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 |
|---|---|---|---|---|---|---|
Surreal.dyadicMap_apply_pow | Mathlib/SetTheory/Surreal/Dyadic.lean | theorem dyadicMap_apply_pow (m : ℤ) (n : ℕ) :
dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m (Submonoid.pow 2 n)) =
m • powHalf n | m : ℤ
n : ℕ
⊢ dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m (Submonoid.pow 2 n)) = m • powHalf n | rw [dyadicMap_apply, @Submonoid.log_pow_int_eq_self 2 one_lt_two] | m : ℤ
n : ℕ
⊢ ↑m * powHalf n = m • powHalf n | 31cee7a46f75546d |
OreLocalization.smul'_char | Mathlib/GroupTheory/OreLocalization/Basic.lean | theorem smul'_char (r₁ : R) (r₂ : X) (s₁ s₂ : S) (u : S) (v : R) (huv : u * r₁ = v * s₂) :
OreLocalization.smul' r₁ s₁ r₂ s₂ = v • r₂ /ₒ (u * s₁) | case h.right
R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type u_2
inst✝ : MulAction R X
r₁ : R
r₂ : X
s₁ s₂ u : ↥S
v : R
huv : ↑u * r₁ = v * ↑s₂
v₀ : R := oreNum r₁ s₂
u₀ : ↥S := oreDenom r₁ s₂
h₀ : ↑u₀ * r₁ = v₀ * ↑s₂
r₃ : R
s₃ : ↥S
h₃ : ↑s₃ * ↑u₀ = r₃ * ↑u
this : r₃ * v * ↑s₂ = ↑s₃ * v₀ * ↑s₂... | simp only [mul_assoc] | no goals | 158523f845373f5e |
RootPairing.coroot_eq_coreflection_of_root_eq' | Mathlib/LinearAlgebra/RootSystem/Basic.lean | private lemma coroot_eq_coreflection_of_root_eq' [CharZero R] [NoZeroSMulDivisors R M]
(p : PerfectPairing R M N)
(root : ι ↪ M)
(coroot : ι ↪ N)
(hp : ∀ i, p (root i) (coroot i) = 2)
(hr : ∀ i, MapsTo (preReflection (root i) (p.flip (coroot i))) (range root) (range root))
(hc : ∀ i, MapsTo (pre... | case intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
p : PerfectPairing R M N
root : ι ↪ M
coroot : ι ↪ N
hp : ∀ (i : ι), (p (root i)... | have hkl : (p.flip (coroot l)) (root k) = 2 := by
simp only [hl, preReflection_apply, hk, PerfectPairing.flip_apply_apply, map_sub, hp j,
map_smul, smul_eq_mul, hp i, mul_sub, sα, α, α', β, mul_two, mul_add, LinearMap.sub_apply,
LinearMap.smul_apply]
rw [mul_comm (p (root i) (coroot j))]
abel | case intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
p : PerfectPairing R M N
root : ι ↪ M
coroot : ι ↪ N
hp : ∀ (i : ι), (p (root i)... | 24145ec892545416 |
EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem cospherical_of_two_zsmul_oangle_eq_of_not_collinear {p₁ p₂ p₃ p₄ : P}
(h : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄) (hn : ¬Collinear ℝ ({p₁, p₂, p₄} : Set P)) :
Cospherical ({p₁, p₂, p₃, p₄} : Set P) | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hn : ¬Collinear ℝ {p₁, p₂, p₄}
hn' : ¬Collinear ℝ {p₁, p₃, p₄}
t₁ : Aff... | decide | no goals | 72f963b4dc2afd6e |
hasSum_mellin_pi_mul_sq' | Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean | /-- Tailored version for odd Jacobi theta functions. -/
lemma hasSum_mellin_pi_mul_sq' {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * r i * rexp (-π * r i ^ 2 * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re) :
HasSum (fun i ↦ Gammaℝ (s + 1) *... | ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
⊢ HasSum (fun i => (s + 1).Gammaℝ * a i * ↑(SignType.sign (r i)) / ↑|r i| ^ s) (mellin F ((s + 1) / 2)) | have hs₁ : s ≠ 0 := fun h ↦ lt_irrefl _ (zero_re ▸ h ▸ hs) | ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
hs₁ : s ≠ 0
⊢ HasSum (fun i => (s + 1).Gammaℝ * a i * ↑(SignType.sign (r i)) / ↑|r i| ^ s) (mellin F ((s +... | 22936c8a231e7fbd |
LinearOrderedField.exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self | Mathlib/Algebra/Order/CompleteField.lean | theorem exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self (ha : 0 < a) (b : β)
(hba : b < inducedMap α β a * inducedMap α β a) : ∃ c ∈ cutMap β (a * a), b < c | case inr.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_3
inst✝² : LinearOrderedField α
inst✝¹ : ConditionallyCompleteLinearOrderedField β
inst✝ : Archimedean α
a : α
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * i... | exact mul_self_lt_mul_self (mod_cast hq.le) (hqa'.trans' <| by assumption_mod_cast) | no goals | 26fca4418a681185 |
MeasureTheory.martingale_martingalePart | Mathlib/Probability/Martingale/Centering.lean | theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
[SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ | Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
μ[martingalePart f ℱ... | intro k hk | Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
μ[martingalePart f ℱ... | c70218dd704192f5 |
MvQPF.liftpPreservation_iff_uniform | Mathlib/Data/QPF/Multivariate/Basic.lean | theorem liftpPreservation_iff_uniform : q.LiftPPreservation ↔ q.IsUniform | n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
⊢ LiftPPreservation ↔ IsUniform | rw [← suppPreservation_iff_liftpPreservation, suppPreservation_iff_isUniform] | no goals | a47fa08b6b739c24 |
Rat.AbsoluteValue.apply_le_sum_digits | Mathlib/NumberTheory/Ostrowski.lean | /-- Given any two integers `n`, `m` with `m > 1`, the absolute value of `n` is bounded by
`m + m * f m + m * (f m) ^ 2 + ... + m * (f m) ^ d` where `d` is the number of digits of the
expansion of `n` in base `m`. -/
lemma apply_le_sum_digits (n : ℕ) {m : ℕ} (hm : 1 < m) :
f n ≤ ((Nat.digits m n).mapIdx fun i _ ↦ m ... | f : AbsoluteValue ℚ ℝ
n m : ℕ
hm : 1 < m
L : List ℕ := m.digits n
L' : List ℚ := List.map Nat.cast (List.mapIdx (fun i a => a * m ^ i) L)
hL' : L' = List.map Nat.cast (List.mapIdx (fun i a => a * m ^ i) L)
hcoef : ∀ {c : ℕ}, c ∈ m.digits n → f ↑c < ↑m
⊢ (List.map (⇑f ∘ Nat.cast ∘ fun x => x.1 * m ^ x.2) L.zipIdx).sum ≤... | refine List.sum_le_sum fun ⟨a, i⟩ hia ↦ ?_ | f : AbsoluteValue ℚ ℝ
n m : ℕ
hm : 1 < m
L : List ℕ := m.digits n
L' : List ℚ := List.map Nat.cast (List.mapIdx (fun i a => a * m ^ i) L)
hL' : L' = List.map Nat.cast (List.mapIdx (fun i a => a * m ^ i) L)
hcoef : ∀ {c : ℕ}, c ∈ m.digits n → f ↑c < ↑m
x✝ : ℕ × ℕ
a i : ℕ
hia : (a, i) ∈ L.zipIdx
⊢ (⇑f ∘ Nat.cast ∘ fun x ... | 0745d5149e053bfa |
FirstOrder.Language.Ultraproduct.term_realize_cast | Mathlib/ModelTheory/Ultraproducts.lean | theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
(t.realize fun i => (x i : (u : Filter α).Product M)) =
(fun a => t.realize fun i => x i a : (u : Filter α).Product M) | case h.e'_3.h.e'_3.h.var
α : Type u_1
M : α → Type u_2
u : Ultrafilter α
L : Language
inst✝ : (a : α) → L.Structure (M a)
β : Type u_3
x : β → (a : α) → M a
a : α
a✝ : β
⊢ Term.realize (fun i => x i a) (var a✝) = Term.realize x (var a✝) a | rfl | no goals | 2e4ccc1963d3c1a2 |
norm_sub_le_of_geometric_bound_of_hasSum | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem norm_sub_le_of_geometric_bound_of_hasSum (hr : r < 1) (hf : ∀ n, ‖f n‖ ≤ C * r ^ n) {a : α}
(ha : HasSum f a) (n : ℕ) : ‖(∑ x ∈ Finset.range n, f x) - a‖ ≤ C * r ^ n / (1 - r) | α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r) | apply dist_le_of_le_geometric_of_tendsto r C hr (dist_partial_sum_le_of_le_geometric hf) | case ha
α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop (𝓝 a) | 9f5f5e32dd93ecb8 |
