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 |
|---|---|---|---|---|---|---|
Real.sqrtTwoAddSeries_two | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem sqrtTwoAddSeries_two : sqrtTwoAddSeries 0 2 = √(2 + √2) | ⊢ sqrtTwoAddSeries 0 2 = √(2 + √2) | simp | no goals | 37f80746f6448103 |
Int.cast_pow | Mathlib/Algebra/Ring/Int/Defs.lean | @[simp, norm_cast] lemma cast_pow {R : Type*} [Ring R] (n : ℤ) (m : ℕ) :
↑(n ^ m) = (n ^ m : R) | R : Type u_1
inst✝ : Ring R
n : ℤ
m : ℕ
⊢ ↑(n ^ m) = ↑n ^ m | induction' m with m ih <;> simp [_root_.pow_succ, *] | no goals | 1c3329c1e4791938 |
StieltjesFunction.measure_Ici | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem measure_Ici {l : ℝ} (hf : Tendsto f atTop (𝓝 l)) (x : ℝ) :
f.measure (Ici x) = ofReal (l - leftLim f x) | f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : ℝ
h_le1 : ∀ (x : ℝ), ↑f (x - 1) ≤ leftLim (↑f) x
h_le2 : ∀ (x : ℝ), leftLim (↑f) x ≤ ↑f x
⊢ Tendsto (fun i => i - 1) atTop atTop | rw [tendsto_atTop_atTop] | f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : ℝ
h_le1 : ∀ (x : ℝ), ↑f (x - 1) ≤ leftLim (↑f) x
h_le2 : ∀ (x : ℝ), leftLim (↑f) x ≤ ↑f x
⊢ ∀ (b : ℝ), ∃ i, ∀ (a : ℝ), i ≤ a → b ≤ a - 1 | 4e0cbe1c1c7072ca |
MeasureTheory.hahn_decomposition | Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | theorem hahn_decomposition (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
∃ s, MeasurableSet s ∧ (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧
∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t | case intro.refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Se... | intro t ht hts | case intro.refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Se... | b74e26e9476d0e19 |
MeasureTheory.MemLp.induction_dense | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem MemLp.induction_dense (hp_ne_top : p ≠ ∞) (P : (α → E) → Prop)
(h0P :
∀ (c : E) ⦃s : Set α⦄,
MeasurableSet s →
μ s < ∞ →
∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g : α → E, eLpNorm (g - s.indicator fun _ => c) p μ ≤ ε ∧ P g)
(h1P : ∀ f g, P f → P g → P (f + g)) (h2P : ∀ f, P f → AEStr... | case intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h0P :
∀ (c : E) ⦃s : Set α⦄,
MeasurableSet s → μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x => c) p μ ≤ ε ∧ ... | refine ⟨g, ?_, Pg⟩ | case intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h0P :
∀ (c : E) ⦃s : Set α⦄,
MeasurableSet s → μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x => c) p μ ≤ ε ∧ ... | 747111af72f956df |
Ordnode.size_balance' | Mathlib/Data/Ordmap/Ordset.lean | theorem size_balance' {l x r} (hl : @Sized α l) (hr : Sized r) :
size (@balance' α l x r) = size l + size r + 1 | α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Sized
hr : r.Sized
⊢ (l.balance' x r).size = l.size + r.size + 1 | unfold balance' | α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Sized
hr : r.Sized
⊢ (if l.size + r.size ≤ 1 then l.node' x r
else
if r.size > delta * l.size then l.rotateL x r
else if l.size > delta * r.size then l.rotateR x r else l.node' x r).size =
l.size + r.size + 1 | 164cc76764f47764 |
PrimeSpectrum.mem_image_comap_zeroLocus_sdiff | Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean | /-- Let `A` be an `R`-algebra.
`𝔭 : Spec R` is in the image of `Z(I) ∩ D(f) ⊆ Spec S`
if and only if `f` is not nilpotent on `κ(𝔭) ⊗ A ⧸ I`. -/
lemma mem_image_comap_zeroLocus_sdiff (f : A) (s : Set A) (x) :
x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f}) ↔
¬ IsNilpotent (algebraMap A ((A ⧸ Idea... | case mpr.intro.intro
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
q : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)
hq : q.IsPrime
hfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q
this : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal.span s... | refine ⟨comap (algebraMap A _) ⟨q, hq⟩, ⟨by simpa [Set.subset_def], by simpa⟩, ?_⟩ | case mpr.intro.intro
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
q : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)
hq : q.IsPrime
hfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q
this : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal.span s... | b5370dee55806ec3 |
MeasureTheory.eLpNorm_one_condExp_le_eLpNorm | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem eLpNorm_one_condExp_le_eLpNorm (f : α → ℝ) : eLpNorm (μ[f|m]) 1 μ ≤ eLpNorm f 1 μ | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ 0 ≤ᶠ[ae μ] μ[|f| |m] | rw [← condExp_zero] | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ ?m.15243[0|?m.15241] ≤ᶠ[ae μ] μ[|f| |m]
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ MeasurableSpace α
α : Ty... | 4c3dc2a5be2f6f42 |
TopCat.isTopologicalBasis_cofiltered_limit | Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean | theorem isTopologicalBasis_cofiltered_limit (hC : IsLimit C) (T : ∀ j, Set (Set (F.obj j)))
(hT : ∀ j, IsTopologicalBasis (T j)) (univ : ∀ i : J, Set.univ ∈ T i)
(inter : ∀ (i) (U1 U2 : Set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i)
(compat : ∀ (i j : J) (f : i ⟶ j) (V : Set (F.obj j)) (_hV : V ∈ T j)... | case h.e'_3.h.mpr.intro.intro.intro.intro.refine_2.h.h.h.h.h.h
J : Type v
inst✝¹ : Category.{w, v} J
inst✝ : IsCofiltered J
F : J ⥤ TopCat
C : Cone F
hC : IsLimit C
T : (j : J) → Set (Set ↑(F.obj j))
hT : ∀ (j : J), IsTopologicalBasis (T j)
univ : ∀ (i : J), Set.univ ∈ T i
inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1... | rw [← coe_comp, D.w] | case h.e'_3.h.mpr.intro.intro.intro.intro.refine_2.h.h.h.h.h.h
J : Type v
inst✝¹ : Category.{w, v} J
inst✝ : IsCofiltered J
F : J ⥤ TopCat
C : Cone F
hC : IsLimit C
T : (j : J) → Set (Set ↑(F.obj j))
hT : ∀ (j : J), IsTopologicalBasis (T j)
univ : ∀ (i : J), Set.univ ∈ T i
inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1... | 8a9de6a458f61cf0 |
LinearMap.span_singleton_inf_orthogonal_eq_bot | Mathlib/LinearAlgebra/SesquilinearForm.lean | theorem span_singleton_inf_orthogonal_eq_bot (B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂) (x : V₁)
(hx : ¬B.IsOrtho x x) : (K₁ ∙ x) ⊓ Submodule.orthogonalBilin (K₁ ∙ x) B = ⊥ | case intro
K : Type u_13
K₁ : Type u_14
V₁ : Type u_17
V₂ : Type u_18
inst✝⁵ : Field K
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J₁ J₁' : K₁ →+* K
B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂
x : V₁
hx : ¬B.IsOrtho x x
μ : V₁ → K₁
h : ∑ i ∈ {x}, μ i • i ∈ Submodule.s... | replace h := h.2 x (by simp [Submodule.mem_span] : x ∈ Submodule.span K₁ ({x} : Finset V₁)) | case intro
K : Type u_13
K₁ : Type u_14
V₁ : Type u_17
V₂ : Type u_18
inst✝⁵ : Field K
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J₁ J₁' : K₁ →+* K
B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂
x : V₁
hx : ¬B.IsOrtho x x
μ : V₁ → K₁
h : B.IsOrtho x (∑ i ∈ {x}, μ i • i)... | b7086190b7929afb |
ENNReal.mul_div_cancel' | Mathlib/Data/ENNReal/Inv.lean | /-- See `ENNReal.mul_div_cancel` for a simpler version assuming `a ≠ 0`, `a ≠ ∞`. -/
protected lemma mul_div_cancel' (ha₀ : a = 0 → b = 0) (ha : a = ∞ → b = 0) : a * (b / a) = b | a b : ℝ≥0∞
ha₀ : a = 0 → b = 0
ha : a = ⊤ → b = 0
⊢ a * (b / a) = b | rw [mul_comm, ENNReal.div_mul_cancel' ha₀ ha] | no goals | 466a2e4a896026ed |
Cardinal.aleph_le_beth | Mathlib/SetTheory/Cardinal/Aleph.lean | theorem aleph_le_beth (o : Ordinal) : ℵ_ o ≤ ℶ_ o | case H₂
o : Ordinal.{u_1}
h : ℵ_ o ≤ ℶ_ o
⊢ ℵ_ o < 2 ^ ℶ_ o | exact (cantor _).trans_le (power_le_power_left two_ne_zero h) | no goals | ddd0314775c52403 |
aux₁ | Mathlib/MeasureTheory/Order/UpperLower.lean | /-- If we can fit a small ball inside a set `sᶜ` intersected with any neighborhood of `x`, then the
density of `s` near `x` is not `1`.
