name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
CategoryTheory.bicategoricalComp_refl | Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean | theorem bicategoricalComp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : η ⊗≫ θ = η ≫ θ | B : Type u
inst✝ : Bicategory B
a b : B
f g h : a ⟶ b
η : f ⟶ g
θ : g ⟶ h
⊢ η ⊗≫ θ = η ≫ θ | dsimp [bicategoricalComp] | B : Type u
inst✝ : Bicategory B
a b : B
f g h : a ⟶ b
η : f ⟶ g
θ : g ⟶ h
⊢ η ≫ 𝟙 g ≫ θ = η ≫ θ | e1671c143dc0d59c |
ProbabilityTheory.comap_cond | Mathlib/Probability/ConditionalProbability.lean | lemma comap_cond {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ ω ∂μ, ω ∈ range i)
(hs : MeasurableSet s) : comap i μ[|s] = (comap i μ)[|i in s] | case h.hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s) | exact hi.measurable hs | no goals | 4f0b4c478fa397f8 |
LieModule.toEnd_pow_lie | Mathlib/Algebra/Lie/OfAssociative.lean | lemma LieModule.toEnd_pow_lie (x y : L) (z : M) (n : ℕ) :
((φ x) ^ n) ⁅y, z⁆ =
∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L x) ^ ij.1) y, ((φ x) ^ ij.2) z⁆ | case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ
ih : (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2)... | simp only [pow_succ', LinearMap.mul_apply, ih, map_sum, map_nsmul,
toEnd_lie, nsmul_add, sum_add_distrib] | case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ
ih : (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2)... | 356832f4698eb33a |
RootPairing.Base.root_sub_root_mem_of_mem_of_mem | Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean | /-- This is Lemma 2.5 (a) from [Geck](Geck2017). -/
lemma root_sub_root_mem_of_mem_of_mem (hk : α k + α i - α j ∈ Φ)
(hkj : k ≠ j) (hk' : α k + α i ∈ Φ) :
α k - α j ∈ Φ | case a
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : Finite ι
inst✝⁴ : CharZero R
inst✝³ : P.IsCrystallographic
inst✝² : NoZeroDivisors R
inst✝¹ : NoZeroSMulDivisors R M
in... | convert (P.coroot' i : M →ₗ[R] R).congr_arg hl using 1 | case h.e'_3
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : Finite ι
inst✝⁴ : CharZero R
inst✝³ : P.IsCrystallographic
inst✝² : NoZeroDivisors R
inst✝¹ : NoZeroSMulDivisors R... | 83118c1462129643 |
Finset.Nat.antidiagonal_filter_le_fst_of_le | Mathlib/Data/Finset/NatAntidiagonal.lean | @[simp] lemma antidiagonal_filter_le_fst_of_le {n k : ℕ} (h : k ≤ n) :
(antidiagonal n).filter (fun a ↦ k ≤ a.fst) = (antidiagonal (n - k)).map
(Embedding.prodMap ⟨_, add_left_injective k⟩ (Embedding.refl ℕ)) | case h.mk
n k : ℕ
h : k ≤ n
i j : ℕ
⊢ (i, j) ∈ filter (fun a => k ≤ a.1) (antidiagonal n) ↔
(i, j) ∈ map ({ toFun := fun x => x + k, inj' := ⋯ }.prodMap (Embedding.refl ℕ)) (antidiagonal (n - k)) | suffices i + j = n ∧ k ≤ i ↔ ∃ a, a + j = n - k ∧ a + k = i by simpa | case h.mk
n k : ℕ
h : k ≤ n
i j : ℕ
⊢ i + j = n ∧ k ≤ i ↔ ∃ a, a + j = n - k ∧ a + k = i | 4d62931494d97781 |
MvPolynomial.transcendental_supported_polynomial_aeval_X_iff | Mathlib/RingTheory/Algebraic/MvPolynomial.lean | theorem transcendental_supported_polynomial_aeval_X_iff
[Nontrivial R] {i : σ} {s : Set σ} {f : R[X]} :
Transcendental (supported R s) (Polynomial.aeval (X i : MvPolynomial σ R) f) ↔
i ∉ s ∧ Transcendental R f | case refine_1
σ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Nontrivial R
i : σ
s : Set σ
f : R[X]
h : i ∈ s
⊢ (Polynomial.aeval (X i)) f ∈ supported R s | exact Algebra.adjoin_mono (Set.singleton_subset_iff.2 (Set.mem_image_of_mem _ h))
(Polynomial.aeval_mem_adjoin_singleton _ _) | no goals | 34eab494257125bf |
AddMonoidAlgebra.mul_of'_modOf | Mathlib/Algebra/MonoidAlgebra/Division.lean | theorem mul_of'_modOf (x : k[G]) (g : G) : x * of' k G g %ᵒᶠ g = 0 | case H.inr.h
k : Type u_1
G : Type u_2
inst✝¹ : Semiring k
inst✝ : AddCancelCommMonoid G
x : k[G]
g g' : G
h : ¬∃ d, g' = g + d
⊢ ¬∃ d, g' = d + g | simpa only [add_comm] using h | no goals | 746a91a884eb3ea6 |
Ring.coe_jacobson_quotient | Mathlib/RingTheory/Jacobson/Radical.lean | theorem coe_jacobson_quotient (I : Ideal R) [I.IsTwoSided] :
(jacobson (R ⧸ I) : Set (R ⧸ I)) = Module.jacobson R (R ⧸ I) | R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
⊢ ↑(jacobson (R ⧸ I)) = ↑(Module.jacobson R (R ⧸ I)) | let f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := ⟨AddHom.id _, fun _ _ ↦ rfl⟩ | R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ ↑(jacobson (R ⧸ I)) = ↑(Module.jacobson R (R ⧸ I)) | 0a5f67c64f9b114a |
let_value_heq | Mathlib/Deprecated/Logic.lean | theorem let_value_heq {α : Sort v} {β : α → Sort u} {a₁ a₂ : α} (b : ∀ x : α, β x)
(h : a₁ = a₂) : HEq (let x : α := a₁; b x) (let x : α := a₂; b x) | α : Sort v
β : α → Sort u
a₁ a₂ : α
b : (x : α) → β x
h : a₁ = a₂
⊢ HEq
(let x := a₁;
b x)
(let x := a₂;
b x) | cases h | case refl
α : Sort v
β : α → Sort u
a₁ : α
b : (x : α) → β x
⊢ HEq
(let x := a₁;
b x)
(let x := a₁;
b x) | 302ea60413d00127 |
LieSubmodule.lie_baseChange | Mathlib/Algebra/Lie/BaseChange.lean | lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆ | case refine_2.intro.intro.intro.intro.mem.intro.intro.mem
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdea... | rintro - ⟨m', hm' : m' ∈ N, rfl⟩ | case refine_2.intro.intro.intro.intro.mem.intro.intro.mem.intro.intro
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A... | c0e93f202362c69d |
MeasureTheory.LevyProkhorov.continuous_equiv_probabilityMeasure | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | /-- The identity map `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
lemma LevyProkhorov.continuous_equiv_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)) | Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω := fun n => ... | refine ⟨0, fun a i hia ↦ le_trans (integral_nonneg f_nn) (hia i le_rfl)⟩ | no goals | e3f8df53302ac07c |
Std.DHashMap.Internal.AssocList.getCast_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/AssocList/Lemmas.lean | theorem getCast_eq [BEq α] [LawfulBEq α] {l : AssocList α β} {a : α} {h} :
l.getCast a h = getValueCast a l.toList (contains_eq.symm.trans h) | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a k : α
v : β k
t : AssocList α β
ih : ∀ {h : contains a t = true}, getCast a t h = getValueCast a t.toList ⋯
h : contains a (cons k v t) = true
⊢ getCast a (cons k v t) h = getValueCast a (cons k v t).toList ⋯ | simp only [getCast, toList_cons, List.getValueCast_cons, ih] | no goals | 3a973134a6887f18 |
MeasureTheory.dist_convolution_le' | Mathlib/Analysis/Convolution.lean | theorem dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : Integrable f μ)
