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 |
|---|---|---|---|---|---|---|
MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff' | Mathlib/GroupTheory/GroupAction/Defs.lean | @[to_additive]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff' {H : Subgroup G} {x : orbitRel.Quotient G α}
{a b : x.orbit} {c : α} (h : (⟦a⟧ : orbitRel.Quotient H x.orbit) = ⟦b⟧) :
(a : α) ∈ MulAction.orbit H c ↔ (b : α) ∈ MulAction.orbit H c | G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
c : α
h : ⟦a⟧ = ⟦b⟧
⊢ ↑a ∈ MulAction.orbit (↥H) c ↔ ↑b ∈ MulAction.orbit (↥H) c | simp_rw [mem_orbit_symm (a₂ := c)] | G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
c : α
h : ⟦a⟧ = ⟦b⟧
⊢ c ∈ MulAction.orbit ↥H ↑a ↔ c ∈ MulAction.orbit ↥H ↑b | aabd1acdfde98a59 |
MeasureTheory.Measure.measure_isMulLeftInvariant_eq_smul_of_ne_top | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | /-- **Uniqueness of left-invariant measures**:
Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
... | G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst... | apply le_antisymm | case a
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTT... | 3a873eec5efe2eb8 |
seminormFromBounded_isNonarchimedean | Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean | theorem seminormFromBounded_isNonarchimedean (f_nonneg : 0 ≤ f)
(f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y)
(hna : IsNonarchimedean f) : IsNonarchimedean (seminormFromBounded' f) | R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
⊢ IsNonarchimedean (seminormFromBounded' f) | refine fun x y ↦ ciSup_le (fun z ↦ ?_) | R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ⊔ seminormFromBounded' f y | 60d60fdeb09113e8 |
Real.deriv_tan | Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | theorem deriv_tan (x : ℝ) : deriv tan x = 1 / cos x ^ 2 :=
if h : cos x = 0 then by
have : ¬DifferentiableAt ℝ tan x := mt differentiableAt_tan.1 (Classical.not_not.2 h)
simp [deriv_zero_of_not_differentiableAt this, h, sq]
else (hasDerivAt_tan h).deriv
| x : ℝ
h : cos x = 0
this : ¬DifferentiableAt ℝ tan x
⊢ deriv tan x = 1 / cos x ^ 2 | simp [deriv_zero_of_not_differentiableAt this, h, sq] | no goals | 32beedff61ac4af4 |
IsDedekindDomain.selmerGroup.valuation_ker_eq | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | theorem valuation_ker_eq :
valuation.ker = K⟮(∅ : Set <| HeightOneSpectrum R),n⟯.subgroupOf (K⟮S,n⟯) | case neg
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
S : Set (HeightOneSpectrum R)
n : ℕ
val✝ : Kˣ ⧸ (powMonoidHom n).range
hx : val✝ ∈ selmerGroup
hx' : ⟨val✝, hx⟩ ∈ valuation.ker
v : HeightOneSpectrum R
x✝ : v ∉ ∅
hv : v ∉ S
⊢ ... | exact hx v hv | no goals | cf0f95b7fd47cf35 |
Std.DHashMap.Internal.Raw₀.getKeyD_eq_fallback | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKeyD_eq_fallback [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {a fallback : α} :
m.contains a = false → m.getKeyD a fallback = fallback | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a fallback : α
⊢ m.contains a = false → m.getKeyD a fallback = fallback | simp_to_model using List.getKeyD_eq_fallback | no goals | db5d4033832e096d |
AnalyticAt.order_mul | Mathlib/Analysis/Analytic/Order.lean | theorem order_mul {f g : 𝕜 → 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
(hf.mul hg).order = hf.order + hg.order | case right.right.intro.intro.intro
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
f g : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
h₂f : ¬hf.order = ⊤
h₂g : ¬hg.order = ⊤
g₁ : 𝕜 → 𝕜
h₁g₁ : AnalyticAt 𝕜 g₁ z₀
h₂g₁ : g₁ z₀ ≠ 0
h₃g₁ : f =ᶠ[𝓝 z₀] fun z => (z - z₀) ^ hf.order.toNat • g₁ z
g₂ : 𝕜 ... | obtain ⟨s, h₁s, h₂s, h₃s⟩ := eventually_nhds_iff.1 h₃g₂ | case right.right.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
f g : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
h₂f : ¬hf.order = ⊤
h₂g : ¬hg.order = ⊤
g₁ : 𝕜 → 𝕜
h₁g₁ : AnalyticAt 𝕜 g₁ z₀
h₂g₁ : g₁ z₀ ≠ 0
h₃g₁ : f =ᶠ[𝓝 z₀] fun z => (z - z₀) ^ hf.order.toN... | 9c09f7b0bc9e51ba |
MvPolynomial.support_esymm'' | Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean | theorem support_esymm'' [DecidableEq σ] [Nontrivial R] (n : ℕ) :
(esymm σ R n).support =
(powersetCard n (univ : Finset σ)).biUnion fun t =>
(Finsupp.single (∑ i ∈ t, Finsupp.single i 1) (1 : R)).support | σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i => (Finsupp.single i 1).support) = s.biUnion fun i => (Finsupp.single i 1).support
hsing... | rw [hs, ht] at this | σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : t.biUnion singleton = s.biUnion singleton