Orientation.map_eq_neg_iff_det_neg | Mathlib/LinearAlgebra/Orientation.lean | theorem map_eq_neg_iff_det_neg (x : Orientation R M ι) (f : M ≃ₗ[R] M)
(h : Fintype.card ι = finrank R M) :
Orientation.map ι f x = -x ↔ LinearMap.det (f : M →ₗ[R] M) < 0 | case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ (map ι f) x = -x ↔ LinearMap.det ↑f < 0 | have H : finrank R M = 0 := h.symm.trans Fintype.card_eq_zero | case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
H : finrank R M = 0
⊢ (map ι f) x = -x ↔ LinearMap.det ↑f < 0 | 4102f0312fa14e8a |
gaugeRescale_gaugeRescale | Mathlib/Analysis/Convex/GaugeRescale.lean | theorem gaugeRescale_gaugeRescale {s t u : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x | case inr
E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
s t u : Set E
hta : Absorbent ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ≠ 0
⊢ gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x | rw [gaugeRescale_def s t x, gaugeRescale_smul, gaugeRescale, gaugeRescale, smul_smul,
div_mul_div_cancel₀] | case inr
E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
s t u : Set E
hta : Absorbent ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ≠ 0
⊢ gauge t x ≠ 0
case inr.hc
E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
s... | 23b7e13d4a7b1b81 |
MeasureTheory.BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | /-- Assuming `levyProkhorovEDist μ ν < ε`, we can bound `∫ f ∂μ` in terms of
`∫ t in (0, ‖f‖], ν (thickening ε {x | f(x) ≥ t}) dt` and `‖f‖`. -/
lemma BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt (μ ν : Measure Ω)
[IsFiniteMeasure μ] [IsFiniteMeasure ν] {ε : ℝ} (ε_pos : 0 < ε)
(hμν : levyProkh... | Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t ... | apply add_le_add_left | case bc
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ... | abad1a50e7cd6063 |
ProbabilityTheory.Kernel.iIndepSets.iIndep | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepSets.iIndep (m : ι → MeasurableSpace Ω)
(h_le : ∀ i, m i ≤ _mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
(h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π κ μ) :
iIndep m κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
m : ι → MeasurableSpace Ω
h_le : ∀ (i : ι), m i ≤ _mΩ
π : ι → Set (Set Ω)
h_pi : ∀ (n : ι), IsPiSystem (π n)
h_generate : ∀ (i : ι), m i = generateFrom (π i)
h_ind : iIndepSets π κ μ
hμ : μ ≠ 0
η : Kernel... | simp [hx] | no goals | 839768f1641edae0 |
Order.wcovBy_add_one | Mathlib/Algebra/Order/SuccPred.lean | theorem wcovBy_add_one (x : α) : x ⩿ x + 1 | α : Type u_1
inst✝³ : Preorder α
inst✝² : Add α
inst✝¹ : One α
inst✝ : SuccAddOrder α
x : α
⊢ x ⩿ succ x | exact wcovBy_succ x | no goals | 538b26d14edeb196 |
Complex.cos_sq | Mathlib/Data/Complex/Trigonometric.lean | theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 | x : ℂ
⊢ cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 | simp [cos_two_mul, div_add_div_same, mul_div_cancel_left₀, two_ne_zero, -one_div] | no goals | 408e5a224394b9fd |
factorPowSucc.isUnit_of_isUnit_image | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | lemma factorPowSucc.isUnit_of_isUnit_image {n : ℕ} (npos : n > 0) {a : R ⧸ I ^ (n + 1)}
(h : IsUnit (factorPow I n.le_succ a)) : IsUnit a | R : Type u_3
inst✝ : CommRing R
I : Ideal R
n : ℕ
npos : n > 0
a : R ⧸ I ^ (n + 1)
h : IsUnit ((factorPow I ⋯) a)
b : R ⧸ I ^ n
right✝ : b * (factorPow I ⋯) a = 1
b' : R ⧸ I ^ n.succ
hb' : (factor ⋯) b' = b
hb : a * b' - 1 ∈ map (mk (I ^ n.succ)) (I ^ n)
c : R
hc : c ∈ ↑(I ^ n)
eq : (mk (I ^ (n + 1))) c = a * b' - 1
⊢ ... | simpa only [← pow_add, Nat.sub_add_cancel npos] using hc | no goals | 2fc8789476860ecf |
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))) | 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 p s x r
hy : y ∈ EMetr... | have A : Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 y) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) := by
apply (hf.tendsto_partialSum hy h'y).comp tendsto_fst | 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 p s x r
hy : y ∈ EMetr... | 84f2046d5fc6e8a5 |
ContinuousMultilinearMap.norm_iteratedFDeriv_le' | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | /-- Controlling the norm of `f.iteratedFDeriv` when `f` is continuous multilinear. For the same
bound on the iterated derivative of `f` in the calculus sense,
see `ContinuousMultilinearMap.norm_iteratedFDeriv_le`. -/
lemma norm_iteratedFDeriv_le' (f : ContinuousMultilinearMap 𝕜 E₁ G) (k : ℕ) (x : (i : ι) → E₁ i) :
... | 𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E₁ G
k : ℕ
x : (i : ι) → E₁ i
⊢... | classical
calc ‖f.iteratedFDeriv k x‖
_ ≤ ∑ e : Fin k ↪ ι, ‖iteratedFDerivComponent f e.toEquivRange (fun i ↦ x i)‖ := norm_sum_le _ _
_ ≤ ∑ _ : Fin k ↪ ι, ‖f‖ * ‖x‖ ^ (Fintype.card ι - k) := by
gcongr with e _
simpa using norm_iteratedFDerivComponent_le f e.toEquivRange x
_ = Nat.descFactorial (Fintype.card ι) k *... | no goals | 63b61560d78f9381 |
MeasureTheory.unifIntegrable_of | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem unifIntegrable_of (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, AEStronglyMeasurable (f i) μ)
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UnifIntegrable f p μ | case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α... | rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem] | case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι →... | b752bffcb51df800 |
LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective | Mathlib/LinearAlgebra/Dual.lean | theorem range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective (f : M →ₗ[R] M')
(hf : Function.Surjective f.range.subtype.dualMap) :
LinearMap.range f.dualMap = f.ker.dualAnnihilator | case h.e'_3
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑(range f).subtype.dualMap
rr_surj : Function.Surjective ⇑f.rangeRestrict
this : range f.rangeRestrict.dualMap = (ker f... | apply congr_arg | case h.e'_3.h
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑(range f).subtype.dualMap
rr_surj : Function.Surjective ⇑f.rangeRestrict
this : range f.rangeRestrict.dualMap = (ker... | fec7dc8cbd988018 |
Lists'.mem_of_subset' | Mathlib/SetTheory/Lists.lean | theorem mem_of_subset' {a} : ∀ {l₁ l₂ : Lists' α true} (_ : l₁ ⊆ l₂) (_ : a ∈ l₁.toList), a ∈ l₂
| nil, _, Lists'.Subset.nil, h => by cases h
| cons' a0 l0, l₂, s, h => by
obtain - | ⟨e, m, s⟩ := s
simp only [toList, Sigma.eta, List.find?, List.mem_cons] at h
rcases h with (rfl | h)
· exact ⟨_, m, e... | α : Type u_1
a : Lists α
b✝ : Bool
a0 : Lists' α b✝
l0 l₂ : Lists' α true
s : a0.cons' l0 ⊆ l₂
h : a ∈ (a0.cons' l0).toList
⊢ a ∈ l₂ | obtain - | ⟨e, m, s⟩ := s | case cons
α : Type u_1
a : Lists α
l0 l₂ : Lists' α true
a✝ a'✝ : Lists α
e : a✝ ~ a'✝
h : a ∈ (a✝.snd.cons' l0).toList
m : a'✝ ∈ l₂.toList
s : l0.Subset l₂
⊢ a ∈ l₂ | 01e260cc3cd68d98 |
SmoothPartitionOfUnity.exists_isSubordinate | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
(hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U | case refine_2.intro.intro
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ ... | exact ⟨f.toSmoothPartitionOfUnity hf, hfU.toSmoothPartitionOfUnity hf⟩ | no goals | bb02907964d5e497 |
Ordinal.log_eq_iff | Mathlib/SetTheory/Ordinal/Exponential.lean | theorem log_eq_iff {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) (y : Ordinal) :
log b x = y ↔ b ^ y ≤ x ∧ x < b ^ succ y | case mpr.intro.a
b x : Ordinal.{u_1}
hb : 1 < b
hx : x ≠ 0
y : Ordinal.{u_1}
hx₁ : b ^ y ≤ x
hx₂ : x < b ^ succ y
⊢ log b x ≤ y | rwa [← lt_succ_iff, ← lt_opow_iff_log_lt hb hx] | no goals | bc76e1c0897bfe2a |
Ordnode.dual_balanceL | Mathlib/Data/Ordmap/Ordset.lean | theorem dual_balanceL (l : Ordnode α) (x : α) (r : Ordnode α) :
dual (balanceL l x r) = balanceR (dual r) x (dual l) | case pos
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ls > delta * rs
⊢ (rec nil
(fun size l x_1 r l_ih r_ih =>
rec nil
(fun size_1 l x_2 r l_ih r_ih =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (... | obtain - | ⟨lls, lll, llx, llr⟩ := ll <;> obtain - | ⟨lrs, lrl, lrx, lrr⟩ := lr <;> try rfl | case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (rec nil
(fun size l x_1 r l_ih r_ih =>
rec nil
(fun size_1 l x_2 r l_ih r_... | 3df822e7a054830f |
AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk | Mathlib/AlgebraicGeometry/Stalk.lean | lemma stalkClosedPointTo_fromSpecStalk (x : X) :
stalkClosedPointTo (X.fromSpecStalk x) =
(X.presheaf.stalkCongr (by rw [fromSpecStalk_closedPoint]; rfl)).hom | X Y : Scheme
f✝ : X ⟶ Y
U V : X.Opens
hU : IsAffineOpen U
hV : IsAffineOpen V
R : CommRingCat
inst✝ : IsLocalRing ↑R
f : Spec R ⟶ X
x : ↑↑X.toPresheafedSpace
⊢ Inseparable x x | rfl | no goals | a51982c434736237 |
MeasureTheory.condExpIndSMul_empty | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | theorem condExpIndSMul_empty {x : G} : condExpIndSMul hm MeasurableSet.empty
((measure_empty (μ := μ)).le.trans_lt ENNReal.coe_lt_top).ne x = 0 | α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
x : G
⊢ (compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condExpL2 ℝ ℝ hm) 0) = 0 | simp only [Submodule.coe_zero, ContinuousLinearMap.map_zero] | no goals | 538016b4c4c1d685 |
Int.natAbs_eq_iff_associated | Mathlib/Data/Int/Associated.lean | theorem Int.natAbs_eq_iff_associated {a b : ℤ} : a.natAbs = b.natAbs ↔ Associated a b | case mp.inr
b : ℤ
⊢ Associated (-b) b | exact ⟨-1, by simp⟩ | no goals | bf707670bc5d3262 |
Polynomial.Monic.eq_one_of_isUnit | Mathlib/Algebra/Polynomial/Monic.lean | theorem Monic.eq_one_of_isUnit (hm : Monic p) (hpu : IsUnit p) : p = 1 | R : Type u
inst✝ : Semiring R
p : R[X]
hm : p.Monic
hpu : IsUnit p
⊢ p = 1 | nontriviality R | R : Type u
inst✝ : Semiring R
p : R[X]
hm : p.Monic
hpu : IsUnit p
a✝ : Nontrivial R
⊢ p = 1 | 61dfc74a6847ac67 |
AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections | Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean | lemma generate_presieveOfSections
{U V : X.AffineZariskiSite} {s : Set Γ(X, U.toOpens)} {f : V ⟶ U} :
Sieve.generate (presieveOfSections U s) f ↔ ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens | case mk.mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f → ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩ | rintro ⟨⟨W, hW⟩, ⟨f₁, hf₁⟩, -, ⟨f₂, hf₂s, rfl⟩, rfl⟩ | case mk.mp.intro.mk.intro.up.up.intro.intro.intro.intro.intro
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f₂ : ↑Γ(X, U.toOpens)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)
hf₁ : X.basicOpen f₁ = toOpens ⟨V, hV⟩
⊢ ∃ f ∈ s, ∃ g,... | f488c05e7e519ddc |
integral_smul_const | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem integral_smul_const {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [CompleteSpace E]
(f : X → 𝕜) (c : E) :
∫ x, f x • c ∂μ = (∫ x, f x ∂μ) • c | case pos
X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
𝕜 : Type u_6
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : X → 𝕜
c : E
hf : ¬Integrable f μ
hc : c = 0
⊢ ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c | simp [hc, integral_zero, smul_zero] | no goals | aac8ef1cc4baf728 |
Ideal.iUnion_minimalPrimes | Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | theorem Ideal.iUnion_minimalPrimes :
⋃ p ∈ I.minimalPrimes, p = { x | ∃ y ∉ I.radical, x * y ∈ I.radical } | case h.mp.intro.intro.intro.intro.intro.intro.mk.intro
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
p : Ideal R
hxp : x ∈ p
hp₃ : ∀ ⦃y : Ideal R⦄, (fun q => q.IsPrime ∧ I ≤ q) y → y ≤ p → p ≤ y
hp₁ : p.IsPrime
hp₂ : I ≤ p
this : map (algebraMap R (Localization.AtPrime p)) p ≤ (map (algebraMap R (Localization.A... | refine ⟨t * b, fun h ↦ (t * b).2 (hp₁.radical_le_iff.mpr hp₂ h), n + 1, ?_⟩ | case h.mp.intro.intro.intro.intro.intro.intro.mk.intro
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
p : Ideal R
hxp : x ∈ p
hp₃ : ∀ ⦃y : Ideal R⦄, (fun q => q.IsPrime ∧ I ≤ q) y → y ≤ p → p ≤ y
hp₁ : p.IsPrime
hp₂ : I ≤ p
this : map (algebraMap R (Localization.AtPrime p)) p ≤ (map (algebraMap R (Localization.A... | c2bb79363c5fb210 |
Polynomial.derivative_X_add_C_pow | Mathlib/Algebra/Polynomial/Derivative.lean | theorem derivative_X_add_C_pow (c : R) (m : ℕ) :
derivative ((X + C c) ^ m) = C (m : R) * (X + C c) ^ (m - 1) | R : Type u
inst✝ : CommSemiring R
c : R
m : ℕ
⊢ derivative ((X + C c) ^ m) = C ↑m * (X + C c) ^ (m - 1) | rw [derivative_pow, derivative_X_add_C, mul_one] | no goals | 5d8c346242b176da |
Padic.exi_rat_seq_conv_cauchy | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
let ⟨N2, hN2⟩ := f.cauchy₂ hε3
exists max N N2
intro j hj
suffices
padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f ... | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε | exists max N N2 | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∀ j ≥ N ⊔ N2, padicNorm p (limSeq f j - limSeq f (N ⊔ N2)) < ε | 5532e11826254be6 |
EuclideanGeometry.OrthocentricSystem.eq_insert_orthocenter | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
{t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
s = insert t.orthocenter (Set.range t.points) | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.... | clear h₂ h₃ | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.... | 727f37cad368951d |
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... | case h
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 : Nor... | filter_upwards with y | case h.h
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 : N... | 2723cd3ae22d6304 |
MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableSpace α}
[TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
(hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
(hf : StronglyMeasurable[m] f) (hf_zero : ∀ x ∉ s, f x = 0) :
StronglyMe... | α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ ... | rw [this] | α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ ... | d81af41568179792 |
