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 |
|---|---|---|---|---|---|---|
FiniteDimensional.of_locallyCompact_manifold | Mathlib/Geometry/Manifold/IsManifold/ExtChartAt.lean | theorem FiniteDimensional.of_locallyCompact_manifold
[CompleteSpace 𝕜] (I : ModelWithCorners 𝕜 E H) [Nonempty M] [LocallyCompactSpace M] :
FiniteDimensional 𝕜 E | E : Type u_8
𝕜 : Type u_9
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_10
inst✝⁵ : TopologicalSpace H
M : Type u_11
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : CompleteSpace 𝕜
I : ModelWithCorners 𝕜 E H
inst✝¹ : Nonempty M
inst✝ : LocallyCo... | exact FiniteDimensional.of_locallyCompactSpace 𝕜 | no goals | 2ad87ba12d29b468 |
MeasureTheory.setIntegral_tilted' | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma setIntegral_tilted' (f : α → ℝ) (g : α → E) {s : Set α} (hs : MeasurableSet s) :
∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ | case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ
g : α → E
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x => rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / 0) • g x ∂μ | simp | no goals | 026f577fdf7186b7 |
ContinuousLinearMap.prod_ext_iff | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | theorem prod_ext_iff {f g : M × M₂ →L[R] M₃} :
f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) | R : Type u_1
inst✝⁹ : Semiring R
M : Type u_2
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
M₂ : Type u_3
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
M₃ : Type u_4
inst✝² : TopologicalSpace M₃
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
f g : M × M₂ →L[R] M₃
... | rfl | no goals | f0c9b8d2b77fd340 |
Set.Icc_add_bij | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | theorem Icc_add_bij : BijOn (· + d) (Icc a b) (Icc (a + d) (b + d)) | M : Type u_1
inst✝¹ : OrderedCancelAddCommMonoid M
inst✝ : ExistsAddOfLE M
a b d : M
⊢ BijOn (fun x => x + d) (Icc a b) (Icc (a + d) (b + d)) | rw [← Ici_inter_Iic, ← Ici_inter_Iic] | M : Type u_1
inst✝¹ : OrderedCancelAddCommMonoid M
inst✝ : ExistsAddOfLE M
a b d : M
⊢ BijOn (fun x => x + d) (Ici a ∩ Iic b) (Ici (a + d) ∩ Iic (b + d)) | 69a7a3dbeaf26ca0 |
Relation.is_graph_iff | Mathlib/Data/Rel.lean | theorem Relation.is_graph_iff (r : Rel α β) : (∃! f, Function.graph f = r) ↔ ∀ x, ∃! y, r x y | case mp
α : Type u_1
β : Type u_2
r : Rel α β
⊢ (∃! f, (fun x y => f x = y) = r) → ∀ (x : α), ∃! y, r x y | rintro ⟨f, rfl, _⟩ x | case mp.intro.intro
α : Type u_1
β : Type u_2
f : α → β
right✝ : ∀ (y : α → β), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y → y = f
x : α
⊢ ∃! y, (fun x y => f x = y) x y | b9c04eef61e821ef |
Polynomial.roots_pow | Mathlib/Algebra/Polynomial/Roots.lean | theorem roots_pow (p : R[X]) (n : ℕ) : (p ^ n).roots = n • p.roots | case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
⊢ (p ^ 0).roots = 0 • p.roots | rw [pow_zero, roots_one, zero_smul, empty_eq_zero] | no goals | 9515e924941f10c8 |
Complex.norm_exp_sub_sum_le_norm_mul_exp | Mathlib/Data/Complex/Exponential.lean | lemma norm_exp_sub_sum_le_norm_mul_exp (x : ℂ) (n : ℕ) :
‖exp x - ∑ m ∈ range n, x ^ m / m.factorial‖ ≤ ‖x‖ ^ n * Real.exp ‖x‖ | x : ℂ
n j : ℕ
hj : j ≥ n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial) | rw [← mul_div_assoc, ← pow_add, add_tsub_cancel_of_le hm.2] | no goals | 0fd8b673128d7210 |
Pell.Solution₁.x_mul_pos | Mathlib/NumberTheory/Pell.lean | theorem x_mul_pos {a b : Solution₁ d} (ha : 0 < a.x) (hb : 0 < b.x) : 0 < (a * b).x | d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < (a * b).x | simp only [x_mul] | d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < a.x * b.x + d * (a.y * b.y) | 5c93ee2e9cc7c71e |
HurwitzZeta.hasSum_int_oddKernel | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | lemma hasSum_int_oddKernel (a : ℝ) {x : ℝ} (hx : 0 < x) :
HasSum (fun n : ℤ ↦ (n + a) * rexp (-π * (n + a) ^ 2 * x)) (oddKernel ↑a x) | a x : ℝ
hx : 0 < x
⊢ 0 < im ?m.80841 | rwa [I_mul_im, ofReal_re] | no goals | 6db94613edf0e05c |
ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | theorem ae_convolution_tendsto_right_of_locallyIntegrable
{ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
(hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
(h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) ... | case h
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.I... | apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (Module.finrank ℝ G)) this | case h.f_int
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝²... | 22b99d3576c4b89d |
InverseSystem.unique_pEquivOn | Mathlib/Order/DirectedInverseSystem.lean | theorem unique_pEquivOn (hs : IsLowerSet s) {e₁ e₂ : PEquivOn f equivSucc s} : e₁ = e₂ | case mk.mk.equiv.h
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e₁
compat₁ : ∀ {i : ι... | refine SuccOrder.prelimitRecOn i.1 (C := fun i ↦ ∀ h : i ∈ s, e₁ ⟨i, h⟩ = e₂ ⟨i, h⟩)
(fun i nmax ih hi ↦ ?_) (fun i lim ih hi ↦ ?_) i.2 | case mk.mk.equiv.h.refine_1
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e₁
compat₁ :... | dcc32ff31e12d17b |
IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_two | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
[hpri : Fact (p : ℕ).Prime] [IsCyclotomicExtension {p ^ (k + 1)} K L]
(hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (hs : s ≤ k)
(htwo : p ^ (k - s + 1) ≠ 2) : norm K (ζ ^ (p : ℕ) ^ s - 1) = (p : K) ^ (... | case refine_2.e_a
p : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hi... | rw [Hex, pow_add] at this | case refine_2.e_a
p : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hi... | 460ee0dff43c909a |
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... | rw [← coe_evalRingHom, ← coeff_map, lieCharpoly_map_eval,
← constantCoeff_apply, LinearMap.charpoly_constantCoeff_eq_zero_iff] | 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 |
List.zipWithAux_toArray_succ' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem zipWithAux_toArray_succ' (as : List α) (bs : List β) (f : α → β → γ) (i : Nat) (cs : Array γ) :
zipWithAux as.toArray bs.toArray f (i + 1) cs = zipWithAux (as.drop (i+1)).toArray (bs.drop (i+1)).toArray f 0 cs | α : Type u_1
β : Type u_2
γ : Type u_3
as : List α
bs : List β
f : α → β → γ
i : Nat
cs : Array γ
⊢ as.toArray.zipWithAux bs.toArray f (i + 1) cs = (drop (i + 1) as).toArray.zipWithAux (drop (i + 1) bs).toArray f 0 cs | induction i generalizing as bs cs with
| zero => simp [zipWithAux_toArray_succ]
| succ i ih =>
rw [zipWithAux_toArray_succ, ih]
simp | no goals | dbd8cc49bcc400e2 |