hsingle : ∀ (s : Finset σ), ∀ x ∈ s, (Finsupp.single x 1).suppo... | 91d19fd65d77a5f5 |
SpectrumRestricts.nnreal_iff_spectralRadius_le | Mathlib/Analysis/Normed/Algebra/Spectrum.lean | lemma nnreal_iff_spectralRadius_le [Algebra ℝ A] {a : A} {t : ℝ≥0} (ht : spectralRadius ℝ a ≤ t) :
SpectrumRestricts a ContinuousMap.realToNNReal ↔
spectralRadius ℝ (algebraMap ℝ A t - a) ≤ t | A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
⊢ ∀ x ∈ spectrum ℝ a, ‖↑t - x‖₊ ≤ t | simpa [spectralRadius, iSup₂_le_iff, ← spectrum.singleton_sub_eq] using h | no goals | 23e88e5a29773e00 |
MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
HaveLebesgueDecomposition μ ν where
lebesgue_decomposition | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | set μ₁ := μ - ν.withDensity ξ with hμ₁ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | 0e340e69a0abebdf |
support_deriv_subset | Mathlib/Analysis/Calculus/Deriv/Support.lean | theorem support_deriv_subset : support (deriv f) ⊆ tsupport f | 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
E : Type v
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
⊢ x ∉ tsupport f → x ∉ support (deriv f) | intro h2x | 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
E : Type v
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h2x : x ∉ tsupport f
⊢ x ∉ support (deriv f) | 523f758b21c1babb |
SimpleGraph.incMatrix_apply_eq_one_iff | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | theorem incMatrix_apply_eq_one_iff : G.incMatrix R a e = 1 ↔ e ∈ G.incidenceSet a | R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝¹ : MulZeroOneClass R
a : α
e : Sym2 α
inst✝ : Nontrivial R
⊢ incMatrix R G a e = 1 ↔ e ∈ G.incidenceSet a | convert one_ne_zero.ite_eq_left_iff | case convert_3
R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝¹ : MulZeroOneClass R
a : α
e : Sym2 α
inst✝ : Nontrivial R
⊢ NeZero 1 | 429919032cabc5a8 |
Set.Nonempty.csSup_mem | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | theorem Set.Nonempty.csSup_mem (h : s.Nonempty) (hs : s.Finite) : sSup s ∈ s | case intro
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
s : Finset α
h : (↑s).Nonempty
⊢ sSup ↑s ∈ ↑s | exact Finset.Nonempty.csSup_mem h | no goals | fab20af230d750d6 |
SimplexCategory.δ_comp_δ | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem δ_comp_δ {n} {i j : Fin (n + 2)} (H : i ≤ j) :
δ i ≫ δ j.succ = δ j ≫ δ i.castSucc | case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin (⦋n⦌.len + 1)
⊢ ↑((Hom.toOrderHom (δ i ≫ δ j.succ)) k) = ↑((Hom.toOrderHom (δ j ≫ δ i.castSucc)) k) | dsimp [δ, Fin.succAbove] | case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin (⦋n⦌.len + 1)
⊢ ↑(if (if k.castSucc < i then k.castSucc else k.succ).castSucc < j.succ then
(if k.castSucc < i then k.castSucc else k.succ).castSucc
else (if k.castSucc < i then k.castSucc else k.succ).succ) =
↑(if (if k.castSucc < j then k.castSu... | 67cee2d3becaf6fb |
Polynomial.factorial_smul_hasseDeriv | Mathlib/Algebra/Polynomial/HasseDeriv.lean | theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] | case succ.h.a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (k + 1) * k ! * (n + k + 1).choose (k + 1) = k ! * (n + k + 1).choose (n + 1) * (n + 1) | rw [mul_comm (k+1) _, mul_assoc, mul_assoc] | case succ.h.a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ k ! * ((k + 1) * (n + k + 1).choose (k + 1)) = k ! * ((n + k + 1).choose (n + 1) * (n + 1)) | 2b6ef2fdee813caa |
CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite | Mathlib/CategoryTheory/Sites/DenseSubsite/SheafEquiv.lean | lemma isIso_ranCounit_app_of_isDenseSubsite (Y : Sheaf J A) (U X) :
IsIso ((yoneda.map ((G.op.ranCounit.app Y.val).app (op U))).app (op X)) | case refine_2.mk.mk.unit.op.mk.mk.unit.op.mk.up.up.refl.refl
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.o... | rcases h with ⟨g, rfl⟩ | case refine_2.mk.mk.unit.op.mk.mk.unit.op.mk.up.up.refl.refl.intro
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow... | b6d9495444a0fb1a |
QuaternionAlgebra.star_smul | Mathlib/Algebra/Quaternion.lean | theorem star_smul [Monoid S] [DistribMulAction S R] [SMulCommClass S R R]
(s : S) (a : ℍ[R,c₁,c₂,c₃]) :
star (s • a) = s • star a :=
QuaternionAlgebra.ext
(by simp [mul_smul_comm]) (smul_neg _ _).symm (smul_neg _ _).symm (smul_neg _ _).symm
| S : Type u_1
R : Type u_3
c₁ c₂ c₃ : R
inst✝³ : CommRing R
inst✝² : Monoid S
inst✝¹ : DistribMulAction S R
inst✝ : SMulCommClass S R R
s : S
a : ℍ[R,c₁,c₂,c₃]
⊢ (star (s • a)).re = (s • star a).re | simp [mul_smul_comm] | no goals | 37b3f95584aa2976 |