Along with `aux₀`, this proves that `x` is a Lebesgue point of `s`. This will be used to prove that
the frontier of an order-connected set is null. -/
private lemma aux₁
(h : ∀ δ,... | case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / ... | on_goal 2 =>
calc
volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))
≤ volume (closedBall x (ε n) \ closedBall (f (ε n) <| hε' n) (ε n / 4)) /
volume (closedBall x (ε n)) := by
gcongr
rw [diff_eq_compl_inter]
refine inter_subset_inter_left _ ?_
rw [subset_c... | case intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / ... | 97816f689cf8a35d |
Int.ofNat_toNat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem ofNat_toNat (a : Int) : (a.toNat : Int) = max a 0 | a : Int
n : Nat
⊢ ↑-[n+1].toNat = max -[n+1] 0 | simp | no goals | 3021bd9f1ab75728 |
EulerSine.integral_sin_mul_sin_mul_cos_pow_eq | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | theorem integral_sin_mul_sin_mul_cos_pow_eq (hn : 2 ≤ n) (hz : z ≠ 0) :
(∫ x in (0 : ℝ)..π / 2, Complex.sin (2 * z * x) * sin x * (cos x : ℂ) ^ (n - 1)) =
(n / (2 * z) * ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) -
(n - 1) / (2 * z) *
∫ x in (0 : ℝ)..π / 2, Complex.cos... | case h.e'_8.h
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
c : HasDerivAt ((fun x => x ^ (n - 1)) ∘ Complex.cos) (↑(n - 1) * Complex.cos ↑x ^ (n - 1 - 1) * -Complex.sin ↑x) ↑x
y : ℝ
⊢ ↑(sin y) * ↑(cos y) ^ (n - 1) = Complex.sin ↑y * ((fun x => x ^ (n - 1)) ∘ Complex.cos) ↑y | simp only [Complex.ofReal_sin, Complex.ofReal_cos, Function.comp] | no goals | aa9c88c94003a4d7 |
AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover | Mathlib/AlgebraicGeometry/Noetherian.lean | theorem isNoetherian_iff_of_finite_affine_openCover {𝒰 : Scheme.OpenCover.{v, u} X}
[Finite 𝒰.J] [∀ i, IsAffine (𝒰.obj i)] :
IsNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing Γ(𝒰.obj i, ⊤) | X : Scheme
𝒰 : X.OpenCover
inst✝¹ : Finite 𝒰.J
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
⊢ IsNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤) | constructor | case mp
X : Scheme
𝒰 : X.OpenCover
inst✝¹ : Finite 𝒰.J
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
⊢ IsNoetherian X → ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
case mpr
X : Scheme
𝒰 : X.OpenCover
inst✝¹ : Finite 𝒰.J
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
⊢ (∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)) → I... | f55c23f20e8dcf72 |
Polynomial.zero_modByMonic | Mathlib/Algebra/Polynomial/Div.lean | theorem zero_modByMonic (p : R[X]) : 0 %ₘ p = 0 | case neg
R : Type u
inst✝ : Ring R
p : R[X]
hp : ¬p.Monic
⊢ (if h : p.Monic then
(if p.degree ≤ ⊥ ∧ ¬0 = 0 then
(C 0 * X ^ (0 - p.natDegree) + ((0 - p * (C 0 * X ^ (0 - p.natDegree))).divModByMonicAux ⋯).1,
((0 - p * (C 0 * X ^ (0 - p.natDegree))).divModByMonicAux ⋯).2)
else 0).2
... | rw [dif_neg hp] | no goals | f0d18cde90b3b083 |
MeasureTheory.uniformIntegrable_finite | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem uniformIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : ∀ i, MemLp (f i) p μ) : UniformIntegrable f p μ | case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), eLpNorm (f i) p μ < ⊤
C : ℝ≥0∞ := (Finset.... | exact Finset.le_max' (α := ℝ≥0∞) _ _ (Finset.mem_image.2 ⟨i, Finset.mem_univ _, rfl⟩) | no goals | 5abac80f706d496a |
Submodule.fg_induction | Mathlib/RingTheory/Finiteness/Basic.lean | theorem fg_induction (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
(P : Submodule R M → Prop) (h₁ : ∀ x, P (Submodule.span R {x}))
(h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : N.FG) : P N | case intro.empty
R : Type u_4
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : Submodule R M → Prop
h₁ : ∀ (x : M), P (span R {x})
h₂ : ∀ (M₁ M₂ : Submodule R M), P M₁ → P M₂ → P (M₁ ⊔ M₂)
⊢ P (span R ↑∅) | rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton] | case intro.empty
R : Type u_4
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : Submodule R M → Prop
h₁ : ∀ (x : M), P (span R {x})
h₂ : ∀ (M₁ M₂ : Submodule R M), P M₁ → P M₂ → P (M₁ ⊔ M₂)
⊢ P (span R {0}) | acaf019d55d4c340 |
HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod {y : E}
(hf : HasFPowerSeriesWithinOnBall f p s x r) (hy : y ∈ EMetric.ball (0 : E) r)
(h'y : x + y ∈ insert x s) :
Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f... | have A : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y,
dist (p.partialSum k z.2) (p.partialSum k y) < ε / 4 := by
have : ContinuousAt (fun z ↦ p.partialSum k z) y := (p.partialSum_continuous k).continuousAt
exact tendsto_snd (Metric.tendsto_nhds.1 this.tendsto (ε / 4) (by linarith)) | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f... | 84f2046d5fc6e8a5 |
Nat.det_vandermonde_id_eq_superFactorial | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | theorem det_vandermonde_id_eq_superFactorial (n : ℕ) :
(Matrix.vandermonde (fun (i : Fin (n + 1)) ↦ (i : R))).det = Nat.superFactorial n | case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
hn : (Matrix.vandermonde fun i => ↑↑i).det = ↑(sf n)
⊢ (∏ j ∈ Ioi 0, (↑↑j - ↑↑0)) * ∏ i : Fin (n + 1), ∏ j ∈ Ioi (Fin.succAbove 0 i), (↑↑j - ↑↑(Fin.succAbove 0 i)) =
↑(n.succ ! * sf n) | push_cast | case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
hn : (Matrix.vandermonde fun i => ↑↑i).det = ↑(sf n)
⊢ (∏ j ∈ Ioi 0, (↑↑j - ↑↑0)) * ∏ i : Fin (n + 1), ∏ j ∈ Ioi (Fin.succAbove 0 i), (↑↑j - ↑↑(Fin.succAbove 0 i)) =
↑n.succ ! * ↑(sf n) | 55fe604c0006566e |
Polynomial.map_restriction | Mathlib/RingTheory/Polynomial/Basic.lean | theorem map_restriction {R : Type u} [CommRing R] (p : R[X]) :
p.restriction.map (algebraMap _ _) = p :=
ext fun n => by rw [coeff_map, Algebra.algebraMap_ofSubring_apply, coeff_restriction]
| R : Type u
inst✝ : CommRing R
p : R[X]
n : ℕ
⊢ (map (algebraMap (↥(Subring.closure ↑p.coeffs)) R) p.restriction).coeff n = p.coeff n | rw [coeff_map, Algebra.algebraMap_ofSubring_apply, coeff_restriction] | no goals | 7be49a5ed8028cf9 |
Module.FaithfullyFlat.trans | Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean | theorem trans : FaithfullyFlat R M | R : Type u_1
inst✝⁸ : CommRing R
S : Type u_2
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FaithfullyFlat R S
inst✝ : FaithfullyFlat S M
⊢ FaithfullyFlat R M | rw [iff_zero_iff_lTensor_zero] | R : Type u_1
inst✝⁸ : CommRing R
S : Type u_2
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FaithfullyFlat R S
inst✝ : FaithfullyFlat S M
⊢ Flat R M ∧
∀ {N : Type (max u_1 u_3)} [inst : AddCommGroup N] [ins... | 67560735acca6665 |
LinearMap.ker_le_iff | Mathlib/Algebra/Module/Submodule/Range.lean | theorem ker_le_iff [RingHomSurjective τ₁₂] {p : Submodule R M} :
ker f ≤ p ↔ ∃ y ∈ range f, f ⁻¹' {y} ⊆ p | R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Ring R
inst✝⁷ : Ring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
f : F
inst✝ : RingHomSurjective τ₁₂
p : Submodule ... | constructor | case mp
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Ring R
inst✝⁷ : Ring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
f : F
inst✝ : RingHomSurjective τ₁₂
p : Su... | 4607f9408a9aa2c2 |
NumberField.hermiteTheorem.finite_of_discr_bdd_of_isReal | Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean | theorem finite_of_discr_bdd_of_isReal :
{K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
haveI : NumberField K := @NumberField.mk _ _ inferInstance K.prop
{w : InfinitePlace K | IsReal w}.Nonempty ∧ |discr K| ≤ N }.Finite | A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈(B ⊔ 1) ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |... | simp | no goals | 258b240e119b72fe |
CategoryTheory.leftAdjoint_preservesTerminal_of_reflective | Mathlib/CategoryTheory/Monad/Limits.lean | /-- The reflector always preserves terminal objects. Note this in general doesn't apply to any other
limit.