(hf : support f ⊆ ball (0 : G) R) (hmg : AEStronglyMeasurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ (‖L‖ * ∫ x, ‖f x‖ ∂μ) * ε | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
... | exact (L.flip z₀).integrable_comp hif | no goals | fddd5e43ba4662a2 |
MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
(h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
(ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) | case inr.intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
... | filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos εpos.ne'), self_mem_nhdsWithin] | case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : Measurabl... | 8e4b11ebb89db50e |
MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem iSup_succ_eq_sup {α} (f : ℕ → α → ℝ≥0∞) (m : ℕ) (a : α) :
⨆ (k : ℕ) (_ : k ≤ m + 1), f k a = f m.succ a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a | α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
hd : d = f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ∧
f m.succ a ⊔ ⨆ k, ⨆... | constructor | case left
α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
hd : d = f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f m.succ a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a
case right
α :... | 2ffdb94e1dd047b7 |
Nat.primeFactorsPiBij_inj | Mathlib/Algebra/Order/Antidiag/Nat.lean | theorem primeFactorsPiBij_inj (d n : ℕ)
(f : (p : ℕ) → p ∈ n.primeFactors → Fin d) (hf : f ∈ pi n.primeFactors fun _ => univ)
(g : (p : ℕ) → p ∈ n.primeFactors → Fin d) (hg : g ∈ pi n.primeFactors fun _ => univ) :
Nat.primeFactorsPiBij d n f hf = Nat.primeFactorsPiBij d n g hg → f = g | case h.a
d n : ℕ
f : (p : ℕ) → p ∈ n.primeFactors → Fin d
hf : f ∈ n.primeFactors.pi fun x => univ
g : (p : ℕ) → p ∈ n.primeFactors → Fin d
hg : g ∈ n.primeFactors.pi fun x => univ
p : ℕ
hp : p ∈ n.primeFactors
hfg : f p hp ≠ g p hp
⊢ ¬p ∣ ∏ p ∈ filter (fun p_1 => g ↑p_1 ⋯ = f p hp) n.primeFactors.attach, ↑p | rw [mem_primeFactors] at hp | case h.a
d n : ℕ
f : (p : ℕ) → p ∈ n.primeFactors → Fin d
hf : f ∈ n.primeFactors.pi fun x => univ
g : (p : ℕ) → p ∈ n.primeFactors → Fin d
hg : g ∈ n.primeFactors.pi fun x => univ
p : ℕ
hp✝ : p ∈ n.primeFactors
hp : Prime p ∧ p ∣ n ∧ n ≠ 0
hfg : f p hp✝ ≠ g p hp✝
⊢ ¬p ∣ ∏ p ∈ filter (fun p_1 => g ↑p_1 ⋯ = f p hp✝) n.p... | 6425189256a15307 |
Std.Sat.AIG.Decl.relabel_atom | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Relabel.lean | theorem relabel_atom {decls : Array (Decl α)} {r : α → β} {hidx : idx < decls.size}
(h : relabel r decls[idx] = .atom a) :
∃ x, decls[idx] = .atom x ∧ a = r x | case h_3
α β : Type
idx : Nat
a : β
decls : Array (Decl α)
r : α → β
hidx : idx < decls.size
decl✝ : Decl α
lhs✝ rhs✝ : Nat
linv✝ rinv✝ : Bool
heq✝ : decls[idx] = gate lhs✝ rhs✝ linv✝ rinv✝
h : gate lhs✝ rhs✝ linv✝ rinv✝ = atom a
⊢ ∃ x, decls[idx] = atom x ∧ a = r x | contradiction | no goals | 6168f9c26b0ec348 |
Set.offDiag_nonempty | Mathlib/Data/Set/Prod.lean | theorem offDiag_nonempty : s.offDiag.Nonempty ↔ s.Nontrivial | α : Type u_1
s : Set α
⊢ s.offDiag.Nonempty ↔ s.Nontrivial | simp [offDiag, Set.Nonempty, Set.Nontrivial] | no goals | ba1a3140e1e92423 |
frequently_nhds_iff | Mathlib/Topology/Basic.lean | theorem frequently_nhds_iff {p : X → Prop} :
(∃ᶠ y in 𝓝 x, p y) ↔ ∀ U : Set X, x ∈ U → IsOpen U → ∃ y ∈ U, p y :=
(nhds_basis_opens x).frequently_iff.trans <| by simp
| X : Type u
x : X
inst✝ : TopologicalSpace X
p : X → Prop
⊢ (∀ (i : Set X), x ∈ i ∧ IsOpen i → ∃ x ∈ i, p x) ↔ ∀ (U : Set X), x ∈ U → IsOpen U → ∃ y ∈ U, p y | simp | no goals | b517c7de817c4064 |
CategoryTheory.IsCodetecting.isIso_iff_of_epi | Mathlib/CategoryTheory/Generator/Basic.lean | lemma IsCodetecting.isIso_iff_of_epi {𝒢 : Set C} (h𝒢 : IsCodetecting 𝒢)
{X Y : C} (f : X ⟶ Y) [Epi f] :
IsIso f ↔ ∀ s ∈ 𝒢, Function.Surjective ((yoneda.obj s).map f.op) | case mpr.refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
h𝒢 : IsCodetecting 𝒢
X Y : C
f : X ⟶ Y
inst✝ : Epi f
hf : ∀ s ∈ 𝒢, Function.Surjective ((yoneda.obj s).map f.op)
A : C
hA : A ∈ 𝒢
g : X ⟶ A
⊢ ∀ (y₁ y₂ : Y ⟶ A), f ≫ y₁ = g → f ≫ y₂ = g → y₁ = y₂ | intro l₁ l₂ h₁ h₂ | case mpr.refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
h𝒢 : IsCodetecting 𝒢
X Y : C
f : X ⟶ Y
inst✝ : Epi f
hf : ∀ s ∈ 𝒢, Function.Surjective ((yoneda.obj s).map f.op)
A : C
hA : A ∈ 𝒢
g : X ⟶ A
l₁ l₂ : Y ⟶ A
h₁ : f ≫ l₁ = g
h₂ : f ≫ l₂ = g
⊢ l₁ = l₂ | 340cf8f3a0501311 |
tendsto_mod_div_atTop_nhds_zero_nat | Mathlib/Analysis/SpecificLimits/Basic.lean | theorem tendsto_mod_div_atTop_nhds_zero_nat {m : ℕ} (hm : 0 < m) :
Tendsto (fun n : ℕ => ((n % m : ℕ) : ℝ) / (n : ℝ)) atTop (𝓝 0) | m : ℕ
hm : 0 < m
⊢ ∀ᶠ (n : ℕ) in atTop, 0 ≤ (fun n => ↑(n % m)) n | aesop | no goals | ebfc4f57575f2f7f |
Matroid.mem_closure_insert | Mathlib/Data/Matroid/Closure.lean | lemma mem_closure_insert (he : e ∉ M.closure X) (hef : e ∈ M.closure (insert f X)) :
f ∈ M.closure (insert e X) | α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e X ∩ M.E) | rw [insert_inter_of_mem heE] | α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e (X ∩ M.E)) | 526bcdba2fecdf54 |
GromovHausdorff.maxVar_bound | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | theorem maxVar_bound [CompactSpace X] [Nonempty X] [CompactSpace Y] [Nonempty Y] :
dist x y ≤ maxVar X Y :=
calc
dist x y ≤ diam (univ : Set (X ⊕ Y)) :=
dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
_ = diam (range inl ∪ range inr : Set (X ⊕ Y)) | X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : MetricSpace Y
x y : X ⊕ Y
inst✝³ : CompactSpace X
inst✝² : Nonempty X
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ 1 * diam univ + 1 + 1 * diam univ ≤ 2 * diam univ + 1 + 2 * diam univ | gcongr <;> norm_num | no goals | 1e13ead7c12daebc |
MeasureTheory.tilted_apply_eq_ofReal_integral | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma tilted_apply_eq_ofReal_integral [SFinite μ] (f : α → ℝ) (s : Set α) :
μ.tilted f s = ENNReal.ofReal (∫ a in s, exp (f a) / ∫ x, exp (f x) ∂μ ∂μ) | case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x => rexp (f x)) μ
⊢ 0 ≤ᶠ[ae (μ.restrict s)] fun a => rexp (f a) / ∫ (x : α), rexp (f x) ∂μ | exact ae_of_all _ (fun _ ↦ by positivity) | no goals | 1b0b9e0fd4cb2ff7 |
Set.exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul | Mathlib/Data/Set/Pointwise/SMul.lean | /-- Any intersection of translates of two sets `s` and `t` can be covered by a single translate of
`(s⁻¹ * s) ∩ (t⁻¹ * t)`.