transGen_wcovBy_of_le | Mathlib/Order/Interval/Finset/Basic.lean | lemma transGen_wcovBy_of_le [Preorder α] [LocallyFiniteOrder α] {x y : α} (hxy : x ≤ y) :
TransGen (· ⩿ ·) x y | case pos
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
x y : α
hxy : x ≤ y
this : #(Ico x y) < #(Icc x y)
hxy' : y ≤ x
⊢ TransGen (fun x1 x2 => x1 ⩿ x2) x y | exact .single <| wcovBy_of_le_of_le hxy hxy' | no goals | 498a00cc0c7bc109 |
Finset.mem_finsuppAntidiag_insert | Mathlib/Algebra/Order/Antidiag/Finsupp.lean | theorem mem_finsuppAntidiag_insert {a : ι} {s : Finset ι}
(h : a ∉ s) (n : μ) {f : ι →₀ μ} :
f ∈ finsuppAntidiag (insert a s) n ↔
∃ m ∈ antidiagonal n, ∃ (g : ι →₀ μ),
f = Finsupp.update g a m.1 ∧ g ∈ finsuppAntidiag s m.2 | case mp.intro
ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
a : ι
s : Finset ι
h : a ∉ s
f : ι →₀ μ
hsupp : f.support ⊆ insert a s
⊢ ∃ a_1 b, a_1 + b = f a + ∑ x ∈ s, f x ∧ ∃ g, f = g.update a a_1 ∧ s.sum ⇑g = b ∧ g.support ⊆ s | refine ⟨_, _, rfl, Finsupp.erase a f, ?_, ?_, ?_⟩ | case mp.intro.refine_1
ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
a : ι
s : Finset ι
h : a ∉ s
f : ι →₀ μ
hsupp : f.support ⊆ insert a s
⊢ f = (Finsupp.erase a f).update a (f a)
case mp.intro.refine_2
ι : Type u_1
μ : Type u_2
inst✝³ : Dec... | 257d51aec0992ace |
ContinuousMap.tendsto_concat | Mathlib/Topology/ContinuousMap/Interval.lean | theorem tendsto_concat {ι : Type*} {p : Filter ι} {F : ι → C(Icc a b, E)} {G : ι → C(Icc b c, E)}
(hfg : ∀ᶠ i in p, (F i) ⊤ = (G i) ⊥) (hfg' : f ⊤ = g ⊥)
(hf : Tendsto F p (𝓝 f)) (hg : Tendsto G p (𝓝 g)) :
Tendsto (fun i => concat (F i) (G i)) p (𝓝 (concat f g)) | α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg : ∀ᶠ (i : ι) in... | rintro K hK U hU hfgU | α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg : ∀ᶠ (i : ι) in... | c35e3215284e9199 |
Ideal.subset_union_prime' | Mathlib/RingTheory/Ideal/Operations.lean | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | case zero
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
s : Finset ι
a b : ι
hp : ∀ i ∈ s, (f i).IsPrime
hn : s.card = 0
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | clear hp | case zero
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
s : Finset ι
a b : ι
hn : s.card = 0
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | fa41770a7d2d9fd9 |
Nat.totient_prime_pow_succ | Mathlib/Data/Nat/Totient.lean | theorem totient_prime_pow_succ {p : ℕ} (hp : p.Prime) (n : ℕ) : φ (p ^ (n + 1)) = p ^ n * (p - 1) :=
calc
φ (p ^ (n + 1)) = #{a ∈ range (p ^ (n + 1)) | (p ^ (n + 1)).Coprime a} :=
totient_eq_card_coprime _
_ = #(range (p ^ (n + 1)) \ (range (p ^ n)).image (· * p)) :=
congr_arg card
(by
... | case H
p : ℕ
hp : Prime p
n : ℕ
⊢ ∀ x < p ^ (n + 1), ¬p ∣ x ↔ ∀ (x_1 : ℕ), ¬(x_1 < p ^ n ∧ x_1 * p = x) | intro a ha | case H
p : ℕ
hp : Prime p
n a : ℕ
ha : a < p ^ (n + 1)
⊢ ¬p ∣ a ↔ ∀ (x : ℕ), ¬(x < p ^ n ∧ x * p = a) | bf91953746dfe806 |
IntermediateField.LinearDisjoint.map' | Mathlib/FieldTheory/LinearDisjoint.lean | theorem map' (H : A.LinearDisjoint L) (K : Type*) [Field K] [Algebra F K] [Algebra L K]
[IsScalarTower F L K] [Algebra E K] [IsScalarTower F E K] [IsScalarTower L E K] :
(A.map (IsScalarTower.toAlgHom F E K)).LinearDisjoint L | F : Type u
E : Type v
inst✝¹³ : Field F
inst✝¹² : Field E
inst✝¹¹ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝¹⁰ : Field L
inst✝⁹ : Algebra F L
inst✝⁸ : Algebra L E
inst✝⁷ : IsScalarTower F L E
H : A.LinearDisjoint L
K : Type u_1
inst✝⁶ : Field K
inst✝⁵ : Algebra F K
inst✝⁴ : Algebra L K
inst✝³ : IsScalarTo... | rw [linearDisjoint_iff] at H ⊢ | F : Type u
E : Type v
inst✝¹³ : Field F
inst✝¹² : Field E
inst✝¹¹ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝¹⁰ : Field L
inst✝⁹ : Algebra F L
inst✝⁸ : Algebra L E
inst✝⁷ : IsScalarTower F L E
H : A.LinearDisjoint (IsScalarTower.toAlgHom F L E).range
K : Type u_1
inst✝⁶ : Field K
inst✝⁵ : Algebra F K
inst✝... | 33ac5cb4b0b56403 |
AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | theorem toStalk_stalkMap_toΓSpec (x : X) :
toStalk _ _ ≫ X.toΓSpecSheafedSpace.stalkMap x = X.presheaf.Γgerm x | X : LocallyRingedSpace
x : ↑X.toTopCat
⊢ (ConcreteCategory.hom X.toΓSpecSheafedSpace.base) x ∈ ⊤ | trivial | no goals | 3dbfcc3ce9f2ac0d |
AffineSubspace.direction_inf_of_mem | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem direction_inf_of_mem {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) :
(s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction | case 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
s₁ s₂ : AffineSubspace k P
p : P
h₁ : p ∈ s₁
h₂ : p ∈ s₂