finrank_vectorSpan_insert_le | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | theorem finrank_vectorSpan_insert_le (s : AffineSubspace k P) (p : P) :
finrank k (vectorSpan k (insert p (s : Set P))) ≤ finrank k s.direction + 1 | case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : ¬FiniteDimensional k ↥s.direction
hf' : ¬FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
⊢ 0 ≤ 1 | exact zero_le_one | no goals | de5f4f906d2ae1ab |
MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀' | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f * g) a ∂μ | let g' := hg.mk g | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f * g) a ∂μ | 8f99ec46bb14da3b |
ModelWithCorners.interior_disjointUnion | Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean | lemma interior_disjointUnion :
ModelWithCorners.interior (I := I) (M ⊕ M') =
Sum.inl '' (ModelWithCorners.interior (I := I) M)
∪ Sum.inr '' (ModelWithCorners.interior (I := I) M') | case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : Chart... | left | case pos.h
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : Cha... | a37cb5a73686bbc7 |
IsUltrametricDist.isUltrametricDist_of_forall_norm_natCast_le_one | Mathlib/Analysis/Normed/Field/Ultra.lean | /-- To prove that a normed division ring is nonarchimedean, it suffices to prove that the norm
of the image of any natural is less than or equal to one. -/
lemma isUltrametricDist_of_forall_norm_natCast_le_one
(h : ∀ n : ℕ, ‖(n : R)‖ ≤ 1) : IsUltrametricDist R | R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
⊢ IsUltrametricDist R | refine isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm (fun x m ↦ ?_) | R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x : R
m : ℕ
⊢ ‖x + 1‖ ^ m ≤ (m + 1) • (1 ⊔ ‖x‖ ^ m) | 065889d95ca2531c |
Nat.multichoose_two | Mathlib/Data/Nat/Choose/Basic.lean | theorem multichoose_two (k : ℕ) : multichoose 2 k = k + 1 | case succ
k : ℕ
IH : multichoose 2 k = k + 1
⊢ multichoose 2 (k + 1) = k + 1 + 1 | rw [multichoose, IH] | case succ
k : ℕ
IH : multichoose 2 k = k + 1
⊢ multichoose 1 (k + 1) + (k + 1) = k + 1 + 1 | 07af28893cc1c434 |
Order.coheight_of_noMaxOrder | Mathlib/Order/KrullDimension.lean | @[simp] lemma coheight_of_noMaxOrder [NoMaxOrder α] (a : α) : coheight a = ⊤ | α : Type u_1
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
a : α
f : ℕ → ↑(Set.Ioi a)
hstrictmono : StrictMono f
m : ℕ
⊢ { length := m, toFun := fun i => if i = 0 then a else ↑(f ↑i), step := ?step }.head = a ∧
{ length := m, toFun := fun i => if i = 0 then a else ↑(f ↑i), step := ?step }.length = m | simp [RelSeries.head] | no goals | f607a58b9d06f9d8 |
LipschitzWith.hasFDerivAt_of_hasLineDerivAt_of_closure | Mathlib/Analysis/Calculus/Rademacher.lean | theorem hasFDerivAt_of_hasLineDerivAt_of_closure
{f : E → F} (hf : LipschitzWith C f) {s : Set E} (hs : sphere 0 1 ⊆ closure s)
{L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v) :
HasFDerivAt f L x | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos :... | rwa [norm_sub_rev] | no goals | b201788e5c940afd |
BitVec.bit_not_add_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem bit_not_add_self (x : BitVec w) :
((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd + x = -1 | w : Nat
x : BitVec w
⊢ ((iunfoldr (fun i c => (c, !x.getLsbD ↑i)) ()).snd.adc x false).snd = -1 | apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl | w : Nat
x : BitVec w
⊢ ∀ (i : Fin w),
adcb ((iunfoldr (fun i c => (c, !x.getLsbD ↑i)) ()).snd.getLsbD ↑i) (x.getLsbD ↑i) false = (false, (-1).getLsbD ↑i) | 1929ded861ebdd30 |
RootPairing.Equiv.coweightHom_injective | Mathlib/LinearAlgebra/RootSystem/Hom.lean | lemma coweightHom_injective (P : RootPairing ι R M N) : Injective (Equiv.coweightHom P) | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
⊢ Injective ⇑(coweightHom P) | refine Injective.of_comp (f := fun a => MulOpposite.op a) fun g g' hgg' => ?_ | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
g g' : P.Aut
hgg' : ((fun a => MulOpposite.op a) ∘ ⇑(coweightHom P)) g = ((fun a => MulOpposite.op a) ∘ ⇑(coweightHom P)) g'
⊢ g = g' | 327d6663427025f5 |
Std.DHashMap.Internal.List.isEmpty_replaceEntry | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem isEmpty_replaceEntry [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} :
(replaceEntry k v l).isEmpty = l.isEmpty | α : Type u
β : α → Type v
inst✝ : BEq α
l : List ((a : α) × β a)
k : α
v : β k
⊢ (replaceEntry k v l).isEmpty = l.isEmpty | induction l using assoc_induction | case nil
α : Type u
β : α → Type v
inst✝ : BEq α
k : α
v : β k
⊢ (replaceEntry k v []).isEmpty = [].isEmpty
case cons
α : Type u
β : α → Type v
inst✝ : BEq α
k : α
v : β k
k✝ : α
v✝ : β k✝
tail✝ : List ((a : α) × β a)
a✝ : (replaceEntry k v tail✝).isEmpty = tail✝.isEmpty
⊢ (replaceEntry k v (⟨k✝, v✝⟩ :: tail✝)).isEmpt... | f9d09d9271745f83 |
List.append_cancel_right_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean | theorem append_cancel_right_eq (as bs cs : List α) : (as ++ bs = cs ++ bs) = (as = cs) | case a.mp
α : Type u_1
as bs cs : List α
⊢ as ++ bs = cs ++ bs → as = cs
case a.mpr
α : Type u_1
as bs cs : List α
⊢ as = cs → as ++ bs = cs ++ bs | next => apply append_cancel_right | case a.mpr
α : Type u_1
as bs cs : List α
⊢ as = cs → as ++ bs = cs ++ bs | 6be9cff41bf607b4 |
Real.Angle.neg_pi_div_two_ne_zero | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : Angle) ≠ 0 | ⊢ -π / 2 ≠ 0 | exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero | no goals | 76d7b0c884677ef3 |
IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
IsDiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P | case mpr.intro.intro.intro.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
this : IsLocalRing R
P : Ideal R
right✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = P
hP1 : P ≠ ⊥
hP2 : P.IsPrime
hPM : P = ⊥
h : maximalIdeal R = ⊥
⊢ False | exact hP1 hPM | no goals | f7521cb0caad85a8 |
Profinite.NobelingProof.projRestricts_eq_id | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem projRestricts_eq_id : ProjRestricts C (fun i (h : J i) ↦ h) = id | I : Type u
C : Set (I → Bool)
J : I → Prop
inst✝ : (i : I) → Decidable (J i)
⊢ ProjRestricts C ⋯ = id | ext ⟨x, y, hy, rfl⟩ i | case h.mk.intro.intro.a.h
I : Type u
C : Set (I → Bool)
J : I → Prop
inst✝ : (i : I) → Decidable (J i)
y : I → Bool
hy : y ∈ C
i : I
⊢ ↑(ProjRestricts C ⋯ ⟨Proj J y, ⋯⟩) i = ↑(id ⟨Proj J y, ⋯⟩) i | ba590648a7f71fc0 |
Batteries.AssocList.toList_eq_toListTR | Mathlib/.lake/packages/batteries/Batteries/Data/AssocList.lean | theorem toList_eq_toListTR : @toList = @toListTR | case h.h.h