-/
lemma leftAdjoint_preservesTerminal_of_reflective (R : D ⥤ C) [Reflective R] :
PreservesLimitsOfShape (Discrete.{v} PEmpty) (monadicLeftAdjoint R) where
preservesLimit {K} | case preserves
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
R : D ⥤ C
inst✝ : Reflective R
K : Discrete PEmpty.{v + 1} ⥤ C
F : Discrete PEmpty.{v + 1} ⥤ D := Functor.empty D
c : Cone (F ⋙ R)
h : IsLimit c
this✝ : HasLimit (F ⋙ R)
this : HasLimit F
⊢ Nonempty (IsLimit ((monadicLeftAd... | constructor | case preserves.val
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
R : D ⥤ C
inst✝ : Reflective R
K : Discrete PEmpty.{v + 1} ⥤ C
F : Discrete PEmpty.{v + 1} ⥤ D := Functor.empty D
c : Cone (F ⋙ R)
h : IsLimit c
this✝ : HasLimit (F ⋙ R)
this : HasLimit F
⊢ IsLimit ((monadicLeftAdjoint ... | 68ed64ef7d4b8b6f |
Lagrange.eq_interpolate_of_eval_eq | Mathlib/LinearAlgebra/Lagrange.lean | theorem eq_interpolate_of_eval_eq {f : F[X]} (hvs : Set.InjOn v s) (degree_f_lt : f.degree < #s)
(eval_f : ∀ i ∈ s, f.eval (v i) = r i) : f = interpolate s v r | F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v r : ι → F
f : F[X]
hvs : Set.InjOn v ↑s
degree_f_lt : f.degree < ↑(#s)
eval_f : ∀ i ∈ s, eval (v i) f = r i
⊢ f = (interpolate s v) r | rw [eq_interpolate hvs degree_f_lt] | F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v r : ι → F
f : F[X]
hvs : Set.InjOn v ↑s
degree_f_lt : f.degree < ↑(#s)
eval_f : ∀ i ∈ s, eval (v i) f = r i
⊢ ((interpolate s v) fun i => eval (v i) f) = (interpolate s v) r | 854f85709b9436f3 |
Finsupp.sum_sum_index' | Mathlib/Algebra/BigOperators/Finsupp.lean | theorem Finsupp.sum_sum_index' (h0 : ∀ i, t i 0 = 0) (h1 : ∀ i x y, t i (x + y) = t i x + t i y) :
(∑ x ∈ s, f x).sum t = ∑ x ∈ s, (f x).sum t | α : Type u_1
ι : Type u_2
A : Type u_4
C : Type u_6
inst✝¹ : AddCommMonoid A
inst✝ : AddCommMonoid C
t : ι → A → C
s✝ : Finset α
f : α → ι →₀ A
h0 : ∀ (i : ι), t i 0 = 0
h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y
a : α
s : Finset α
has : a ∉ s
ih : (∑ x ∈ s, f x).sum t = ∑ x ∈ s, (f x).sum t
⊢ (∑ x ∈ insert ... | simp_rw [Finset.sum_insert has, Finsupp.sum_add_index' h0 h1, ih] | no goals | 23af0593f3118115 |
SimplexCategoryGenRel.hom_induction | Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/Basic.lean | /-- An unrolled version of the induction principle obtained in the previous lemma. -/
@[elab_as_elim, cases_eliminator, induction_eliminator]
lemma hom_induction (P : MorphismProperty SimplexCategoryGenRel)
(id : ∀ {n : ℕ}, P (𝟙 (mk n)))
(comp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ... | P : MorphismProperty SimplexCategoryGenRel
id : ∀ {n : ℕ}, P (𝟙 (mk n))
comp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ i)
comp_σ : ∀ {n m : ℕ} (u : mk n ⟶ mk (m + 1)) (i : Fin (m + 1)), P u → P (u ≫ σ i)
a b : SimplexCategoryGenRel
f : a ⟶ b
⊢ generators.multiplicativeClosure ≤ P | intro _ _ f hf | P : MorphismProperty SimplexCategoryGenRel
id : ∀ {n : ℕ}, P (𝟙 (mk n))
comp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ i)
comp_σ : ∀ {n m : ℕ} (u : mk n ⟶ mk (m + 1)) (i : Fin (m + 1)), P u → P (u ≫ σ i)
a b : SimplexCategoryGenRel
f✝ : a ⟶ b
X✝ Y✝ : SimplexCategoryGenRel
f : X✝ ⟶ Y✝
hf : gen... | 5b459ccac98f64d1 |
CategoryTheory.Limits.PullbackCone.mono_of_isLimitMkIdId | Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean | theorem mono_of_isLimitMkIdId (f : X ⟶ Y) (t : IsLimit (mk (𝟙 X) (𝟙 X) rfl : PullbackCone f f)) :
Mono f :=
⟨fun {Z} g h eq => by
rcases PullbackCone.IsLimit.lift' t _ _ eq with ⟨_, rfl, rfl⟩
rfl⟩
| C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : X ⟶ Y
t : IsLimit (mk (𝟙 X) (𝟙 X) ⋯)
Z : C
g h : Z ⟶ X
eq : g ≫ f = h ≫ f
⊢ g = h | rcases PullbackCone.IsLimit.lift' t _ _ eq with ⟨_, rfl, rfl⟩ | case mk.intro
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : X ⟶ Y
t : IsLimit (mk (𝟙 X) (𝟙 X) ⋯)
Z : C
val✝ : Z ⟶ (mk (𝟙 X) (𝟙 X) ⋯).pt
eq : (val✝ ≫ (mk (𝟙 X) (𝟙 X) ⋯).fst) ≫ f = (val✝ ≫ (mk (𝟙 X) (𝟙 X) ⋯).snd) ≫ f
⊢ val✝ ≫ (mk (𝟙 X) (𝟙 X) ⋯).fst = val✝ ≫ (mk (𝟙 X) (𝟙 X) ⋯).snd | 98d4f8fb735a5040 |
Int.cast_mul_eq_zsmul_cast | Mathlib/Algebra/Ring/Int/Defs.lean | /-- Note this holds in marginally more generality than `Int.cast_mul` -/
lemma cast_mul_eq_zsmul_cast {α : Type*} [AddCommGroupWithOne α] :
∀ m n : ℤ, ↑(m * n) = m • (n : α) :=
fun m ↦ Int.induction_on m (by simp) (fun _ ih ↦ by simp [add_mul, add_zsmul, ih]) fun _ ih ↦ by
simp only [sub_mul, one_mul, cast_su... | α : Type u_1
inst✝ : AddCommGroupWithOne α
m : ℤ
x✝ : ℕ
ih : ∀ (n : ℤ), ↑(-↑x✝ * n) = -↑x✝ • ↑n
⊢ ∀ (n : ℤ), ↑((-↑x✝ - 1) * n) = (-↑x✝ - 1) • ↑n | simp only [sub_mul, one_mul, cast_sub, ih, sub_zsmul, one_zsmul, ← sub_eq_add_neg, forall_const] | no goals | 581b90f0b94a9546 |
Nat.add_lt_of_lt_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean | theorem add_lt_of_lt_sub {a b c : Nat} (h : a < c - b) : a + b < c | case h
a b c : Nat
h : a < c - b
hgt : c < b
⊢ False | apply Nat.not_lt_zero a | case h
a b c : Nat
h : a < c - b
hgt : c < b
⊢ a < 0 | f12023f17c28ef23 |
AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | lemma toSpec_base_apply_eq_comap {f} (x : Proj| pbo f) :
(toSpec 𝒜 f).base x = PrimeSpectrum.comap (mapId 𝒜 (Submonoid.powers_le.mpr x.2))
(closedPoint (AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal)) | R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑(Proj.restrict ⋯).toTopCat
⊢ ((PrimeSpectrum.comap (CommRingCat.Hom.hom (CommRingCat.ofHom (mapId 𝒜 ⋯)))).comp
(PrimeSpectrum.comap
(CommRingCat.Hom.hom ((... | exact congr(PrimeSpectrum.comap _ $(@IsLocalRing.comap_closedPoint
(HomogeneousLocalization.AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal) _ _
((Proj| pbo f).presheaf.stalk x) _ _ _ (isLocalHom_of_isIso _))) | no goals | 0f648351cda17755 |
Nat.digits_add | Mathlib/Data/Nat/Digits.lean | theorem digits_add (b : ℕ) (h : 1 < b) (x y : ℕ) (hxb : x < b) (hxy : x ≠ 0 ∨ y ≠ 0) :
digits b (x + b * y) = x :: digits b y | case intro.succ.e_tail.e_a
x b : ℕ
h : 1 < b + 2
hxb : x < b + 2
n✝ : ℕ
hxy : x ≠ 0 ∨ n✝ + 1 ≠ 0
⊢ (x + (b + 2) * (n✝ + 1)) / (b + 2) = n✝ + 1 | simp [add_mul_div_left, div_eq_of_lt hxb] | no goals | 5707ca6647bd00ed |
CategoryTheory.HomOrthogonal.equiv_of_iso | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | theorem equiv_of_iso (o : HomOrthogonal s) {α β : Type} [Finite α] [Finite β] {f : α → ι}
{g : β → ι} (i : (⨁ fun a => s (f a)) ≅ ⨁ fun b => s (g b)) :
∃ e : α ≃ β, ∀ a, g (e a) = f a | case h
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
inst✝² : ∀ (i : ι), InvariantBasisNumber (End (s i))
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
i : (⨁ fun a => s (f a)) ≅ ⨁ fun b => s (g b)
c : ι
⊢ Nonem... | apply Cardinal.eq.1 | case h
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
inst✝² : ∀ (i : ι), InvariantBasisNumber (End (s i))
o : HomOrthogonal s
α β : Type
inst✝¹ : Finite α
inst✝ : Finite β
f : α → ι
g : β → ι
i : (⨁ fun a => s (f a)) ≅ ⨁ fun b => s (g b)
c : ι
⊢ Cardi... | 24935b447fa3d747 |
LinearIndependent.map_of_isPurelyInseparable_of_isSeparable | Mathlib/FieldTheory/PurelyInseparable/Tower.lean | theorem LinearIndependent.map_of_isPurelyInseparable_of_isSeparable [IsPurelyInseparable F E]
{ι : Type*} {v : ι → K} (hsep : ∀ i : ι, IsSeparable F (v i))
(h : LinearIndependent F v) : LinearIndependent E v | case intro
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : Exp... | let n := l.support.sup f | case intro