Antitone.alternating_series_le_tendsto | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem Antitone.alternating_series_le_tendsto
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfa : Antitone f) (k : ℕ) : ∑ i ∈ range (2 * k), (-1) ^ i * f i ≤ l | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
hm : Monotone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
n : ℕ
⊢ id n ≤ 2 * n | dsimp | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
hm : Monotone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
n : ℕ
⊢ n ≤ 2 * n | 74802fbd069e02ac |
Affine.Simplex.dist_circumcenter_sq_eq_sq_sub_circumradius | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem dist_circumcenter_sq_eq_sq_sub_circumradius {n : ℕ} {r : ℝ} (s : Simplex ℝ P n) {p₁ : P}
(h₁ : ∀ i : Fin (n + 1), dist (s.points i) p₁ = r)
(h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter)
(h : s.points 0 ∈ affineSpan ℝ (Set.range s.points)) :
dist p₁ s.circumcenter * dist p₁ s.circumce... | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
r : ℝ
s : Simplex ℝ P n
p₁ : P
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter
h : s.points 0 ∈ affineSpan ℝ (Set.ran... | simp only [h₁', dist_comm p₁, add_sub_cancel_left, Simplex.dist_circumcenter_eq_circumradius] | no goals | a172e99f2584bdc4 |
CategoryTheory.Limits.inv_prodComparison_map_fst | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | theorem inv_prodComparison_map_fst [IsIso (prodComparison F A B)] :
inv (prodComparison F A B) ≫ F.map prod.fst = prod.fst | C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u₂
inst✝³ : Category.{w, u₂} D
F : C ⥤ D
A B : C
inst✝² : HasBinaryProduct A B
inst✝¹ : HasBinaryProduct (F.obj A) (F.obj B)
inst✝ : IsIso (prodComparison F A B)
⊢ inv (prodComparison F A B) ≫ F.map prod.fst = prod.fst | simp [IsIso.inv_comp_eq] | no goals | 9bf7e403217b9d49 |
ProbabilityTheory.comap_cond | Mathlib/Probability/ConditionalProbability.lean | lemma comap_cond {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ ω ∂μ, ω ∈ range i)
(hs : MeasurableSet s) : comap i μ[|s] = (comap i μ)[|i in s] | case h.hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s) | first
| exact hi.injective
| exact hi.measurableSet_image'
| exact hs
| exact ht
| exact hi.measurable hs
| exact (hi.measurable hs).inter ht | no goals | 4f0b4c478fa397f8 |
Primrec.list_ofFn | Mathlib/Computability/Primrec.lean | theorem list_ofFn :
∀ {n} {f : Fin n → α → σ}, (∀ i, Primrec (f i)) → Primrec fun a => List.ofFn fun i => f i a
| 0, _, _ => by simp only [List.ofFn_zero]; exact const []
| n + 1, f, hf => by
simpa [List.ofFn_succ] using list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ)
| α : Type u_1
σ : Type u_3
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
x✝¹ : Fin 0 → α → σ
x✝ : ∀ (i : Fin 0), Primrec (x✝¹ i)
⊢ Primrec fun a => [] | exact const [] | no goals | 08a4ab3a3904ba4f |
CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | theorem compatible_iff (x : FirstObj P X) : (Arrows.Compatible P π ((Types.productIso _).hom x)) ↔
firstMap P X π x = secondMap P X π x | case mp
C : Type u
inst✝¹ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (Presieve.ofArrows X π).hasPullbacks
x : FirstObj P X
⊢ Arrows.PullbackCompatible P π ((Types.productIso fun i => P.obj (op (X i))).hom x) →
firstMap P X π x = secondMap P X π x | intro t | case mp
C : Type u
inst✝¹ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (Presieve.ofArrows X π).hasPullbacks
x : FirstObj P X
t : Arrows.PullbackCompatible P π ((Types.productIso fun i => P.obj (op (X i))).hom x)
⊢ firstMap P X π x = secondMap P X π x | 2fc6ec72d3282c45 |
Real.strictAnti_rpow_of_base_lt_one | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | lemma strictAnti_rpow_of_base_lt_one {b : ℝ} (hb₀ : 0 < b) (hb₁ : b < 1) :
StrictAnti (b ^ · : ℝ → ℝ) | b : ℝ
hb₀ : 0 < b
hb₁ : b < 1
⊢ StrictAnti fun x => b ^ x | simp_rw [Real.rpow_def_of_pos hb₀] | b : ℝ
hb₀ : 0 < b
hb₁ : b < 1
⊢ StrictAnti fun x => rexp (log b * x) | 7dc5d0774fecfa85 |
Array.size_swapAt! | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem size_swapAt! (a : Array α) (i : Nat) (v : α) :
(a.swapAt! i v).2.size = a.size | α : Type u_1
a : Array α
i : Nat
v : α
⊢ (if h : i < a.size then a.swapAt i v ⋯
else
panicWithPosWithDecl "Init.Data.Array.Basic" "Array.swapAt!" 275 4
("index " ++ toString i ++ " out of bounds")).snd.size =
a.size | split | case isTrue
α : Type u_1
a : Array α
i : Nat
v : α
h✝ : i < a.size
⊢ (a.swapAt i v ⋯).snd.size = a.size
case isFalse
α : Type u_1
a : Array α
i : Nat
v : α
h✝ : ¬i < a.size
⊢ (panicWithPosWithDecl "Init.Data.Array.Basic" "Array.swapAt!" 275 4
("index " ++ toString i ++ " out of bounds")).snd.size =
a.siz... | 281d82ebf6ef2cda |
List.prev_getElem | Mathlib/Data/List/Cycle.lean | theorem prev_getElem (l : List α) (h : Nodup l) (i : Nat) (hi : i < l.length) :
prev l l[i] (get_mem _ _) =
(l[(i + (l.length - 1)) % l.length]'(Nat.mod_lt _ (by omega))) :=
match l with
| [] => by simp at hi
| x::l => by
induction l generalizing i x with
| nil => simp
| cons y l hl =>
... | case cons.succ.succ.hy.a
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
y : α
l : List α
hl :
∀ (i : ℕ) (x : α),
(x :: l).Nodup →
∀ (hi : i < (x :: l).length),
(x :: l).prev (x :: l)[i] ⋯ = (x :: l)[(i + ((x :: l).length - 1)) % (x :: l).length]
x : α
h : Function.Injective (x :: y :: l).get
i : ℕ
h... | rw [← H] | case cons.succ.succ.hy.a
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
y : α
l : List α
hl :
∀ (i : ℕ) (x : α),
(x :: l).Nodup →
∀ (hi : i < (x :: l).length),
(x :: l).prev (x :: l)[i] ⋯ = (x :: l)[(i + ((x :: l).length - 1)) % (x :: l).length]
x : α
h : Function.Injective (x :: y :: l).get
i : ℕ
h... | 9ab0f8d6d9da20dc |
intervalIntegral.integral_unitInterval_deriv_eq_sub | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | /-- A variant of `intervalIntegral.integral_deriv_eq_sub`, the Fundamental theorem
of calculus, involving integrating over the unit interval. -/
lemma integral_unitInterval_deriv_eq_sub [RCLike 𝕜] [NormedSpace 𝕜 E] [IsScalarTower ℝ 𝕜 E]
{f f' : 𝕜 → E} {z₀ z₁ : 𝕜}
(hcont : ContinuousOn (fun t : ℝ ↦ f' (z₀ +... | 𝕜 : Type u_2
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
f f' : 𝕜 → E
z₀ z₁ : 𝕜
hcont : ContinuousOn (fun t => f' (z₀ + t • z₁)) (Icc 0 1)
hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁))... | exact this.const_add z₀ | no goals | 33b334f9c09cf4b0 |
Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measure α)
[IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), close... | α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (Satelli... | intro n | α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (Satelli... | fb6f6078be13e9db |
Complex.norm_exp_eq_iff_re_eq | Mathlib/Data/Complex/Trigonometric.lean | theorem norm_exp_eq_iff_re_eq {x y : ℂ} : ‖exp x‖ = ‖exp y‖ ↔ x.re = y.re | x y : ℂ
⊢ ‖cexp x‖ = ‖cexp y‖ ↔ x.re = y.re | rw [norm_exp, norm_exp, Real.exp_eq_exp] | no goals | 30047c529b2e88ee |