α : Type u_2
β : Type u_1
as : AssocList α β
⊢ as.toList = (List.foldl (fun d x => d.push (x.fst, x.snd)) #[] as.toList).toList | exact .symm <| (Array.foldl_toList_eq_map (toList as) _ id).trans (List.map_id _) | no goals | 8b1a4ef5b4991f78 |
CStarAlgebra.span_nonneg_inter_ball | Mathlib/Analysis/CStarAlgebra/SpecialFunctions/PosPart.lean | /-- A C⋆-algebra is spanned by nonnegative elements of norm less than `r`. -/
lemma span_nonneg_inter_ball {r : ℝ} (hr : 0 < r) :
span ℂ ({x : A | 0 ≤ x} ∩ Metric.ball 0 r) = ⊤ | A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 (r / 2)) ≤ span ℂ ({x | 0 ≤ x} ∩ Metric.ball 0 r) | gcongr | case h.H
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ Metric.closedBall 0 (r / 2) ⊆ Metric.ball 0 r | eed7d32d179fb6f5 |
Fin.inv_partialProd_mul_eq_contractNth | Mathlib/Algebra/BigOperators/Fin.lean | theorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)
(j : Fin (n + 1)) (k : Fin n) :
(partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =
j.contractNth (· * ·) g k | case inr.inr
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ (partialProd g (j.succ.succAbove k.castSucc))⁻¹ * partialProd g (j.succAbove k).succ =
j.contractNth (fun x1 x2 => x1 * x2) g k | rwa [succAbove_of_le_castSucc, succAbove_of_le_castSucc, partialProd_succ, partialProd_succ,
castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt] | case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ j ≤ k.castSucc
case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ j.succ ≤ k.castSucc.castSucc | f84a17ef32860b19 |
ProbabilityTheory.Kernel.compProd_fst_condKernelReal | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | lemma compProd_fst_condKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
fst κ ⊗ₖ condKernelReal κ = κ | α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
⊢ κ.fst ⊗ₖ κ.condKernelReal = κ | rw [condKernelReal, compProd_toKernel] | no goals | 22f942ab3ebf60df |
GenContFract.compExactValue_correctness_of_stream_eq_some | Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | theorem compExactValue_correctness_of_stream_eq_some :
∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n →
v = compExactValue ((of v).contsAux n) ((of v).contsAux <| n + 1) ifp_n.fr | case succ.intro.intro.intro.inr
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
nth_fr... | obtain ⟨ifp_n', nth_stream_eq', ifp_n_fract_ne_zero, ⟨refl⟩⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧
ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n :=
IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq | case succ.intro.intro.intro.inr.intro.intro.intro.refl
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
nth_fract_ne_zero : ifp_n.fr ≠ 0
conts : Pair K := g.contsAux (n + 2)
pconts : P... | 434c0d1df2cfd41d |
AnalyticAt.fderiv | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem AnalyticAt.fderiv [CompleteSpace F] (h : AnalyticAt 𝕜 f x) :
AnalyticAt 𝕜 (fderiv 𝕜 f) x | case intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p x r
... | exact hp.fderiv.analyticAt | no goals | 414431133816d537 |
HNNExtension.NormalWord.unitsSMul_one_group_smul | Mathlib/GroupTheory/HNNExtension.lean | theorem unitsSMul_one_group_smul (g : A) (w : NormalWord d) :
unitsSMul φ 1 ((g : G) • w) = (φ g : G) • (unitsSMul φ 1 w) | case pos.cons
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g : ↥A
g✝ : G
u✝ : ℤˣ
w✝ : NormalWord d
h1✝ : w✝.head ∈ d.set u✝
h2✝ : ∀ u' ∈ Option.map Prod.fst w✝.toList.head?, w✝.head ∈ toSubgroup A B u✝ → u✝ = u'
this : Cancels 1 (↑g • cons g✝ u✝ w✝ h1✝ h2✝) ↔ Cancels 1 (cons g✝ u... | rfl | no goals | 17f5302f5114464d |
ENNReal.Lp_add_le | Mathlib/Analysis/MeanInequalities.lean | theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p) | case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑((∑ i ∈ s, ((fun i => (f i).toNNReal) i + (fun i => (g i).toNNReal) i) ^ p) ^ (1 / p)) ≤
↑((∑ i ∈ s, (fun i => (f i).toNNReal) i ^ p) ^ (1 / p) + (∑ i ∈ s, (fun i => (g i).toNNReal) i ^ ... | push_cast [ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this | case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x ∈ s, (↑(f x).toNNReal + ↑(g x).toNNReal) ^ p) ^ (1 / p) ≤
(∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ (1 / p) + (∑ x ∈ s, ↑(g x).toNNReal ^ p) ^ (1 / p)
⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤... | cfe53d149b9d93ea |
Set.PartiallyWellOrderedOn.partiallyWellOrderedOn_sublistForall₂ | Mathlib/Order/WellFoundedSet.lean | theorem partiallyWellOrderedOn_sublistForall₂ (r : α → α → Prop) [IsRefl α r] [IsTrans α r]
{s : Set α} (h : s.PartiallyWellOrderedOn r) :
{ l : List α | ∀ x, x ∈ l → x ∈ s }.PartiallyWellOrderedOn (List.SublistForall₂ r) | case inr
α : Type u_2
r : α → α → Prop
inst✝¹ : IsRefl α r
inst✝ : IsTrans α r
s : Set α
h : s.PartiallyWellOrderedOn r
h✝ : Nonempty α
⊢ {l | ∀ x ∈ l, x ∈ s}.PartiallyWellOrderedOn (List.SublistForall₂ r) | inhabit α | case inr
α : Type u_2
r : α → α → Prop
inst✝¹ : IsRefl α r
inst✝ : IsTrans α r
s : Set α
h : s.PartiallyWellOrderedOn r
h✝ : Nonempty α
inhabited_h : Inhabited α
⊢ {l | ∀ x ∈ l, x ∈ s}.PartiallyWellOrderedOn (List.SublistForall₂ r) | 642b3f587e356afd |
Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem | Mathlib/RingTheory/GradedAlgebra/Radical.lean | theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)
(I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem :
∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_con... | ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousEle... | rw [mem_filter] at mem_max₁ mem_max₂ | ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousEle... | 7b2fd2fea167fa2f |
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
a b : ↑(orbit G x)
g : ↥H
h : ↑g • b = a
⊢ a ∈ orbit (↥H) b | subst h | 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 | 455130db4c4a28b8 |
LinearMap.liftBaseChange_one_tmul | Mathlib/RingTheory/TensorProduct/Basic.lean | lemma liftBaseChange_one_tmul (l : M →ₗ[R] N) (y) : l.liftBaseChange A (1 ⊗ₜ y) = l y | R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
y : M
⊢ (liftBaseChange A l) (1 ⊗ₜ[R] y) = l y | simp | no goals | ab42b9224ea6fc89 |
Ordnode.Valid'.node4L | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
(hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
(H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
0 < size l ∧
ratio * size r ≤ size m ∧
delta * size l ≤ size m + size r ∧
3 * (size m... | α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < (Ordnode.node s ml z mr).size