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : Exp... | e47247a3e0a1480f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.limplies_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem limplies_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n)
(f_AssignmentsInvariant : AssignmentsInvariant f) :
Limplies (PosFin n) f f.assignments | case pos
n : Nat
f : DefaultFormula n
p : PosFin n → Bool
pf : p ⊨ f
hsize : f.assignments.size = n
i : PosFin n
f_AssignmentsInvariant :
hasAssignment (decide (p i = false)) f.assignments[i.val] = true → Limplies (PosFin n) f (i, decide (p i = false))
h✝ : hasAssignment (decide (p i = false)) f.assignments[i.val] = ... | next h =>
specialize f_AssignmentsInvariant h p pf
by_cases hpi : p i <;> simp [hpi, Entails.eval] at f_AssignmentsInvariant | no goals | 324db374b482bd66 |
Polynomial.comp_eq_zero_iff | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | lemma comp_eq_zero_iff [Semiring R] [NoZeroDivisors R] {p q : R[X]} :
p.comp q = 0 ↔ p = 0 ∨ p.eval (q.coeff 0) = 0 ∧ q = C (q.coeff 0) | R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
h : p.comp q = 0
⊢ p = 0 ∨ eval (q.coeff 0) p = 0 ∧ q = C (q.coeff 0) | have key : p.natDegree = 0 ∨ q.natDegree = 0 := by
rw [← mul_eq_zero, ← natDegree_comp, h, natDegree_zero] | R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
h : p.comp q = 0
key : p.natDegree = 0 ∨ q.natDegree = 0
⊢ p = 0 ∨ eval (q.coeff 0) p = 0 ∧ q = C (q.coeff 0) | 3c93f890bf6dfb1a |
TopCat.GlueData.fromOpenSubsetsGlue_isOpenMap | Mathlib/Topology/Gluing.lean | theorem fromOpenSubsetsGlue_isOpenMap : IsOpenMap (fromOpenSubsetsGlue U) | α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : IsOpen s
⊢ IsOpen (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s) | rw [(ofOpenSubsets U).isOpen_iff] at hs | α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
⊢ IsOpen (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s) | 65b5b2de14fddbed |
StieltjesFunction.length_subadditive_Icc_Ioo | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem length_subadditive_Icc_Ioo {a b : ℝ} {c d : ℕ → ℝ} (ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)) :
ofReal (f b - f a) ≤ ∑' i, ofReal (f (d i) - f (c i)) | f : StieltjesFunction
a b : ℝ
c d : ℕ → ℝ
ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)
this :
∀ (s : Finset ℕ) (b : ℝ),
Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i)) | rcases isCompact_Icc.elim_finite_subcover_image
(fun (i : ℕ) (_ : i ∈ univ) => @isOpen_Ioo _ _ _ _ (c i) (d i)) (by simpa using ss) with
⟨s, _, hf, hs⟩ | case intro.intro.intro
f : StieltjesFunction
a b : ℝ
c d : ℕ → ℝ
ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)
this :
∀ (s : Finset ℕ) (b : ℝ),
Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
s : Set ℕ
left✝ : s ⊆ univ
hf : s.Finite
hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i)
... | 16e01a7b132038e4 |
HomologicalComplex.restrictionMap_f' | Mathlib/Algebra/Homology/Embedding/Restriction.lean | @[reassoc]
lemma restrictionMap_f' {i : ι} {i' : ι'} (hi : e.f i = i') :
(restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫
φ.f i' ≫ (L.restrictionXIso e hi).inv | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
⊢ (restrictionMap φ e).f i = (K.restrictionXIso e ⋯).hom ≫ φ.f (e.f i) ≫ (L.restrictionXIso e ⋯).... | simp [restrictionXIso] | no goals | e78462e2308f811b |
CategoryTheory.Presheaf.isLocallySurjective_of_whisker | Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean | lemma isLocallySurjective_of_whisker (hH : CoverPreserving J K H)
[H.IsCoverDense K] [IsLocallySurjective J (whiskerLeft H.op f)] : IsLocallySurjective K f where
imageSieve_mem {X} a | case h
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{u_6, u_1} C
inst✝⁵ : Category.{u_7, u_2} D
inst✝⁴ : Category.{u_8, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
i... | apply K.pullback_stable | case h.hS
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{u_6, u_1} C
inst✝⁵ : Category.{u_7, u_2} D
inst✝⁴ : Category.{u_8, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y... | 6db4c45a55a27efe |
IsCoprime.intCast | Mathlib/RingTheory/Coprime/Basic.lean | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) | case intro.intro
R : Type u_1
inst✝ : CommRing R
a b u v : ℤ
H : u * a + v * b = 1
⊢ IsCoprime ↑a ↑b | use u, v | case h
R : Type u_1
inst✝ : CommRing R
a b u v : ℤ
H : u * a + v * b = 1
⊢ ↑u * ↑a + ↑v * ↑b = 1 | 0b43dd1d4f9f208c |
normEDS_three | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | @[simp]
lemma normEDS_three : normEDS b c d 3 = c | R : Type u
inst✝ : CommRing R
b c d : R
⊢ ¬Even 3 | decide | no goals | 1d48b070f930378a |
CStarRing.norm_coe_unitary_mul | Mathlib/Analysis/CStarAlgebra/Basic.lean | theorem norm_coe_unitary_mul (U : unitary E) (A : E) : ‖(U : E) * A‖ = ‖A‖ | E : Type u_2
inst✝² : NormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
U : ↥(unitary E)
A : E
a✝ : Nontrivial E
⊢ ‖↑U * A‖ = ‖A‖ | refine le_antisymm ?_ ?_ | case refine_1
E : Type u_2
inst✝² : NormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
U : ↥(unitary E)
A : E
a✝ : Nontrivial E
⊢ ‖↑U * A‖ ≤ ‖A‖
case refine_2
E : Type u_2
inst✝² : NormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
U : ↥(unitary E)
A : E
a✝ : Nontrivial E
⊢ ‖A‖ ≤ ‖↑U * A‖ | f09abe48df8a1a8c |
Filter.EventuallyEq.lineDerivWithin_eq | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | theorem Filter.EventuallyEq.lineDerivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₁ : E → F
s : Set E
x v : E
hs : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ Continuous fun t => x + t • v | fun_prop | no goals | c1bb0926c6962436 |
DistribLattice.prime_ideal_of_disjoint_filter_ideal | Mathlib/Order/PrimeSeparator.lean | theorem prime_ideal_of_disjoint_filter_ideal (hFI : Disjoint (F : Set α) (I : Set α)) :
∃ J : Ideal α, (IsPrime J) ∧ I ≤ J ∧ Disjoint (F : Set α) J | case h
α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : BoundedOrder α
F : PFilter α
I : Ideal α
hFI : Disjoint ↑F ↑I
S : Set (Set α) := {J | IsIdeal J ∧ ↑I ≤ J ∧ Disjoint (↑F) J}
Jset : Set α
left✝ : ↑I ⊆ Jset
hmax : Maximal (fun x => x ∈ S) Jset
Jidl : IsIdeal Jset
IJ : ↑I ≤ Jset
J : Ideal α := Jidl.toIdeal
IJ' : I ≤ J... | use b ⊔ (a₁ ⊓ a₂) | case h
α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : BoundedOrder α
F : PFilter α
I : Ideal α
hFI : Disjoint ↑F ↑I
S : Set (Set α) := {J | IsIdeal J ∧ ↑I ≤ J ∧ Disjoint (↑F) J}
Jset : Set α
left✝ : ↑I ⊆ Jset
hmax : Maximal (fun x => x ∈ S) Jset
Jidl : IsIdeal Jset
IJ : ↑I ≤ Jset
J : Ideal α := Jidl.toIdeal
IJ' : I ≤ J... | 689f1ec2649ea4e9 |
TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure_nonunital | Mathlib/RingTheory/TwoSidedIdeal/Operations.lean | lemma mem_span_iff_mem_addSubgroup_closure_nonunital {s : Set R} {z : R} :
z ∈ span s ↔ z ∈ AddSubgroup.closure (s ∪ s * univ ∪ univ * s ∪ univ * s * univ) | case refine_1.inl.inr.intro.intro.intro.intro
R : Type u_1
inst✝ : NonUnitalRing R
s : Set R
z x r y : R
hy : y ∈ s
⊢ x * (fun x1 x2 => x1 * x2) r y ∈ s ∪ s * univ ∪ univ * s ∪ univ * s * univ | exact .inl <| .inr <| ⟨x * r, mem_univ _, y, hy, mul_assoc ..⟩ | no goals | a217d500043db5e0 |
Balanced.sub | Mathlib/Analysis/LocallyConvex/Basic.lean | theorem Balanced.sub (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s - t) | 𝕜 : Type u_1
E : Type u_3
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Balanced 𝕜 s
ht : Balanced 𝕜 t
