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 |
|---|---|---|---|---|---|---|
DFinsupp.support_update | Mathlib/Data/DFinsupp/Defs.lean | theorem support_update (f : Π₀ i, β i) (i : ι) (b : β i) [Decidable (b = 0)] :
support (f.update i b) = if b = 0 then support (f.erase i) else insert i f.support | case h
ι : Type u
β : ι → Type v
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
f : Π₀ (i : ι), β i
i : ι
b : β i
inst✝ : Decidable (b = 0)
j : ι
⊢ j ∈ (f.update i b).support ↔ j ∈ if b = 0 then (erase i f).support else insert i f.support | split_ifs with hb | case pos
ι : Type u
β : ι → Type v
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
f : Π₀ (i : ι), β i
i : ι
b : β i
inst✝ : Decidable (b = 0)
j : ι
hb : b = 0
⊢ j ∈ (f.update i b).support ↔ j ∈ (erase i f).support
case neg
ι : Type u
β : ι → Type v
inst✝³ : Decida... | 2470c241078527e0 |
ContinuousLinearMap.coe_ofIsClosedGraph | Mathlib/Analysis/Normed/Operator/Banach.lean | theorem coe_ofIsClosedGraph (hg : IsClosed (g.graph : Set <| E × F)) :
↑(ContinuousLinearMap.ofIsClosedGraph hg) = g | case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed ↑g.graph
x✝ : E
⊢ ↑(ofIsClosedGraph hg) x✝ = g ... | rfl | no goals | f67b8ce3b00b5c1c |
LieAlgebra.engel_isBot_of_isMin | Mathlib/Algebra/Lie/CartanExists.lean | /-- Let `L` be a Lie algebra of dimension `n` over a field `K` with at least `n` elements.
Given a Lie subalgebra `U` of `L`, and an element `x ∈ U` such that `U ≤ engel K x`.
Suppose that `engel K x` is minimal amongst the Engel subalgebras `engel K y` for `y ∈ U`.
Then `engel K x ≤ engel K y` for all `y ∈ U`.
Lemma ... | 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
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex... | have hz' : ∃ n : ℕ, (toEnd K U Q v ^ n) z' = 0 := by
rw [mem_engel_iff] at hz
obtain ⟨n, hn⟩ := hz
use n
apply_fun LieSubmodule.Quotient.mk' E at hn
rw [LieModuleHom.map_zero] at hn
rw [← hn]
clear hn
induction n with
| zero => simp only [z', pow_zero, LinearMap.one_apply]
| succ n ih => rw [pow_suc... | 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
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex... | 53ed38f54130aaf6 |
ProbabilityTheory.strong_law_ae_real | Mathlib/Probability/StrongLaw.lean | theorem strong_law_ae_real {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω}
(X : ℕ → Ω → ℝ) (hint : Integrable (X 0) μ)
(hindep : Pairwise ((IndepFun · · μ) on X))
(hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :
∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => (∑ i ∈ range n, X i ω) / n) atTop (𝓝 μ[X 0]) | case neg
Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
mΩ : MeasureSpace Ω := MeasureSpace.mk μ
h : ¬∀ᵐ (ω : Ω), X 0 ω = 0
this : IsProbabilityMeasure μ
⊢ ∀ᵐ (ω : Ω) ∂μ, T... | let pos : ℝ → ℝ := fun x => max x 0 | case neg
Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
mΩ : MeasureSpace Ω := MeasureSpace.mk μ
h : ¬∀ᵐ (ω : Ω), X 0 ω = 0
this : IsProbabilityMeasure μ
pos : ℝ → ℝ := fun... | 9e772f0b624ac943 |
List.head_dropLast | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem head_dropLast (xs : List α) (h) :
xs.dropLast.head h = xs.head (by rintro rfl; simp at h) | case cons.nil
α : Type u_1
x : α
h : [x].dropLast ≠ []
⊢ [x].dropLast.head h = [x].head ⋯ | simp at h | no goals | 28b1b7583cb0f295 |
CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f = f ≫ biprod.inr | ext | case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lif... | 70e51031cc3995fa |
CategoryTheory.Bicategory.inv_whiskerRight | Mathlib/CategoryTheory/Bicategory/Basic.lean | theorem inv_whiskerRight {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] :
inv (η ▷ h) = inv η ▷ h | B : Type u
inst✝¹ : Bicategory B
a b c : B
f g : a ⟶ b
η : f ⟶ g
h : b ⟶ c
inst✝ : IsIso η
⊢ inv (η ▷ h) = inv η ▷ h | apply IsIso.inv_eq_of_hom_inv_id | case hom_inv_id
B : Type u
inst✝¹ : Bicategory B
a b c : B
f g : a ⟶ b
η : f ⟶ g
h : b ⟶ c
inst✝ : IsIso η
⊢ η ▷ h ≫ inv η ▷ h = 𝟙 (f ≫ h) | dbbcca10cd4ed4a4 |
dependsOn_iff_factorsThrough | Mathlib/Logic/Function/DependsOn.lean | lemma dependsOn_iff_factorsThrough {f : (Π i, α i) → β} {s : Set ι} :
DependsOn f s ↔ FactorsThrough f s.restrict | ι : Type u_1
α : ι → Type u_2
β : Type u_3
f : ((i : ι) → α i) → β
s : Set ι
⊢ DependsOn f s ↔ FactorsThrough f s.restrict | rw [DependsOn, FactorsThrough] | ι : Type u_1
α : ι → Type u_2
β : Type u_3
f : ((i : ι) → α i) → β
s : Set ι
⊢ (∀ ⦃x y : (i : ι) → α i⦄, (∀ i ∈ s, x i = y i) → f x = f y) ↔
∀ ⦃a b : (i : ι) → α i⦄, s.restrict a = s.restrict b → f a = f b | cc8f3687e9cefdc3 |
ruzsaSzemerediNumberNat_asymptotic_lower_bound | Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean | theorem ruzsaSzemerediNumberNat_asymptotic_lower_bound :
(fun n ↦ n ^ 2 * exp (-4 * sqrt (log n)) : ℕ → ℝ) =O[atTop]
fun n ↦ (ruzsaSzemerediNumberNat n : ℝ) | n : ℕ
hn : 6 ≤ ↑n
⊢ 0 ≤ ↑n / 3 - 2 | linarith | no goals | 72990b3a12cc99c5 |
MeasureTheory.lmarginal_update_of_not_mem | Mathlib/MeasureTheory/Integral/Marginal.lean | theorem lmarginal_update_of_not_mem {i : δ}
{f : (∀ i, π i) → ℝ≥0∞} (hf : Measurable f) (hi : i ∉ s) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∘ (Function.update · i y) ∂μ) x | case empty
δ : Type u_1
π : δ → Type u_3
inst✝² : (x : δ) → MeasurableSpace (π x)
μ : (i : δ) → Measure (π i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
i : δ
f : ((i : δ) → π i) → ℝ≥0∞
hf : Measurable f
y : π i
hi : i ∉ ∅
x : (i : δ) → π i
⊢ (∫⋯∫⁻_∅, f ∂μ) (update x i y) = (∫⋯∫⁻_∅, f ∘ fu... | simp | no goals | b8b86d6841ffd789 |
contDiffWithinAt_localInvariantProp_of_le | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
(contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
(ContDiffWithinAtProp I I' m) where
is_local {s x u f} u_open xu | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | simpa only [hy, mfld_simps] using hs hy.1 | no goals | 841928255da4a913 |
ProbabilityTheory.Kernel.withDensity_tsum | Mathlib/Probability/Kernel/WithDensity.lean | theorem withDensity_tsum [Countable ι] (κ : Kernel α β) [IsSFiniteKernel κ] {f : ι → α → β → ℝ≥0∞}
(hf : ∀ i, Measurable (Function.uncurry (f i))) :
withDensity κ (∑' n, f n) = Kernel.sum fun n => withDensity κ (f n) | α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : Meas... | fun_prop | no goals | 4ff375ce758fb3f0 |