l0 : 0 < l.size
mr₁ : ratio * r.size ≤ ml.size + mr.si... | rw [← Nat.succ_mul] at this | α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < (Ordnode.node s ml z mr).size
l0 : 0 < l.size
mr₁ : ratio * r.size ≤ ml.size + mr.si... | 4d0cc9fa23e9caa2 |
Vector.getElem_zero_flatten | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Find.lean | theorem getElem_zero_flatten {L : Vector (Vector α m) n} (h : 0 < n * m) :
(flatten L)[0] = (L.findSome? fun l => l[0]?).get (getElem_zero_flatten.proof h) | α : Type u_1
m n : Nat
L : Vector (Vector α m) n
h : 0 < n * m
t : L.flatten[0]? = findSome? (fun l => l[0]?) L
⊢ L.flatten[0] = (findSome? (fun l => l[0]?) L).get ⋯ | simp [getElem?_eq_getElem, h] at t | α : Type u_1
m n : Nat
L : Vector (Vector α m) n
h : 0 < n * m
t : some L[0][0] = findSome? (fun l => l[0]?) L
⊢ L.flatten[0] = (findSome? (fun l => l[0]?) L).get ⋯ | 65e6e193ff3f3283 |
List.foldl_argAux_eq_none | Mathlib/Data/List/MinMax.lean | theorem foldl_argAux_eq_none : l.foldl (argAux r) o = none ↔ l = [] ∧ o = none :=
List.reverseRecOn l (by simp) fun tl hd => by
simp only [foldl_append, foldl_cons, argAux, foldl_nil, append_eq_nil_iff, and_false, false_and,
iff_false]
cases foldl (argAux r) o tl
· simp
· simp only [false_iff, n... | α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o : Option α
⊢ foldl (argAux r) o [] = none ↔ [] = [] ∧ o = none | simp | no goals | 82e69bedad8e20e2 |
RelSeries.append_apply_right | Mathlib/Order/RelSeries.lean | lemma append_apply_right (p q : RelSeries r) (connect : r p.last q.head)
(i : Fin (q.length + 1)) :
p.append q connect (i.natAdd p.length + 1) = q i | case h.e'_2.h.e'_6.h.h
α : Type u_1
r : Rel α α
p q : RelSeries r
connect : r p.last q.head
i : Fin (q.length + 1)
⊢ p.length + ↑i + 1 < (p.length + q.length + 1).succ | omega | no goals | 81a20ecaf5e7e778 |
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_1.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✝ : Sigm... | exact ⟨f, f.contMDiff, hf⟩ | no goals | bb02907964d5e497 |
AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback | Mathlib/AlgebraicGeometry/Morphisms/SurjectiveOnStalks.lean | /-- If `Y ⟶ S` is surjective on stalks, then for every `X ⟶ S`, `X ×ₛ Y` is a subset of
`X × Y` (cartesian product as topological spaces) with the induced topology. -/
lemma isEmbedding_pullback {X Y S : Scheme.{u}} (f : X ⟶ S) (g : Y ⟶ S) [SurjectiveOnStalks g] :
IsEmbedding (fun x ↦ ((pullback.fst f g).base x, (p... | case h.e'_5.h.h.intro.snd
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spe... | simp only [L, ← Scheme.comp_base_apply, pullback.lift_snd, Iso.symm_hom,
Iso.inv_hom_id] | case h.e'_5.h.h.intro.snd
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spe... | 9509c1734d72b9a6 |
exists_dist_le_le | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z ≤ ε | E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ 0
h : dist x z ≤ 0 + δ
⊢ dist x z ≤ δ | rwa [zero_add] at h | no goals | 59238fa715097bae |
TopologicalSpace.IsOpenCover.quasiSober_iff_forall | Mathlib/Topology/Sober.lean | lemma TopologicalSpace.IsOpenCover.quasiSober_iff_forall {ι : Type*} {U : ι → Opens α}
(hU : TopologicalSpace.IsOpenCover U) : QuasiSober α ↔ ∀ i, QuasiSober (U i) | case h.a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t)) | refine (subset_closure_inter_of_isPreirreducible_of_isOpen h (U i).isOpen ⟨x, ⟨hx, hi⟩⟩).trans
(closure_mono ?_) | case h.a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ∩ ↑(U i) ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t) | d49af6a604b6ce82 |
AlgebraicTopology.AlternatingFaceMapComplex.d_squared | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 | case hi
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
ij : P
hij : ij ∈ S
⊢ φ ij hij ∈ Sᶜ | simp only [S, φ, Finset.mem_univ, Finset.compl_filter, Finset.mem_filter, true_and,
Fin.val_succ, Fin.coe_castLT] at hij ⊢ | case hi
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
ij : P
hij : ↑ij.2 ≤ ↑ij.1
⊢ ¬↑ij.1 + 1 ≤... | 795ab48b2006c7af |
ZMod.inv_coe_unit | Mathlib/Data/ZMod/Basic.lean | theorem inv_coe_unit {n : ℕ} (u : (ZMod n)ˣ) : (u : ZMod n)⁻¹ = (u⁻¹ : (ZMod n)ˣ) | n : ℕ
u : (ZMod n)ˣ
this : ↑((↑u).val.gcd n) = ↑1
⊢ (↑u)⁻¹ = ↑u⁻¹ | rw [← mul_inv_eq_gcd, Nat.cast_one] at this | n : ℕ
u : (ZMod n)ˣ
this : ↑u * (↑u)⁻¹ = 1
⊢ (↑u)⁻¹ = ↑u⁻¹ | 18898df54cabae2d |
List.dropInfix?_go_eq_some_iff | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l ... | case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ ... | rw [append_assoc, ← h₁] at h | case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i'' s'' : List α
h : (i'' ++ s'').drop... | 2a30116d5cc9db54 |
Submodule.mapQ_pow | Mathlib/LinearAlgebra/Quotient/Basic.lean | theorem mapQ_pow {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ)
(h' : p ≤ p.comap (f ^ k) := p.le_comap_pow_of_le_comap h k) :
p.mapQ p (f ^ k) h' = p.mapQ p f h ^ k | case zero
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : Submodule R M
f : M →ₗ[R] M
h : p ≤ comap f p
h' : optParam (p ≤ comap (f ^ 0) p) ⋯
⊢ p.mapQ p (f ^ 0) h' = p.mapQ p f h ^ 0 | simp [LinearMap.one_eq_id] | no goals | f678a150cf10dce9 |
CoalgebraCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_right | Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean | theorem comul_tensorObj_tensorObj_right :
Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗
(CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
= Coalgebra.comul (A := M ⊗[R] N ⊗[R] P) | R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom
((comonEquivalence R).symm.inverse.obj (of R M) ... | dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
instCoalgebraStruct_comul] | R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom ((of R M).toComonObj ⊗ (of R N ⊗ of R P).toComonObj).c... | f4373b2177e1dea4 |
MeasureTheory.exists_upperSemicontinuous_le_integral_le | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0,
(∀ x, g x ≤ f x) ∧
UpperSemicontinuous g ∧
Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, ↑(g x) ∂μ | case intro.intro.intro.intro.refine_2.hfm
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
fint : Integrable (fun x => ↑(f x)) μ
ε : ℝ≥0
εpos : 0 < ↑ε
If : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤
g : α → ℝ≥0
gf : ∀ (x : α), g x ≤ f x
gcont : U... | exact fint.aestronglyMeasurable | no goals | e7f3a2333862318c |
ProbabilityTheory.Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self | Mathlib/Probability/Independence/ZeroOne.lean | theorem Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