⊢ Balanced 𝕜 (s - t) | simp_rw [sub_eq_add_neg] | 𝕜 : Type u_1
E : Type u_3
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Balanced 𝕜 s
ht : Balanced 𝕜 t
⊢ Balanced 𝕜 (s + -t) | 609a724600ad6e81 |
Ordinal.veblenWith_eq_veblenWith_iff | Mathlib/SetTheory/Ordinal/Veblen.lean | theorem veblenWith_eq_veblenWith_iff :
veblenWith f o₁ a = veblenWith f o₂ b ↔
o₁ = o₂ ∧ a = b ∨ o₁ < o₂ ∧ a = veblenWith f o₂ b ∨ o₂ < o₁ ∧ veblenWith f o₁ a = b | f : Ordinal.{u} → Ordinal.{u}
o₁ o₂ a b : Ordinal.{u}
hf : IsNormal f
⊢ (match cmp o₁ o₂ with
| Ordering.eq => cmp a b
| Ordering.lt => cmp a (veblenWith f o₂ b)
| Ordering.gt => cmp (veblenWith f o₁ a) b) =
Ordering.eq ↔
o₁ = o₂ ∧ a = b ∨ o₁ < o₂ ∧ a = veblenWith f o₂ b ∨ o₂ < o₁ ∧ veblenWi... | aesop (add simp lt_asymm) | no goals | 0d52f83a3c4edba7 |
Nat.add_le_mul | Mathlib/Data/Nat/Init.lean | protected lemma add_le_mul {a : ℕ} (ha : 2 ≤ a) : ∀ {b : ℕ} (_ : 2 ≤ b), a + b ≤ a * b
| 2, _ => by omega
| b + 3, _ => by have := Nat.add_le_mul ha (Nat.le_add_left _ b); rw [mul_succ]; omega
| a : ℕ
ha : 2 ≤ a
x✝ : 2 ≤ 2
⊢ a + 2 ≤ a * 2 | omega | no goals | 13e04ab1add282ca |
Set.disjoint_ordT5Nhd | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | theorem disjoint_ordT5Nhd : Disjoint (ordT5Nhd s t) (ordT5Nhd t s) | case intro.intro.inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
ha : [[a, x]] ⊆ tᶜ
hb : [[b, x]] ⊆ sᶜ
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
hax : a ≤ x
hxb : x ≤ b
⊢ x ∈ ⊥ | have h' : x ∈ ordSeparatingSet s t := ⟨mem_iUnion₂.2 ⟨a, has, ha⟩, mem_iUnion₂.2 ⟨b, hbt, hb⟩⟩ | case intro.intro.inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
ha : [[a, x]] ⊆ tᶜ
hb : [[b, x]] ⊆ sᶜ
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
hax : a ≤ x
hxb : x ≤ b
h' : x ∈ s.ordSeparat... | cc7be6509b6c5432 |
Array.all_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem all_eq {xs : Array α} {p : α → Bool} : xs.all p = decide (∀ i, (_ : i < xs.size) → p xs[i]) | case neg
α : Type u_1
xs : Array α
p : α → Bool
h : ¬xs.all p = true
⊢ xs.all p = decide (∀ (i : Nat) (x : i < xs.size), p xs[i] = true) | simp only [Bool.not_eq_true] at h | case neg
α : Type u_1
xs : Array α
p : α → Bool
h : xs.all p = false
⊢ xs.all p = decide (∀ (i : Nat) (x : i < xs.size), p xs[i] = true) | 906b6db12be6dc9b |
MeasureTheory.Measure.integrable_measure_prod_mk_left | Mathlib/MeasureTheory/Integral/Prod.lean | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | case h
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ⊤
x : α
hx : ν (Prod.mk x ⁻¹' s) < ⊤
⊢ ENNReal.ofReal (ν (Prod.mk x ⁻¹' s)).toReal = ν (Prod.mk x ⁻¹' s) | rw [lt_top_iff_ne_top] at hx | case h
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ⊤
x : α
hx : ν (Prod.mk x ⁻¹' s) ≠ ⊤
⊢ ENNReal.ofReal (ν (Prod.mk x ⁻¹' s)).toReal = ν (Prod.mk x ⁻¹' s) | cda3d9546eac8d79 |
AddGroupWithOne.ext | Mathlib/Algebra/Ring/Ext.lean | theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ | case h_mul
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : HAdd.hAdd = HAdd.hAdd
h_one : One.one = One.one
this✝ : toAddMonoidWithOne = toAddMonoidWithOne
this : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast
⊢ Add.add = Add.add | exact h_add | no goals | 3b41a61ef664adfb |
MeasureTheory.Measure.singularPart_eq_zero | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | lemma singularPart_eq_zero (μ ν : Measure α) [μ.HaveLebesgueDecomposition ν] :
μ.singularPart ν = 0 ↔ μ ≪ ν | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
h_dec : μ = ν.withDensity (μ.rnDeriv ν)
h : μ.singularPart ν = 0
⊢ ν.withDensity (μ.rnDeriv ν) ≪ ν | exact withDensity_absolutelyContinuous ν _ | no goals | 4a91c07e6c9d50fd |
DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply | Mathlib/Analysis/InnerProductSpace/PiL2.lean | theorem DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply [DecidableEq ι]
{V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V)
(hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (w : PiLp 2 fun i => V i) :
(hV.isometryL2OfOrthogonalFamily hV').symm w = ∑ i, (w i : E) | ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ
w : PiLp 2 fun i => ↥(V i)
e₁ : (⨁ (i : ι), ↥(V ... | exact this (e₁.symm w) | no goals | 8f71fc0c58ddeeed |
List.drop_eq_getElem?_toList_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem drop_eq_getElem?_toList_append {l : List α} {n : Nat} :
l.drop n = l[n]?.toList ++ l.drop (n + 1) | case cons
α : Type u_1
hd : α
tl : List α
ih : ∀ {n : Nat}, drop n tl = tl[n]?.toList ++ drop (n + 1) tl
n : Nat
⊢ drop n (hd :: tl) = (hd :: tl)[n]?.toList ++ drop (n + 1) (hd :: tl) | cases n | case cons.zero
α : Type u_1
hd : α
tl : List α
ih : ∀ {n : Nat}, drop n tl = tl[n]?.toList ++ drop (n + 1) tl
⊢ drop 0 (hd :: tl) = (hd :: tl)[0]?.toList ++ drop (0 + 1) (hd :: tl)
case cons.succ
α : Type u_1
hd : α
tl : List α
ih : ∀ {n : Nat}, drop n tl = tl[n]?.toList ++ drop (n + 1) tl
n✝ : Nat
⊢ drop (n✝ + 1) (hd... | 8b64a3b2a111c7eb |
CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.prod_uniq | Mathlib/CategoryTheory/Localization/Prod.lean | lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
(h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
F₁ = F₂ | case h.h
C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F₁ F₂ : W₁.Localization × W₂.Localization ⥤ E
h : W₁.Q.prod W₂.Q ⋙ F₁ = W₁.Q.prod W₂.Q ⋙ F₂
⊢ W₁.Q ⋙ curry.obj F₁ = W₁.Q ⋙ curry.obj F₂ | apply Functor.flip_injective | case h.h.h
C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F₁ F₂ : W₁.Localization × W₂.Localization ⥤ E
h : W₁.Q.prod W₂.Q ⋙ F₁ = W₁.Q.prod W₂.Q ⋙ F₂
⊢ (W₁.Q ⋙ curry.obj F₁).flip = (W₁.Q ⋙ cu... | 77a9b99162fee2d1 |
AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero | Mathlib/Analysis/Analytic/Uniqueness.lean | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero {f : E → F} {U : Set E}
(hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U)
{z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
EqOn f 0 U | 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
f : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfz₀ : f =ᶠ[𝓝 z₀] 0
F' : Type u_3 := UniformS... | have : e (f z) = e 0 := by simpa only using A hz | 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
f : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfz₀ : f =ᶠ[𝓝 z₀] 0
F' : Type u_3 := UniformS... | fc36c195452effd3 |
IsLocallyConstant.range_finite | Mathlib/Topology/LocallyConstant/Basic.lean | theorem range_finite [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f) :
(Set.range f).Finite | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
f : X → Y
hf : IsLocallyConstant f
this✝ : TopologicalSpace Y := ⊥
this : DiscreteTopology Y
⊢ (range f).Finite | exact (isCompact_range hf.continuous).finite_of_discrete | no goals | a1204405fd1910fe |
CategoryTheory.MorphismProperty.IsInvertedBy.iff_map_le_isomorphisms | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | lemma IsInvertedBy.iff_map_le_isomorphisms (W : MorphismProperty C) (F : C ⥤ D) :
W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D | C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
F : C ⥤ D
⊢ W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D | rw [iff_le_inverseImage_isomorphisms, map_le_iff] | no goals | 33fdfd2662ec8270 |
CochainComplex.HomComplex.Cochain.rightShift_smul | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | @[simp]
lemma rightShift_smul (a n' : ℤ) (hn' : n' + a = n) (x : R) :
