique which is order-isomorphic to `β`.
* There is a blue clique of cardinality `c`.
-/ |
Multipliable.tendsto_tprod_one (hf : Multipliable f) (hs : ∀ b, ∀ᶠ i in l, b ∉ s i) : Tendsto (fun i ↦ ∏' b : s i, f b) l (𝓝 1) := by rw [tendsto_iff_eventually] rintro p hp obtain ⟨t, ht⟩ := hf.tprod_vanishing hp filter_upwards [t.eventually_all.2 fun b _ ↦ hs b] with i hi exact ht _ <| by simpa [Set.disjoint_right] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.Topology.Algebra.InfiniteSum.Group"
] | FormalConjecturesForMathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tendsto_tprod_one | null |
Multipliable.prod_le_tprod_set (hst : ↑s ⊆ t) (hfst : ∀ i ∉ s, i ∈ t → 1 ≤ f i) (hf : Multipliable f) : ∏ i ∈ s, f i ≤ ∏' i : t, f i := by rw [tprod_subtype, Finset.prod_congr rfl] refine (hf.mulIndicator t).prod_le_tprod _ (by simpa [Set.mulIndicator, apply_ite]) rintro i hi simp [Set.mulIndicator, hst hi] | lemma | FormalConjecturesForMathlib | [
"import Mathlib.Topology.Algebra.InfiniteSum.Order"
] | FormalConjecturesForMathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.prod_le_tprod_set | null |
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