v : V
⊢ v ∈ (s₁ ⊓ s₂).direction ↔ v ∈ s₁.direction ⊓ s₂.direction | rw [Submodule.mem_inf, ← vadd_mem_iff_mem_direction v h₁, ← vadd_mem_iff_mem_direction v h₂, ←
vadd_mem_iff_mem_direction v ((mem_inf_iff p s₁ s₂).2 ⟨h₁, h₂⟩), mem_inf_iff] | no goals | 4acd2c2dfdaebcb6 |
AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isEquivalent_smoothingFn_sub_self (i : α) :
(fun (n : ℕ) => ε (b i * n) - ε n) ~[atTop] fun n => -log (b i) / (log n)^2 | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
⊢ (fun n => log (b i) + log ↑n) = fun n => log ↑n + log (b i) | ext | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
x✝ : ℕ
⊢ log (b i) + log ↑x✝ = log ↑x✝ + log (b i) | 058fd0dc7e6df047 |
DeltaGeneratedSpace.sup | Mathlib/Topology/Compactness/DeltaGeneratedSpace.lean | /-- Suprema of delta-generated topologies are delta-generated. -/
protected lemma DeltaGeneratedSpace.sup {X : Type*} {t₁ t₂ : TopologicalSpace X}
(h₁ : @DeltaGeneratedSpace X t₁) (h₂ : @DeltaGeneratedSpace X t₂) :
@DeltaGeneratedSpace X (t₁ ⊔ t₂) | X : Type u_3
t₁ t₂ : TopologicalSpace X
h₁ : DeltaGeneratedSpace X
h₂ : DeltaGeneratedSpace X
⊢ DeltaGeneratedSpace X | exact .iSup <| Bool.forall_bool.2 ⟨h₂, h₁⟩ | no goals | b0ffdc09140382a3 |
SSet.horn.hom_ext | Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean | /-- Two morphisms from a horn are equal if they are equal on all suitable faces. -/
protected
lemma hom_ext {n : ℕ} {i : Fin (n+2)} {S : SSet} (σ₁ σ₂ : Λ[n+1, i] ⟶ S)
(h : ∀ (j) (h : j ≠ i), σ₁.app _ (face i j h) = σ₂.app _ (face i j h)) :
σ₁ = σ₂ | case app.h.op.h
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op ⦋n⦌) (face i j h) = σ₂.app (op ⦋n⦌) (face i j h)
⊢ ∀ (n_1 : ℕ), σ₁.app (op ⦋n_1⦌) = σ₂.app (op ⦋n_1⦌) | intro m | case app.h.op.h
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op ⦋n⦌) (face i j h) = σ₂.app (op ⦋n⦌) (face i j h)
m : ℕ
⊢ σ₁.app (op ⦋m⦌) = σ₂.app (op ⦋m⦌) | 81aed7a2f2a51d1a |
smul_neg_of_neg_of_pos | Mathlib/Algebra/Order/Module/Defs.lean | lemma smul_neg_of_neg_of_pos [SMulPosStrictMono α β] (ha : a < 0) (hb : 0 < b) : a • b < 0 | α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁵ : Zero α
inst✝⁴ : Zero β
inst✝³ : SMulWithZero α β
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : SMulPosStrictMono α β
ha : a < 0
hb : 0 < b
⊢ a • b < 0 | simpa only [zero_smul] using smul_lt_smul_of_pos_right ha hb | no goals | 7ec7082171bbc7a7 |
PartialHomeomorph.isLocalStructomorphWithinAt_iff | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H}
[ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H}
(hx : x ∈ f.source ∪ sᶜ) :
G.IsLocalStructomorphWithinAt (⇑f) s x ↔
x ∈ s → ∃ e : PartialHomeomorph H H,
e ∈ G ∧ e.source ⊆ f.sour... | case mp.intro.intro.intro.refine_3
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : PartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : PartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartial... | rw [f.open_source.interior_eq] | case mp.intro.intro.intro.refine_3
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : PartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : PartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartial... | 8a619387a2fde1b4 |
EisensteinSeries.summable_norm_eisSummand | Mathlib/NumberTheory/ModularForms/EisensteinSeries/IsBoundedAtImInfty.lean | lemma summable_norm_eisSummand {k : ℤ} (hk : 3 ≤ k) (z : ℍ) :
Summable fun (x : Fin 2 → ℤ) ↦ ‖(eisSummand k x z)‖ | k : ℤ
hk : 3 ≤ k
z : ℍ
hk' : 2 < ↑k
b : Fin 2 → ℤ
⊢ ‖eisSummand k b z‖ ≤ r z ^ (-↑k) * ‖b‖ ^ (-↑k) | simp only [eisSummand, norm_zpow] | k : ℤ
hk : 3 ≤ k
z : ℍ
hk' : 2 < ↑k
b : Fin 2 → ℤ
⊢ ‖↑(b 0) * ↑z + ↑(b 1)‖ ^ (-k) ≤ r z ^ (-↑k) * ‖b‖ ^ (-↑k) | e467256a43f343e5 |
MeasureTheory.measurableSet_exists_tendsto | Mathlib/MeasureTheory/Constructions/Polish/Basic.lean | theorem measurableSet_exists_tendsto [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ]
[hγ : OpensMeasurableSpace γ] [Countable ι] {l : Filter ι}
[l.IsCountablyGenerated] {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) :
MeasurableSet { x | ∃ c, Tendsto (fun n => f n x) l (𝓝 c) } | case inr.intro
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁵ : MeasurableSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : PolishSpace γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this✝ : ... | refine MeasurableSet.biInter Set.countable_univ fun K _ => ?_ | case inr.intro
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁵ : MeasurableSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : PolishSpace γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this✝ : ... | 7e9dea099a7e1607 |