MeasureTheory.integral_integral_swap_of_hasCompactSupport | Mathlib/MeasureTheory/Integral/Prod.lean | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasC... | apply setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_) | E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasC... | ab5aab42565633fa |
Mathlib.Tactic.Monoidal.naturality_comp | Mathlib/Tactic/CategoryTheory/Monoidal/PureCoherence.lean | theorem naturality_comp {p f g h pf : C} {η : f ≅ g} {θ : g ≅ h}
(η_f : p ⊗ f ≅ pf) (η_g : p ⊗ g ≅ pf) (η_h : p ⊗ h ≅ pf)
(ih_η : p ◁ η ≪≫ η_g = η_f) (ih_θ : p ◁ θ ≪≫ η_h = η_g) :
p ◁ (η ≪≫ θ) ≪≫ η_h = η_f | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
p f g h pf : C
η : f ≅ g
θ : g ≅ h
η_f : p ⊗ f ≅ pf
η_g : p ⊗ g ≅ pf
η_h : p ⊗ h ≅ pf
ih_η : p ◁ η ≪≫ η_g = η_f
ih_θ : p ◁ θ ≪≫ η_h = η_g
⊢ p ◁ (η ≪≫ θ) ≪≫ η_h = η_f | simp_all | no goals | 0d5be8673b45bcac |
Order.succ_le_succ_iff_of_not_isMax | Mathlib/Order/SuccPred/Basic.lean | theorem succ_le_succ_iff_of_not_isMax (ha : ¬IsMax a) (hb : ¬IsMax b) :
succ a ≤ succ b ↔ a ≤ b | α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : SuccOrder α
a b : α
ha : ¬IsMax a
hb : ¬IsMax b
⊢ succ a ≤ succ b ↔ a ≤ b | rw [succ_le_iff_of_not_isMax ha, lt_succ_iff_of_not_isMax hb] | no goals | e7e08b388b1a61ed |
MvPolynomial.comap_comp_apply | Mathlib/Algebra/MvPolynomial/Comap.lean | theorem comap_comp_apply (f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R)
(g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R) (x : υ → R) :
comap (g.comp f) x = comap f (comap g x) | σ : Type u_1
τ : Type u_2
υ : Type u_3
R : Type u_4
inst✝ : CommSemiring R
f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R
g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R
x : υ → R
i : σ
⊢ g = aeval fun i => g (X i) | exact aeval_unique g | no goals | 5fa2901e7925d316 |
Real.ediam_Icc | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem ediam_Icc (a b : ℝ) : EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) | case inl
a b : ℝ
h : a ≤ b
⊢ EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) | rw [Real.ediam_eq (isBounded_Icc _ _), csSup_Icc h, csInf_Icc h] | no goals | 79ca1c3df5e45d2d |
Finset.sum_card_inter_le | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | theorem sum_card_inter_le (h : ∀ a ∈ s, #{b ∈ B | a ∈ b} ≤ n) : (∑ t ∈ B, #(s ∩ t)) ≤ #s * n | α : Type u_2
inst✝ : DecidableEq α
s : Finset α
B : Finset (Finset α)
n : ℕ
h : ∀ a ∈ s, #(filter (fun b => a ∈ b) B) ≤ n
⊢ (∑ x ∈ B, ∑ a ∈ s, if a ∈ x then 1 else 0) ≤ ∑ x ∈ s, ∑ a ∈ B, if x ∈ a then 1 else 0 | exact sum_comm.le | no goals | 476a03b2f17a093d |
Turing.TM2to1.addBottom_map | Mathlib/Computability/TuringMachine.lean | theorem addBottom_map (L : ListBlank (∀ k, Option (Γ k))) :
(addBottom L).map ⟨Prod.snd, by rfl⟩ = L | case h.e'_2.h.e'_4
K : Type u_1
Γ : K → Type u_2
L L' : ListBlank ((k : K) → Option (Γ k))
l : List ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } (ListBlank.mk l)) =
ListBlank.mk l | simp | no goals | a6b51c31a82f1722 |
jacobiSym.sq_one | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | theorem sq_one {a : ℤ} {b : ℕ} (h : a.gcd b = 1) : J(a | b) ^ 2 = 1 | a : ℤ
b : ℕ
h : a.gcd ↑b = 1
⊢ J(a | b) ^ 2 = 1 | rcases eq_one_or_neg_one h with h₁ | h₁ <;> rw [h₁] <;> rfl | no goals | 7f77c9d28313920c |
LightCondensed.isoFinYonedaComponents_inv_comp | Mathlib/Condensed/Discrete/Colimit.lean | lemma isoFinYonedaComponents_inv_comp {X Y : LightProfinite.{u}} [Finite X] [Finite Y]
(f : Y → F.obj ⟨LightProfinite.of PUnit⟩) (g : X ⟶ Y) :
(isoFinYonedaComponents F X).inv (f ∘ g) = F.map g.op ((isoFinYonedaComponents F Y).inv f) | case a.h
F : LightProfiniteᵒᵖ ⥤ Type u
inst✝² : PreservesFiniteProducts F
X Y : LightProfinite
inst✝¹ : Finite ↑X.toTop
inst✝ : Finite ↑Y.toTop
f : ↑Y.toTop → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))
g : X ⟶ Y
x : ↑X.toTop
⊢ (f ∘ ⇑(ConcreteCategory.hom g)) x =
(isoFinYonedaComponents F X).hom (F.map g.... | rw [isoFinYonedaComponents_hom_apply] | case a.h
F : LightProfiniteᵒᵖ ⥤ Type u
inst✝² : PreservesFiniteProducts F
X Y : LightProfinite
inst✝¹ : Finite ↑X.toTop
inst✝ : Finite ↑Y.toTop
f : ↑Y.toTop → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))
g : X ⟶ Y
x : ↑X.toTop
⊢ (f ∘ ⇑(ConcreteCategory.hom g)) x =
F.map (CompHausLike.const (LightProfinite.... | 2e6571f5e83fd29b |
CochainComplex.shiftFunctorAdd'_inv_app_f' | Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | lemma shiftFunctorAdd'_inv_app_f' (K : CochainComplex C ℤ) (a b ab : ℤ) (h : a + b = ab) (n : ℤ) :
((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).inv.app K).f n =
(K.XIsoOfEq (by dsimp; rw [← h, add_assoc, add_comm a])).hom | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + { as := b }.as + { as := a }.as = n + { as := ab }.as | dsimp | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + b + a = n + ab | bba1a45a8e5d6554 |
WeierstrassCurve.Jacobian.polynomialX_eq | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma polynomialX_eq : W'.polynomialX =
C W'.a₁ * X 1 * X 2 - (C 3 * X 0 ^ 2 + C (2 * W'.a₂) * X 0 * X 2 ^ 2 + C W'.a₄ * X 2 ^ 4) | R : Type r
inst✝ : CommRing R
W' : Jacobian R
⊢ W'.polynomialX = C W'.a₁ * X y * X z - (C 3 * X x ^ 2 + C (2 * W'.a₂) * X x * X z ^ 2 + C W'.a₄ * X z ^ 4) | rw [polynomialX, polynomial] | R : Type r
inst✝ : CommRing R
W' : Jacobian R
⊢ (pderiv x)
(X y ^ 2 + C W'.a₁ * X x * X y * X z + C W'.a₃ * X y * X z ^ 3 -
(X x ^ 3 + C W'.a₂ * X x ^ 2 * X z ^ 2 + C W'.a₄ * X x * X z ^ 4 + C W'.a₆ * X z ^ 6)) =
C W'.a₁ * X y * X z - (C 3 * X x ^ 2 + C (2 * W'.a₂) * X x * X z ^ 2 + C W'.a₄ * X z ^ 4) | 967ee92e03d89bb0 |
hyperoperation_two | Mathlib/Data/Nat/Hyperoperation.lean | theorem hyperoperation_two : hyperoperation 2 = (· * ·) | case h.h
m k : ℕ
⊢ hyperoperation 2 m k = m * k | induction' k with bn bih | case h.h.zero
m : ℕ
⊢ hyperoperation 2 m 0 = m * 0
case h.h.succ
m bn : ℕ
bih : hyperoperation 2 m bn = m * bn
⊢ hyperoperation 2 m (bn + 1) = m * (bn + 1) | 2701f86d00b32a67 |
sign_neg | Mathlib/Data/Sign.lean | theorem sign_neg (ha : a < 0) : sign a = -1 | α : Type u_1
inst✝² : Zero α
inst✝¹ : Preorder α
inst✝ : DecidableRel fun x1 x2 => x1 < x2
a : α
ha : a < 0
⊢ sign a = -1 | rwa [sign_apply, if_neg <| asymm ha, if_pos] | no goals | 6d84f335f724158c |
AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inter | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lemma image_preimage_eq_opensRange_inter (U : Y.Opens) : f ''ᵁ f ⁻¹ᵁ U = f.opensRange ⊓ U | case h