(h_indep : Kernel.IndepSet t t κ μα) :
∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 ∨ κ a t = ∞ | case neg
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
t : Set Ω
h_indep : ∀ᵐ (a : α) ∂μα, (κ a) (t ∩ t) = (κ a) t * (κ a) t
a : α
ha : 1 = (κ a) t
h0 : ¬(κ a) t = 0
h_top : ¬(κ a) t = ⊤
⊢ (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤ | exact Or.inr (Or.inl ha.symm) | no goals | 8b17ec0015226de1 |
QPF.Cofix.bisim_aux | Mathlib/Data/QPF/Univariate/Basic.lean | theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
(h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
∀ x y, r x y → x = y | F : Type u → Type u
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> x.dest = Quot.mk r <$> y.dest
x✝ : Cofix F
x : (P F).M
y✝ : Cofix F
y : (P F).M
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M → (P F).M → Prop := fun x y => r (Quot.mk Mcongr ... | have h₀ :
Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
h _ _ r'ab | F : Type u → Type u
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> x.dest = Quot.mk r <$> y.dest
x✝ : Cofix F
x : (P F).M
y✝ : Cofix F
y : (P F).M
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M → (P F).M → Prop := fun x y => r (Quot.mk Mcongr ... | ee671efbc83b977d |
Finset.weightedVSubOfPoint_eq_of_weights_eq | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂ | case e_f.h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
i : ι
⊢ w₁ i • (p i -ᵥ p j) = w₂ i • (p i -ᵥ p j) | rcases eq_or_ne i j with h | h | case e_f.h.inl
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
i : ι
h : i = j
⊢ w₁ i • (p i -ᵥ p j) = w₂ i • (p i -ᵥ p j)
case e_f.h.inr
k : Type u_1
V : Ty... | 8a064655adca0142 |
Real.hasDerivAt_fourierChar | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | lemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x | x y : ℝ
⊢ cexp (↑(2 * π * y) * I) = cexp (2 * ↑π * I * ↑1 * ↑y / ↑1) | push_cast | x y : ℝ
⊢ cexp (2 * ↑π * ↑y * I) = cexp (2 * ↑π * I * 1 * ↑y / 1) | 09c33dc599e467bd |
Turing.TM1to0.tr_supports | Mathlib/Computability/PostTuringMachine.lean | theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
TM0.Supports (tr M) ↑(trStmts M S) | case right.mk.some.mk.some
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : TM1.Stmt Γ Λ σ
h₁ : ((some val✝, v'), s) ∈ tr M ... | simp only [tr, Option.mem_def] at h₁ | case right.mk.some.mk.some
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : TM1.Stmt Γ Λ σ
h₁ : some (trAux M a q v) = some ... | c5e09604fd205550 |
Equiv.Perm.cycle_induction_on | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycle_induction_on [Finite β] (P : Perm β → Prop) (σ : Perm β) (base_one : P 1)
(base_cycles : ∀ σ : Perm β, σ.IsCycle → P σ)
(induction_disjoint : ∀ σ τ : Perm β,
Disjoint σ τ → IsCycle σ → P σ → P τ → P (σ * τ)) : P σ | β : Type u_3
inst✝ : Finite β
P : Perm β → Prop
σ : Perm β
base_one : P 1
base_cycles : ∀ (σ : Perm β), σ.IsCycle → P σ
induction_disjoint : ∀ (σ τ : Perm β), σ.Disjoint τ → σ.IsCycle → P σ → P τ → P (σ * τ)
⊢ P σ | cases nonempty_fintype β | case intro
β : Type u_3
inst✝ : Finite β
P : Perm β → Prop
σ : Perm β
base_one : P 1
base_cycles : ∀ (σ : Perm β), σ.IsCycle → P σ
induction_disjoint : ∀ (σ τ : Perm β), σ.Disjoint τ → σ.IsCycle → P σ → P τ → P (σ * τ)
val✝ : Fintype β
⊢ P σ | a7d4f63bf9ff66c2 |
Matroid.mapEmbedding_isBasis_iff | Mathlib/Data/Matroid/Map.lean | @[simp] lemma mapEmbedding_isBasis_iff {f : α ↪ β} {I X : Set β} :
(M.mapEmbedding f).IsBasis I X ↔ M.IsBasis (f ⁻¹' I) (f ⁻¹' X) ∧ I ⊆ X ∧ X ⊆ range f | case refine_2.intro
α : Type u_1
β : Type u_2
M : Matroid α
f : α ↪ β
I : Set β
X : Set α
x✝ : M.IsBasis (⇑f ⁻¹' I) (⇑f ⁻¹' (⇑f '' X)) ∧ I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f
hb : M.IsBasis (⇑f ⁻¹' I) (⇑f ⁻¹' (⇑f '' X))
hIX : I ⊆ ⇑f '' X
hX : ⇑f '' X ⊆ range ⇑f
⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ I = ⇑f '' I₀ ∧ ⇑f '' X = ⇑f '' X₀ | obtain ⟨I, -, rfl⟩ := subset_image_iff.1 hIX | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
M : Matroid α
f : α ↪ β
X : Set α
hX : ⇑f '' X ⊆ range ⇑f
I : Set α
x✝ : M.IsBasis (⇑f ⁻¹' (⇑f '' I)) (⇑f ⁻¹' (⇑f '' X)) ∧ ⇑f '' I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f
hb : M.IsBasis (⇑f ⁻¹' (⇑f '' I)) (⇑f ⁻¹' (⇑f '' X))
hIX : ⇑f '' I ⊆ ⇑f '' X
⊢ ∃ I₀ X₀, M.IsBasis I₀... | aceb5823b23814d0 |
GromovHausdorff.totallyBounded | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
(#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
TotallyBounded t | t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N ... | simp only [F, (E q).symm_apply_apply] | no goals | c10025d34debaff9 |
DividedPowers.coincide_on_smul | Mathlib/RingTheory/DividedPowers/Basic.lean | theorem coincide_on_smul {J : Ideal A} (hJ : DividedPowers J) {n : ℕ} (ha : a ∈ I • J) :
hI.dpow n a = hJ.dpow n a | case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
a : A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
k : ℕ × ℕ
a✝ : k ∈ antidiagonal n
⊢ hI.dpow k.1 x * hI.dpow k.2 y = h... | rw [hx', hy'] | no goals | 404c3d86efc5a9c7 |
Real.geom_mean_weighted_of_constant | Mathlib/Analysis/MeanInequalities.lean | theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = x :=
calc
∏ i ∈ s, z i ^ w i = ∏ i ∈ s, x ^ w i | case intro.intro
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
this : ∑ i ∈ s, w i ≠ 0
i : ι
his : i ∈ s
hi : w i ≠ 0
⊢ 0 ≤ z i | exact hz i his | no goals | 332f81523d8d5681 |
AlgebraicClosure.spanCoeffs_ne_top | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | theorem spanCoeffs_ne_top : spanCoeffs k ≠ ⊤ | case intro.intro
k : Type u
inst✝ : Field k
v : Monics k × ℕ →₀ MvPolynomial (Vars k) k
left✝ : v ∈ Finsupp.supported (MvPolynomial (Vars k) k) (MvPolynomial (Vars k) k) Set.univ
hv : ∑ x ∈ v.support, (toSplittingField (Finset.image Prod.fst v.support)) (v x • (subProdXSubC x.1).coeff x.2) = 1
j : Monics k × ℕ
hj : j ∈... | rw [smul_eq_mul, map_mul, toSplittingField_coeff (Finset.mem_image_of_mem _ hj), mul_zero] | no goals | 7ddf9f30cf89e546 |
Polynomial.monic_restriction | Mathlib/RingTheory/Polynomial/Basic.lean | theorem monic_restriction {p : R[X]} : Monic (restriction p) ↔ Monic p | R : Type u
inst✝ : Ring R
p : R[X]
⊢ p.restriction.coeff p.natDegree = 1 ↔ ↑(p.restriction.coeff p.natDegree) = 1 | exact ⟨fun H => by rw [H, OneMemClass.coe_one], fun H => Subtype.coe_injective H⟩ | no goals | 7b55e8e0a768e58b |
MeasureTheory.volume_sum_rpow_lt_one | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | theorem MeasureTheory.volume_sum_rpow_lt_one (hp : 1 ≤ p) :
volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) | ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-x‖ = ‖x‖
nm_add : ∀ (x y : ι → ℝ... | simp_rw [eq_norm] at eq_zero nm_zero nm_neg nm_add | ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, |0 x| ^ p) ^ (1 / p) = 0
... | 412f34b3848867a2 |
IsClosed.ae_eq_univ_iff_eq | Mathlib/MeasureTheory/Measure/OpenPos.lean | theorem _root_.IsClosed.ae_eq_univ_iff_eq (hF : IsClosed F) :
F =ᵐ[μ] univ ↔ F = univ | X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed F
h : F =ᶠ[ae μ] univ
⊢ F = univ | rwa [ae_eq_univ, hF.isOpen_compl.measure_eq_zero_iff μ, compl_empty_iff] at h | no goals | bd0f56f21b466be3 |
Complex.integral_cpow_mul_exp_neg_mul_Ioi | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | /-- Expresses the integral over `Ioi 0` of `t ^ (a - 1) * exp (-(r * t))` in terms of the Gamma
function, for complex `a`. -/
lemma integral_cpow_mul_exp_neg_mul_Ioi {a : ℂ} {r : ℝ} (ha : 0 < a.re) (hr : 0 < r) :
∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * exp (-(r * t)) = (1 / r) ^ a * Gamma a | a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
⊢ (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1) | nth_rewrite 2 [← cpow_one (1 / r : ℂ)] | a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
⊢ (1 / ↑r) ^ a = (1 / ↑r) ^ 1 * (1 / ↑r) ^ (a - 1) | 5d070e1c4207a464 |
Lean.Order.List.monotone_forIn'_loop | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_forIn'_loop {α : Type uu}
(as : List α) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (as' : List α) (b : β)
(p : Exists (fun bs => bs ++ as' = as)) (hmono : monotone f) :
monotone (fun x => List.forIn'.loop as (f x) as' b p) | case nil
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
β : Type u
γ : Type w
inst✝ : PartialOrder γ
α : Type uu
as : List α
f : γ → (a : α) → a ∈ as → β → m (ForInStep β)
hmono : monotone f
b : β
p : ∃ bs, bs ++ [] = as
⊢ monotone fun x => List.forIn'.loop as (f x) ... | apply monotone_const | no goals | 54f63850339d8283 |
VectorField.leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt | Mathlib/Analysis/Calculus/VectorField.lean | /-- The Lie bracket of vector fields in vector spaces satisfies the Leibniz identity
`[U, [V, W]] = [[U, V], W] + [V, [U, W]]`. -/
lemma leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt
{U V W : E → E} {s : Set E} {x : E} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
(hU : ContDiffWithinAt 𝕜 2 U s x) (hV :... | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U V W : E → E
s : Set E
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hU : ContDiffWithinAt 𝕜 2 U s x
hV : ContDiffWithinAt 𝕜 2 V s x
hW : ContDiffWithinAt 𝕜 2 W s x
h'U : IsSymmSndFDerivWithinAt 𝕜 U s x
... | simp only [lieBracketWithin_eq, map_sub] | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U V W : E → E
s : Set E
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hU : ContDiffWithinAt 𝕜 2 U s x
hV : ContDiffWithinAt 𝕜 2 V s x
hW : ContDiffWithinAt 𝕜 2 W s x
h'U : IsSymmSndFDerivWithinAt 𝕜 U s x
... | a8ebc5b229cab643 |
deriv_const_smul' | Mathlib/Analysis/Calculus/Deriv/Mul.lean | /-- A variant of `deriv_const_smul` without differentiability assumption when the scalar
multiplication is by field elements. -/
lemma deriv_const_smul' {f : 𝕜 → F} {x : 𝕜} {R : Type*} [Field R] [Module R F] [SMulCommClass 𝕜 R F]
[ContinuousConstSMul R F] (c : R) :
deriv (fun y ↦ c • f y) x = c • deriv f x | 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
R : Type u_3
inst✝³ : Field R
inst✝² : Module R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
c : R
hc : c ≠ 0
hf : DifferentiableAt 𝕜 (fun y => c • f y) x
⊢ Differenti... | exact DifferentiableAt.const_smul hf c⁻¹ | no goals | 6906aeb992f7f336 |
LieAlgebra.IsSemisimple.finitelyAtomistic | Mathlib/Algebra/Lie/Semisimple/Basic.lean | /--
In a semisimple Lie algebra,
Lie ideals that are contained in the supremum of a finite collection of atoms
are themselves the supremum of a finite subcollection of those atoms.
By a compactness argument, this statement can be extended to arbitrary sets of atoms.
See `atomistic`.
The proof is by induction on the f... | R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieId... | have _inst := isSimple_of_isAtom J (hs hJs) | R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieId... | 9feaa310a91f7482 |
LaurentSeries.valuation_le_iff_coeff_lt_eq_zero | Mathlib/RingTheory/LaurentSeries.lean | theorem valuation_le_iff_coeff_lt_eq_zero {D : ℤ} {f : K⸨X⸩} :
Valued.v f ≤ ↑(Multiplicative.ofAdd (-D : ℤ)) ↔ ∀ n : ℤ, n < D → f.coeff n = 0 | case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : ¬D + ↑s ≤ 0
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ ↑(Multiplicative.ofAdd (-(D + ↑s))) | obtain ⟨d, hd⟩ := Int.eq_ofNat_of_zero_le (le_of_lt <| not_le.mp hDs) | case neg.intro
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : ¬D + ↑s ≤ 0
d : ℕ
hd : D + ↑s = ↑d
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ ↑(Multiplicative.ofAdd (-(D + ↑s... | 34ebed25af8874e3 |
edist_lt_coe | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c | α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y < ↑c ↔ nndist x y < c | rw [edist_nndist, ENNReal.coe_lt_coe] | no goals | b3c5e1c1c01dd681 |
εNFA.isPath_append | Mathlib/Computability/EpsilonNFA.lean | theorem isPath_append {x y : List (Option α)} :
M.IsPath s u (x ++ y) ↔ ∃ t, M.IsPath s t x ∧ M.IsPath t u y where
mp | α : Type u
σ : Type v
M : εNFA α σ
s u : σ
x y : List (Option α)
⊢ M.IsPath s u (x ++ y) → ∃ t, M.IsPath s t x ∧ M.IsPath t u y | induction' x with x a ih generalizing s | case nil
α : Type u
σ : Type v
M : εNFA α σ
u : σ
y : List (Option α)
s : σ
⊢ M.IsPath s u ([] ++ y) → ∃ t, M.IsPath s t [] ∧ M.IsPath t u y
case cons
α : Type u
σ : Type v
M : εNFA α σ
u : σ
y : List (Option α)
x : Option α
a : List (Option α)
ih : ∀ {s : σ}, M.IsPath s u (a ++ y) → ∃ t, M.IsPath s t a ∧ M.IsPath t u... | e5aadd4797a0d54e |
HomologicalComplex.mapBifunctorAssociatorX_hom_D₂ | Mathlib/Algebra/Homology/BifunctorAssociator.lean | @[reassoc]
lemma mapBifunctorAssociatorX_hom_D₂ (j j' : ι₄) :
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j).hom ≫
mapBifunctor₂₃.D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' =
mapBifunctor₁₂.D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' ≫
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom | case hfg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{u_16, u_1} C₁
inst✝³² : Category.{u_17, u_2} C₂
inst✝³¹ : Category.{u_15, u_5} C₃