MeasureTheory.ergodicSMul_iterateMulAct | Mathlib/Dynamics/Ergodic/Action/Basic.lean | theorem ergodicSMul_iterateMulAct {f : α → α} (hf : Measurable f) :
ErgodicSMul (IterateMulAct f) α μ ↔ Ergodic f μ | case refine_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → α
hf : Measurable f
x✝ :
MeasurePreserving f μ μ ∧
∀ {s : Set α},
MeasurableSet s → (∀ (g : IterateMulAct f), (fun x => g • x) ⁻¹' s =ᶠ[ae μ] s) → EventuallyConst s (ae μ)
h₁ : MeasurePreserving f μ μ
h₂ :
∀ {s : Set α},
Measurable... | refine h₂ hm fun n ↦ ?_ | case refine_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → α
hf : Measurable f
x✝ :
MeasurePreserving f μ μ ∧
∀ {s : Set α},
MeasurableSet s → (∀ (g : IterateMulAct f), (fun x => g • x) ⁻¹' s =ᶠ[ae μ] s) → EventuallyConst s (ae μ)
h₁ : MeasurePreserving f μ μ
h₂ :
∀ {s : Set α},
Measurable... | 17bba3a79d18d80f |
Asymptotics.isLittleO_norm_pow_id | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem isLittleO_norm_pow_id {n : ℕ} (h : 1 < n) :
(fun x : E' => ‖x‖ ^ n) =o[𝓝 0] fun x => x | E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
n : ℕ
h : 1 < n
⊢ (fun x => ‖x‖ ^ n) =o[𝓝 0] fun x => x | have := @isLittleO_norm_pow_norm_pow E' _ _ _ h | E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
n : ℕ
h : 1 < n
this : (fun x => ‖x‖ ^ n) =o[𝓝 0] fun x => ‖x‖ ^ 1
⊢ (fun x => ‖x‖ ^ n) =o[𝓝 0] fun x => x | 2973b7c8a10e88c9 |
Subgroup.isOpen_of_isClosed_of_finiteIndex | Mathlib/Topology/Algebra/ClosedSubgroup.lean | @[to_additive]
lemma isOpen_of_isClosed_of_finiteIndex (H : Subgroup G) [H.FiniteIndex]
(h : IsClosed (H : Set G)) : IsOpen (H : Set G) | G : Type u
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : ContinuousMul G
H : Subgroup G
inst✝ : H.FiniteIndex
h✝ : IsClosed ↑H
x : G
h : x ∈ (↑H)ᶜ
⊢ 1⁻¹ * x ∉ H | simpa only [inv_one, one_mul, ne_eq] | no goals | 5ef2fd5bf30d07f9 |
AEMeasurable.isLUB | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | theorem AEMeasurable.isLUB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
(hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsLUB { a | ∃ i, f i b = a } (g b)) :
AEMeasurable g μ | case inl
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) ... | simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg | case inl
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) ... | 22b5a03f49a80425 |
Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain | Mathlib/RingTheory/Ideal/GoingUp.lean | theorem exists_ideal_over_prime_of_isIntegral_of_isDomain [Algebra.IsIntegral R S] (P : Ideal R)
[IsPrime P] (hP : RingHom.ker (algebraMap R S) ≤ P) :
∃ Q : Ideal S, IsPrime Q ∧ Q.comap (algebraMap R S) = P | R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P | let Rₚ := Localization P.primeCompl | R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
⊢ ∃ Q, Q.IsPrime ... | ef69a42214a052c2 |
Nat.mem_properDivisors_prime_pow | Mathlib/NumberTheory/Divisors.lean | theorem mem_properDivisors_prime_pow {p : ℕ} (pp : p.Prime) (k : ℕ) {x : ℕ} :
x ∈ properDivisors (p ^ k) ↔ ∃ (j : ℕ) (_ : j < k), x = p ^ j | case h
p : ℕ
pp : Prime p
k x : ℕ
⊢ ∀ (a : ℕ), a ≤ k ∧ x = p ^ a ∧ x < p ^ k ↔ a < k ∧ x = p ^ a | intro a | case h
p : ℕ
pp : Prime p
k x a : ℕ
⊢ a ≤ k ∧ x = p ^ a ∧ x < p ^ k ↔ a < k ∧ x = p ^ a | b547ebe2f9034e32 |
MvPolynomial.eval_indicator_apply_eq_one | Mathlib/FieldTheory/Finite/Polynomial.lean | theorem eval_indicator_apply_eq_one (a : σ → K) : eval a (indicator a) = 1 | K : Type u_1
σ : Type u_2
inst✝² : Fintype K
inst✝¹ : Fintype σ
inst✝ : CommRing K
a : σ → K
a✝ : Nontrivial K
this : 0 < Fintype.card K - 1
⊢ (eval a) (indicator a) = 1 | simp only [indicator, map_prod, map_sub, map_one, map_pow, eval_X, eval_C, sub_self,
zero_pow this.ne', sub_zero, Finset.prod_const_one] | no goals | 2a99aadb9c32fee5 |
Convex.isLittleO_alternate_sum_square | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
(h4w : x + (4 : ℝ) • w ∈ interior s) :
(fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
- f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
fun h => h ^ 2 | E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior... | convert TA1.sub TA2 using 1 | case h.e'_7
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f... | 1b1627b2cea2c23c |
MeasureTheory.AddContent.measure_eq | Mathlib/MeasureTheory/OuterMeasure/OfAddContent.lean | theorem measure_eq [mα : MeasurableSpace α] (m : AddContent C) (hC : IsSetSemiring C)
(hC_gen : mα = MeasurableSpace.generateFrom C) (m_sigma_subadd : m.IsSigmaSubadditive)
(hs : s ∈ C) :
m.measure hC hC_gen.le m_sigma_subadd s = m s | α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ (m.measure hC ⋯ m_sigma_subadd) s = m s | rw [measure, trim_measurableSet_eq] | α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ (m.measureCaratheodory hC m_sigma_subadd) s = m s
case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m :... | 0f2be7b641ace502 |
MeasureTheory.levyProkhorovEDist_triangle | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | lemma levyProkhorovEDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure Ω) :
levyProkhorovEDist μ κ ≤ levyProkhorovEDist μ ν + levyProkhorovEDist ν κ | Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ⊤
LPνκ_finite : ¬levyProkhorovEDist ν κ = ⊤
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
half_ε_lt_top : ε /... | rw [← hs_add_r, ← ENNReal.toReal_add] | case ha
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ⊤
LPνκ_finite : ¬levyProkhorovEDist ν κ = ⊤
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
half_ε_lt_t... | afd23a58f0bdc27d |
ConcaveOn.smul' | Mathlib/Analysis/Convex/Mul.lean | lemma ConcaveOn.smul' [OrderedSMul 𝕜 E] (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g)
(hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) :
ConcaveOn 𝕜 s (f • g) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : LinearOrderedCommRing 𝕜
inst✝⁹ : LinearOrderedCommRing E
inst✝⁸ : LinearOrderedAddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : OrderedSMul 𝕜 F
inst✝¹ : OrderedSMul E F
s ... | simp only [mul_add, add_smul, smul_add] | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : LinearOrderedCommRing 𝕜
inst✝⁹ : LinearOrderedCommRing E
inst✝⁸ : LinearOrderedAddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : OrderedSMul 𝕜 F
inst✝¹ : OrderedSMul E F
s ... | 4d4df82f123ffdd1 |
Holor.cprankMax_sum | Mathlib/Data/Holor.lean | theorem cprankMax_sum [Ring α] {β} {n : ℕ} (s : Finset β) (f : β → Holor α ds) :
(∀ x ∈ s, CPRankMax n (f x)) → CPRankMax (s.card * n) (∑ x ∈ s, f x) :=
letI := Classical.decEq β
Finset.induction_on s (by simp [CPRankMax.zero])
(by
intro x s (h_x_notin_s : x ∉ s) ih h_cprank
simp only [Finset.su... | α : Type
ds : List ℕ
inst✝ : Ring α
β : Type u_1
n : ℕ
s✝ : Finset β
f : β → Holor α ds
this : DecidableEq β := Classical.decEq β
x : β
s : Finset β
h_x_notin_s : x ∉ s
ih : (∀ x ∈ s, CPRankMax n (f x)) → CPRankMax (s.card * n) (∑ x ∈ s, f x)
h_cprank : ∀ x_1 ∈ insert x s, CPRankMax n (f x_1)