Nat.Partrec.Code.eval_prec_zero | Mathlib/Computability/PartrecCode.lean | theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a | cf cg : Code
a : ℕ
⊢ Nat.rec (cf.eval (a, 0).1)
(fun y IH => do
let i ← IH
cg.eval (Nat.pair (a, 0).1 (Nat.pair y i)))
(a, 0).2 =
cf.eval a | simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only [] | cf cg : Code
a : ℕ
⊢ Nat.rec (cf.eval a)
(fun y IH => do
let i ← IH
cg.eval (Nat.pair a (Nat.pair y i)))
0 =
cf.eval a | d72357165d18b045 |
Finset.iSup_option_toFinset | Mathlib/Order/CompleteLattice/Finset.lean | theorem iSup_option_toFinset (o : Option α) (f : α → β) : ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x | α : Type u_2
β : Type u_3
inst✝ : CompleteLattice β
o : Option α
f : α → β
⊢ ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x | simp | no goals | 9441f6bac2bb150d |
BitVec.cons_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem cons_append (x : BitVec w₁) (y : BitVec w₂) (a : Bool) :
(cons a x) ++ y = (cons a (x ++ y)).cast (by omega) | w₁ w₂ : Nat
x : BitVec w₁
y : BitVec w₂
a : Bool
⊢ w₁ + w₂ + 1 = w₁ + 1 + w₂ | omega | no goals | e7f41fbac28bb709 |
LieAlgebra.IsKilling.reflectRoot_isNonZero | Mathlib/Algebra/Lie/Weights/RootSystem.lean | lemma reflectRoot_isNonZero (α β : Weight K H L) (hβ : β.IsNonZero) :
(reflectRoot α β).IsNonZero | case pos
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hβ : β.IsNonZero
e : (reflectRoot α β).Is... | simp [coroot_eq_zero_iff.mpr hα] | no goals | adbaae50ded5a86b |
LinearMap.det_restrictScalars | Mathlib/RingTheory/Norm/Transitivity.lean | theorem LinearMap.det_restrictScalars [AddCommGroup A] [Module R A] [Module S A]
[IsScalarTower R S A] [Module.Free S A] {f : A →ₗ[S] A} :
(f.restrictScalars R).det = Algebra.norm R f.det | case inr.inl.inl
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : No... | classical
rw [Algebra.norm_eq_matrix_det bS, ← AlgHom.coe_toRingHom, ← det_toMatrix bA, det_det,
← det_toMatrix (bS.smulTower' bA), restrictScalars_toMatrix]
rfl | no goals | e61a35e4b25ff2ca |
Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap | Mathlib/Dynamics/Newton.lean | theorem aeval_pow_two_pow_dvd_aeval_iterate_newtonMap
(h : IsNilpotent (aeval x P)) (h' : IsUnit (aeval x <| derivative P)) (n : ℕ) :
(aeval x P) ^ (2 ^ n) ∣ aeval (P.newtonMap^[n] x) P | case succ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
(aeval
(P.newtonMap^[n] x +
-Ring.inverse ((ae... | refine dvd_add ?_ (dvd_mul_of_dvd_right ?_ _) | case succ.refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
(aeval
(P.newtonMap^[n] x +
-Ring.inv... | e6bdc0f95c76630a |
Equiv.Perm.subgroup_eq_top_of_swap_mem | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem subgroup_eq_top_of_swap_mem [DecidableEq α] {H : Subgroup (Perm α)}
[d : DecidablePred (· ∈ H)] {τ : Perm α} (h0 : (Fintype.card α).Prime)
(h1 : Fintype.card α ∣ Fintype.card H) (h2 : τ ∈ H) (h3 : IsSwap τ) : H = ⊤ | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
H : Subgroup (Perm α)
d : DecidablePred fun x => x ∈ H
τ : Perm α
h0 : Nat.Prime (Fintype.card α)
h1 : Fintype.card α ∣ Fintype.card ↥H
h2 : τ ∈ H
h3 : τ.IsSwap
this : Fact (Nat.Prime (Fintype.card α))
σ : ↥H
hσ : orderOf σ = Fintype.card α
hσ1 : orderOf ... | have hσ3 : (σ : Perm α).support = ⊤ :=
Finset.eq_univ_of_card (σ : Perm α).support (hσ2.orderOf.symm.trans hσ1) | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
H : Subgroup (Perm α)
d : DecidablePred fun x => x ∈ H
τ : Perm α
h0 : Nat.Prime (Fintype.card α)
h1 : Fintype.card α ∣ Fintype.card ↥H
h2 : τ ∈ H
h3 : τ.IsSwap
this : Fact (Nat.Prime (Fintype.card α))
σ : ↥H
hσ : orderOf σ = Fintype.card α
hσ1 : orderOf ... | 813409dc950119c3 |
Finset.fold_union_empty_singleton | Mathlib/Data/Finset/Fold.lean | theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
⊢ fold (fun x1 x2 => x1 ∪ x2) ∅ singleton s = s | induction' s using Finset.induction_on with a s has ih | case empty
α : Type u_1
inst✝ : DecidableEq α
⊢ fold (fun x1 x2 => x1 ∪ x2) ∅ singleton ∅ = ∅
case insert
α : Type u_1
inst✝ : DecidableEq α
a : α
s : Finset α
has : a ∉ s
ih : fold (fun x1 x2 => x1 ∪ x2) ∅ singleton s = s
⊢ fold (fun x1 x2 => x1 ∪ x2) ∅ singleton (insert a s) = insert a s | ae70382d7b18ec39 |
measurableSet_of_differentiableWithinAt_Ici_of_isComplete | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem measurableSet_of_differentiableWithinAt_Ici_of_isComplete {K : Set F} (hK : IsComplete K) :
MeasurableSet { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } | case h