inst✝³⁰ : Category.{u_13, u_6} C₄
inst✝²⁹ : Category.{u_14, u_3} C₁₂
inst✝²⁸ : Category.{u_18, u_4} C₂₃
inst✝²⁷ : HasZeroMo... | by_cases h₁ : c₂.Rel i₂ (c₂.next i₂) | case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{u_16, u_1} C₁
inst✝³² : Category.{u_17, u_2} C₂
inst✝³¹ : Category.{u_15, u_5} C₃
inst✝³⁰ : Category.{u_13, u_6} C₄
inst✝²⁹ : Category.{u_14, u_3} C₁₂
inst✝²⁸ : Category.{u_18, u_4} C₂₃
inst✝²⁷ : HasZeroMo... | 9e9b646514c4ff68 |
Ring.DirectLimit.lift_injective | Mathlib/Algebra/Colimit/Ring.lean | lemma lift_injective [Nonempty ι] [IsDirected ι (· ≤ ·)]
(injective : ∀ i, Function.Injective <| g i) :
Function.Injective (lift G f P g Hg) | ι : Type u_1
inst✝⁴ : Preorder ι
G : ι → Type u_2
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝² : CommRing P
g : (i : ι) → G i →+* P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
injective :... | induction z using DirectLimit.induction_on with
| ih _ g => rw [lift_of] at hz; rw [injective _ g hz, _root_.map_zero] | no goals | ec9feee4de21e3d5 |
Nat.sub_le_of_le_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean | theorem sub_le_of_le_add {a b c : Nat} (h : a ≤ c + b) : a - b ≤ c | a b c : Nat
h : a ≤ c + b
d : Nat
hd : a + d = c + b
hge : b ≤ a
⊢ a - b ≤ c | apply @le.intro _ _ d | a b c : Nat
h : a ≤ c + b
d : Nat
hd : a + d = c + b
hge : b ≤ a
⊢ a - b + d = c | 738662ec716bb284 |
AlgebraicGeometry.HasRingHomProperty.isStableUnderBaseChange | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | lemma isStableUnderBaseChange (hP : RingHom.IsStableUnderBaseChange Q) :
P.IsStableUnderBaseChange | case hP'.H.inr
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
hP : RingHom.IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => Q
this✝ : (sourceAffineLocally fun {R S} [CommRing R] [CommRing S] => Q).IsLocal :=
... | apply this _ (comp_of_isOpenImmersion _ _ _ H) inferInstance | no goals | 12b3b3ff9d099e59 |
Real.sSup_smul_of_nonneg | Mathlib/Data/Real/Pointwise.lean | theorem Real.sSup_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sSup (a • s) = a • sSup s | case inr.inr
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s | by_cases h : BddAbove s | case pos
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : BddAbove s
⊢ sSup (a • s) = a • sSup s
case neg
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
h... | 87e20576dbe946d5 |
Subgroup.fg_iff_add_fg | Mathlib/GroupTheory/Finiteness.lean | theorem Subgroup.fg_iff_add_fg (P : Subgroup G) : P.FG ↔ P.toAddSubgroup.FG | G : Type u_3
inst✝ : Group G
P : Subgroup G
⊢ P.FG ↔ (toAddSubgroup P).FG | exact (Subgroup.toSubmonoid P).fg_iff_add_fg | no goals | df9b2c8f40405e93 |
Multiset.replicate_le_replicate | Mathlib/Data/Multiset/Replicate.lean | theorem replicate_le_replicate (a : α) {k n : ℕ} : replicate k a ≤ replicate n a ↔ k ≤ n :=
_root_.trans (by rw [← replicate_le_coe, coe_replicate]) (List.replicate_sublist_replicate a)
| α : Type u_1
a : α
k n : ℕ
⊢ replicate k a ≤ replicate n a ↔ List.replicate k a <+ List.replicate n a | rw [← replicate_le_coe, coe_replicate] | no goals | f94b5364c732b628 |
TopologicalSpace.ext_iff_isClosed | Mathlib/Topology/Basic.lean | theorem TopologicalSpace.ext_iff_isClosed {X} {t₁ t₂ : TopologicalSpace X} :
t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s | X : Type u_3
t₁ t₂ : TopologicalSpace X
⊢ t₁ = t₂ ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s | rw [TopologicalSpace.ext_iff, compl_surjective.forall] | X : Type u_3
t₁ t₂ : TopologicalSpace X
⊢ (∀ (x : Set X), IsOpen xᶜ ↔ IsOpen xᶜ) ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s | 9a9b5b388220d5cd |
BitVec.setWidth_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem setWidth_succ (x : BitVec w) :
setWidth (i+1) x = cons (getLsbD x i) (setWidth i x) | case pred
w i : Nat
x : BitVec w
j : Nat
h : j < i + 1
⊢ x.getLsbD j = if j = i then x.getLsbD i else decide (j < i) && x.getLsbD j | if j_eq : j = i then
simp [j_eq]
else
have j_lt : j < i := Nat.lt_of_le_of_ne (Nat.le_of_succ_le_succ h) j_eq
simp [j_eq, j_lt] | no goals | 127c2c3d6ebd3d56 |
polynomial_expand_eq | Mathlib/FieldTheory/Perfect.lean | lemma polynomial_expand_eq (f : R[X]) :
expand R p f = (f.map (frobeniusEquiv R p).symm) ^ p | R : Type u_1
p : ℕ
inst✝² : CommSemiring R
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
f : R[X]
⊢ (expand R p) f = map (↑(frobeniusEquiv R p).symm) f ^ p | rw [← (f.map (S := R) (frobeniusEquiv R p).symm).expand_char p, map_expand, map_map,
frobenius_comp_frobeniusEquiv_symm, map_id] | no goals | 2cb846c588a95df2 |
Equiv.refl_trans | Mathlib/Logic/Equiv/Defs.lean | theorem refl_trans (e : α ≃ β) : (Equiv.refl α).trans e = e | case mk
α : Sort u
β : Sort v
toFun✝ : α → β
invFun✝ : β → α
left_inv✝ : LeftInverse invFun✝ toFun✝
right_inv✝ : RightInverse invFun✝ toFun✝
⊢ (Equiv.refl α).trans { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } =
{ toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, ri... | rfl | no goals | 9d117295fb2de6db |
Ideal.spanIntNorm_localization | Mathlib/RingTheory/Ideal/Norm/RelNorm.lean | theorem spanIntNorm_localization (I : Ideal S) (M : Submonoid R) (hM : M ≤ R⁰)
{Rₘ : Type*} (Sₘ : Type*) [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ]
[Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
[IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid ... | R : Type u_1
inst✝²⁶ : CommRing R
inst✝²⁵ : IsDomain R
S : Type u_3
inst✝²⁴ : CommRing S
inst✝²³ : IsDomain S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsIntegrallyClosed S
inst✝²⁰ : Algebra R S
inst✝¹⁹ : Module.Finite R S
inst✝¹⁸ : NoZeroSMulDivisors R S
inst✝¹⁷ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
I... | have : IsScalarTower Rₘ Sₘ L := by
apply IsScalarTower.of_algebraMap_eq'
apply IsLocalization.ringHom_ext M
rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Sₘ), RingHom.comp_assoc,
RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq R S Sₘ,
IsLocalization.ma... | R : Type u_1
inst✝²⁶ : CommRing R
inst✝²⁵ : IsDomain R
S : Type u_3
inst✝²⁴ : CommRing S
inst✝²³ : IsDomain S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsIntegrallyClosed S
inst✝²⁰ : Algebra R S
inst✝¹⁹ : Module.Finite R S
inst✝¹⁸ : NoZeroSMulDivisors R S
inst✝¹⁷ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
I... | c51e5741dd2367a5 |
List.mem_mapFinIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean | theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] h = b | case mpr
β : Type u_1
α : Type u_2
b : β
l : List α
f : (i : Nat) → α → i < l.length → β