This is useful to show that the intersection of approximate subgroups is an approximate subgroup. -/
@[to_additive
"Any intersection of translates of two sets `s` and `t` can be covered by a sing... | case inr.intro.intro
α : Type u_2
inst✝ : Group α
s t : Set α
a b z : α
hzA : z ∈ a • s
hzB : z ∈ b • t
⊢ ∃ z, a • s ∩ b • t ⊆ z • (s⁻¹ * s ∩ (t⁻¹ * t)) | refine ⟨z, ?_⟩ | case inr.intro.intro
α : Type u_2
inst✝ : Group α
s t : Set α
a b z : α
hzA : z ∈ a • s
hzB : z ∈ b • t
⊢ a • s ∩ b • t ⊆ z • (s⁻¹ * s ∩ (t⁻¹ * t)) | 1519a0b605a044b8 |
Units.continuous_iff | Mathlib/Topology/Algebra/Constructions.lean | theorem continuous_iff {f : X → Mˣ} :
Continuous f ↔ Continuous (val ∘ f) ∧ Continuous (fun x => ↑(f x)⁻¹ : X → M) | M : Type u_1
X : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : TopologicalSpace X
f : X → Mˣ
⊢ Continuous f ↔ Continuous (val ∘ f) ∧ Continuous fun x => ↑(f x)⁻¹ | simp only [isInducing_embedProduct.continuous_iff, embedProduct_apply, Function.comp_def,
continuous_prod_mk, opHomeomorph.symm.isInducing.continuous_iff, opHomeomorph_symm_apply,
unop_op] | no goals | f19a20f38489efca |
norm_eq_iInf_iff_real_inner_eq_zero | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) :
(‖u - v‖ = ⨅ w : (↑K : Set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 :=
Iff.intro
(by
intro h
have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
w : F
hw : w ∈ K
w' : F := w - v
this : w' ∈ K
h₁ : ⟪u - v, w'⟫_ℝ = 0
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0 | exact le_of_eq h₁ | no goals | cbfd395ad9083d60 |
HasStrictFDerivAt.sum | Mathlib/Analysis/Calculus/FDeriv/Add.lean | theorem HasStrictFDerivAt.sum (h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x) :
HasStrictFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, (fun p => A i p.1 - A i p.2 - (A' i) (p.1 - p.2)) =o[n... | convert IsLittleO.sum h | case h.e'_7.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, (fun p => A i p.1 - A i p.2 - (A' i) (p.... | 408925ebfdc15526 |
AlgebraicGeometry.IsSeparated.of_valuativeCriterion | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | /-- The **valuative criterion** for separated morphisms. -/
@[stacks 01L0]
lemma IsSeparated.of_valuativeCriterion [QuasiSeparated f]
(hf : ValuativeCriterion.Uniqueness f) : IsSeparated f where
diagonal_isClosedImmersion | case exists_lift.val.refine_2.h₀
X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiSeparated f
hf : ValuativeCriterion.Uniqueness f
S : ValuativeCommSq (pullback.diagonal f)
hc : CommSq S.i₁ (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))) f (S.i₂ ≫ pullback.fst f f ≫ f)
S' : ValuativeCommSq f := ValuativeCommSq.mk S.R S.K S.i₁ (... | simp | no goals | e5bcb69fa70ee31c |
ProbabilityTheory.complexMGF_undef | Mathlib/Probability/Moments/ComplexMGF.lean | lemma complexMGF_undef (hX : AEMeasurable X μ) (h : ¬ Integrable (fun ω ↦ rexp (z.re * X ω)) μ) :
complexMGF X μ z = 0 | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hX : AEMeasurable X μ
h : ¬Integrable (fun ω => rexp (z.re * X ω)) μ
⊢ complexMGF X μ z = 0 | rw [complexMGF, integral_undef] | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hX : AEMeasurable X μ
h : ¬Integrable (fun ω => rexp (z.re * X ω)) μ
⊢ ¬Integrable (fun ω => cexp (z * ↑(X ω))) μ | b95cb73c1e792b3e |
cardQuot_pow_of_prime | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem cardQuot_pow_of_prime [IsDedekindDomain S] (hP : P ≠ ⊥) {i : ℕ} :
cardQuot (P ^ i) = cardQuot P ^ i | case succ.refine_2
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_not_mem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ... | rw [hf, add_zero] | case succ.refine_2
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_not_mem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ... | bc62aa8e9d3aa451 |
CategoryTheory.IsFiltered.sup_objs_exists | Mathlib/CategoryTheory/Filtered/Basic.lean | theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | case h.inr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X : C
O' : Finset C
nm : X ∉ O'
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
Y : C
mY : Y ∈ insert X O'
h : Y ≠ X
⊢ Nonempty (Y ⟶ max X S') | exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToMax _ _⟩ | no goals | 6340905fdf09a34b |
Complex.ofReal_sin_ofReal_re | Mathlib/Data/Complex/Trigonometric.lean | theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=
conj_eq_iff_re.1 <| by rw [← sin_conj, conj_ofReal]
| x : ℝ
⊢ (starRingEnd ℂ) (sin ↑x) = sin ↑x | rw [← sin_conj, conj_ofReal] | no goals | c3993ac5addee1e8 |
ConvexOn.leftDeriv_le_rightDeriv_of_mem_interior | Mathlib/Analysis/Convex/Deriv.lean | lemma leftDeriv_le_rightDeriv_of_mem_interior (hfc : ConvexOn ℝ S f) (hxs : x ∈ interior S) :
derivWithin f (Iio x) x ≤ derivWithin f (Ioi x) x | case intro.intro.intro.refine_2.intro.intro.intro
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ slope f x z ≤ sInf (slope f x '' {y | y ∈ S ∧ x < y}) | refine le_csInf ?_ ?_ | case intro.intro.intro.refine_2.intro.intro.intro.refine_1
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
z : ℝ
hzs : z ∈ S
hzx : z < x
⊢ (slope f x '' {y | y ∈ S ∧ x < y}).Nonempty
case intro.intro.intro.refine_2.intro.intro.intro.refine_2
S : Set ℝ
f... | ee20e04546739631 |
Affine.Triangle.mem_circumsphere_of_two_zsmul_oangle_eq | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem mem_circumsphere_of_two_zsmul_oangle_eq {t : Triangle ℝ P} {p : P} {i₁ i₂ i₃ : Fin 3}
(h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
(h : (2 : ℤ) • ∡ (t.points i₁) p (t.points i₃) =
(2 : ℤ) • ∡ (t.points i₁) (t.points i₂) (t.points i₃)) : p ∈ t.circumsphere | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) =... | have h₂' : t'.points i₂ = p := h₂ | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) =... | c3015bbdfd99892f |
ENNReal.tsum_one | Mathlib/Topology/Algebra/InfiniteSum/ENNReal.lean | @[simp]
lemma tsum_one : ∑' _ : α, (1 : ℝ≥0∞) = ENat.card α | α : Type u_1
⊢ ∑' (x : ↑univ), 1 = ↑(ENat.card α) | simpa [encard_univ] using tsum_set_one univ | no goals | 986741dd2fc21f23 |
Submodule.ClosedComplemented.of_quotient_finiteDimensional | Mathlib/Analysis/Normed/Module/Complemented.lean | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
[FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented | case intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
p : Subspace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 (E ⧸ p)
hp : IsClosed ↑p
q : Subspace 𝕜 E
hq : IsCompl p q
this : FiniteDimensional 𝕜 ↥q
⊢... | exact .of_isCompl_isClosed hq hp q.closed_of_finiteDimensional | no goals | e57481270e3fef5a |
MeasureTheory.measurableSet_generateFrom_singleton_iff | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ | case neg.«_@».Mathlib.MeasureTheory.MeasurableSpace.Basic._hyg.15042.«1»
α : Type u_1
s : Set α
x : Set Prop
hT : True ∉ x
hF : False ∉ x
hp : False ∈ x
⊢ False ∈ ∅ | contradiction | no goals | 39127bdfcef76ebf |
MvPolynomial.weightedHomogeneousComponent_of_mem | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | theorem weightedHomogeneousComponent_of_mem [DecidableEq M] {m n : M}
{p : MvPolynomial σ R} (h : p ∈ weightedHomogeneousSubmodule R w n) :
weightedHomogeneousComponent w m p = if m = n then p else 0 | case pos
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0) | split_ifs <;>
simp only [zero_coeff, coeff_zero] | no goals | 1004a57e5935800b |
WittVector.mul_polyOfInterest_aux1 | Mathlib/RingTheory/WittVector/MulCoeff.lean | theorem mul_polyOfInterest_aux1 (n : ℕ) :
∑ i ∈ range (n + 1), (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n | case h.e'_2.e_f.h.a
p : ℕ
hp : Fact (Nat.Prime p)
n i : ℕ