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f
U : Y.Opens
⊢ ↑(f ''ᵁ f ⁻¹ᵁ U) = ↑(f.opensRange ⊓ U) | simp [Set.image_preimage_eq_range_inter] | no goals | a002d14a9ba52c96 |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos' | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
(hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 | case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ ↑s j < 0 | set k := Nat.find hn | case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ ↑s j < 0 | 37ca223bdee11c33 |
Matrix.isDiag_fromBlocks_iff | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | theorem isDiag_fromBlocks_iff [Zero α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α}
{D : Matrix n n α} : (A.fromBlocks B C D).IsDiag ↔ A.IsDiag ∧ B = 0 ∧ C = 0 ∧ D.IsDiag | case mpr.intro.intro.intro
α : Type u_1
n : Type u_4
m : Type u_5
inst✝ : Zero α
A : Matrix m m α
B : Matrix m n α
C : Matrix n m α
D : Matrix n n α
ha : A.IsDiag
hb : B = 0
hc : C = 0
hd : D.IsDiag
⊢ (fromBlocks A B C D).IsDiag | convert IsDiag.fromBlocks ha hd | no goals | a80791093c043daf |
Topology.IsClosedEmbedding.sigmaCompactSpace | Mathlib/Topology/Compactness/SigmaCompact.lean | protected lemma Topology.IsClosedEmbedding.sigmaCompactSpace {e : Y → X}
(he : IsClosedEmbedding e) : SigmaCompactSpace Y :=
⟨⟨fun n => e ⁻¹' compactCovering X n, fun _ =>
he.isCompact_preimage (isCompact_compactCovering _ _), by
rw [← preimage_iUnion, iUnion_compactCovering, preimage_univ]⟩⟩
| X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : SigmaCompactSpace X
e : Y → X
he : IsClosedEmbedding e
⊢ ⋃ n, (fun n => e ⁻¹' compactCovering X n) n = univ | rw [← preimage_iUnion, iUnion_compactCovering, preimage_univ] | no goals | 9652d5e084f0a860 |
Polynomial.card_support_eq_one | Mathlib/Algebra/Polynomial/EraseLead.lean | theorem card_support_eq_one : #f.support = 1 ↔
∃ (k : ℕ) (x : R) (_ : x ≠ 0), f = C x * X ^ k | case refine_1.intro.intro.intro.intro
R : Type u_1
inst✝ : Semiring R
k : Fin 1 → ℕ
x : Fin 1 → R
w✝ : StrictMono k
hx : ∀ (i : Fin 1), x i ≠ 0
h : #(∑ i : Fin 1, C (x i) * X ^ k i).support = 1
⊢ ∃ k_1 x_1, ∃ (_ : x_1 ≠ 0), ∑ i : Fin 1, C (x i) * X ^ k i = C x_1 * X ^ k_1 | exact ⟨k 0, x 0, hx 0, Fin.sum_univ_one _⟩ | no goals | d8ddef9fa67e60c8 |
Real.compact_inner_le_weight_mul_Lp_of_nonneg | Mathlib/Analysis/MeanInequalities.lean | /-- **Weighted Hölder inequality** in terms of `Finset.expect`. -/
lemma compact_inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) {w f : ι → ℝ}
(hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 ≤ f i) :
𝔼 i ∈ s, w i * f i ≤ (𝔼 i ∈ s, w i) ^ (1 - p⁻¹) * (𝔼 i ∈ s, w i * f i ^ p) ^ p⁻¹ | ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
i : ι
x✝ : i ∈ s
this✝ : 0 ≤ w i
this : 0 ≤ f i
⊢ 0 ≤ w i * f i ^ p | positivity | no goals | 03820a3c7b910519 |
Ordering.compares_iff_of_compares_impl | Mathlib/Order/Compare.lean | theorem compares_iff_of_compares_impl [LinearOrder α] [Preorder β] {a b : α} {a' b' : β}
(h : ∀ {o}, Compares o a b → Compares o a' b') (o) : Compares o a b ↔ Compares o a' b' | case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a b : α
a' b' : β
h : ∀ {o : Ordering}, o.Compares a b → o.Compares a' b'
o : Ordering
ho : o.Compares a' b'
hab : a < b
⊢ o.Compares a b | have hab : Compares Ordering.lt a b := hab | case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a b : α
a' b' : β
h : ∀ {o : Ordering}, o.Compares a b → o.Compares a' b'
o : Ordering
ho : o.Compares a' b'
hab✝ : a < b
hab : lt.Compares a b
⊢ o.Compares a b | d371f30ca44f3226 |
Valuation.isEquiv_iff_val_sub_one_lt_one | Mathlib/RingTheory/Valuation/Basic.lean | theorem isEquiv_iff_val_sub_one_lt_one [LinearOrderedCommGroupWithZero Γ₀]
[LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
v.IsEquiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1 | K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_4
Γ'₀ : Type u_5
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ (∀ {x : K}, v x < 1 ↔ v' x < 1) ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1 | exact (Equiv.subRight 1).surjective.forall | no goals | 1bf75026ba24bd43 |
comap_map_eq_map_of_isLocalization_algebraMapSubmonoid | Mathlib/RingTheory/Trace/Quotient.lean | lemma comap_map_eq_map_of_isLocalization_algebraMapSubmonoid :
(Ideal.map (algebraMap R Sₚ) p).comap (algebraMap S Sₚ) = pS | case intro.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
Sₚ : Type u_4
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScal... | refine (Submodule.add_mem_iff_left _ ?_).mp this | case intro.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
Sₚ : Type u_4
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScal... | d6f88c93445b65e1 |
continuousOn_integral_bilinear_of_locally_integrable_of_compact_support | Mathlib/MeasureTheory/Integral/SetIntegral.lean | /-- Consider a parameterized integral `x ↦ ∫ y, L (g y) (f x y)` where `L` is bilinear,
`g` is locally integrable and `f` is continuous and uniformly compactly supported. Then the
integral depends continuously on `x`. -/
lemma continuousOn_integral_bilinear_of_locally_integrable_of_compact_support
[NormedSpace 𝕜 E... | Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddC... | gcongr | no goals | 2723cd3ae22d6304 |
Submodule.span_range_update_add_smul | Mathlib/LinearAlgebra/Span/Defs.lean | lemma span_range_update_add_smul (hij : i ≠ j) (v : ι → M) (r : R) :
span R (Set.range (Function.update v j (v j + r • v i))) = span R (Set.range v) | case refine_1.inr
R : Type u_1
M : Type u_4
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_8
inst✝ : DecidableEq ι
i j : ι
hij : i ≠ j
v : ι → M
r : R
k : ι
hjk : j ≠ k
⊢ update v j (v j + r • v i) k ∈ span R (range v) | exact subset_span ⟨k, (update_of_ne hjk.symm ..).symm⟩ | no goals | bfb26e8467ba0321 |
VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) | case h
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
⊢ ∀ (a : α),
Tendsto (fun a => (μ.restrict s)... | intro x hx h'x | case h
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
x : α
hx : Tendsto (fun a => (μ.restrict s) a / μ... | af5f846186fd28e4 |
NNReal.inner_le_weight_mul_Lp | Mathlib/Analysis/MeanInequalities.lean | /-- **Weighted Hölder inequality**. -/
lemma inner_le_weight_mul_Lp (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0) :
∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ | ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