(x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn' | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n' + a = n
x : R
⊢ (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn' | ext p q hpq | case h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n' + a = n
x : R
p q : ℤ
hpq : p + n' = q
⊢ ((x • γ).rightShift a n' hn').v p q hpq = (x • γ.rightShift a n' hn').v p q hpq | ee8e7d1e3839df76 |
geom_sum_eq_zero_iff_neg_one | Mathlib/Algebra/GeomSum.lean | theorem geom_sum_eq_zero_iff_neg_one [LinearOrderedRing α] (hn : n ≠ 0) :
∑ i ∈ range n, x ^ i = 0 ↔ x = -1 ∧ Even n | α : Type u
n : ℕ
x : α
inst✝ : LinearOrderedRing α
hn : n ≠ 0
h : x = -1 → ¬Even n
hx : x = -1 ∨ x ≠ -1
⊢ ∑ i ∈ range n, x ^ i ≠ 0 | rcases hx with hx | hx | case inl
α : Type u
n : ℕ
x : α
inst✝ : LinearOrderedRing α
hn : n ≠ 0
h : x = -1 → ¬Even n
hx : x = -1
⊢ ∑ i ∈ range n, x ^ i ≠ 0
case inr
α : Type u
n : ℕ
x : α
inst✝ : LinearOrderedRing α
hn : n ≠ 0
h : x = -1 → ¬Even n
hx : x ≠ -1
⊢ ∑ i ∈ range n, x ^ i ≠ 0 | b77f74310391edb9 |
List.sublist_flatten_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem sublist_flatten_iff {L : List (List α)} {l} :
l <+ L.flatten ↔
∃ L' : List (List α), l = L'.flatten ∧ ∀ i (_ : i < L'.length), L'[i] <+ L[i]?.getD [] | case cons.mp
α : Type u_1
l' : List α
L : List (List α)
ih : ∀ {l : List α}, l <+ L.flatten ↔ ∃ L', l = L'.flatten ∧ ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
l : List α
⊢ (∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁ <+ l' ∧ ∃ L', l₂ = L'.flatten ∧ ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []) →
∃ L', l = L... | rintro ⟨l₁, l₂, rfl, s, L', rfl, h⟩ | case cons.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
l' : List α
L : List (List α)
ih : ∀ {l : List α}, l <+ L.flatten ↔ ∃ L', l = L'.flatten ∧ ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
l₁ : List α
s : l₁ <+ l'
L' : List (List α)
h : ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
⊢ ∃ L'_... | 2d054bb50b16d9ba |
TopCat.GlueData.ι_eq_iff_rel | Mathlib/Topology/Gluing.lean | theorem ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
𝖣.ι i x = 𝖣.ι j y ↔ D.Rel ⟨i, x⟩ ⟨j, y⟩ | case mpr.intro.intro
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
z : ↑(D.V (⟨i, x⟩.fst, ⟨j, y⟩.fst))
e₁ : (ConcreteCategory.hom (D.f i j)) z = (ConcreteCategory.hom (D.f i j)) z
e₂ :
(ConcreteCategory.hom (D.f j i)) ((ConcreteCategory.hom (D.t i j)) z) =
(ConcreteCategory.hom (D.f j i)) ((ConcreteCategory.ho... | rw [D.glue_condition_apply] | no goals | e0a110dda282edcf |
Int.even_iff_not_odd | Mathlib/Algebra/Ring/Int/Parity.lean | @[deprecated not_odd_iff_even (since := "2024-08-21")]
lemma even_iff_not_odd : Even n ↔ ¬Odd n | n : ℤ
⊢ Even n ↔ ¬Odd n | rw [not_odd_iff, even_iff] | no goals | 659139c98549e4e5 |
frontier_le_subset_eq | Mathlib/Topology/Order/OrderClosed.lean | theorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b ≤ g b } ⊆ { b | f b = g b } | case intro
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
f g : β → α
inst✝ : TopologicalSpace β
hf : Continuous f
hg : Continuous g
b : β
hb₁ : b ∈ {b | f b ≤ g b}
hb₂ : b ∈ closure {b | f b ≤ g b}ᶜ
⊢ b ∈ {b | f b = g b} | refine le_antisymm hb₁ (closure_lt_subset_le hg hf ?_) | case intro
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
f g : β → α
inst✝ : TopologicalSpace β
hf : Continuous f
hg : Continuous g
b : β
hb₁ : b ∈ {b | f b ≤ g b}
hb₂ : b ∈ closure {b | f b ≤ g b}ᶜ
⊢ b ∈ closure {b | g b < f b} | 0ce32949b8808351 |
Order.height_eq_krullDim_Iic | Mathlib/Order/KrullDimension.lean | lemma height_eq_krullDim_Iic (x : α) : (height x : ℕ∞) = krullDim (Set.Iic x) | case a.h.h
α : Type u_1
inst✝ : Preorder α
x : α
p : LTSeries ↑(Set.Iic x)
⊢ RelSeries.last p ≤ ⊤ → ↑p.length ≤ ⨆ p, ⨆ (_ : RelSeries.last p ≤ x), ↑p.length | intro _ | case a.h.h
α : Type u_1
inst✝ : Preorder α
x : α
p : LTSeries ↑(Set.Iic x)
i✝ : RelSeries.last p ≤ ⊤
⊢ ↑p.length ≤ ⨆ p, ⨆ (_ : RelSeries.last p ≤ x), ↑p.length | b4cd0b25c1815825 |
isApproximateSubgroup_one | Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean | /-- A `1`-approximate subgroup is the same thing as a subgroup. -/
@[to_additive (attr := simp)
"A `1`-approximate subgroup is the same thing as a subgroup."]
lemma isApproximateSubgroup_one {A : Set G} :
IsApproximateSubgroup 1 (A : Set G) ↔ ∃ H : Subgroup G, H = A where
mp hA | case intro
G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
hx_inv : x⁻¹ ∈ A
⊢ A * A ⊆ A | have hx_sq : x * x ∈ A := by
rw [← hA.inv_eq_self]
simpa using hx' (smul_mem_smul_set (mul_mem_mul hx_inv hA.one_mem)) | case intro
G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
hx_inv : x⁻¹ ∈ A
hx_sq : x * x ∈ A
⊢ A * A ⊆ A | 1d755e438e264b32 |
Matroid.IsStrictRestriction.ssubset | Mathlib/Data/Matroid/Restrict.lean | theorem IsStrictRestriction.ssubset (h : N <r M) : N.E ⊂ M.E | case intro.intro
α : Type u_1
M : Matroid α
R : Set α
h : M ↾ R <r M
⊢ (M ↾ R).E ⊂ M.E | refine h.isRestriction.subset.ssubset_of_ne (fun h' ↦ h.2 ⟨R, Subset.rfl, ?_⟩) | case intro.intro
α : Type u_1
M : Matroid α
R : Set α
h : M ↾ R <r M
h' : (M ↾ R).E = M.E
⊢ M = M ↾ R ↾ R | e50c5ed5ba55dc85 |
LieAlgebra.InvariantForm.atomistic | Mathlib/Algebra/Lie/InvariantForm.lean | lemma atomistic : ∀ I : LieIdeal K L, sSup {J : LieIdeal K L | IsAtom J ∧ J ≤ I} = I | case hst
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
Φ : LinearMap.BilinForm K L
hΦ_nondeg : Φ.Nondegenerate
hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ
hΦ_refl : Φ.IsRefl
hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I
I : LieIdeal K L
a✝ :
... | rw [eq_bot_iff] | case hst
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
Φ : LinearMap.BilinForm K L
hΦ_nondeg : Φ.Nondegenerate
hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ
hΦ_refl : Φ.IsRefl
hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I
I : LieIdeal K L
a✝ :
... | 9856eccb3c27c11b |
CategoryTheory.MorphismProperty.isStableUnderCobaseChange_iff_pushouts_le | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | lemma isStableUnderCobaseChange_iff_pushouts_le :
P.IsStableUnderCobaseChange ↔ P.pushouts ≤ P | case mpr.of_isPushout
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
h : P.pushouts ≤ P
A✝ A'✝ B✝ B'✝ : C
f✝ : A✝ ⟶ A'✝
g✝ : A✝ ⟶ B✝
f'✝ : B✝ ⟶ B'✝
g'✝ : A'✝ ⟶ B'✝
h₁ : IsPushout g✝ f✝ f'✝ g'✝
h₂ : P f✝
⊢ P f'✝ | exact h _ ⟨_, _, _, _, _, h₂, h₁⟩ | no goals | d27fb1816868ef70 |
StrictConvex.add_smul_sub_mem | Mathlib/Analysis/Convex/Strict.lean | theorem StrictConvex.add_smul_sub_mem (h : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y)
{t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • (y - x) ∈ interior s | case a
𝕜 : Type u_1
E : Type u_3
inst✝³ : OrderedRing 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x y : E
h : StrictConvex 𝕜 s
hx : x ∈ s
hy : y ∈ s
hxy : x ≠ y
t : 𝕜
ht₀ : 0 < t
ht₁ : t < 1
⊢ x + t • (y - x) ∈ openSegment 𝕜 x y | rw [openSegment_eq_image'] | case a
𝕜 : Type u_1
E : Type u_3
inst✝³ : OrderedRing 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x y : E
h : StrictConvex 𝕜 s
hx : x ∈ s
hy : y ∈ s
hxy : x ≠ y
t : 𝕜
ht₀ : 0 < t
ht₁ : t < 1
⊢ x + t • (y - x) ∈ (fun θ => x + θ • (y - x)) '' Ioo 0 1 | 4d2f0ab6105dbf60 |
MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
∫⁻ x, ‖u x‖ₑ ^ p ∂μ ≤
lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ∂μ) ^ p | F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