m✝ : Fin 2 × ℕ →₀ ℕ
⊢ MvPolynomial.coeff m✝ (↑p ^ i * wittStructureInt p (X 0 * X 1) i ^ p ^ (n - i)) =
MvPolynomial.coeff m✝
((bind₁ (wittStructureInt p (X 0 * X 1))) ((monomial (Finsupp.single i (p ^ (n - i)))) (↑p ^ i))) | congr 1 | case h.e'_2.e_f.h.a.e_p
p : ℕ
hp : Fact (Nat.Prime p)
n i : ℕ
m✝ : Fin 2 × ℕ →₀ ℕ
⊢ ↑p ^ i * wittStructureInt p (X 0 * X 1) i ^ p ^ (n - i) =
(bind₁ (wittStructureInt p (X 0 * X 1))) ((monomial (Finsupp.single i (p ^ (n - i)))) (↑p ^ i)) | 7791cdeef9b1aacd |
BitVec.lawful_divSubtractShift | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem lawful_divSubtractShift (qr : DivModState w) (h : qr.Poised args) :
DivModState.Lawful args (divSubtractShift args qr) | case pos.hdiv
w : Nat
qr : DivModState w
n d : BitVec w
h : DivModState.Poised { n := n, d := d } qr
hrwn : qr.wr + qr.wn = w
hd : 0 < { n := n, d := d }.d
hrd : qr.r.toNat < { n := n, d := d }.d.toNat
hr : qr.r.toNat < 2 ^ qr.wr
hn : qr.q.toNat < 2 ^ qr.wr
hrnd : { n := n, d := d }.n.toNat >>> qr.wn = { n := n, d := d... | case pos.hdiv =>
simp [qr.toNat_shiftRight_sub_one_eq h, h.hdiv, this,
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hrWidth,
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hqWidth]
omega | no goals | 3089d1385aadec96 |
ProbabilityTheory.lintegral_toKernel_mem | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
(a : α) {s : Set (β × ℝ)} (hs : MeasurableSet s) :
∫⁻ b, hf.toKernel f (a, b) {y | (b, y) ∈ s} ∂(ν a) = κ a s | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set (β × ℝ)
t₁ : Set β
ht₁ : MeasurableSet t₁
t₂ : Set ℝ
ht₂ : MeasurableSet t₂
⊢ ∀ a ∈ t₁, {x | (a, x) ∈ t₁ ×ˢ t₂} = t₂ | intro a ha | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a✝ : α
s : Set (β × ℝ)
t₁ : Set β
ht₁ : MeasurableSet t₁
t₂ : Set ℝ
ht₂ : MeasurableSet t₂
a : β
ha : a ∈ t₁
⊢ {x | (a, x) ∈ t₁ ×ˢ... | 57dad7b07efe58fe |
MeasureTheory.StronglyMeasurable.measurableSet_mulSupport | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
[MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) | α : Type u_1
β : Type u_2
f : α → β
m : MeasurableSpace α
inst✝² : One β
inst✝¹ : TopologicalSpace β
inst✝ : MetrizableSpace β
hf : StronglyMeasurable f
⊢ MeasurableSet (mulSupport f) | borelize β | α : Type u_1
β : Type u_2
f : α → β
m : MeasurableSpace α
inst✝² : One β
inst✝¹ : TopologicalSpace β
inst✝ : MetrizableSpace β
hf : StronglyMeasurable f
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ MeasurableSet (mulSupport f) | c381be9691767642 |
centralBinom_le_of_no_bertrand_prime | Mathlib/NumberTheory/Bertrand.lean | theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_large : 2 < n)
(no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) :
centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (... | rw [Finset.prod_const] | case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (... | 94a497c22436f8fd |
CategoryTheory.regularTopology.equalizerCondition_precomp_of_preservesPullback | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | theorem equalizerCondition_precomp_of_preservesPullback (P : Cᵒᵖ ⥤ D) (F : E ⥤ C)
[∀ {X B} (π : X ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) F]
[F.PreservesEffectiveEpis] (hP : EqualizerCondition P) : EqualizerCondition (F.op ⋙ P) | case refine_2
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
inst✝³ : Category.{u_6, u_3} E
P : Cᵒᵖ ⥤ D
F : E ⥤ C
inst✝² : ∀ {X B : E} (π : X ⟶ B) [inst : EffectiveEpi π], PreservesLimit (cospan π π) F
inst✝¹ : F.PreservesEffectiveEpis
hP : EqualizerCondition P
X B ... | refine (hP (F.map π) (PullbackCone.mk (F.map c.fst) (F.map c.snd) ?_) ?_).some | case refine_2.refine_1
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
inst✝³ : Category.{u_6, u_3} E
P : Cᵒᵖ ⥤ D
F : E ⥤ C
inst✝² : ∀ {X B : E} (π : X ⟶ B) [inst : EffectiveEpi π], PreservesLimit (cospan π π) F
inst✝¹ : F.PreservesEffectiveEpis
hP : EqualizerConditi... | 648132f16508b883 |
union_support_maximal_linearIndependent_eq_range_basis | Mathlib/LinearAlgebra/Basis/Cardinality.lean | theorem union_support_maximal_linearIndependent_eq_range_basis {ι : Type w} (b : Basis ι R M)
{κ : Type w'} (v : κ → M) (ind : LinearIndependent R v) (m : ind.Maximal) :
⋃ k, ((b.repr (v k)).support : Set ι) = Set.univ | case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)... | let v' (o : Option κ) : M := o.elim (b i) v | case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)... | 21593d2c43401dd3 |
RatFunc.num_dvd | Mathlib/FieldTheory/RatFunc/Basic.lean | theorem num_dvd {x : RatFunc K} {p : K[X]} (hp : p ≠ 0) :
num x ∣ p ↔ ∃ q : K[X], q ≠ 0 ∧ x = algebraMap _ _ p / algebraMap _ _ q | case h
K : Type u
inst✝ : Field K
x : RatFunc K
q : K[X]
hp : x.num * q ≠ 0
_hx : x.num ≠ 0
hq : q ≠ 0
⊢ x.denom * q ≠ 0 ∧ x = (algebraMap K[X] (RatFunc K)) (x.num * q) / (algebraMap K[X] (RatFunc K)) (x.denom * q) | rw [RingHom.map_mul, RingHom.map_mul, ← div_mul_div_comm, div_self, mul_one, num_div_denom] | case h
K : Type u
inst✝ : Field K
x : RatFunc K
q : K[X]
hp : x.num * q ≠ 0
_hx : x.num ≠ 0
hq : q ≠ 0
⊢ x.denom * q ≠ 0 ∧ x = x
case h
K : Type u
inst✝ : Field K
x : RatFunc K
q : K[X]
hp : x.num * q ≠ 0
_hx : x.num ≠ 0
hq : q ≠ 0
⊢ (algebraMap K[X] (RatFunc K)) q ≠ 0 | 81e0952079e69fa9 |
Ordinal.lsub_le_iff | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem lsub_le_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} :
lsub.{_, v} f ≤ a ↔ ∀ i, f i < a | case h.e'_2.h.a
ι : Type u
f : ι → Ordinal.{max u v}
a : Ordinal.{max v u}
a✝ : ι
⊢ f a✝ < a ↔ (succ ∘ f) a✝ ≤ a | simp only [comp_apply, succ_le_iff] | no goals | 58794809dafd5879 |
AlgebraicGeometry.IsAffineOpen.isLocalization_stalk | Mathlib/AlgebraicGeometry/AffineScheme.lean | theorem isLocalization_stalk (x : U) :
IsLocalization.AtPrime (X.presheaf.stalk x) (hU.primeIdealOf x).asIdeal | case mk
X : Scheme
U : X.Opens
hU : IsAffineOpen U
y : PrimeSpectrum ↑Γ(X, U)
hx : (ConcreteCategory.hom hU.fromSpec.base) y ∈ U
hy : y = hU.primeIdealOf ⟨(ConcreteCategory.hom hU.fromSpec.base) y, hx⟩
⊢ IsLocalization.AtPrime (↑(X.presheaf.stalk ↑⟨(ConcreteCategory.hom hU.fromSpec.base) y, hx⟩)) y.asIdeal | exact hU.isLocalization_stalk' y hx | no goals | 1c62227211372277 |
PerfectPairing.exists_basis_basis_of_span_eq_top_of_mem_algebraMap | Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean | /-- If a perfect pairing over a field `L` takes values in a subfield `K` along two `K`-subspaces
whose `L` span is full, then these subspaces induce a `K`-structure in the sense of
[*Algebra I*, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989]. -/
lemma exists_basis_basis_of_span_eq_top_of_mem_algebraMap
(M'... | case intro.intro.intro.intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairi... | have h_span : span K v = span K (Set.range b) := by simp [b] | case intro.intro.intro.intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairi... | 28e714660261030d |
Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycleFactorsFinset_mul_inv_mem_eq_sdiff [DecidableEq α] [Fintype α] {f g : Perm α}
(h : f ∈ cycleFactorsFinset g) : cycleFactorsFinset (g * f⁻¹) = cycleFactorsFinset g \ {f} | case refine_3.inr
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleFacto... | rcases hd x with hx | hx | case refine_3.inr.inl
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleF... | 3d77fd0de031f082 |
GenContFract.of_correctness_of_nth_stream_eq_none | Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | theorem of_correctness_of_nth_stream_eq_none (nth_stream_eq_none : IntFractPair.stream v n = none) :
v = (of v).convs (n - 1) | case succ.inl
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
n : ℕ