⊢ p⁻¹ + (1 - p⁻¹)⁻¹⁻¹ = 1 | simp | no goals | 4eef6f52055e7b19 |
Valued.continuous_valuation | Mathlib/Topology/Algebra/Valued/ValuedField.lean | theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) | case inl
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
⊢ ∀ (γ₀ : Γ₀), γ₀ ≠ 0 → ∀ᶠ (x : K) in 𝓝 0, v x < γ₀ | intro γ hγ | case inl
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
γ : Γ₀
hγ : γ ≠ 0
⊢ ∀ᶠ (x : K) in 𝓝 0, v x < γ | 40ad4a371f000982 |
IsMaxChain.image | Mathlib/Order/Chain.lean | lemma IsMaxChain.image {s : β → β → Prop} (e : r ≃r s) {c : Set α} (hc : IsMaxChain r c) :
IsMaxChain s (e '' c) where
left := hc.isChain.image _ _ _ fun _ _ ↦ by exact e.map_rel_iff.2
right t ht hf | α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
e : r ≃r s
c : Set α
hc : IsMaxChain r c
t : Set β
ht : IsChain s t
hf : ⇑e '' c ⊆ t
⊢ ⇑e '' c = t | rw [← e.coe_fn_toEquiv, ← e.toEquiv.eq_preimage_iff_image_eq, preimage_equiv_eq_image_symm] | α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
e : r ≃r s
c : Set α
hc : IsMaxChain r c
t : Set β
ht : IsChain s t
hf : ⇑e '' c ⊆ t
⊢ c = ⇑e.symm '' t | 4860b894cab14a5c |
Equiv.Perm.IsCycle.pow_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.pow_iff [Finite β] {f : Perm β} (hf : IsCycle f) {n : ℕ} :
IsCycle (f ^ n) ↔ n.Coprime (orderOf f) | case intro.mp
β : Type u_3
inst✝ : Finite β
f : Perm β
hf : f.IsCycle
n : ℕ
val✝ : Fintype β
⊢ (f ^ n).IsCycle → n.Coprime (orderOf f) | intro h | case intro.mp
β : Type u_3
inst✝ : Finite β
f : Perm β
hf : f.IsCycle
n : ℕ
val✝ : Fintype β
h : (f ^ n).IsCycle
⊢ n.Coprime (orderOf f) | 66cd6cfb46d266c4 |
List.forall_mem_ofFn_iff | Mathlib/Data/List/OfFn.lean | theorem forall_mem_ofFn_iff {n : ℕ} {f : Fin n → α} {P : α → Prop} :
(∀ i ∈ ofFn f, P i) ↔ ∀ j : Fin n, P (f j) | α : Type u
n : ℕ
f : Fin n → α
P : α → Prop
⊢ (∀ i ∈ ofFn f, P i) ↔ ∀ (j : Fin n), P (f j) | simp | no goals | e343e33f4f309901 |
CategoryTheory.Iso.comp_inv_eq | Mathlib/CategoryTheory/Iso.lean | theorem comp_inv_eq (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : f ≫ α.inv = g ↔ f = g ≫ α.hom :=
⟨fun H => by simp [H.symm], fun H => by simp [H]⟩
| C : Type u
inst✝ : Category.{v, u} C
X Y Z : C
α : X ≅ Y
f : Z ⟶ Y
g : Z ⟶ X
H : f ≫ α.inv = g
⊢ f = g ≫ α.hom | simp [H.symm] | no goals | c1f43a133801aee1 |
Finset.filter_product | Mathlib/Data/Finset/Prod.lean | theorem filter_product (p : α → Prop) (q : β → Prop) [DecidablePred p] [DecidablePred q] :
((s ×ˢ t).filter fun x : α × β => p x.1 ∧ q x.2) = s.filter p ×ˢ t.filter q | α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
p : α → Prop
q : β → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
⊢ filter (fun x => p x.1 ∧ q x.2) (s ×ˢ t) = filter p s ×ˢ filter q t | ext ⟨a, b⟩ | case h.mk
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
p : α → Prop
q : β → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
a : α
b : β
⊢ (a, b) ∈ filter (fun x => p x.1 ∧ q x.2) (s ×ˢ t) ↔ (a, b) ∈ filter p s ×ˢ filter q t | 90c7b19611f0fabd |
DirichletCharacter.BadChar.F_differentiable | Mathlib/NumberTheory/LSeries/Nonvanishing.lean | /-- If `χ` is a bad character, then `F` is an entire function. -/
private lemma F_differentiable (B : BadChar N) : Differentiable ℂ B.F | case h.inr
N : ℕ
inst✝ : NeZero N
B : DirichletCharacter.BadChar N
G : ℂ → ℂ := Function.update (fun s => (s - 1) * riemannZeta s) 1 1
H : ℂ → ℂ := Function.update (fun s => (LFunction B.χ s - LFunction B.χ 1) / (s - 1)) 1 (deriv (LFunction B.χ) 1)
t : ℂ
ht : t ≠ 1
⊢ B.F t = (G * H) t | simp only [F, G, H, Function.update_of_ne ht, mul_comm _ (riemannZeta _), B.hχ, sub_zero,
Pi.mul_apply, mul_assoc, mul_div_cancel₀ _ (sub_ne_zero.mpr ht)] | no goals | a1693c7d84c8fc2a |
Algebra.FormallyUnramified.lift_unique_of_ringHom | Mathlib/RingTheory/Unramified/Basic.lean | theorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type*} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
sim... | case H
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : FormallyUnramified R A
C : Type u_1
inst✝ : CommRing C
f : B →+* C
hf : IsNilpotent (RingHom.ker f)
g₁ g₂ : A →ₐ[R] B
h : f.comp ↑g₁ = f.comp ↑g₂
x : A
⊢ ((Ideal.Quotien... | have := RingHom.congr_fun h x | case H
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : FormallyUnramified R A
C : Type u_1
inst✝ : CommRing C
f : B →+* C
hf : IsNilpotent (RingHom.ker f)
g₁ g₂ : A →ₐ[R] B
h : f.comp ↑g₁ = f.comp ↑g₂
x : A
this : (f.comp ↑g... | da42b270224e7bee |
OnePoint.not_continuous_cofiniteTopology_of_symm | Mathlib/Topology/Compactification/OnePoint.lean | theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm | X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
⊢ ¬Continuous ⇑CofiniteTopology.of.symm | inhabit X | X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ¬Continuous ⇑CofiniteTopology.of.symm | 847652b7cd833a2e |
HomologicalComplex.mapBifunctorMapHomotopy.comm₁ | Mathlib/Algebra/Homology/BifunctorHomotopy.lean | lemma comm₁ (j : J) :
(mapBifunctorMap f₁ f₂ F c).f j =
(mapBifunctor K₁ K₂ F c).d j (c.next j) ≫
mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c (c.next j) j +
mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j (c.prev j) ≫
(mapBifunctor L₁ L₂ F c).d (c.prev j) j +
(mapBifunctorMap f₁' f₂ F c).f... | case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{u_8, u_1} C₁
inst✝¹⁰ : Category.{u_9, u_2} C₂
inst✝⁹ : Category.{u_7, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : Homological... | rw [HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄, comp_zero, comp_zero, smul_zero,
HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄, zero_comp, neg_zero] | no goals | a0c0f521a8c24f60 |
Orientation.oangle_sub_right_smul_rotation_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem oangle_sub_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
o.oangle (r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x - x)
= Real.arctan r⁻¹ | case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)
⊢ o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x - x) = ↑(... | rw [sub_eq_add_neg, hx, o.oangle_add_right_smul_rotation_pi_div_two] | case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)
⊢ r • (o.rotation ↑(π / 2)) x ≠ 0 | e8cec3656d73d3d2 |