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
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
h... | rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← mul_assoc, ← ENNReal.coe_rpow_of_ne_zero hc.ne'] | F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
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
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
h... | 7e7c86c82b1c72d8 |
SimpleGraph.regularityReduced_edges_card_aux | Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean | lemma regularityReduced_edges_card_aux [Nonempty α] (hε : 0 < ε) (hP : P.IsEquipartition)
(hPε : P.IsUniform G (ε/8)) (hP' : 4 / ε ≤ #P.parts) :
2 * (#G.edgeFinset - #(G.regularityReduced P (ε/8) (ε/4)).edgeFinset : ℝ)
< 2 * ε * (card α ^ 2 : ℕ) | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hε : 0 < ε
hP : P.IsEquipartition
hPε : P.IsUniform G (ε / 8)
hP' : 4 / ε ≤ ↑(#P.parts)
A : Finset (α × α) :=
(P.nonUniforms G (ε / 8)).biUnion fun x =>
match x with
... | gcongr | case h₁.bc
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hε : 0 < ε
hP : P.IsEquipartition
hPε : P.IsUniform G (ε / 8)
hP' : 4 / ε ≤ ↑(#P.parts)
A : Finset (α × α) :=
(P.nonUniforms G (ε / 8)).biUnion fun x =>
ma... | 945c877c5af388f8 |
List.prod_int_mod | Mathlib/Algebra/BigOperators/Group/List/Basic.lean | lemma prod_int_mod (l : List ℤ) (n : ℤ) : l.prod % n = (l.map (· % n)).prod % n | l : List ℤ
n : ℤ
⊢ l.prod % n = (map (fun x => x % n) l).prod % n | induction l <;> simp [Int.mul_emod, *] | no goals | 607a556d2d79ada6 |
disjoint_ball_ball_iff | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y | case intro.intro
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : Disjoint (ball x δ) (ball y ε)
hxy : dist x y < ε + δ
z : E
hxz : dist x z < δ
hzy : dist z y < ε
⊢ False | rw [dist_comm] at hxz | case intro.intro
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : Disjoint (ball x δ) (ball y ε)
hxy : dist x y < ε + δ
z : E
hxz : dist z x < δ
hzy : dist z y < ε
⊢ False | f423a604f95f49b3 |
Basis.coe_sumCoords_of_fintype | Mathlib/LinearAlgebra/Basis/Defs.lean | theorem coe_sumCoords_of_fintype [Fintype ι] : (b.sumCoords : M → R) = ∑ i, b.coord i | ι : Type u_1
R : Type u_3
M : Type u_6
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
b : Basis ι R M
inst✝ : Fintype ι
⊢ ⇑b.sumCoords = ⇑(∑ i : ι, b.coord i) | ext m | case h
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
b : Basis ι R M
inst✝ : Fintype ι
m : M
⊢ b.sumCoords m = (∑ i : ι, b.coord i) m | 3d97f1b91390e088 |
Std.Sat.CNF.Clause.eval_congr | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean | theorem eval_congr (a1 a2 : α → Bool) (c : Clause α) (hw : ∀ i, Mem i c → a1 i = a2 i) :
eval a1 c = eval a2 c | α : Type u_1
a1 a2 : α → Bool
i : Literal α
c : List (Literal α)
ih : (∀ (i : α), Mem i c → a1 i = a2 i) → eval a1 c = eval a2 c
hw : ∀ (i_1 : α), Mem i_1 (i :: c) → a1 i_1 = a2 i_1
⊢ ∀ (i : α), Mem i c → a1 i = a2 i | intro j h | α : Type u_1
a1 a2 : α → Bool
i : Literal α
c : List (Literal α)
ih : (∀ (i : α), Mem i c → a1 i = a2 i) → eval a1 c = eval a2 c
hw : ∀ (i_1 : α), Mem i_1 (i :: c) → a1 i_1 = a2 i_1
j : α
h : Mem j c
⊢ a1 j = a2 j | d3bd42fd519c28d6 |
Nat.exists_prime_lt_and_le_two_mul | Mathlib/NumberTheory/Bertrand.lean | theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :
∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n | n : ℕ
hn0 : n ≠ 0
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | rcases lt_or_le 511 n with h | h | case inl
n : ℕ
hn0 : n ≠ 0
h : 511 < n
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n
case inr
n : ℕ
hn0 : n ≠ 0
h : n ≤ 511
⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n | 481aaedc4f9db56d |
Submodule.inf_iSup_genEigenspace | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) (k : ℕ∞) :
p ⊓ ⨆ μ, f.genEigenspace μ k = ⨆ μ, p ⊓ f.genEigenspace μ k | K : Type u_1
V : Type u_2
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
f : End K V
inst✝ : FiniteDimensional K V
h : ∀ x ∈ p, f x ∈ p
k : ℕ∞
m : K →₀ V
hm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k
hm₀ : (m.sum fun _i xi => xi) ∈ p
hm₁ : (m.sum fun _i xi => xi) ∈ ⨆ μ, (f.genEigenspace μ... | apply (f.genEigenspace μ').mono | case a
K : Type u_1
V : Type u_2
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
f : End K V
inst✝ : FiniteDimensional K V
h : ∀ x ∈ p, f x ∈ p
k : ℕ∞
m : K →₀ V
hm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k
hm₀ : (m.sum fun _i xi => xi) ∈ p
hm₁ : (m.sum fun _i xi => xi) ∈ ⨆ μ, (f.genEigen... | a9d126d678497dfd |
SimpleGraph.card_cliqueFinset_le | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem card_cliqueFinset_le : #(G.cliqueFinset n) ≤ (card α).choose n | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
n : ℕ
⊢ #(G.cliqueFinset n) ≤ #(powersetCard n univ) | refine card_mono fun s => ?_ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
n : ℕ
s : Finset α
⊢ s ∈ G.cliqueFinset n → s ∈ powersetCard n univ | e2e0f29c0cf57a6f |
Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto | Mathlib/Analysis/Complex/CauchyIntegral.lean | theorem circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto {c : ℂ}
{R : ℝ} (h0 : 0 < R) {f : ℂ → E} {y : E} {s : Set ℂ} (hs : s.Countable)
(hc : ContinuousOn f (closedBall c R \ {c}))
(hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z) (hy : Tendsto f (𝓝[{c}ᶜ] c) (𝓝 y))... | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0... | simp only [smul_sub] | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0... | 161c13e075bc65e7 |
Con.comap_conGen_equiv | Mathlib/GroupTheory/Congruence/Basic.lean | theorem comap_conGen_equiv {M N : Type*} [Mul M] [Mul N] (f : MulEquiv M N) (rel : N → N → Prop) :
Con.comap f (map_mul f) (conGen rel) = conGen (fun x y ↦ rel (f x) (f y)) | case mul
M : Type u_4
N : Type u_5
inst✝¹ : Mul M
inst✝ : Mul N
f : M ≃* N
rel : N → N → Prop
a✝² b✝ : M
h : (conGen rel) (f a✝²) (f b✝)
n1 n2 w x y z : N
a✝¹ : ConGen.Rel rel w x
a✝ : ConGen.Rel rel y z
ih : ∀ (a b : M), f a = w → f b = x → (conGen fun x y => rel (f x) (f y)) a b
ih1 : ∀ (a b : M), f a = y → f b = z →... | rw [fa, fb] | case mul
M : Type u_4
N : Type u_5
inst✝¹ : Mul M
inst✝ : Mul N
f : M ≃* N
rel : N → N → Prop
a✝² b✝ : M
h : (conGen rel) (f a✝²) (f b✝)
n1 n2 w x y z : N
a✝¹ : ConGen.Rel rel w x
a✝ : ConGen.Rel rel y z
ih : ∀ (a b : M), f a = w → f b = x → (conGen fun x y => rel (f x) (f y)) a b
ih1 : ∀ (a b : M), f a = y → f b = z →... | d315948435eb07b2 |
List.filterMap_eq_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filterMap_eq_cons_iff {l} {b} {bs} :
filterMap f l = b :: bs ↔
∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ (∀ x, x ∈ l₁ → f x = none) ∧ f a = some b ∧
filterMap f l₂ = bs | case mp.cons.some
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
a : α✝¹
l : List α✝¹
b : α✝
h : f a = some b
ih :
filterMap f l = b :: filterMap f l →
∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝¹), x ∈ l₁ → f x = none) ∧ f a = some b ∧ filterMap f l₂ = filterMap f l
⊢ ∃ l₁ a_1 l₂,
a :: l = l₁ ++ a_1 :: l₂ ∧ ... | refine ⟨[], a, l, by simp [h]⟩ | no goals | a63f5fdfcdb83803 |
AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_support | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma zeroLocus_inter_subset_support (I : IdealSheafData X) (U : X.affineOpens) :
X.zeroLocus (U := U.1) (I.ideal U) ∩ U ⊆ I.support | case intro.intro
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑↑X.toPresheafedSpace