IH : IntFractPair.stream v n = none → v = (of v).convs (n - 1)
nth_stream_eq_none✝ : IntFractPair.stream v (n + 1) = none
g : GenContFract K := of v
nth_stream_eq_none : IntFractPair.stream v n = none
⊢ v = g.convs n | cases n with
| zero => contradiction
| succ n' =>
have : g.TerminatedAt n' :=
of_terminatedAt_n_iff_succ_nth_intFractPair_stream_eq_none.2
nth_stream_eq_none
have : g.convs (n' + 1) = g.convs n' :=
convs_stable_of_terminated n'.le_succ this
rw [this]
exact IH nth_stream_eq_none | no goals | ee5931253fd2c40a |
FermatLastTheoremForThreeGen.Solution.lambda_not_dvd_x | Mathlib/NumberTheory/FLT/Three.lean | private lemma lambda_not_dvd_x : ¬ λ ∣ S.x := fun h ↦ by
replace h := mul_dvd_mul_left (λ ^ (3 * S.multiplicity - 2)) h
rw [mul_comm, ← x_spec] at h
replace h :=
mul_dvd_mul (mul_dvd_mul h S.lambda_dvd_a_add_eta_mul_b) S.lambda_dvd_a_add_eta_sq_mul_b
simp only [← a_cube_add_b_cube_eq_mul, S.H, w_spec, Units... | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ (λ ^ S.multiplicity * FermatLastTheoremForThreeGen.Solution.w S) ^ 3
⊢ False | have := S.two_le_multiplicity | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ (λ ^ S.multiplicity * FermatLastTheoremForThreeGen.Solution.w S) ^ 3
this : 2 ≤ S.multiplicity
⊢ False | 2126d926ee812016 |
HasConstantSpeedOnWith.union | Mathlib/Analysis/ConstantSpeed.lean | theorem HasConstantSpeedOnWith.union {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l)
(hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x) :
HasConstantSpeedOnWith f (s ∪ t) l | case h.mp.intro.inl.intro
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l... | exacts [Or.inl ⟨ws, zw, hs.2 ws⟩, Or.inr ⟨wt, ht.2 wt, wy⟩] | no goals | 34e577ecfbaa9f49 |
MvPolynomial.support_symmDiff_support_subset_support_add | Mathlib/Algebra/MvPolynomial/Basic.lean | theorem support_symmDiff_support_subset_support_add [DecidableEq σ] (p q : MvPolynomial σ R) :
p.support ∆ q.support ⊆ (p + q).support | case hs
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p q : MvPolynomial σ R
⊢ p.support \ q.support ⊆ (p + q).support | exact support_sdiff_support_subset_support_add p q | no goals | 414beece3a48f089 |
Orientation.volumeForm_neg_orientation | Mathlib/Analysis/InnerProductSpace/Orientation.lean | theorem volumeForm_neg_orientation : (-o).volumeForm = -o.volumeForm | case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
⊢ (-o).volumeForm = -o.volumeForm | let e : OrthonormalBasis (Fin n.succ) ℝ E := o.finOrthonormalBasis n.succ_pos Fact.out | case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ (-o).volumeForm = -o.volumeForm | b018b6f98e344b85 |
hasDerivAt_tsum_of_isPreconnected | Mathlib/Analysis/Calculus/SmoothSeries.lean | theorem hasDerivAt_tsum_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
(h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun n => g n y₀)
(hy : y ∈ t) : HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y | α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g g' : α → 𝕜 → F
t : Set 𝕜
y₀ y : 𝕜
hu : Summable u
ht : IsOpen t
h't : IsPreconnected t
hg' : ∀ (n : α), ∀ y ∈ t, ‖g' ... | simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul] | no goals | d575ff3867ea78db |
PrimeSpectrum.toPiLocalization_surjective_of_discreteTopology | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem toPiLocalization_surjective_of_discreteTopology :
Function.Surjective (toPiLocalization R) := fun x ↦ by
have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm ... | R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
⊢ ∃ a, (toPiLocalization R) a = x | have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm <| Set.singleton_subset_iff.mpr hpf⟩ | R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
this : ∀ (p : PrimeSpectrum R), ∃ f, ↑(basicOpen f) = {p}
⊢ ∃ a, (toPiLocalization R) a = x | 680284c76fd78d8c |
LSeries.abscissaOfAbsConv_logMul | Mathlib/NumberTheory/LSeries/Deriv.lean | /-- The abscissa of absolute convergence of the point-wise product of `log` and `f`
is the same as that of `f`. -/
@[simp]
lemma LSeries.abscissaOfAbsConv_logMul {f : ℕ → ℂ} :
abscissaOfAbsConv (logMul f) = abscissaOfAbsConv f | case h
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < ↑s
n : ℕ
hn : 1 ⊔ ⌈Real.exp 1⌉₊ ≤ n
⊢ ‖f n / ↑n ^ ↑s‖ ≤ ‖Real.log ↑n‖ * ‖f n / ↑n ^ ↑s‖ | refine le_mul_of_one_le_left (norm_nonneg _) (.trans ?_ <| Real.le_norm_self _) | case h
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < ↑s
n : ℕ
hn : 1 ⊔ ⌈Real.exp 1⌉₊ ≤ n
⊢ 1 ≤ Real.log ↑n | d6d82b44ec397e36 |
Batteries.TransCmp.compareOfLessAndEq | Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean | theorem TransCmp.compareOfLessAndEq
[LT α] [DecidableRel (LT.lt (α := α))] [DecidableEq α]
(lt_irrefl : ∀ x : α, ¬x < x)
(lt_trans : ∀ {x y z : α}, x < y → y < z → x < z)
(lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y) :
TransCmp (α := α) (compareOfLessAndEq · ·) | α : Type u_1
inst✝² : LT α
inst✝¹ : DecidableRel LT.lt
inst✝ : DecidableEq α
lt_irrefl : ∀ (x : α), ¬x < x
lt_trans : ∀ {x y z : α}, x < y → y < z → x < z
lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y
this : OrientedCmp fun x1 x2 => compareOfLessAndEq x1 x2
x y z : α
yx : ¬y < x
zy : ¬z < y
zx : z < x
xy : x < y
⊢... | exact zy (lt_trans zx xy) | no goals | 265051c307a250c5 |
AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen | Mathlib/AlgebraicGeometry/AffineScheme.lean | theorem basicOpen_basicOpen_is_basicOpen (g : Γ(X, X.basicOpen f)) :
∃ f' : Γ(X, U), X.basicOpen f' = X.basicOpen g | case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
x : ↑Γ(X, U)
n : ℕ
⊢ X.basicOpen f ⊓ X.basicOpen x =
X.basicOpen f ⊓ X.basicOpen x ⊓
X.basicOpen ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑Γ(X, X.basicOpen f)) ⟨(fun x => f ^ x) n, ⋯⟩) | refine (inf_eq_left.mpr (inf_le_left.trans_eq (Scheme.basicOpen_of_isUnit _ ?_).symm)).symm | case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
x : ↑Γ(X, U)
n : ℕ
⊢ IsUnit ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑Γ(X, X.basicOpen f)) ⟨(fun x => f ^ x) n, ⋯⟩) | 32b1bf2d9457c577 |
eigenvalue_pos_of_pos | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
(hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : 0 < μ | case intro
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : ℝ
T : E →ₗ[𝕜] E
hμ : HasEigenvalue T ↑μ
hnn : ∀ (x : E), 0 < RCLike.re (inner x (T x))
v : E
hv : HasEigenvector T (↑μ) v
hpos : 0 < ‖v‖ ^ 2
this : RCLike.re (inner v (T v)) = μ * ‖v‖ ^ 2
⊢ 0 < μ | exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v) | no goals | bb16797dca3f431f |
Turing.mem_eval | Mathlib/Computability/PostTuringMachine.lean | theorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none | case refine_1.none
σ : Type u_1
f : σ → Option σ
a✝ b : σ
h✝ : b ∈ eval f a✝
a : σ
h : b ∈ eval f a
IH : ∀ (a' : σ), f a = some a' → Reaches f a' b ∧ f b = none
e : f a = none
⊢ Reaches f a b ∧ f b = none | rw [Part.mem_unique h
(PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e]; rfl)] | case refine_1.none
σ : Type u_1
f : σ → Option σ
a✝ b : σ
h✝ : b ∈ eval f a✝
a : σ
h : b ∈ eval f a
IH : ∀ (a' : σ), f a = some a' → Reaches f a' b ∧ f b = none
e : f a = none
⊢ Reaches f a a ∧ f a = none | f40442f1b60c5272 |
MeasureTheory.IsSetSemiring.sUnion_union_disjointOfDiffUnion_of_subset | Mathlib/MeasureTheory/SetSemiring.lean | lemma sUnion_union_disjointOfDiffUnion_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
(hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) [DecidableEq (Set α)] :
⋃₀ ↑(I ∪ hC.disjointOfDiffUnion hs hI) = s | α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
inst✝ : DecidableEq (Set α)