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ MeasurableSet (B f K ((1 / 2) ^ b✝³) ((1 / 2) ^ b✝¹) ((1 / 2) ^ b✝⁵)) | exact measurableSet_B | no goals | 87b779006b1e4c5f |
Std.DHashMap.Raw.Const.mem_insertManyIfNewUnit_list | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem mem_insertManyIfNewUnit_list [EquivBEq α] [LawfulHashable α] (h : m.WF)
{l : List α} {k : α} :
k ∈ insertManyIfNewUnit m l ↔ k ∈ m ∨ l.contains k | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
m : Raw α fun x => Unit
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List α
k : α
⊢ k ∈ insertManyIfNewUnit m l ↔ k ∈ m ∨ l.contains k = true | simp [mem_iff_contains, contains_insertManyIfNewUnit_list h] | no goals | ca8842b141ed857c |
CategoryTheory.Limits.IsZero.iff_id_eq_zero | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | theorem iff_id_eq_zero (X : C) : IsZero X ↔ 𝟙 X = 0 :=
⟨fun h => h.eq_of_src _ _, fun h =>
⟨fun Y => ⟨⟨⟨0⟩, fun f => by
rw [← id_comp f, ← id_comp (0 : X ⟶ Y), h, zero_comp, zero_comp]; simp only⟩⟩,
fun Y => ⟨⟨⟨0⟩, fun f => by
rw [← comp_id f, ← comp_id (0 : Y ⟶ X), h, comp_zero, comp_zero]; ... | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
X : C
h : 𝟙 X = 0
Y : C
f : Y ⟶ X
⊢ 0 = default | simp only | no goals | 3042a8b113054138 |
Submodule.iSup_induction' | Mathlib/LinearAlgebra/Span/Basic.lean | theorem iSup_induction' {ι : Sort*} (p : ι → Submodule R M) {C : ∀ x, (x ∈ ⨆ i, p i) → Prop}
(mem : ∀ (i) (x) (hx : x ∈ p i), C x (mem_iSup_of_mem i hx)) (zero : C 0 (zero_mem _))
(add : ∀ x y hx hy, C x hx → C y hy → C (x + y) (add_mem ‹_› ‹_›)) {x : M}
(hx : x ∈ ⨆ i, p i) : C x hx | case refine_3.intro.intro
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Sort u_9
p : ι → Submodule R M
C : (x : M) → x ∈ ⨆ i, p i → Prop
mem : ∀ (i : ι) (x : M) (hx : x ∈ p i), C x ⋯
zero : C 0 ⋯
add : ∀ (x y : M) (hx : x ∈ ⨆ i, p i) (hy : y ∈ ⨆ i, p i), C x hx → C y hy →... | exact ⟨_, add _ _ _ _ Cx Cy⟩ | no goals | d12c9e2abdb26d9e |
isZGroup_of_coprime | Mathlib/GroupTheory/SpecificGroups/ZGroup.lean | theorem isZGroup_of_coprime [Finite G] [IsZGroup G] [IsZGroup G'']
(h_le : f'.ker ≤ f.range) (h_cop : (Nat.card G).Coprime (Nat.card G'')) :
IsZGroup G' | case inl
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
f : G →* G'
f' : G' →* G''
inst✝² : Finite G
inst✝¹ : IsZGroup G
inst✝ : IsZGroup G''
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
this : Fact (Nat.Prime p)
h_cop : (Nat.card ↥f'.ker).Coprime f'.ker.index
h : ↑P ≤ f'.ker
h... | obtain ⟨Q, hQ⟩ := Sylow.mapSurjective_surjective f.rangeRestrict_surjective p (P.subtype h_le) | case inl.intro
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
f : G →* G'
f' : G' →* G''
inst✝² : Finite G
inst✝¹ : IsZGroup G
inst✝ : IsZGroup G''
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
this : Fact (Nat.Prime p)
h_cop : (Nat.card ↥f'.ker).Coprime f'.ker.index
h : ↑P ≤ f'... | 8e6ebf384c14685f |
List.modifyTailIdx_modifyTailIdx_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem modifyTailIdx_modifyTailIdx_le {f g : List α → List α} (m n : Nat) (l : List α)
(h : n ≤ m) :
(l.modifyTailIdx f n).modifyTailIdx g m =
l.modifyTailIdx (fun l => (f l).modifyTailIdx g (m - n)) n | case intro
α : Type u_1
f g : List α → List α
n : Nat
l : List α
m : Nat
h : n ≤ n + m
⊢ modifyTailIdx g (n + m) (modifyTailIdx f n l) = modifyTailIdx (fun l => modifyTailIdx g (n + m - n) (f l)) n l | rw [Nat.add_comm, modifyTailIdx_modifyTailIdx, Nat.add_sub_cancel] | no goals | bad6b859ff5254bb |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_subset | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_subset (f : DefaultFormula n) (id : Nat) (c : DefaultClause n) :
c ∈ toList (deleteOne f id) → c ∈ toList f | case h_3.inl.h.isTrue
n : Nat
f : DefaultFormula n
id : Nat
c : DefaultClause n
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
(nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup :... | rcases List.getElem_of_mem h1 with ⟨i, h, h4⟩ | case h_3.inl.h.isTrue.intro.intro
n : Nat
f : DefaultFormula n
id : Nat
c : DefaultClause n
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
(nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodup... | 548982dcd427efd2 |
Polynomial.mul_eq_sum_sum | Mathlib/Algebra/Polynomial/Basic.lean | theorem mul_eq_sum_sum :
p * q = ∑ i ∈ p.support, q.sum fun j a => (monomial (i + j)) (p.coeff i * a) | case a
R : Type u
inst✝ : Semiring R
p q : R[X]
⊢ (p * q).toFinsupp = (∑ i ∈ p.support, q.sum fun j a => (monomial (i + j)) (p.coeff i * a)).toFinsupp | rcases p with ⟨⟩ | case a.ofFinsupp
R : Type u
inst✝ : Semiring R