HallMarriageTheorem.hall_hard_inductive_step_B | Mathlib/Combinatorics/Hall/Finite.lean | theorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧... | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Inje... | let t' : s → Finset α := fun x' => t x' | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Inje... | de2e47a85ff6c58c |
DFinsupp.coprodMap_apply_single | Mathlib/LinearAlgebra/DFinsupp.lean | theorem coprodMap_apply_single (f : ∀ i : ι, M i →ₗ[R] N) (i : ι) (x : M i) :
coprodMap f (single i x) = f i x | ι : Type u_1
R : Type u_2
M : ι → Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommMonoid (M i)
inst✝³ : (i : ι) → Module R (M i)
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : DecidableEq ι
f : (i : ι) → M i →ₗ[R] N
i : ι
x : M i
⊢ (coprodMap f) (single i x) = (f i) x | simp [coprodMap] | no goals | 1e4b19ef7076cd82 |
Lean.Omega.Int.add_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/Int.lean | theorem add_congr {a b c d : Int} (h₁ : a = b) (h₂ : c = d) : a + c = b + d | a c d : Int
h₂ : c = d
⊢ a + c = a + d | subst h₂ | a c : Int
⊢ a + c = a + c | 83b8fb0efd34c125 |
IsBoundedBilinearMap.continuous | Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean | theorem IsBoundedBilinearMap.continuous (h : IsBoundedBilinearMap 𝕜 f) : Continuous f | case hf
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
... | rw [← isLittleO_one_iff ℝ, ← one_mul 1] | case hf
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
... | e893f50e1a714d43 |
MeasureTheory.FiniteMeasure.average_eq_integral_normalize | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | theorem average_eq_integral_normalize {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(nonzero : μ ≠ 0) (f : Ω → E) :
average (μ : Measure Ω) f = ∫ ω, f ω ∂(μ.normalize : Measure Ω) | Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
nonzero : μ ≠ 0
f : Ω → E
⊢ ∫ (x : Ω), f x ∂(↑μ univ)⁻¹ • ↑μ = ∫ (ω : Ω), f ω ∂μ.mass⁻¹ • ↑μ | congr | case e_μ.e_a
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
nonzero : μ ≠ 0
f : Ω → E
⊢ (↑μ univ)⁻¹ = ↑ENNReal.ofNNRealHom.toMonoidWithZeroHom μ.mass⁻¹ | 19e353c48b59d551 |
PadicInt.appr_spec | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem appr_spec (n : ℕ) : ∀ x : ℤ_[p], x - appr x n ∈ Ideal.span {(p : ℤ_[p]) ^ n} | case neg.intro
p : ℕ
hp_prime : Fact (Nat.Prime p)
n : ℕ
ih : ∀ (x : ℤ_[p]), ↑p ^ n ∣ x - ↑(x.appr n)
x : ℤ_[p]
h : ¬x - ↑(x.appr n) = 0
c : ℤ_[p]
hc : x - ↑(x.appr n) = ↑p ^ n * c
hc' : c ≠ 0
⊢ ↑p ^ (n + 1) ∣ ↑p ^ n * c - ↑p ^ n * (toZMod ↑(unitCoeff h) * 0 ^ (↑(↑p ^ n * c).valuation - ↑n).natAbs).cast | rw [valuation_p_pow_mul _ _ hc', Nat.cast_add, add_sub_cancel_left, _root_.pow_succ, ← mul_sub] | case neg.intro
p : ℕ
hp_prime : Fact (Nat.Prime p)
n : ℕ
ih : ∀ (x : ℤ_[p]), ↑p ^ n ∣ x - ↑(x.appr n)
x : ℤ_[p]
h : ¬x - ↑(x.appr n) = 0
c : ℤ_[p]
hc : x - ↑(x.appr n) = ↑p ^ n * c
hc' : c ≠ 0
⊢ ↑p ^ n * ↑p ∣ ↑p ^ n * (c - (toZMod ↑(unitCoeff h) * 0 ^ (↑c.valuation).natAbs).cast) | 6d18cd4591ade642 |
CategoryTheory.Pretriangulated.exists_iso_of_arrow_iso | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma exists_iso_of_arrow_iso (T₁ T₂ : Triangle C) (hT₁ : T₁ ∈ distTriang C)
(hT₂ : T₂ ∈ distTriang C) (e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁) :
∃ (e' : T₁ ≅ T₂), e'.hom.hom₁ = e.hom.left ∧ e'.hom.hom₂ = e.hom.right | C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := ... | infer_instance | no goals | 9637a6fb8661e804 |
LinearMap.range_prod_eq | Mathlib/LinearAlgebra/Prod.lean | theorem range_prod_eq {f : M →ₗ[R] M₂} {g : M →ₗ[R] M₃} (h : ker f ⊔ ker g = ⊤) :
range (prod f g) = (range f).prod (range g) | R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
h : ker f ⊔ ker g = ⊤
⊢ range (f.prod g) = (range f).prod (range g) | refine le_antisymm (f.range_prod_le g) ?_ | R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
h : ker f ⊔ ker g = ⊤
⊢ (range f).prod (range g) ≤ range (f.prod g) | 8c8d54532dff4081 |
Real.binEntropy_lt_log_two | Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean | /-- For probability `p ≠ 0.5`, `binEntropy p < log 2`. -/
lemma binEntropy_lt_log_two : binEntropy p < log 2 ↔ p ≠ 2⁻¹ | p✝ p : ℝ
h : p ≠ 2⁻¹
hp : p < 2⁻¹
hp₀ : p ≤ 0
⊢ 1 < 2 | norm_num | no goals | eecbe0dab536b9d5 |
MonovaryOn.sum_smul_comp_perm_le_sum_smul | Mathlib/Algebra/Order/Rearrangement.lean | theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
(hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i | case step.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrderedSemiring α
inst✝³ : ExistsAddOfLE α
inst✝² : LinearOrderedCancelAddCommMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
s✝ : Finset ι
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
σ :... | simp only [← s.sum_erase_add _ h1s, add_comm] | case step.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrderedSemiring α
inst✝³ : ExistsAddOfLE α
inst✝² : LinearOrderedCancelAddCommMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
s✝ : Finset ι
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
σ :... | 4fd7a15670de10aa |
Cardinal.mk_eq_two_iff' | Mathlib/SetTheory/Cardinal/Basic.lean | theorem mk_eq_two_iff' (x : α) : #α = 2 ↔ ∃! y, y ≠ x | case mp.intro.intro.intro
α : Type u
x a b : α
hne : a ≠ b
h : ∀ (x : α), x = a ∨ x = b
⊢ ∃! y, y ≠ x | rcases h x with (rfl | rfl) | case mp.intro.intro.intro.inl
α : Type u
x b : α
hne : x ≠ b
h : ∀ (x_1 : α), x_1 = x ∨ x_1 = b
⊢ ∃! y, y ≠ x
case mp.intro.intro.intro.inr
α : Type u
x a : α
hne : a ≠ x
h : ∀ (x_1 : α), x_1 = a ∨ x_1 = x
⊢ ∃! y, y ≠ x | 8af01dfac8c6082d |
BitVec.getMsbD_ushiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getMsbD_ushiftRight {x : BitVec w} {i n : Nat} :
(x >>> n).getMsbD i = (decide (i < w) && (!decide (i < n) && x.getMsbD (i - n))) | case pos
w : Nat
x : BitVec w
i n : Nat
h : i < n
⊢ (decide (i < w) && x.getLsbD (n + (w - 1 - i))) =
(decide (i < w) && (!decide (i < n) && (decide (i - n < w) && x.getLsbD (w - 1 - (i - n))))) | simp [getLsbD_ge, show w ≤ (n + (w - 1 - i)) by omega] | case pos
w : Nat
x : BitVec w
i n : Nat
h : i < n
⊢ i < w → n ≤ i → i - n < w → x.getLsbD (w - 1 - (i - n)) = false | 9cf7cade4c8c9d82 |
ProbabilityTheory.strong_law_ae | Mathlib/Probability/StrongLaw.lean | theorem strong_law_ae (X : ℕ → Ω → E) (hint : Integrable (X 0) μ)
(hindep : Pairwise ((IndepFun · · μ) on X))
(hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :
∀ᵐ ω ∂μ, Tendsto (fun n : ℕ ↦ (n : ℝ) ⁻¹ • (∑ i ∈ range n, X i ω)) atTop (𝓝 μ[X 0]) | case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), Ide... | have Yint : Integrable (Y 0) μ := Integrable.congr hint (AEStronglyMeasurable.ae_eq_mk (A 0).1) | case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ), Ide... | 0d66fa53c2ecc53a |
Finsupp.mk_mem_graph_iff | Mathlib/Data/Finsupp/Basic.lean | theorem mk_mem_graph_iff {a : α} {m : M} {f : α →₀ M} : (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0 | case mp.intro.intro.refl
α : Type u_1
M : Type u_5
inst✝ : Zero M
a : α
f : α →₀ M
ha : f a ≠ 0
⊢ f a = f a ∧ f a ≠ 0 | exact ⟨rfl, ha⟩ | no goals | 763a85cf05b11a6a |
ContinuousLinearMap.opNorm_extend_le | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | theorem opNorm_extend_le :
‖f.extend e h_dense (isUniformEmbedding_of_bound _ h_e).isUniformInducing‖ ≤ N * ‖f‖ | case refine_1.inr
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedS... | simp | no goals | 1e363b3609cf7221 |
Finset.inf_sdiff_right | Mathlib/Data/Finset/Lattice/Fold.lean | theorem inf_sdiff_right (hs : s.Nonempty) (f : ι → α) (a : α) :
(s.inf fun b => f b \ a) = s.inf f \ a | case singleton
α : Type u_2
ι : Type u_5
inst✝ : BooleanAlgebra α
s : Finset ι
f : ι → α
a : α
a✝ : ι
⊢ ({a✝}.inf fun b => f b \ a) = {a✝}.inf f \ a | rw [inf_singleton, inf_singleton] | no goals | ce808222bb39ab02 |
Batteries.RBNode.Path.zoom_ins | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem zoom_ins {t : RBNode α} {cmp : α → α → Ordering} :
t.zoom (cmp v) path = (t', path') →
path.ins (t.ins cmp v) = path'.ins (t'.setRoot v) | α : Type u_1
v : α
path : Path α
t' : RBNode α
path' : Path α
cmp : α → α → Ordering
x✝² a✝ : RBNode α
y✝ : α
b✝ : RBNode α
x✝¹ : Ordering
heq✝ : cmp v y✝ = Ordering.eq
x✝ : (node black a✝ y✝ b✝, path) = (t', path')
⊢ path.ins (node black a✝ v b✝) = path.ins (setRoot v (node black a✝ y✝ b✝)) | rfl | no goals | 01692d19938891c2 |
CategoryTheory.Limits.HasBiproduct.of_hasProduct | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | theorem HasBiproduct.of_hasProduct (f : J → C) [HasProduct f] : HasBiproduct f | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Finite J
f : J → C
inst✝ : HasProduct f
⊢ HasBiproduct f | cases nonempty_fintype J | case intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Finite J
f : J → C
inst✝ : HasProduct f
val✝ : Fintype J
⊢ HasBiproduct f | 0999d3267f227ba7 |
Module.End.genEigenspace_div | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma genEigenspace_div (f : End K V) (a b : K) (hb : b ≠ 0) :
genEigenspace f (a / b) 1 = LinearMap.ker (b • f - a • 1) :=
calc
genEigenspace f (a / b) 1 = genEigenspace f (b⁻¹ * a) 1 | K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
a b : K
hb : b ≠ 0
⊢ (f.genEigenspace (b⁻¹ * a)) 1 = LinearMap.ker (f - (b⁻¹ * a) • 1) | rw [genEigenspace_one] | no goals | 3794ff0de35221f1 |
MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left' | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToSimpleFunc_mono_left' (T T' : Set α → E →L[ℝ] G'')
(hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →ₛ E)
(hf : Integrable f μ) : setToSimpleFunc T f ≤ setToSimpleFunc T' f | α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝¹ : NormedLatticeAddCommGroup G''
inst✝ : NormedSpace ℝ G''
T T' : Set α → E →L[ℝ] G''
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α →ₛ E
hf : ... | refine sum_le_sum fun i _ => ?_ | α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝¹ : NormedLatticeAddCommGroup G''
inst✝ : NormedSpace ℝ G''
T T' : Set α → E →L[ℝ] G''
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α →ₛ E
hf : ... | 324cf30398c3ee84 |
Multiset.Subset.dedup_add_left | Mathlib/Data/Multiset/Dedup.lean | theorem Subset.dedup_add_left {s t : Multiset α} (h : t ⊆ s) :
dedup (s + t) = dedup s | α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
h : t ⊆ s
⊢ (s + t).dedup = s.dedup | rw [s.add_comm, Subset.dedup_add_right h] | no goals | e58090401294e18f |
Class.congToClass_empty | Mathlib/SetTheory/ZFC/Basic.lean | theorem congToClass_empty : congToClass ∅ = ∅ | ⊢ congToClass ∅ = ∅ | ext z | case a
z : ZFSet.{u_1}
⊢ congToClass ∅ z ↔ ∅ z | 6a8c86a7aa2551c5 |
Finset.Nat.antidiagonal_succ | Mathlib/Data/Finset/NatAntidiagonal.lean | theorem antidiagonal_succ (n : ℕ) :
antidiagonal (n + 1) =
cons (0, n + 1)
((antidiagonal n).map
(Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩ (Embedding.refl _)))
(by simp) | n : ℕ
⊢ (0, n + 1) ∉ map ({ toFun := Nat.succ, inj' := Nat.succ_injective }.prodMap (Embedding.refl ℕ)) (antidiagonal n) | simp | no goals | 6469a5d9be6684ac |
SetTheory.PGame.Identical.add_right | Mathlib/SetTheory/Game/PGame.lean | lemma Identical.add_right {x₁ x₂ y} : x₁ ≡ x₂ → x₁ + y ≡ x₂ + y :=
match x₁, x₂, y with
| mk x₁l x₁r x₁L x₁R, mk x₂l x₂r x₂L x₂R, mk yl yr yL yR => by
intro h
refine ⟨⟨?_, ?_⟩, ⟨?_, ?_⟩⟩ <;> rintro (_ | _) <;> try exact ⟨.inr _, h.add_right⟩
· exact (h.1.1 _).elim (⟨.inl ·, ·.add_right⟩)
· exact (h.... | case refine_4.inr
x₁ x₂ y : PGame
x₁l x₁r : Type u_1
x₁L : x₁l → PGame
x₁R : x₁r → PGame
x₂l x₂r : Type u_1
x₂L : x₂l → PGame
x₂R : x₂r → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
h : mk x₁l x₁r x₁L x₁R ≡ mk x₂l x₂r x₂L x₂R
val✝ : yr
⊢ ∃ a,
(fun i j =>
(fun t =>
Sum.rec
... | exact ⟨.inr _, h.add_right⟩ | no goals | 818fb0dc88ef3055 |
LieGroup.of_le | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | theorem LieGroup.of_le {m n : WithTop ℕ∞} (hmn : m ≤ n)
[h : LieGroup I n G] : LieGroup I m G | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : Group G
m n : WithTop ℕ∞
hmn : m ≤ n
h : LieGroup I n G
t... | exact ⟨h.contMDiff_inv.of_le hmn⟩ | no goals | 524301a4b565be64 |
Multiset.powersetCardAux_perm | Mathlib/Data/Multiset/Powerset.lean | theorem powersetCardAux_perm {n} {l₁ l₂ : List α} (p : l₁ ~ l₂) :
powersetCardAux n l₁ ~ powersetCardAux n l₂ | case zero
α : Type u_1
l₁ l₂ : List α
p : l₁ ~ l₂
⊢ powersetCardAux 0 l₁ ~ powersetCardAux 0 l₂ | simp | no goals | 94d2fdcb8d19de11 |
Real.rpow_le_rpow_of_nonpos | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | lemma rpow_le_rpow_of_nonpos (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0) : y ^ z ≤ x ^ z | case hb
x y z : ℝ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
⊢ 0 < y ^ z | positivity | no goals | ecccdc1e1d2eb039 |
supPrime_iff_supIrred | Mathlib/Order/Irreducible.lean | theorem supPrime_iff_supIrred : SupPrime a ↔ SupIrred a :=
⟨SupPrime.supIrred,