⊢ ⋃₀ ↑(I ∪ hC.disjointOfDiffUnion hs hI) = s | conv_rhs => rw [← sUnion_union_sUnion_disjointOfDiffUnion_of_subset hC hs hI hI_ss] | α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
inst✝ : DecidableEq (Set α)
⊢ ⋃₀ ↑(I ∪ hC.disjointOfDiffUnion hs hI) = ⋃₀ ↑I ∪ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI) | cee69316b4a21487 |
padicValNat.div_of_dvd | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem div_of_dvd (h : b ∣ a) :
padicValNat p (a / b) = padicValNat p a - padicValNat p b | p a b : ℕ
hp : Fact (Nat.Prime p)
h : b ∣ a
⊢ padicValNat p (a / b) = padicValNat p a - padicValNat p b | rcases eq_or_ne a 0 with (rfl | ha) | case inl
p b : ℕ
hp : Fact (Nat.Prime p)
h : b ∣ 0
⊢ padicValNat p (0 / b) = padicValNat p 0 - padicValNat p b
case inr
p a b : ℕ
hp : Fact (Nat.Prime p)
h : b ∣ a
ha : a ≠ 0
⊢ padicValNat p (a / b) = padicValNat p a - padicValNat p b | b9ef4a5d0d954b5d |
nontrivial_of_hasRankNullity | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | theorem nontrivial_of_hasRankNullity : Nontrivial R | R : Type u_1
inst✝¹ : Ring R
inst✝ : HasRankNullity.{u, u_1} R
H : Subsingleton R
⊢ False | have := rank_quotient_add_rank (R := R) (M := PUnit) ⊥ | R : Type u_1
inst✝¹ : Ring R
inst✝ : HasRankNullity.{u, u_1} R
H : Subsingleton R
this : Module.rank R (PUnit.{u + 1} ⧸ ⊥) + Module.rank R ↥⊥ = Module.rank R PUnit.{u + 1}
⊢ False | d0036cc4f503eac7 |
Vector.toList_eq_empty_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem toList_eq_empty_iff (v : Vector α n) : v.toList = [] ↔ n = 0 | α : Type u_1
n : Nat
h : #[].size = n
⊢ n = 0 | simp_all | no goals | b929db816e818466 |
Array.filterMap_attachWith | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem filterMap_attachWith {q : α → Prop} {l : Array α} {f : {x // q x} → Option β} (H)
(w : stop = (l.attachWith q H).size) :
(l.attachWith q H).filterMap f 0 stop = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) | case mk
α : Type u_1
β : Type u_2
q : α → Prop
f : { x // q x } → Option β
toList✝ : List α
H : ∀ (x : α), x ∈ { toList := toList✝ } → q x
⊢ filterMap f ({ toList := toList✝ }.attachWith q H) =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => f ⟨x, ⋯⟩)
{ toList := toList✝ }.attach | simp [Function.comp_def] | no goals | 5db662eb1b9957d4 |
MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) :
DenseRange (coeSimpleFuncNonnegToLpNonneg p μ G) := fun g ↦ by
borelize G
rw [mem_closure_iff_seq_limit]
have hg_memLp : MemLp (g : α → G) p μ := Lp.memLp (g : Lp G p μ)
have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ ... | α : Type u_1
inst✝¹ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝ : NormedLatticeAddCommGroup G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : Separab... | filter_upwards [Lp.simpleFunc.coeFn_zero p μ G, h_toLp n] with a ha0 ha_toLp | case h
α : Type u_1
inst✝¹ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝ : NormedLatticeAddCommGroup G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : ... | 1fc70ad144f2653f |
MeasureTheory.tendsto_measure_of_le_liminf_measure_of_limsup_measure_le | Mathlib/MeasureTheory/Measure/Portmanteau.lean | theorem tendsto_measure_of_le_liminf_measure_of_limsup_measure_le {ι : Type*} {L : Filter ι}
{μ : Measure Ω} {μs : ι → Measure Ω} {E₀ E E₁ : Set Ω} (E₀_subset : E₀ ⊆ E) (subset_E₁ : E ⊆ E₁)
(nulldiff : μ (E₁ \ E₀) = 0) (h_E₀ : μ E₀ ≤ L.liminf fun i ↦ μs i E₀)
(h_E₁ : (L.limsup fun i ↦ μs i E₁) ≤ μ E₁) : L.T... | case hinf
Ω : Type u_1
inst✝ : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
E₀ E E₁ : Set Ω
E₀_subset : E₀ ⊆ E
subset_E₁ : E ⊆ E₁
nulldiff : μ (E₁ \ E₀) = 0
h_E₀ : μ E₀ ≤ liminf (fun i => (μs i) E₀) L
h_E₁ : limsup (fun i => (μs i) E₁) L ≤ μ E₁
E₀_ae_eq_E : E₀ =ᶠ[ae μ] E
⊢ μ E ≤ liminf (... | calc
μ E = μ E₀ := measure_congr E₀_ae_eq_E.symm
_ ≤ L.liminf fun i ↦ μs i E₀ := h_E₀
_ ≤ L.liminf fun i ↦ μs i E :=
liminf_le_liminf (.of_forall fun _ ↦ measure_mono E₀_subset) | no goals | 5cf451a541c4adf1 |
factorization_mul | Mathlib/RingTheory/UniqueFactorizationDomain/Finsupp.lean | theorem factorization_mul {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) :
factorization (a * b) = factorization a + factorization b | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ factorization (a * b) = factorization a + factorization b | simp [factorization, normalizedFactors_mul ha hb] | no goals | d7f1916bc7a3f9f4 |
LinearMap.polyCharpolyAux_basisIndep | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | /-- `LinearMap.polyCharpolyAux` is independent of the choice of basis of the target module.
Proof strategy:
1. Rewrite `polyCharpolyAux` as the (honest, ordinary) characteristic polynomial
of the basechange of `φ` to the multivariate polynomial ring `MvPolynomial ι R`.
2. Use that the characteristic polynomial of a... | R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R ... | apply hf | case a
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basi... | e0d11818785cd00c |
CategoryTheory.Sieve.pullback_functorPushforward_equivalence_eq | Mathlib/CategoryTheory/Sites/Sieves.lean | lemma pullback_functorPushforward_equivalence_eq {X : C} (S : Sieve X) :
Sieve.pullback (e.unit.app X) (Sieve.functorPushforward e.inverse
(Sieve.functorPushforward e.functor S)) = S | case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
X : C
S : Sieve X
Y✝ : C
f✝ : Y✝ ⟶ (𝟭 C).obj X
⊢ (pullback (e.unit.app X) (functorPushforward e.inverse (functorPushforward e.functor S))).arrows f✝ ↔ S.arrows f✝ | simp | no goals | 01c0d4922eecab06 |
List.sym_map | Mathlib/Data/List/Sym.lean | theorem sym_map {β : Type*} (f : α → β) (n : ℕ) (xs : List α) :
(xs.map f).sym n = (xs.sym n).map (Sym.map f) :=
match n, xs with
| 0, _ => by simp only [List.sym]; rfl
| n + 1, [] => by simp [List.sym]
| n + 1, x :: xs => by
rw [map_cons, List.sym, ← map_cons, sym_map f n (x :: xs), sym_map f (n + 1) x... | α : Type u_1
β : Type u_3
f : α → β
n✝ : ℕ
xs✝ : List α
n : ℕ
x : α
xs : List α
⊢ List.sym (n + 1) (map f (x :: xs)) = map (Sym.map f) (List.sym (n + 1) (x :: xs)) | rw [map_cons, List.sym, ← map_cons, sym_map f n (x :: xs), sym_map f (n + 1) xs] | α : Type u_1
β : Type u_3
f : α → β
n✝ : ℕ
xs✝ : List α
n : ℕ
x : α
xs : List α
⊢ map (fun p => f x ::ₛ p) (map (Sym.map f) (List.sym n (x :: xs))) ++ map (Sym.map f) (List.sym (n + 1) xs) =
map (Sym.map f) (List.sym (n + 1) (x :: xs)) | 9c4a921cc06ff9a6 |
CategoryTheory.Functor.isTriangulated_iff_of_iso | Mathlib/CategoryTheory/Triangulated/Functor.lean | lemma isTriangulated_iff_of_iso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
[NatTrans.CommShift e.hom ℤ] : F₁.IsTriangulated ↔ F₂.IsTriangulated | case mp
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{u_4, u_1} C
inst✝¹³ : Category.{u_5, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunct... | exact isTriangulated_of_iso e | no goals | 80e58c9c0d65fc3c |
Nat.digits_eq_cons_digits_div | Mathlib/Data/Nat/Digits.lean | theorem digits_eq_cons_digits_div {b n : ℕ} (h : 1 < b) (w : n ≠ 0) :
digits b n = (n % b) :: digits b (n / b) | case succ.succ.zero
b : ℕ
h : 1 < b + 1 + 1
w : 0 ≠ 0
⊢ (b + 1 + 1).digits 0 = 0 % (b + 1 + 1) :: (b + 1 + 1).digits (0 / (b + 1 + 1)) | norm_num at w | no goals | 7cabbb56ad772e66 |
cauchySeq_of_controlled | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem cauchySeq_of_controlled [SemilatticeSup β] [Nonempty β] (U : β → Set (α × α))
(hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) {f : β → α}
(hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N) : CauchySeq f :=
-- Porting note: changed to semi-implicit arguments
cauchySeq_iff_tendsto.2
(by
intro s hs
... | α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝¹ : SemilatticeSup β