⊢ CPRankMax (s.card * n + ... | simp only [Nat.one_mul, Nat.add_comm] | α : Type
ds : List ℕ
inst✝ : Ring α
β : Type u_1
n : ℕ
s✝ : Finset β
f : β → Holor α ds
this : DecidableEq β := Classical.decEq β
x : β
s : Finset β
h_x_notin_s : x ∉ s
ih : (∀ x ∈ s, CPRankMax n (f x)) → CPRankMax (s.card * n) (∑ x ∈ s, f x)
h_cprank : ∀ x_1 ∈ insert x s, CPRankMax n (f x_1)
⊢ CPRankMax (n + s.card * ... | 0c565ca3da967211 |
IsUpperSet.div_left | Mathlib/Algebra/Order/UpperLower.lean | theorem IsUpperSet.div_left (ht : IsUpperSet t) : IsLowerSet (s / t) | α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
ht : IsUpperSet t
⊢ IsLowerSet (s * t⁻¹) | exact ht.inv.mul_left | no goals | 843eaf181c6d2c88 |
Real.arctan_inv_of_pos | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | theorem arctan_inv_of_pos {x : ℝ} (h : 0 < x) : arctan x⁻¹ = π / 2 - arctan x | x : ℝ
h : 0 < x
⊢ arctan x⁻¹ = π / 2 - arctan x | rw [← arctan_tan (x := _ - _), tan_pi_div_two_sub, tan_arctan] | case hx₁
x : ℝ
h : 0 < x
⊢ -(π / 2) < π / 2 - arctan x
case hx₂
x : ℝ
h : 0 < x
⊢ π / 2 - arctan x < π / 2 | fa2a609837cbeb83 |
Module.End.independent_iInf_maxGenEigenspace_of_forall_mapsTo | Mathlib/LinearAlgebra/Eigenspace/Pi.lean | lemma independent_iInf_maxGenEigenspace_of_forall_mapsTo
(h : ∀ i j φ, MapsTo (f i) ((f j).maxGenEigenspace φ) ((f j).maxGenEigenspace φ)) :
iSupIndep fun χ : ι → R ↦ ⨅ i, (f i).maxGenEigenspace (χ i) | case empty
ι : Type u_1
R : Type u_2
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
f : ι → End R M
inst✝ : NoZeroSMulDivisors R M
h : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))
χ₁ : ι → R
⊢ χ₁ ∉ ∅ → Disjoint (⨅ i, (... | simp | no goals | 4dd72b81f13de433 |
MvPolynomial.aeval_sumElim_pderiv_inl | Mathlib/Algebra/MvPolynomial/PDeriv.lean | lemma aeval_sumElim_pderiv_inl {S τ : Type*} [CommRing S] [Algebra R S]
(p : MvPolynomial (σ ⊕ τ) R) (f : τ → S) (j : σ) :
aeval (Sum.elim X (C ∘ f)) ((pderiv (Sum.inl j)) p) =
(pderiv j) ((aeval (Sum.elim X (C ∘ f))) p) | case h_X
R : Type u
σ : Type v
inst✝² : CommSemiring R
S : Type u_1
τ : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : τ → S
j : σ
p : MvPolynomial (σ ⊕ τ) R
q : σ ⊕ τ
h : (aeval (Sum.elim X (⇑C ∘ f))) ((pderiv (Sum.inl j)) p) = (pderiv j) ((aeval (Sum.elim X (⇑C ∘ f))) p)
⊢ (aeval (Sum.elim X (⇑C ∘ f))) p * (aev... | cases q <;> simp [Pi.single_apply] | no goals | cea34ff12d9793df |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Operations/Udiv.lean | theorem go_decl_eq (aig : AIG α) (curr : Nat) (falseRef trueRef : AIG.Ref aig)
(n d : AIG.RefVec aig w) (wn wr : Nat) (q r : AIG.RefVec aig w) :
∀ (idx : Nat) (h1) (h2),
(go aig curr falseRef trueRef n d wn wr q r).aig.decls[idx]'h2 = aig.decls[idx]'h1 | case h_1
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
res : BlastUdivOutput aig w
curr✝ : Nat
hgo : { aig := aig, q := q, r := r, hle := ⋯ } = res
⊢ ∀ (idx : Nat) (h1 : idx < aig.decls.size) (h2 : idx < res.aig.decls.... | simp [← hgo] | no goals | 5789cda59f77ea8b |
List.fst_eq_of_mem_zipIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem fst_eq_of_mem_zipIdx {x : α × Nat} {l : List α} {k : Nat} (h : x ∈ zipIdx l k) :
x.1 = l[x.2 - k]'(by have := le_snd_of_mem_zipIdx h; have := snd_lt_add_of_mem_zipIdx h; omega) | case cons.tail
α : Type u_1
x : α × Nat
hd : α
tl : List α
k : Nat
m : Mem x (tl.zipIdx (k + 1))
ih : x.fst = tl[x.snd - (k + 1)]
this : x.snd - k = x.snd - (k + 1) + 1
⊢ x.fst = (hd :: tl)[x.snd - k] | simp [this, ih] | no goals | 6b68bcf3cd90098c |
Nat.pow_pow_add_primeFactors_one_lt | Mathlib/NumberTheory/Fermat.lean | /-- Prime factors of `a ^ (2 ^ n) + 1` are of form `k * 2 ^ (n + 1) + 1`. -/
lemma pow_pow_add_primeFactors_one_lt {a n p : ℕ} (hp : p.Prime) (hp2 : p ≠ 2)
(hpdvd : p ∣ a ^ (2 ^ n) + 1) :
∃ k, p = k * 2 ^ (n + 1) + 1 | a n p : ℕ
hp : Prime p
hp2 : p ≠ 2
hpdvd : p ∣ a ^ 2 ^ n + 1
this✝ : Fact (2 < p)
this : Fact (Prime p)
ha1 : ↑a ^ 2 ^ n = -1
⊢ ∃ k, p = k * 2 ^ (n + 1) + 1 | have ha0 : (a : ZMod p) ≠ 0 := by
intro h
rw [h, zero_pow (pow_ne_zero n two_ne_zero), zero_eq_neg] at ha1
exact one_ne_zero ha1 | a n p : ℕ
hp : Prime p
hp2 : p ≠ 2
hpdvd : p ∣ a ^ 2 ^ n + 1
this✝ : Fact (2 < p)
this : Fact (Prime p)
ha1 : ↑a ^ 2 ^ n = -1
ha0 : ↑a ≠ 0
⊢ ∃ k, p = k * 2 ^ (n + 1) + 1 | 6ed2a2d9400f8630 |
IsPrimitiveRoot.toInteger_sub_one_dvd_prime | Mathlib/NumberTheory/Cyclotomic/Rat.lean | /-- In a `p ^ (k + 1)`-th cyclotomic extension of `ℚ`, we have that
`ζ - 1` divides `p` in `𝓞 K`. -/
lemma toInteger_sub_one_dvd_prime [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) : ((hζ.toInteger - 1)) ∣ p | p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
htwo : ↑p ^ (k + 1) = 2
hp2 : ↑p = 2
hk : k = 0
hζ' : ζ = -1
⊢ hζ.toInteger = -1 | ext | case h
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
htwo : ↑p ^ (k + 1) = 2
hp2 : ↑p = 2
hk : k = 0
hζ' : ζ = -1
⊢ ↑hζ.toInteger = ↑(-1) | 7b3f289dfd402da6 |
CategoryTheory.Limits.colimitLimitToLimitColimit_surjective | Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean | theorem colimitLimitToLimitColimit_surjective :
Function.Surjective (colimitLimitToLimitColimit F) | case intro.intro.h.w
J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e : ∀ (j : J), colimit.ι ((curry.obj F).obj j) (k j) (y j) = l... | simp only [id, ← e, Limits.ι_colimitLimitToLimitColimit_π_apply,
colimit_eq_iff, Bifunctor.map_id_comp, types_comp_apply, curry_obj_obj_map,
Functor.comp_obj, colim_obj, Limit.π_mk] | case intro.intro.h.w
J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e : ∀ (j : J), colimit.ι ((curry.obj F).obj j) (k j) (y j) = l... | 1be585fb623fc3d9 |
ClassGroup.equiv_mk0 | Mathlib/RingTheory/ClassGroup.lean | theorem ClassGroup.equiv_mk0 [IsDedekindDomain R] (I : (Ideal R)⁰) :
ClassGroup.equiv K (ClassGroup.mk0 I) =
QuotientGroup.mk' (toPrincipalIdeal R K).range (FractionalIdeal.mk0 K I) | R : Type u_1
K : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : ↥(Ideal R)⁰
⊢ (QuotientGroup.mk' (toPrincipalIdeal R K).range)
((Units.mapEquiv ↑(canonicalEquiv R⁰ (FractionRing R) K)) ((FractionalIdeal.mk0 (Fractio... | congr 1 | case h.e_6.h
R : Type u_1
K : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : ↥(Ideal R)⁰
⊢ (Units.mapEquiv ↑(canonicalEquiv R⁰ (FractionRing R) K)) ((FractionalIdeal.mk0 (FractionRing R)) I) =
(FractionalIdeal.mk0 K) ... | 7d59bf1c821655a1 |
derivWithin_smul_const | Mathlib/Analysis/Calculus/Deriv/Mul.lean | theorem derivWithin_smul_const (hc : DifferentiableWithinAt 𝕜 c s x) (f : F) :
derivWithin (fun y => c y • f) s x = derivWithin c s x • f | 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s... | rcases uniqueDiffWithinAt_or_nhdsWithin_eq_bot s x with hxs | hxs | case inl
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithin... | 3895cc9acf01c4d6 |
NNRat.num_div_den | Mathlib/Algebra/Field/Rat.lean | @[simp]
lemma num_div_den (q : ℚ≥0) : (q.num : ℚ≥0) / q.den = q | case a
q : ℚ≥0
⊢ ↑(↑q.num / ↑q.den) = ↑q | rw [coe_div, coe_natCast, coe_natCast, num, ← Int.cast_natCast] | case a