q : R[X]
toFinsupp✝ : R[ℕ]
⊢ ({ toFinsupp := toFinsupp✝ } * q).toFinsupp =
(∑ i ∈ { toFinsupp := toFinsupp✝ }.support,
q.sum fun j a => (monomial (i + j)) ({ toFinsupp := toFinsupp✝ }.coeff i * a)).toFinsupp | 21c553fc40a042b8 |
LinearMap.continuous_of_nonzero_on_open | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | theorem LinearMap.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : Set E) (hs₁ : IsOpen s)
(hs₂ : s.Nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0) : Continuous l | case intro
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
s : Set E
hs₁ : IsOpen s
hs₃ : ∀ x ∈ s, l x ≠ 0
hl : Dense ↑(ker l)
x : E
hx : x ∈ s
⊢ False | have : x ∈ interior (LinearMap.ker l : Set E)ᶜ := by
rw [mem_interior_iff_mem_nhds]
exact mem_of_superset (hs₁.mem_nhds hx) hs₃ | case intro
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
s : Set E
hs₁ : IsOpen s
hs₃ : ∀ x ∈ s, l x ≠ 0
hl : Dense ↑(ker l)
x : E
hx : x ∈ s
this : x ∈ inter... | efc7813d4d7a7cd6 |
MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm' | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem ae_tendsto_of_cauchy_eLpNorm' [CompleteSpace E] {f : ℕ → α → E} {p : ℝ}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
p : ℝ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N
h1 : ∀ (n : ℕ), eL... | have h3 : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i :=
lintegral_rpow_tsum_coe_enorm_sub_le_tsum hf hp1 h2 | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
p : ℝ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N
h1 : ∀ (n : ℕ), eL... | c5de247928b5733b |
LieModule.weightSpaceOfIsLieTower_aux | Mathlib/Algebra/Lie/LieTheorem.lean | /-- An auxiliary lemma used only in the definition `LieModule.weightSpaceOfIsLieTower` below. -/
private lemma weightSpaceOfIsLieTower_aux (z : L) (v : V) (hv : v ∈ weightSpace V χ) :
⁅z, v⁆ ∈ weightSpace V χ | R : Type u_1
L : Type u_2
A : Type u_3
V : Type u_4
inst✝¹⁹ : CommRing R
inst✝¹⁸ : IsPrincipalIdealRing R
inst✝¹⁷ : IsDomain R
inst✝¹⁶ : CharZero R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : LieRing A
inst✝¹² : LieAlgebra R A
inst✝¹¹ : Bracket L A
inst✝¹⁰ : Bracket A L
inst✝⁹ : AddCommGroup V
inst✝⁸ : Modul... | induction n generalizing w | case zero
R : Type u_1
L : Type u_2
A : Type u_3
V : Type u_4
inst✝¹⁹ : CommRing R
inst✝¹⁸ : IsPrincipalIdealRing R
inst✝¹⁷ : IsDomain R
inst✝¹⁶ : CharZero R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : LieRing A
inst✝¹² : LieAlgebra R A
inst✝¹¹ : Bracket L A
inst✝¹⁰ : Bracket A L
inst✝⁹ : AddCommGroup V
inst... | 6a6e6c619a907741 |
CategoryTheory.Functor.initial_of_adjunction | Mathlib/CategoryTheory/Limits/Final.lean | theorem initial_of_adjunction {L : C ⥤ D} {R : D ⥤ C} (adj : L ⊣ R) : Initial L :=
{ out := fun d =>
let u : CostructuredArrow L d := CostructuredArrow.mk (adj.counit.app d)
@zigzag_isConnected _ _ ⟨u⟩ fun f g =>
Relation.ReflTransGen.trans
(Relation.ReflTransGen.single
(show... | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
adj : L ⊣ R
d : D
u : CostructuredArrow L d := CostructuredArrow.mk (adj.counit.app d)
f g : CostructuredArrow L d
⊢ L.map ((adj.homEquiv g.left d) g.hom) ≫ u.hom = g.hom | simp [u] | no goals | 49df1e34dec46552 |
MvPolynomial.C_mul_monomial | Mathlib/Algebra/MvPolynomial/Basic.lean | theorem C_mul_monomial : C a * monomial s a' = monomial s (a * a') | R : Type u
σ : Type u_1
a a' : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ C a * (monomial s) a' = (monomial s) (a * a') | show AddMonoidAlgebra.single _ _ * AddMonoidAlgebra.single _ _ = AddMonoidAlgebra.single _ _ | R : Type u
σ : Type u_1
a a' : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
⊢ AddMonoidAlgebra.single 0 a * AddMonoidAlgebra.single s a' = AddMonoidAlgebra.single s (a * a') | 0e4575f03e193348 |
Std.DHashMap.Internal.List.getKeyD_insertListIfNewUnit_of_contains | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getKeyD_insertListIfNewUnit_of_contains [BEq α] [EquivBEq α]
{l : List ((_ : α) × Unit)} {toInsert : List α}
{k fallback : α}
(contains : containsKey k l = true) :
getKeyD k (insertListIfNewUnit l toInsert) fallback = getKeyD k l fallback | α : Type u
inst✝¹ : BEq α
inst✝ : EquivBEq α
l : List ((_ : α) × Unit)
toInsert : List α
k fallback : α
contains : containsKey k l = true
⊢ getKeyD k (insertListIfNewUnit l toInsert) fallback = getKeyD k l fallback | rw [getKeyD_eq_getKey?,
getKey?_insertListIfNewUnit_of_contains contains, getKeyD_eq_getKey?] | no goals | af22f4e72830f613 |