inst✝ : Nonempty β
U : β → Set (α × α)
hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s
f : β → α
hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N
s : Set (α × α)
hs : s ∈ 𝓤 α
⊢ s ∈ map (Prod.map f f) atTop | rw [mem_map, mem_atTop_sets] | α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝¹ : SemilatticeSup β
inst✝ : Nonempty β
U : β → Set (α × α)
hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s
f : β → α
hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N
s : Set (α × α)
hs : s ∈ 𝓤 α
⊢ ∃ a, ∀ b ≥ a, b ∈ Prod.map f f ⁻¹' s | 6d28c0917de53b42 |
Matrix.linfty_opNNNorm_eq_opNNNorm | Mathlib/Analysis/Matrix.lean | lemma linfty_opNNNorm_eq_opNNNorm (A : Matrix m n α) :
‖A‖₊ = ‖ContinuousLinearMap.mk (Matrix.mulVecLin A)‖₊ | m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
⊢ ∑ j : n, ‖A i j‖₊ ≤ N | cases isEmpty_or_nonempty n | case inl
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : IsEmpty n
⊢ ∑ j : n, ‖A i j‖₊ ≤ N
cas... | e5f792842b2dab2d |
HomologicalComplex.to_double_hom_ext | Mathlib/Algebra/Homology/Double.lean | @[ext]
lemma to_double_hom_ext {K : HomologicalComplex C c} {φ φ' : K ⟶ double f hi₀₁}
(h₀ : φ.f i₀ = φ'.f i₀) (h₁ : φ.f i₁ = φ'.f i₁) : φ = φ' | case neg
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ i₁ : ι
hi₀₁ : c.Rel i₀ i₁
K : HomologicalComplex C c
φ φ' : K ⟶ double f hi₀₁
h₀ : φ.f i₀ = φ'.f i₀
h₁ : φ.f i₁ = φ'.f i₁
k : ι
h : ¬(k = i₀ ∨ k = i₁)
⊢ φ.f k... | simp only [not_or] at h | case neg
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ i₁ : ι
hi₀₁ : c.Rel i₀ i₁
K : HomologicalComplex C c
φ φ' : K ⟶ double f hi₀₁
h₀ : φ.f i₀ = φ'.f i₀
h₁ : φ.f i₁ = φ'.f i₁
k : ι
h : ¬k = i₀ ∧ ¬k = i₁
⊢ φ.f k ... | c4f6965891aa734b |
Doset.mk_out_eq_mul | Mathlib/GroupTheory/DoubleCoset.lean | theorem mk_out_eq_mul (H K : Subgroup G) (g : G) :
∃ h k : G, h ∈ H ∧ k ∈ K ∧ (mk H K g : Quotient ↑H ↑K).out = h * g * k | G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K g = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
⊢ ∃ h k, h ∈ H ∧ k ∈ K ∧ Quotient.out (mk H K g) = h * g * k | obtain ⟨h, h_h, k, hk, T⟩ := this.1 rfl | case intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K g = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
h : G
h_h : h ∈ H
k : G
hk : k ∈ K
T : g = h * Quotient.out (mk H K g) * k
⊢ ∃ h k, h ∈ H ∧ k ∈ K ∧ Quotient.out (mk H K g) = h * g * k | 02594208f2533041 |
DFA.evalFrom_comap | Mathlib/Computability/DFA.lean | theorem evalFrom_comap (f : α' → α) (s : σ) (x : List α') :
(M.comap f).evalFrom s x = M.evalFrom s (x.map f) | α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
f : α' → α
s : σ
x : List α'
⊢ (comap f M).evalFrom s x = M.evalFrom s (List.map f x) | induction x using List.reverseRecOn with
| nil => simp
| append_singleton x a ih => simp [ih] | no goals | 4d608835fbb525ea |
MeasureTheory.IsStoppingTime.measurableSpace_const | Mathlib/Probability/Process/Stopping.lean | theorem measurableSpace_const (f : Filtration ι m) (i : ι) :
(isStoppingTime_const f i).measurableSpace = f i | case pos
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : i ≤ j
⊢ MeasurableSet s | exact f.mono hij _ h | no goals | cae152d379995cda |
isNoetherian_of_finite_isArtinian | Mathlib/RingTheory/HopkinsLevitzki.lean | theorem isNoetherian_of_finite_isArtinian {R} [CommRing R] [Module R M]
[Module.Finite R M] [IsArtinian R M] : IsNoetherian R M | case intro.intro
M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : s.Finite
span : Submodule.span R s = ⊤
⊢ IsNoetherian R M | rw [← s.iUnion_of_singleton_coe, Submodule.span_iUnion] at span | case intro.intro
M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : s.Finite
span : ⨆ i, Submodule.span R {↑i} = ⊤
⊢ IsNoetherian R M | 0be4039c290118ba |
Quiver.heq_of_homOfEq_ext | Mathlib/Combinatorics/Quiver/Basic.lean | lemma heq_of_homOfEq_ext {X Y X' Y' : V} (hX : X = X') (hY : Y = Y') {f : X ⟶ Y} {f' : X' ⟶ Y'}
(e : Quiver.homOfEq f hX hY = f') : HEq f f' | V : Type u_1
inst✝ : Quiver V
X Y : V
f f' : X ⟶ Y
e : homOfEq f ⋯ ⋯ = f'
⊢ HEq f f' | rw [Quiver.homOfEq_rfl] at e | V : Type u_1
inst✝ : Quiver V
X Y : V
f f' : X ⟶ Y
e : f = f'
⊢ HEq f f' | 6b00b4f8fb3e49a1 |
cfc_tsub | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | theorem cfc_tsub {A : Type*} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A]
[StarOrderedRing A] [Algebra ℝ A] [IsTopologicalRing A] [T2Space A]
[ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
[NonnegSpectrumClass ℝ A] (f g : ℝ≥0 → ℝ≥0)
(a : A) (hfg : ∀ x ∈ spectrum ℝ≥0 a, g x ≤ ... | A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f g : ℝ≥0 → ℝ≥0
a : A
hfg : ∀ x ∈ spe... | have ha' := SpectrumRestricts.nnreal_of_nonneg ha | A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f g : ℝ≥0 → ℝ≥0
a : A
hfg : ∀ x ∈ spe... | bfcc5c40ea36abec |
MeasureTheory.IsTightMeasureSet.of_compactSpace | Mathlib/MeasureTheory/Measure/Tight.lean | /-- In a compact space, every set of measures is tight. -/
lemma of_compactSpace [CompactSpace α] : IsTightMeasureSet S | α : Type u_1
inst✝¹ : TopologicalSpace α
mα : MeasurableSpace α
S : Set (Measure α)
inst✝ : CompactSpace α
⊢ IsTightMeasureSet S | simp only [IsTightMeasureSet, cocompact_eq_bot, smallSets_bot, tendsto_pure_left, iSup_apply,
measure_empty, ENNReal.iSup_zero, ciSup_const] | α : Type u_1
inst✝¹ : TopologicalSpace α
mα : MeasurableSpace α
S : Set (Measure α)
inst✝ : CompactSpace α
⊢ ∀ s ∈ 𝓝 0, 0 ∈ s | 21154a5a66e41064 |
List.rotate_eq_self_iff_eq_replicate | Mathlib/Data/List/Rotate.lean | theorem rotate_eq_self_iff_eq_replicate [hα : Nonempty α] :
∀ {l : List α}, (∀ n, l.rotate n = l) ↔ ∃ a, l = replicate l.length a
| [] => by simp
| a :: l => ⟨fun h => ⟨a, ext_getElem (length_replicate _ _).symm fun n h₁ h₂ => by
rw [getElem_replicate, ← Option.some_inj, ← getElem?_eq_getElem, ← head?_rot... | α : Type u
hα : Nonempty α
a : α
l : List α
h : ∀ (n : ℕ), (a :: l).rotate n = a :: l
n : ℕ
h₁ : n < (a :: l).length
h₂ : n < (replicate (a :: l).length a).length
⊢ (a :: l)[n] = (replicate (a :: l).length a)[n] | rw [getElem_replicate, ← Option.some_inj, ← getElem?_eq_getElem, ← head?_rotate h₁, h,
head?_cons] | no goals | db31b7be5d2403ad |
ProbabilityTheory.IndepFun.integrable_mul | Mathlib/Probability/Integration.lean | theorem IndepFun.integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
[NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (hX : Integrable X μ)
(hY : Integrable Y μ) : Integrable (X * Y) μ | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
β : Type u_2
inst✝² : MeasurableSpace β
X Y : Ω → β
inst✝¹ : NormedDivisionRing β
inst✝ : BorelSpace β
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
nX : Ω → ℝ≥0∞ := fun a => ‖X a‖ₑ
⊢ Integrable (X * Y) μ | let nY : Ω → ℝ≥0∞ := fun a => ‖Y a‖ₑ | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
β : Type u_2
inst✝² : MeasurableSpace β
X Y : Ω → β
inst✝¹ : NormedDivisionRing β
inst✝ : BorelSpace β
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
nX : Ω → ℝ≥0∞ := fun a => ‖X a‖ₑ
nY : Ω → ℝ≥0∞ := fun a => ‖Y a‖ₑ
⊢ Integrable (X * Y) μ | d40345ba044b444c |
MeasureTheory.tilted_neg_same' | Mathlib/MeasureTheory/Measure/Tilted.lean | @[simp]
lemma tilted_neg_same' (hf : Integrable (fun x ↦ exp (f x)) μ) :
(μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x => rexp (f x)) μ
⊢ (μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ | rw [tilted_tilted hf] | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x => rexp (f x)) μ
⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ | 0162737593442fec |
Array.eraseP_of_forall_getElem_not | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem eraseP_of_forall_getElem_not {l : Array α} (h : ∀ i, (h : i < l.size) → ¬p l[i]) : l.eraseP p = l :=