q : ℚ≥0
⊢ ↑↑(↑q).num.natAbs / ↑q.den = ↑q | e61d31e94aa6be48 |
Profinite.exists_locallyConstant_finite_aux | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom | case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyCon... | clear this | case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyCon... | ad55d9835f0fad2e |
Complex.cos_eq_zero_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | theorem cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2 | rw [cos, h, ← exp_pi_mul_I, exp_eq_exp_iff_exists_int, mul_right_comm] | θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
⊢ (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2 | ea1f630e1fe1d549 |
Stream'.Seq.of_mem_append | Mathlib/Data/Seq/Seq.lean | theorem of_mem_append {s₁ s₂ : Seq α} {a : α} (h : a ∈ append s₁ s₂) : a ∈ s₁ ∨ a ∈ s₂ | case cons.inl
α : Type u
s₂ : Seq α
a : α
ss : Seq α
h : a ∈ ss
b : α
s' : Seq α
o : a = b ∨ ∀ {s₁ : Seq α}, a ∈ s₁.append s₂ → s₁.append s₂ = s' → a ∈ s₁ ∨ a ∈ s₂
c : α
t₁ : Seq α
m : a ∈ (cons c t₁).append s₂
e : (cons c t₁).append s₂ = cons b s'
this : ((cons c t₁).append s₂).destruct = (cons b s').destruct
e' : a =... | exact Or.inl (mem_cons _ _) | no goals | 616ea8064ee30260 |
hnot_hnot_sdiff_distrib | Mathlib/Order/Heyting/Basic.lean | theorem hnot_hnot_sdiff_distrib (a b : α) : ￢￢(a \ b) = ￢￢a \ ￢￢b | case a
α : Type u_2
inst✝ : CoheytingAlgebra α
a b : α
⊢ ￢￢(a \ b) ≤ ￢￢a \ ￢￢b | refine hnot_le_comm.1 ((hnot_anti sdiff_le_inf_hnot).trans' ?_) | case a
α : Type u_2
inst✝ : CoheytingAlgebra α
a b : α
⊢ ￢(￢￢a \ ￢￢b) ≤ ￢(a ⊓ ￢b) | e5490f530f49ead6 |
Array.cons_lex_cons | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lex/Lemmas.lean | theorem cons_lex_cons [BEq α] {lt : α → α → Bool} {a b : α} {xs ys : Array α} :
(#[a] ++ xs).lex (#[b] ++ ys) lt =
(lt a b || a == b && xs.lex ys lt) | case true
α : Type u_1
inst✝ : BEq α
lt : α → α → Bool
a b : α
xs ys : Array α
⊢ (match
(match
if true = true then ForInStep.done ⟨some true, PUnit.unit⟩
else
if (a != b) = true then ForInStep.done ⟨some false, PUnit.unit⟩ else ForInStep.yield ⟨none, PUnit.unit⟩ with
| ForI... | simp | no goals | af913f9ccd5e7085 |
DedekindDomain.ProdAdicCompletions.IsFiniteAdele.mul | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | theorem mul {x y : K_hat R K} (hx : x.IsFiniteAdele) (hy : y.IsFiniteAdele) :
(x * y).IsFiniteAdele | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : x.IsFiniteAdele
hy : y.IsFiniteAdele
⊢ (x * y).IsFiniteAdele | rw [IsFiniteAdele, Filter.eventually_cofinite] at hx hy ⊢ | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : {x_1 | x x_1 ∉ adicCompletionIntegers K x_1}.Finite
hy : {x | y x ∉ adicCompletionIntegers K x}.Finite
⊢ {x_1 | (x * y) x_1 ∉ adicCompletionIntegers K x_1}.Fini... | 9956a2ee34d7ebb6 |
exists_associated_pow_of_mul_eq_pow | Mathlib/Algebra/GCDMonoid/Basic.lean | theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
{k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d... | apply (mul_right_inj' h0₂).mp | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d... | 4887d5e0abe68a27 |
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite | Mathlib/MeasureTheory/Measure/SeparableMeasure.lean | theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
(h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ ... | exact toReal_mono (ne_of_lt <| sum_lt_top.2 fun _ _ ↦ measure_lt_top μ _)
(measure_biUnion_finset_le _ _) | no goals | 92d6fe7992cc346b |
exists_spanning_measurableSet_le | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | theorem exists_spanning_measurableSet_le {f : α → ℝ≥0} (hf : Measurable f) (μ : Measure α)
[SigmaFinite μ] :
∃ s : ℕ → Set α,
(∀ n, MeasurableSet (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, f x ≤ n) ∧
⋃ i, s i = Set.univ | α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n => {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n => sigma_finite_sets n ∩ norm_sets n
h_meas : ∀ (n... | refine fun hif => hif.trans ?_ | α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n => {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n => sigma_finite_sets n ∩ norm_sets n
h_meas : ∀ (n... | 447eb51f3f11a510 |
MeasureTheory.MemLp.prod | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | /-- See `MemLp.prod'` for the applied version. -/
protected lemma MemLp.prod (hf : ∀ i ∈ s, MemLp (f i) (p i) μ) :
MemLp (∏ i ∈ s, f i) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ | ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ s, MemLp (f i) (p i) μ
⊢ MemLp (∏ i ∈ s, f i) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ | induction s using cons_induction with
| empty =>
by_cases hμ : μ = 0 <;>
simp [MemLp, eLpNormEssSup_const, hμ, aestronglyMeasurable_const, Pi.one_def]
| cons i s hi ih =>
rw [prod_cons]
exact (ih <| forall_of_forall_cons hf).mul (hf i <| mem_cons_self ..) (hpqr := ⟨by simp⟩) | no goals | f26d6d3f2cdcf189 |
Nat.pow_lt_pow_iff_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem pow_lt_pow_iff_right {a n m : Nat} (h : 1 < a) :
a ^ n < a ^ m ↔ n < m | case mp.a
a n m : Nat
h : 1 < a
⊢ ¬(a ^ n < a ^ m → n < m) → False | intros w | case mp.a
a n m : Nat
h : 1 < a
w : ¬(a ^ n < a ^ m → n < m)
⊢ False | e0769ff7083a3d17 |
Stream'.WSeq.LiftRel.trans | Mathlib/Data/Seq/WSeq.lean | theorem LiftRel.trans (R : α → α → Prop) (H : Transitive R) : Transitive (LiftRel R) :=
fun s t u h1 h2 => by
refine ⟨fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u, ⟨t, h1, h2⟩, fun {s u} h => ?_⟩
rcases h with ⟨t, h1, h2⟩
have h1 := liftRel_destruct h1
have h2 := liftRel_destruct h2
refine
Computation.... | case intro.intro.intro.intro.some.some.mk.none
α : Type u
R : α → α → Prop
H : Transitive R
s✝¹ t✝ u✝ : WSeq α
h1✝¹ : LiftRel R s✝¹ t✝
h2✝¹ : LiftRel R t✝ u✝
s✝ u t : WSeq α
h1✝ : LiftRel R s✝ t
h2✝ : LiftRel R t u
h1 : Computation.LiftRel (LiftRelO R (LiftRel R)) s✝.destruct t.destruct
h2 : Computation.LiftRel (LiftRe... | cases t1 | no goals | edcc30ddce7afaad |
ArithmeticFunction.vonMangoldt_mul_zeta | Mathlib/NumberTheory/VonMangoldt.lean | theorem vonMangoldt_mul_zeta : Λ * ζ = log | ⊢ Λ * ↑ζ = log | ext n | case h
n : ℕ
⊢ (Λ * ↑ζ) n = log n | 0cbf011d894da1b2 |
Mathlib.Tactic.Ring.add_pf_add_overlap_zero | Mathlib/Tactic/Ring/Basic.lean | theorem add_pf_add_overlap_zero
(h : IsNat (a₁ + b₁) (nat_lit 0)) (h₄ : a₂ + b₂ = c) : (a₁ + a₂ : R) + (b₁ + b₂) = c | R : Type u_1
inst✝ : CommSemiring R
a₁ a₂ b₁ b₂ c : R
h : IsNat (a₁ + b₁) 0
h₄ : a₂ + b₂ = c
⊢ a₁ + a₂ + (b₁ + b₂) = c | subst_vars | R : Type u_1
inst✝ : CommSemiring R
a₁ a₂ b₁ b₂ : R
h : IsNat (a₁ + b₁) 0
⊢ a₁ + a₂ + (b₁ + b₂) = a₂ + b₂ | eb5a9b3617dce6da |
IsGreatest.nnnorm_cfcₙ_nnreal | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean | lemma IsGreatest.nnnorm_cfcₙ_nnreal (f : ℝ≥0 → ℝ≥0) (a : A)
(hf : ContinuousOn f (σₙ ℝ≥0 a) | A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a ... | exact continuous_subtype_val.comp_continuousOn <|
ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image .. | no goals | ef206d17d4f97fdb |
FDRep.char_one | Mathlib/RepresentationTheory/Character.lean | theorem char_one (V : FDRep k G) : V.character 1 = Module.finrank k V | k : Type u
inst✝¹ : Field k
G : Type u
inst✝ : Monoid G
V : FDRep k G
⊢ V.character 1 = ↑(finrank k ↑V.V) | simp only [character, map_one, trace_one] | no goals | 2ebbc1a121f1e3f7 |
CompHaus.epi_iff_surjective | Mathlib/Topology/Category/CompHaus/Basic.lean | theorem epi_iff_surjective {X Y : CompHaus.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f | case mp.intro
X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
D : Set ↑Y.toTop := {y}
hD : IsClosed D
⊢ False | have hCD : Disjoint C D := by
rw [Set.disjoint_singleton_right]
rintro ⟨y', hy'⟩
exact hy y' hy' | case mp.intro
X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
D : Set ↑Y.toTop := {y}
hD : IsClosed D
hCD : Disjoint C D
⊢ False | ff981d5c42859c8e |
Algebra.Norm.Transitivity.det_det_aux | Mathlib/RingTheory/Norm/Transitivity.lean | lemma det_det_aux
(ih : ∀ M, (f (det M)).det = ((M.map f).comp {a // (a = k) = False} _ n n R).det) :
((f M.det).det - ((M.map f).comp m m n n R).det) *
(f (M k k)).det ^ (Fintype.card m - 1) = 0 | R : Type u_1
S : Type u_2
n : Type u_4
m : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Matrix m m S
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
k : m
inst✝¹ : Fintype m
inst✝ : Fintype n
f : S →+* Matrix n n R
ih :
∀ (M : Matrix { a // (a = k) = False } { a // (a = k) = False } S),
(f M.det).det = ((Ma... | rw [sub_mul, comp_det_mul_pow, ← det_pow, ← map_pow, ← det_mul, ← _root_.map_mul,
det_mul_corner_pow, _root_.map_mul, det_mul, ih, sub_self] | no goals | 73bc4885c91c9ed7 |
norm_add_lt_of_not_sameRay | Mathlib/Analysis/Convex/StrictConvexSpace.lean | theorem norm_add_lt_of_not_sameRay (h : ¬SameRay ℝ x y) : ‖x + y‖ < ‖x‖ + ‖y‖ | case intro.intro
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
hx : 0 < ‖x‖
hy : 0 < ‖y‖
hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y
hxy : 0 < ‖x‖ + ‖y‖
⊢ ‖x + y‖ < ‖x‖ + ‖y‖ | have :=
combo_mem_ball_of_ne (inv_norm_smul_mem_unitClosedBall x)
(inv_norm_smul_mem_unitClosedBall y) hne (div_pos hx hxy) (div_pos hy hxy)
(by rw [← add_div, div_self hxy.ne']) | case intro.intro
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
hx : 0 < ‖x‖
hy : 0 < ‖y‖
hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y
hxy : 0 < ‖x‖ + ‖y‖
this : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1
⊢ ‖x + y‖ < ‖x‖ + ‖y‖ | 98b452b77c170acf |
Matrix.det_ne_zero_of_sum_col_pos | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | /-- A matrix whose nondiagonal entries are negative with the sum of the entries of each
column positive has nonzero determinant. -/
lemma det_ne_zero_of_sum_col_pos [DecidableEq n] {S : Type*} [LinearOrderedCommRing S]
{A : Matrix n n S} (h1 : Pairwise fun i j => A i j < 0) (h2 : ∀ j, 0 < ∑ i, A i j) :
A.det ≠ ... | case intro.intro
n✝ : Type u_1
inst✝³ : Fintype n✝
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
h_sup : 0 < Finset.univ.sup' ⋯ v
j₀ : n
h_j₀... | refine ⟨j₀, ?_⟩ | case intro.intro
n✝ : Type u_1
inst✝³ : Fintype n✝
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
h_sup : 0 < Finset.univ.sup' ⋯ v
j₀ : n
h_j₀... | 3754ab49091301b2 |
Submodule.existsUnique_from_graph | Mathlib/LinearAlgebra/LinearPMap.lean | theorem existsUnique_from_graph {g : Submodule R (E × F)}
(hg : ∀ {x : E × F} (_hx : x ∈ g) (_hx' : x.fst = 0), x.snd = 0) {a : E}
(ha : a ∈ g.map (LinearMap.fst R E F)) : ∃! b : F, (a, b) ∈ g | case a
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
g : Submodule R (E × F)
hg : ∀ {x : E × F}, x ∈ g → x.1 = 0 → x.2 = 0
a : E
ha : a ∈ map (LinearMap.fst R E F) g
⊢ (∃ x, (a, x) ∈ g) ↔ a ∈ map (LinearMap.fst R E F) g | simp | no goals | bd5e4ac4a1e6692d |
QuadraticMap.posDef_pi_iff | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | theorem posDef_pi_iff {P} [Fintype ι] [OrderedAddCommMonoid P] [Module R P]
{Q : ∀ i, QuadraticMap R (Mᵢ i) P} : (pi Q).PosDef ↔ ∀ i, (Q i).PosDef | case mp.intro
ι : Type u_1
R : Type u_2
Mᵢ : ι → Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝³ : (i : ι) → Module R (Mᵢ i)
P : Type u_10
inst✝² : Fintype ι
inst✝¹ : OrderedAddCommMonoid P
inst✝ : Module R P
Q : (i : ι) → QuadraticMap R (Mᵢ i) P
hle : ∀ (i : ι) (x : Mᵢ i), 0 ≤ (Q i) x
h... | exact ⟨hle i, anisotropic_of_pi ha i⟩ | no goals | 77c7beb36c84b56a |
RightDerivMeasurableAux.A_mem_nhdsGT | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem A_mem_nhdsGT {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) : A f L r ε ∈ 𝓝[>] x | case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : I... | intro y hy z hz | case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : I... | dbdd2fc84afd393e |
Finset.sum_card_slice_div_choose_le_one | Mathlib/Combinatorics/SetFamily/LYM.lean | theorem sum_card_slice_div_choose_le_one [Fintype α]
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
(∑ r ∈ range (Fintype.card α + 1), (#(𝒜 # r) : 𝕜) / (Fintype.card α).choose r) ≤ 1 | 𝕜 : Type u_1
α : Type u_2
inst✝¹ : LinearOrderedField 𝕜
𝒜 : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsAntichain (fun x1 x2 => x1 ⊆ x2) ↑𝒜
⊢ 0 < (Fintype.card α).choose (Fintype.card α - Fintype.card α) | rw [tsub_self, choose_zero_right] | 𝕜 : Type u_1
α : Type u_2
inst✝¹ : LinearOrderedField 𝕜
𝒜 : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsAntichain (fun x1 x2 => x1 ⊆ x2) ↑𝒜
⊢ 0 < 1 | 7bf2cd66ee834671 |
Asymptotics.isTheta_const_mul_right | Mathlib/Analysis/Asymptotics/Theta.lean | theorem isTheta_const_mul_right {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0) :
(f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g | α : Type u_1
E : Type u_3
𝕜 : Type u_14
inst✝¹ : Norm E
inst✝ : NormedField 𝕜
f : α → E
l : Filter α
c : 𝕜
g : α → 𝕜
hc : c ≠ 0
⊢ (f =Θ[l] fun x => c * g x) ↔ f =Θ[l] g | simpa only [← smul_eq_mul] using isTheta_const_smul_right hc | no goals | 88557470b82a8b7f |
CategoryTheory.isTriangulated_of_essSurj_mapComposableArrows_two | Mathlib/CategoryTheory/Triangulated/Functor.lean | /-- If `F : C ⥤ D` is a triangulated functor from a triangulated category, then `D`
is also triangulated if tuples of composables arrows in `D` can be lifted to `C`. -/
lemma isTriangulated_of_essSurj_mapComposableArrows_two
(F : C ⥤ D) [F.CommShift ℤ] [F.IsTriangulated]
[(F.mapComposableArrows 2).EssSurj] [IsT... | case octahedron_axiom.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{u_3, u_1} C
inst✝¹⁴ : Category.{u_4, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst... | obtain ⟨_, _, _, h₂₃'⟩ := distinguished_cocone_triangle g | case octahedron_axiom.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{u_3, u_1} C
inst✝¹⁴ : Category.{u_4, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : ... | b514da98fd8dcf0f |
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