And.imp_right fun h b c => by simp_rw [← inf_eq_left, inf_sup_left]; exact @h _ _⟩
| α : Type u_2
inst✝ : DistribLattice α
a : α
h : ∀ ⦃b c : α⦄, b ⊔ c = a → b = a ∨ c = a
b c : α
⊢ a ⊓ b ⊔ a ⊓ c = a → a ⊓ b = a ∨ a ⊓ c = a | exact @h _ _ | no goals | 83b91ce76975c660 |
Std.Sat.AIG.relabelNat_unsat_iff | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RelabelNat.lean | theorem relabelNat_unsat_iff [Nonempty α] {aig : AIG α} {hidx1} {hidx2} :
(aig.relabelNat).UnsatAt idx hidx1 ↔ aig.UnsatAt idx hidx2 | α : Type
inst✝² : DecidableEq α
inst✝¹ : Hashable α
idx : Nat
inst✝ : Nonempty α
aig : AIG α
hidx1 : idx < aig.relabelNat.decls.size
hidx2 : idx < aig.decls.size
x y : α
hx : x ∈ aig
hy : y ∈ aig
heq :
(match (RelabelNat.State.ofAIG aig)[x]? with
| some var => var
| none => 0) =
match (RelabelNat.State.of... | split at heq | case h_1
α : Type
inst✝² : DecidableEq α
inst✝¹ : Hashable α
idx : Nat
inst✝ : Nonempty α
aig : AIG α
hidx1 : idx < aig.relabelNat.decls.size
hidx2 : idx < aig.decls.size
x y : α
hx : x ∈ aig
hy : y ∈ aig
x✝ : Option Nat
var✝ : Nat
heq✝ : (RelabelNat.State.ofAIG aig)[x]? = some var✝
heq :
var✝ =
match (RelabelNat... | 8afc7232d41c16dc |
List.forIn_yield_eq_foldl | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn_yield_eq_foldl
(l : List α) (f : α → β → β) (init : β) :
forIn (m := Id) l init (fun a b => .yield (f a b)) =
l.foldl (fun b a => f a b) init | α : Type u_1
β : Type u_2
l : List α
f : α → β → β
init : β
⊢ ForInStep.value <$>
List.foldlM
(fun b a =>
match b with
| ForInStep.yield b => ForInStep.yield (f a b)
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield init) l =
foldl (fun b a => f a b)... | induction l generalizing init <;> simp_all | no goals | 9fb9777bbb27c054 |
HomologicalComplex.mapBifunctor₁₂.d_eq | Mathlib/Algebra/Homology/BifunctorAssociator.lean | lemma d_eq (j j' : ι₄) [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄] :
(mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).d j j' =
D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' +
D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' | case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{u_13, u_1} C₁
inst✝¹⁹ : Category.{u_14, u_2} C₂
inst✝¹⁸ : Category.{u_15, u_5} C₃
inst✝¹⁷ : Category.{u_16, u_6} C₄
inst✝¹⁶ : Category.{u_17, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ :... | rw [mapBifunctor.d₁_eq_zero' _ _ _ _ h₁ _ _ h₂, comp_zero] | case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{u_13, u_1} C₁
inst✝¹⁹ : Category.{u_14, u_2} C₂
inst✝¹⁸ : Category.{u_15, u_5} C₃
inst✝¹⁷ : Category.{u_16, u_6} C₄
inst✝¹⁶ : Category.{u_17, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ :... | bcc237c029dad5d6 |
CategoryTheory.GrothendieckTopology.Plus.sep | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | theorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.Cover X) (x y : ToType ((J.plusObj P).obj (op X)))
(h : ∀ I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y | case intro.intro.intro.intro.intro.a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ... | exact he1 | no goals | dc0e1800e09060e0 |
ProbabilityTheory.Kernel.iIndepFun.cond_iInter | Mathlib/Probability/Independence/Kernel.lean | /-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
lemma iIndepFun.cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i))
(hindep : iIndepFun (fun _ ↦ mα.prod mβ) (fun i ω ↦ (X i ω, Y i ω)) κ μ)
(hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i))
... | ι : Type u_4
Ω : Type u_5
α : Type u_6
β : Type u_7
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α Ω
μ : Measure α
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω =... | rw [cond_apply] | case hms
ι : Type u_4
Ω : Type u_5
α : Type u_6
β : Type u_7
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α Ω
μ : Measure α
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (... | 31a9bf09c9f649df |
StarConvex.preimage_add_left | Mathlib/Analysis/Convex/Star.lean | theorem StarConvex.preimage_add_left (hs : StarConvex 𝕜 (x + z) s) :
StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s) | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x z : E
s : Set E
hs : StarConvex 𝕜 (x + z) s
⊢ StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s) | rw [add_comm] at hs | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x z : E
s : Set E
hs : StarConvex 𝕜 (z + x) s
⊢ StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s) | 33cfd68ced1e9361 |
ProbabilityTheory.gaussianReal_map_const_add | Mathlib/Probability/Distributions/Gaussian.lean | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_const_add (y : ℝ) :
(gaussianReal μ v).map (y + ·) = gaussianReal (μ + y) v | μ : ℝ
v : ℝ≥0
y : ℝ
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | exact gaussianReal_map_add_const y | no goals | 9b06087afbd8663d |
StieltjesFunction.eq_of_measure_of_eq | Mathlib/MeasureTheory/Measure/Stieltjes.lean | lemma eq_of_measure_of_eq (g : StieltjesFunction) {y : ℝ}
(hfg : f.measure = g.measure) (hy : f y = g y) :
f = g | case h.inr
f g : StieltjesFunction
y : ℝ
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : ℝ
hxy : y ≤ x
⊢ ↑f x = ↑g x | have hf := measure_Ioc f y x | case h.inr
f g : StieltjesFunction
y : ℝ
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : ℝ
hxy : y ≤ x
hf : f.measure (Ioc y x) = ofReal (↑f x - ↑f y)
⊢ ↑f x = ↑g x | 87bb9caced0194d7 |
MeasureTheory.StronglyMeasurable.integral_kernel | Mathlib/Probability/Kernel/MeasurableIntegral.lean | theorem StronglyMeasurable.integral_kernel ⦃f : β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x ↦ ∫ y, f y ∂κ x | case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : ¬CompleteSpace E
⊢ StronglyMeasurable fun x => ∫ (y : β), f y ∂κ x | simp [integral, hE, stronglyMeasurable_const] | no goals | 294012712b5f3f53 |
CategoryTheory.finrank_hom_simple_simple_eq_one_iff | Mathlib/CategoryTheory/Preadditive/Schur.lean | theorem finrank_hom_simple_simple_eq_one_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
[FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
finrank 𝕜 (X ⟶ Y) = 1 ↔ Nonempty (X ≅ Y) | case mp
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ∃ v, v ≠ 0 ∧ ∀ (w : X ⟶... | obtain ⟨f, nz, -⟩ := h | case mp.intro.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
nz : ... | 222dff5965f721df |
MulAction.mem_subgroup_orbit_iff | Mathlib/GroupTheory/GroupAction/Defs.lean | @[to_additive]
lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) | case refine_2.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : α
b : ↑(orbit G x)
g : ↥H
⊢ ↑g • b ∈ orbit (↥H) b | exact MulAction.mem_orbit _ g | no goals | 455130db4c4a28b8 |
MeromorphicOn.isClopen_setOf_order_eq_top | Mathlib/Analysis/Meromorphic/Order.lean | theorem isClopen_setOf_order_eq_top : IsClopen { u : U | (hf u.1 u.2).order = ⊤ } | case h.left
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | ⋯.order = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y ≠ 0
h₂t' : IsOpen t'
h₃t' : ... | simp only [mem_compl_iff, mem_setOf_eq] | case h.left
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | ⋯.order = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y ≠ 0
h₂t' : IsOpen t'
h₃t' : ... | 7349d8844a909f0d |
AlgebraicTopology.DoldKan.Q_succ | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | theorem Q_succ (q : ℕ) : (Q (q + 1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ Q (q + 1) = Q q - P q ≫ Hσ q | simp only [Q, P_succ, comp_add, comp_id] | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ 𝟙 K[X] - (P q + P q ≫ Hσ q) = 𝟙 K[X] - P q - P q ≫ Hσ q | ddb29a49b4bcd503 |
MeasureTheory.lintegral_iSup_ae | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n))
(h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ | case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a :... | simp only [not_forall, not_le, mem_setOf_eq, not_exists, not_lt] at this | case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a :... | 82bab590dae19ea3 |
MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_eLpNorm | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem completeSpace_lp_of_cauchy_complete_eLpNorm [hp : Fact (1 ≤ p)]
(H :
∀ (f : ℕ → α → E) (_ : ∀ n, MemLp (f n) p μ) (B : ℕ → ℝ≥0∞) (_ : ∑' i, B i < ∞)
(_ : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N),
∃ (f_lim : α → E), MemLp f_lim p μ ∧
atTop.Tendsto (fun n => ... | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
∀ (f : ℕ → α → E),
(∀ (n : ℕ), MemLp (f n) p μ) →
∀ (B : ℕ → ℝ≥0∞),
∑' (i : ℕ), B i < ⊤ →
(∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
... | rw [← hB.tsum_eq] | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
∀ (f : ℕ → α → E),
(∀ (n : ℕ), MemLp (f n) p μ) →
∀ (B : ℕ → ℝ≥0∞),
∑' (i : ℕ), B i < ⊤ →
(∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
... | 3439de49b0148c06 |
TensorProduct.gradedMul_assoc | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | theorem gradedMul_assoc (x y z : DirectSum _ 𝒜 ⊗[R] DirectSum _ ℬ) :
gradedMul R 𝒜 ℬ (gradedMul R 𝒜 ℬ x y) z = gradedMul R 𝒜 ℬ x (gradedMul R 𝒜 ℬ y z) | R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : Dir... | let mA := gradedMul R 𝒜 ℬ | R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : Dir... | 6f9ba3157be931f0 |
Vector.toArray_mapM_go | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem toArray_mapM_go [Monad m] [LawfulMonad m] (f : α → m β) (v : Vector α n) (i h r) :
toArray <$> mapM.go f v i h r = Array.mapM.map f v.toArray i r.toArray | case isFalse
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
n : Nat
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
v : Vector α n
i : Nat
h : i ≤ n
r : Vector β i
h✝ : ¬i < n
⊢ toArray <$> pure (Vector.cast ⋯ r) = pure r.toArray | simp | no goals | b8bb1e3d17c268d2 |
Polynomial.le_trailingDegree_monomial | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
letI := Classical.decEq R
if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge
| R : Type u
a : R
n : ℕ
inst✝ : Semiring R
this : DecidableEq R := Classical.decEq R
ha : a = 0
⊢ ↑n ≤ ((monomial n) a).trailingDegree | simp [ha] | no goals | 59c5f44efc726995 |
Ideal.finite_mulSupport | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem finite_mulSupport {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => v.maxPowDividing I).Finite :=
haveI h_subset : {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} ⊆
{v : HeightOneSpectrum R |
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) ≠ 0} | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
v : HeightOneSpectrum R
hv : v ∈ {v | v.maxPowDividing I ≠ 1}
h_zero : ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0
⊢ v.maxPowDividing I = 1 | rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
pow_zero _] | no goals | 9a120bd46701260d |
NumberField.InfinitePlace.isUnramified_mk_iff_forall_isConj | Mathlib/NumberTheory/NumberField/Embeddings.lean | lemma isUnramified_mk_iff_forall_isConj [IsGalois k K] {φ : K →+* ℂ} :
IsUnramified k (mk φ) ↔ ∀ σ : K ≃ₐ[k] K, ComplexEmbedding.IsConj φ σ → σ = 1 | k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ : K →+* ℂ
H : ∀ (σ : K ≃ₐ[k] K), ComplexEmbedding.IsConj φ σ → σ = 1
hφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
t... | simpa using RingHom.congr_fun hφ.2 r | no goals | b5747d3e7e0b9e81 |
ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x | case h.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝... | have (l) : 1 + l ≠ Fintype.card ι := by
rw [add_comm, Fintype.card_eq_zero]; exact Nat.succ_ne_zero _ | case h.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝... | d7dded01cf2a4e1b |
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✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
le : ⟪u - v, w⟫_ℝ ≤ 0
w'' : F := -w + v
this : w'' ∈ K
h₁ : ⟪u - v, w'' - v⟫_ℝ ≤ 0
⊢ ⟪u - v, w⟫_ℝ ≥ 0 | have h₂ : w'' - v = -w := by
simp only [w'', neg_inj, add_neg_cancel_right, sub_eq_add_neg] | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
le : ⟪u - v, w⟫_ℝ ≤ 0
w'' : F := -w + v
this : w'' ∈ K
h₁ : ⟪u - v, w'' - v⟫_ℝ ≤ 0
h₂ : w'' - v = -w
⊢ ⟪u - v, w⟫_ℝ ≥ 0 | cbfd395ad9083d60 |
LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft | Mathlib/LinearAlgebra/SesquilinearForm.lean | theorem IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft [Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁} {v : Basis n R M} (h : B.IsOrthoᵢ v) (hB : B.SeparatingLeft)
(i : n) : ¬B.IsOrtho (v i) (v i) | case intro.h.inr
R : Type u_1
M : Type u_5
M₁ : Type u_6
n : Type u_19
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₁
inst✝¹ : Module R M₁
I I' : R →+* R
inst✝ : Nontrivial R
B : M →ₛₗ[I] M →ₛₗ[I'] M₁
v : Basis n R M
h : B.IsOrthoᵢ ⇑v
hB : B.SeparatingLeft
i : n
ho : B.IsOrtho ... | exact h hij | no goals | 8eec3a006ef84a04 |
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