eraseP_of_forall_mem_not fun a m => by
rw [mem_iff_getElem] at m
obtain ⟨i, w, rfl⟩ := m
exact h i w
| α : Type u_1
p : α → Bool
l : Array α
h : ∀ (i : Nat) (h : i < l.size), ¬p l[i] = true
a : α
m : a ∈ l
⊢ ¬p a = true | rw [mem_iff_getElem] at m | α : Type u_1
p : α → Bool
l : Array α
h : ∀ (i : Nat) (h : i < l.size), ¬p l[i] = true
a : α
m : ∃ i h, l[i] = a
⊢ ¬p a = true | 880db80905b9f130 |
SetTheory.PGame.mul_neg | Mathlib/SetTheory/Game/Basic.lean | /-- `x * -y` and `-(x * y)` have the same moves. -/
@[simp]
lemma mul_neg (x y : PGame) : x * -y = -(x * y) :=
match x, y with
| mk xl xr xL xR, mk yl yr yL yR => by
refine ext rfl rfl ?_ ?_ <;> rintro (⟨i, j⟩ | ⟨i, j⟩) _ ⟨rfl⟩
all_goals
dsimp
rw [PGame.neg_sub', PGame.neg_add]
congr
... | case refine_2.inr.mk.refl
x y : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
i : xr
j : yr
β : Type u_1
⊢ (xR i * mk yr yl (fun x => -yR x) fun x => -yL x) + mk xl xr xL xR * -yR j - xR i * -yR j =
-(xR i * mk yl yr yL yR) + -(mk xl xr xL xR * yR j) - -(xR ... | congr | case refine_2.inr.mk.refl.e_a.e_a
x y : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
i : xr
j : yr
β : Type u_1
⊢ (xR i * mk yr yl (fun x => -yR x) fun x => -yL x) = -(xR i * mk yl yr yL yR)
case refine_2.inr.mk.refl.e_a.e_a
x y : PGame
xl xr : Type u_1
xL : x... | 4dd968479e14eff2 |
IsPreconnected.biUnion_of_reflTransGen | Mathlib/Topology/Connected/Basic.lean | theorem IsPreconnected.biUnion_of_reflTransGen {ι : Type*} {t : Set ι} {s : ι → Set α}
(H : ∀ i ∈ t, IsPreconnected (s i))
(K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
IsPreconnected (⋃ n ∈ t, s n) | α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
t : Set ι
s : ι → Set α
H : ∀ i ∈ t, IsPreconnected (s i)
K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j
R : ι → ι → Prop := fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t
P : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen R i j → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ Is... | obtain ⟨i : ι, hi : i ∈ t, hxi : x ∈ s i⟩ := mem_iUnion₂.1 hx | case intro.intro
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
t : Set ι
s : ι → Set α
H : ∀ i ∈ t, IsPreconnected (s i)
K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j
R : ι → ι → Prop := fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t
P : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen R i j → ∃ p ⊆ t, i... | bfbb6da4007cc299 |
Padic.norm_le_pow_iff_norm_lt_pow_add_one | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_le_pow_iff_norm_lt_pow_add_one (x : ℚ_[p]) (n : ℤ) :
‖x‖ ≤ (p : ℝ) ^ n ↔ ‖x‖ < (p : ℝ) ^ (n + 1) | p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
n : ℤ
aux : ∀ (n : ℤ), 0 < ↑p ^ n
⊢ ‖x‖ ≤ ↑p ^ n ↔ ‖x‖ < ↑p ^ (n + 1) | by_cases hx0 : x = 0 | case pos
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
n : ℤ
aux : ∀ (n : ℤ), 0 < ↑p ^ n
hx0 : x = 0
⊢ ‖x‖ ≤ ↑p ^ n ↔ ‖x‖ < ↑p ^ (n + 1)
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
n : ℤ
aux : ∀ (n : ℤ), 0 < ↑p ^ n
hx0 : ¬x = 0
⊢ ‖x‖ ≤ ↑p ^ n ↔ ‖x‖ < ↑p ^ (n + 1) | a9f8ab08c755e528 |
IsPGroup.card_center_eq_prime_pow | Mathlib/GroupTheory/PGroup.lean | theorem card_center_eq_prime_pow (hGpn : Nat.card G = p ^ n) (hn : 0 < n) :
∃ k > 0, Nat.card (center G) = p ^ k | p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Fact (Nat.Prime p)
n : ℕ
hGpn : Nat.card G = p ^ n
hn : 0 < n
this : Finite G
⊢ ∃ k > 0, Nat.card ↥(center G) = p ^ k | have hcG := to_subgroup (of_card hGpn) (center G) | p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Fact (Nat.Prime p)
n : ℕ
hGpn : Nat.card G = p ^ n
hn : 0 < n
this : Finite G
hcG : IsPGroup p ↥(center G)
⊢ ∃ k > 0, Nat.card ↥(center G) = p ^ k | 5aec44ab13b1ce62 |
RootPairing.mem_range_root_of_mem_range_reflection_of_mem_range_root | Mathlib/LinearAlgebra/RootSystem/Defs.lean | lemma mem_range_root_of_mem_range_reflection_of_mem_range_root
{r : M ≃ₗ[R] M} {α : M} (hr : r ∈ range P.reflection) (hα : α ∈ range P.root) :
r • α ∈ range P.root | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
r : M ≃ₗ[R] M
α : M
hr : r ∈ range P.reflection
hα : α ∈ range ⇑P.root
⊢ r • α ∈ range ⇑P.root | obtain ⟨i, rfl⟩ := hr | case intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
α : M
hα : α ∈ range ⇑P.root
i : ι
⊢ P.reflection i • α ∈ range ⇑P.root | 5f0c2cf55edf4d44 |
Equiv.Perm.signAux3_mul_and_swap | Mathlib/GroupTheory/Perm/Sign.lean | theorem signAux3_mul_and_swap [Finite α] (f g : Perm α) (s : Multiset α) (hs : ∀ x, x ∈ s) :
signAux3 (f * g) hs = signAux3 f hs * signAux3 g hs ∧
Pairwise fun x y => signAux3 (swap x y) hs = -1 | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Finite α
f g : Perm α
s : Multiset α
hs : ∀ (x : α), x ∈ s
⊢ (f * g).signAux3 hs = f.signAux3 hs * g.signAux3 hs ∧ Pairwise fun x y => (swap x y).signAux3 hs = -1 | obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin α | case intro.intro
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Finite α
f g : Perm α
s : Multiset α
hs : ∀ (x : α), x ∈ s
n : ℕ
e : α ≃ Fin n
⊢ (f * g).signAux3 hs = f.signAux3 hs * g.signAux3 hs ∧ Pairwise fun x y => (swap x y).signAux3 hs = -1 | e47fa89a1e6e7e6b |
Set.prod_mul_prod_comm | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | @[to_additive (attr := simp)]
lemma prod_mul_prod_comm [Mul β] (s₁ s₂: Set α) (t₁ t₂ : Set β) :
(s₁ ×ˢ t₁) * (s₂ ×ˢ t₂) = (s₁ * s₂) ×ˢ (t₁ * t₂) | α : Type u_2
β : Type u_3
inst✝¹ : Mul α
inst✝ : Mul β
s₁ s₂ : Set α
t₁ t₂ : Set β
⊢ s₁ ×ˢ t₁ * s₂ ×ˢ t₂ = (s₁ * s₂) ×ˢ (t₁ * t₂) | ext | case h
α : Type u_2
β : Type u_3
inst✝¹ : Mul α
inst✝ : Mul β
s₁ s₂ : Set α
t₁ t₂ : Set β
x✝ : α × β
⊢ x✝ ∈ s₁ ×ˢ t₁ * s₂ ×ˢ t₂ ↔ x✝ ∈ (s₁ * s₂) ×ˢ (t₁ * t₂) | 726912104abc2586 |
Finset.prod_image_le_of_one_le | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | @[to_additive]
lemma prod_image_le_of_one_le
{g : ι → ι'} {f : ι' → N} (hf : ∀ u ∈ s.image g, 1 ≤ f u) :
∏ u ∈ s.image g, f u ≤ ∏ u ∈ s, f (g u) | ι : Type u_1
N : Type u_5
inst✝¹ : OrderedCommMonoid N
s : Finset ι
ι' : Type u_9
inst✝ : DecidableEq ι'
g : ι → ι'
f : ι' → N
hf : ∀ u ∈ image g s, 1 ≤ f u
a : ι'
hag : a ∈ image g s
⊢ f a ≤ f a ^ #(filter (fun a_1 => g a_1 = a) s) | obtain ⟨i, hi, hig⟩ := Finset.mem_image.mp hag | case intro.intro
ι : Type u_1
N : Type u_5
inst✝¹ : OrderedCommMonoid N
s : Finset ι
ι' : Type u_9
inst✝ : DecidableEq ι'
g : ι → ι'
f : ι' → N
hf : ∀ u ∈ image g s, 1 ≤ f u
a : ι'
hag : a ∈ image g s
i : ι
hi : i ∈ s
hig : g i = a
⊢ f a ≤ f a ^ #(filter (fun a_1 => g a_1 = a) s) | 8b434b5772b56f02 |
SpectrumRestricts.nnreal_iff | Mathlib/Analysis/Normed/Algebra/Spectrum.lean | lemma nnreal_iff [Algebra ℝ A] {a : A} :
SpectrumRestricts a ContinuousMap.realToNNReal ↔ ∀ x ∈ spectrum ℝ a, 0 ≤ x | case refine_1
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
h : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ spectrum ℝ a
⊢ 0 ≤ x | obtain ⟨x, -, rfl⟩ := h.algebraMap_image.symm ▸ hx | case refine_1.intro.intro
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
h : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ≥0
hx : (algebraMap ℝ≥0 ℝ) x ∈ spectrum ℝ a
⊢ 0 ≤ (algebraMap ℝ≥0 ℝ) x | 81f2156a3335f496 |
Stream'.WSeq.head_some_of_get?_some | Mathlib/Data/Seq/WSeq.lean | theorem head_some_of_get?_some {s : WSeq α} {a n} (h : some a ∈ get? s n) :
∃ a', some a' ∈ head s | case succ
α : Type u
s : WSeq α
n : ℕ
IH : ∀ {a : α}, some a ∈ s.get? n → ∃ a', some a' ∈ s.head
a : α
h : some a ∈ s.get? (n + 1)
a' : α
h' : some a' ∈ (s.drop n).head
⊢ ∃ a', some a' ∈ s.head | exact IH h' | no goals | 3bd1fa37e58b287a |
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