⊢ (∃ i h, f i l[i] h = b) → b ∈ l.mapFinIdx f | rintro ⟨i, h, rfl⟩ | case mpr.intro.intro
β : Type u_1
α : Type u_2
l : List α
f : (i : Nat) → α → i < l.length → β
i : Nat
h : i < l.length
⊢ f i l[i] h ∈ l.mapFinIdx f | 00d1fdd38a729f1a |
Polynomial.sum_fin | Mathlib/Algebra/Polynomial/Degree/Support.lean | theorem sum_fin [AddCommMonoid S] (f : ℕ → R → S) (hf : ∀ i, f i 0 = 0) {n : ℕ} {p : R[X]}
(hn : p.degree < n) : (∑ i : Fin n, f i (p.coeff i)) = p.sum f | case pos
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : AddCommMonoid S
f : ℕ → R → S
hf : ∀ (i : ℕ), f i 0 = 0
n : ℕ
p : R[X]
hn : p.degree < ↑n
hp : p = 0
⊢ ∀ x ∈ univ, f (↑x) (coeff 0 ↑x) = 0 | intro i _ | case pos
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : AddCommMonoid S
f : ℕ → R → S
hf : ∀ (i : ℕ), f i 0 = 0
n : ℕ
p : R[X]
hn : p.degree < ↑n
hp : p = 0
i : Fin n
a✝ : i ∈ univ
⊢ f (↑i) (coeff 0 ↑i) = 0 | 15253ffb89262ddf |
Polynomial.isIntegral_isLocalization_polynomial_quotient | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Idea... | case h.left
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
IsLo... | apply monic_X_sub_C | no goals | 4f255f198e9b3f92 |
eq_pos_convex_span_of_mem_convexHull | Mathlib/Analysis/Convex/Caratheodory.lean | theorem eq_pos_convex_span_of_mem_convexHull {x : E} (hx : x ∈ convexHull 𝕜 s) :
∃ (ι : Sort (u + 1)) (_ : Fintype ι),
∃ (z : ι → E) (w : ι → 𝕜), Set.range z ⊆ s ∧ AffineIndependent 𝕜 z ∧ (∀ i, 0 < w i) ∧
∑ i, w i = 1 ∧ ∑ i, w i • z i = x | case intro.intro.intro.intro.intro.intro.refine_5
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' :... | rw [Finset.sum_attach (f := fun e => w e • e), Finset.sum_filter_of_ne] | case intro.intro.intro.intro.intro.intro.refine_5
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' :... | c36d0bcdbb3360db |
IsCauSeq.bounded | Mathlib/Algebra/Order/CauSeq/Basic.lean | lemma bounded (hf : IsCauSeq abv f) : ∃ r, ∀ i, abv (f i) < r | case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
hf : IsCauSeq abv f
i : ℕ
h : ∀ j ≥ i, abv (f j - f i) < 1
⊢ ∃ r, ∀ (i : ℕ), abv (f i) < r | set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR | case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
hf : IsCauSeq abv f
i : ℕ
h : ∀ j ≥ i, abv (f j - f i) < 1
R : ℕ → α := Nat.rec (abv (f 0)) fun i c => c ⊔ abv (f i.succ)
hR : R = Nat.rec (abv (f 0)) fun i c => c ⊔ abv (f i.succ)
⊢ ∃ r,... | 9032991fc685129f |
EulerSine.tendsto_integral_cos_pow_mul_div | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | theorem tendsto_integral_cos_pow_mul_div {f : ℝ → ℂ} (hf : ContinuousOn f (Icc 0 (π / 2))) :
Tendsto
(fun n : ℕ => (∫ x in (0 : ℝ)..π / 2, (cos x : ℂ) ^ n * f x) /
(∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ))
atTop (𝓝 <| f 0) | f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
⊢ 0 < 1 | exact zero_lt_one | no goals | 6e4d26a12747c680 |
Array.findIdx_subtype | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean | theorem findIdx_subtype {p : α → Prop} {l : Array { x // p x }}
{f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
l.findIdx f = l.unattach.findIdx g | case mk
α : Type u_1
p : α → Prop
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
toList✝ : List { x // p x }
⊢ findIdx f { toList := toList✝ } = findIdx g { toList := toList✝ }.unattach | simp [hf] | no goals | ea378e9c0e6d7dec |
SimpleGraph.FarFromTriangleFree.le_card_cliqueFinset | Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean | /-- **Triangle Removal Lemma**. If not all triangles can be removed by removing few edges (on the
order of `(card α)^2`), then there were many triangles to start with (on the order of
`(card α)^3`). -/
lemma FarFromTriangleFree.le_card_cliqueFinset (hG : G.FarFromTriangleFree ε) :
triangleRemovalBound ε * card α ^ ... | case inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hG : G.FarFromTriangleFree ε
h✝ : Nonempty α
hε : 0 < ε
l : ℕ := ⌈4 / ε⌉₊
hl : 4 / ε ≤ ↑l
hl' : l ≤ Fintype.card α
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : l ≤ #P... | obtain ⟨t, ht⟩ := hG.cliqueFinset_nonempty' regularityReduced_le k | case inr.inr.inr.intro.intro.intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hG : G.FarFromTriangleFree ε
h✝ : Nonempty α
hε : 0 < ε
l : ℕ := ⌈4 / ε⌉₊
hl : 4 / ε ≤ ↑l
hl' : l ≤ Fintype.card α
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : ... | 68b85a2891fa9866 |
preNormEDS_odd | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma preNormEDS_odd (m : ℤ) : preNormEDS b c d (2 * m + 1) =
preNormEDS b c d (m + 2) * preNormEDS b c d m ^ 3 * (if Even m then b else 1) -
preNormEDS b c d (m - 1) * preNormEDS b c d (m + 1) ^ 3 * (if Even m then 1 else b) | case nat.succ.zero
R : Type u
inst✝ : CommRing R
b c d : R
⊢ preNormEDS b c d (2 * ↑(0 + 1) + 1) =
(preNormEDS b c d (↑(0 + 1) + 2) * preNormEDS b c d ↑(0 + 1) ^ 3 * if Even ↑(0 + 1) then b else 1) -
preNormEDS b c d (↑(0 + 1) - 1) * preNormEDS b c d (↑(0 + 1) + 1) ^ 3 * if Even ↑(0 + 1) then 1 else b | simp | no goals | a81418ac2a260128 |
Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁' | Mathlib/Analysis/Complex/Hadamard.lean | /-- **Hadamard three-line theorem** on `re ⁻¹' [0, 1]` (Variant in simpler terms): Let `f` be a
bounded function, continuous on the closed strip `re ⁻¹' [0, 1]` and differentiable on open strip
`re ⁻¹' (0, 1)`. If, for all `z.re = 0`, `‖f z‖ ≤ a` for some `a ∈ ℝ` and, similarly, for all
`z.re = 1`, `‖f z‖ ≤ b` for some... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormI... | use ‖(f 1)‖, 1 | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sS... | 0fab8ad239caab3f |
Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
{i₁ i₂ : Fin (n + 3)} :
⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
s.points i₁ -ᵥ s.points i₂⟫ =
0 | case right
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
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumc... | simp [fs, ← hj] at hi | no goals | 4dd62aff1c506de0 |
Complex.HadamardThreeLines.scale_diffContOnCl | Mathlib/Analysis/Complex/Hadamard.lean | /-- The function `scale f l u` is `diffContOnCl`. -/
lemma scale_diffContOnCl {f : ℂ → E} {l u : ℝ} (hul : l < u)
(hd : DiffContOnCl ℂ f (verticalStrip l u)) :
DiffContOnCl ℂ (scale f l u) (verticalStrip 0 1) | case hg.hf.hc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
l u : ℝ
hul : l < u
hd : DiffContOnCl ℂ f (verticalStrip l u)
⊢ DiffContOnCl ℂ (fun x => x) (verticalStrip 0 1) | exact Differentiable.diffContOnCl differentiable_id' | no goals | 3fd41ceba2ac6041 |
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