hxV : x ∈ ↑↑V
hxU : x ∈ ↑↑U
hx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f
s : ↑Γ(X, ↑V)
hfU : s ∈ I.ideal V
hxs : x ∈ X.basicOpen s
⊢ False | obtain ⟨f, g, hfg, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 x ⟨hxU, hxV⟩ | case intro.intro.intro.intro.intro
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑↑X.toPresheafedSpace
hxV : x ∈ ↑↑V
hxU : x ∈ ↑↑U
hx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f
s : ↑Γ(X, ↑V)
hfU : s ∈ I.ideal V
hxs : x ∈ X.basicOpen s
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hxf : x ∈ X.basi... | ac346fd7f1a4fa9f |
CommRingCat.KaehlerDifferential.ext | Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.lean | @[ext]
lemma ext {M : ModuleCat B} {α β : KaehlerDifferential f ⟶ M}
(h : ∀ (b : B), α (d b) = β (d b)) : α = β | A B : CommRingCat
f : A ⟶ B
M : ModuleCat ↑B
α β : KaehlerDifferential f ⟶ M
h : ∀ (b : ↑B), (ConcreteCategory.hom α) (d b) = (ConcreteCategory.hom β) (d b)
⊢ ⊤ ≤ LinearMap.ker (ModuleCat.Hom.hom (α - β)) | rw [← KaehlerDifferential.span_range_derivation, Submodule.span_le] | A B : CommRingCat
f : A ⟶ B
M : ModuleCat ↑B
α β : KaehlerDifferential f ⟶ M
h : ∀ (b : ↑B), (ConcreteCategory.hom α) (d b) = (ConcreteCategory.hom β) (d b)
⊢ Set.range ⇑(_root_.KaehlerDifferential.D ↑A ↑B) ⊆ ↑(LinearMap.ker (ModuleCat.Hom.hom (α - β))) | 3a02a1e0fa8a646a |
MeasureTheory.Content.innerContent_exists_compact | Mathlib/MeasureTheory/Measure/Content.lean | theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
(hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε | G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε | rcases le_or_lt (μ.innerContent U) ε with h | h | case inl
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : μ.innerContent U ≤ ↑ε
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0... | 1912c8ba0130552e |
Set.countable_setOf_finite_subset | Mathlib/Data/Set/Countable.lean | theorem countable_setOf_finite_subset {s : Set α} (hs : s.Countable) :
{ t | Set.Finite t ∧ t ⊆ s }.Countable | case intro.intro
α : Type u
s : Set α
hs : s.Countable
this : Countable ↑s
t : Set ↑s
ht : (Subtype.val '' t).Finite
⊢ Subtype.val '' t ∈ range fun t => Subtype.val '' ↑t | lift t to Finset s using ht.of_finite_image Subtype.val_injective.injOn | case intro.intro.intro
α : Type u
s : Set α
hs : s.Countable
this : Countable ↑s
t : Finset ↑s
ht : (Subtype.val '' ↑t).Finite
⊢ Subtype.val '' ↑t ∈ range fun t => Subtype.val '' ↑t | ac95a49c211f784f |
Convex.is_const_of_fderivWithin_eq_zero | Mathlib/Analysis/Calculus/MeanValue.lean | theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
(hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
x y : E
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
hf'... | have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by
simp only [hf' x hx, norm_zero, le_rfl] | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
x y : E
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
hf'... | 9913d0fe64b83cc3 |
Odd.strictMono_pow | Mathlib/Algebra/Order/Ring/Basic.lean | lemma Odd.strictMono_pow (hn : Odd n) : StrictMono fun a : R => a ^ n | R : Type u_3
inst✝¹ : LinearOrderedSemiring R
inst✝ : ExistsAddOfLE R
hn : Odd 0
⊢ False | simp [Odd, eq_comm (a := 0)] at hn | no goals | 789fc8d7e45ac2d6 |
LinearPMap.left_le_sup | Mathlib/LinearAlgebra/LinearPMap.lean | theorem left_le_sup (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : f ≤ f.sup g h | R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
z₁ : ↥f.domain
z₂ : ↥(f.sup g h).domain
hz : ↑z₁ = ↑z₂
⊢ ↑f z₁ + ↑g 0 = ↑(f.sup g h) z₂ | refine (sup_apply h _ _ _ ?_).symm | R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
z₁ : ↥f.domain
z₂ : ↥(f.sup g h).domain
hz : ↑z₁ = ↑z₂
⊢ ↑z₁ + ↑0 = ↑z₂ | 9fa85484af1849de |
Complex.isPrimitiveRoot_iff | Mathlib/RingTheory/RootsOfUnity/Complex.lean | theorem isPrimitiveRoot_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :
IsPrimitiveRoot ζ n ↔ ∃ i < n, ∃ _ : i.Coprime n, exp (2 * π * I * (i / n)) = ζ | case mpr.intro.intro.intro
n : ℕ
hn : n ≠ 0
hn0 : ↑n ≠ 0
i : ℕ
hi : i.Coprime n
⊢ IsPrimitiveRoot (cexp (2 * ↑π * I * (↑i / ↑n))) n | exact isPrimitiveRoot_exp_of_coprime i n hn hi | no goals | 3ccb57b63ef696b5 |
Stream'.WSeq.liftRel_dropn_destruct | Mathlib/Data/Seq/WSeq.lean | theorem liftRel_dropn_destruct {R : α → β → Prop} {s t} (H : LiftRel R s t) :
∀ n, Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s n)) (destruct (drop t n))
| 0 => liftRel_destruct H
| n + 1 => by
simp only [LiftRelO, drop, Nat.add_eq, Nat.add_zero, destruct_tail, tail.aux]
apply liftRel_... | α : Type u
β : Type v
R : α → β → Prop
s : WSeq α
t : WSeq β
H : LiftRel R s t
n : ℕ
a : Option (α × WSeq α)
b : Option (β × WSeq β)
o : LiftRelO R (LiftRel R) a b
x✝ : LiftRelO R (LiftRel R) none none
⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (tail.aux none) (tail.aux none) | simp [-liftRel_pure_left, -liftRel_pure_right] | no goals | 1a57f81f307e696b |
MeasureTheory.withDensity_absolutelyContinuous | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) :
μ.withDensity f ≪ μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs₁ : MeasurableSet s
hs₂ : μ s = 0
⊢ ∫⁻ (a : α) in s, f a ∂μ = 0 | exact setLIntegral_measure_zero _ _ hs₂ | no goals | f394db0412a9858c |
GenContFract.succ_nth_conv'_eq_squashGCF_nth_conv' | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | theorem succ_nth_conv'_eq_squashGCF_nth_conv' :
g.convs' (n + 1) = (squashGCF g n).convs' n | K : Type u_1
n : ℕ
g : GenContFract K
inst✝ : DivisionRing K
⊢ g.convs' (n + 1) = (g.squashGCF n).convs' n | cases n with
| zero =>
cases g_s_head_eq : g.s.get? 0 <;>
simp [g_s_head_eq, squashGCF, convs', convs'Aux, Stream'.Seq.head]
| succ =>
simp only [succ_succ_nth_conv'Aux_eq_succ_nth_conv'Aux_squashSeq, convs',
squashGCF] | no goals | c535dd95cf67eb71 |
MeasureTheory.aeconst_of_forall_preimage_smul_ae_eq | Mathlib/Dynamics/Ergodic/Action/Basic.lean | theorem aeconst_of_forall_preimage_smul_ae_eq [SMul G α] [ErgodicSMul G α μ] {s : Set α}
(hm : NullMeasurableSet s μ) (h : ∀ g : G, (g • ·) ⁻¹' s =ᵐ[μ] s) :
EventuallyConst s (ae μ) | case intro.intro
G : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SMul G α
inst✝ : ErgodicSMul G α μ
s : Set α
h : ∀ (g : G), (fun x => g • x) ⁻¹' s =ᶠ[ae μ] s
t : Set α
htm : MeasurableSet t
hst : s =ᶠ[ae μ] t
g : G
⊢ (fun x => g • x) ⁻¹' t =ᶠ[ae μ] t | refine .trans (.trans ?_ (h g)) hst | case intro.intro
G : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SMul G α
inst✝ : ErgodicSMul G α μ
s : Set α
h : ∀ (g : G), (fun x => g • x) ⁻¹' s =ᶠ[ae μ] s
t : Set α
htm : MeasurableSet t
hst : s =ᶠ[ae μ] t
g : G
⊢ (fun x => g • x) ⁻¹' t =ᶠ[ae μ] (fun x => g • x) ⁻¹' s | f6caf7b1f8b21950 |
ENNReal.limsup_const_mul | Mathlib/Order/Filter/ENNReal.lean | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : u =ᶠ[f] 0
x : α
hx : u x = 0 x
⊢ (fun x => a * u x) x = 0 x | simp [hx] | no goals | 821a30d44957705c |
LinearMap.isSymmetric_adjoint_mul_self | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | theorem isSymmetric_adjoint_mul_self (T : E →ₗ[𝕜] E) : IsSymmetric (LinearMap.adjoint T * T) | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
⊢ (adjoint T * T).IsSymmetric | intro x y | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
x y : E
⊢ ⟪(adjoint T * T) x, y⟫_𝕜 = ⟪x, (adjoint T * T) y⟫_𝕜 | 5056180349104259 |
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