Nat.Simproc.add_eq_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Simproc.lean | theorem add_eq_le (a : Nat) {b c : Nat} (h : b ≤ c) : (a + b = c) = (a = c - b) | a b c : Nat
h : b ≤ c
r : (a + b = c) = (a = c - b)
⊢ (a + b = c) = (a = c - b) | exact r | no goals | ebf50dc4e205a512 |
MeasureTheory.SimpleFunc.memLp_approxOn | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem memLp_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
(hf : MemLp f p μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s]
(hi₀ : MemLp (fun _ => y₀) p μ) (n : ℕ) : MemLp (approxOn f fmeas s y₀ h₀ n) p μ | β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x => y₀) p μ
n : ℕ
⊢ eLpNorm (fun x => (appro... | have hf' : MemLp (fun x => ‖f x - y₀‖) p μ := by
have h_meas : Measurable fun x => ‖f x - y₀‖ := by
simp only [← dist_eq_norm]
exact (continuous_id.dist continuous_const).measurable.comp fmeas
refine ⟨h_meas.aemeasurable.aestronglyMeasurable, ?_⟩
rw [eLpNorm_norm]
convert eLpNorm_add_lt_top hf hi₀.neg w... | β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x => y₀) p μ
n : ℕ
hf' : MemLp (fun x => ‖f x... | 04f51db9394e766d |
Mathlib.Meta.NormNum.minFacHelper_0 | Mathlib/Tactic/NormNum/Prime.lean | theorem minFacHelper_0 (n : ℕ)
(h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
MinFacHelper n (nat_lit 3) | n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
⊢ 3 % 2 = 1 | norm_num | no goals | dc6f77f66bc00faa |
IsGalois.card_aut_eq_finrank | Mathlib/FieldTheory/Galois/Basic.lean | theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
Fintype.card (E ≃ₐ[F] E) = finrank F E | case intro.f
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
{ toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
commutes' := ... | apply Equiv.mk (fun ϕ => iso.trans (ϕ.trans iso.symm)) fun ϕ => iso.symm.trans (ϕ.trans iso) | case intro.f.left_inv
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
{ toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
comm... | 9073114732e73abf |
TopCat.Presheaf.isSheaf_of_isTerminal_of_indiscrete | Mathlib/Topology/Sheaves/PUnit.lean | theorem isSheaf_of_isTerminal_of_indiscrete {X : TopCat.{w}} (hind : X.str = ⊤) (F : Presheaf C X)
(it : IsTerminal <| F.obj <| op ⊥) : F.IsSheaf := fun c U s hs => by
obtain rfl | hne := eq_or_ne U ⊥
· intro _ _
rw [@existsUnique_iff_exists _ ⟨fun _ _ => _⟩]
· refine ⟨it.from _, fun U hU hs => IsTermin... | case h.e'_5.h.h.e'_4.inl
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
hind : X.str = ⊤
F : Presheaf C X
it : IsTerminal (F.obj (op ⊥))
c : C
s : Sieve ⊤
hs : s ∈ (Opens.grothendieckTopology ↑X) ⊤
hne : ⊤ ≠ ⊥
he : IsEmpty ↑X
⊢ s.arrows (𝟙 ⊤) | exact (hne <| SetLike.ext'_iff.2 <| Set.univ_eq_empty_iff.2 he).elim | no goals | 461b3f0d8c2711cb |
Algebra.IsPushout.symm | Mathlib/RingTheory/IsTensorProduct.lean | theorem Algebra.IsPushout.symm (h : Algebra.IsPushout R S R' S') : Algebra.IsPushout R R' S S' | R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S... | constructor | case out
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTo... | ee0cd1db0e501e91 |
Set.isAtom_iff | Mathlib/Order/Atoms.lean | theorem isAtom_iff {s : Set α} : IsAtom s ↔ ∃ x, s = {x} | case intro
α : Type u_2
x : α
⊢ IsAtom {x} | exact isAtom_singleton x | no goals | 3a49d3b38234aece |
integrableOn_Icc_iff_integrableOn_Ico' | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ | case neg
α : Type u_1
E : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : μ {b} ≠ ⊤
hab : ¬a ≤ b
⊢ ¬a < b | contrapose! hab | case neg
α : Type u_1
E : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : μ {b} ≠ ⊤
hab : a < b
⊢ a ≤ b | 2b0c2ded062a2f7f |
Std.Tactic.BVDecide.BVExpr.bitblast.blastAdd.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Add.lean | theorem go_denote_eq (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (cin : Ref aig)
(s : AIG.RefVec aig curr) (lhs rhs : AIG.RefVec aig w) (assign : α → Bool)
(lhsExpr rhsExpr : BitVec w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, lhs.get idx hidx, assign⟧ = lhsExpr.getLsbD idx)
(hright : ∀ (idx : Na... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s✝ : aig.RefVec curr
lhs rhs : aig.RefVec w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := lhs.get idx hidx }⟧ = lhsExpr.getLsbD idx
hright... | rw [heq] | no goals | d0919cd06ff6dbb0 |
Ideal.isPrimary_decomposition_pairwise_ne_radical | Mathlib/RingTheory/Lasker.lean | lemma isPrimary_decomposition_pairwise_ne_radical {I : Ideal R}
{s : Finset (Ideal R)} (hs : s.inf id = I) (hs' : ∀ ⦃J⦄, J ∈ s → J.IsPrimary) :
∃ t : Finset (Ideal R), t.inf id = I ∧ (∀ ⦃J⦄, J ∈ t → J.IsPrimary) ∧
(t : Set (Ideal R)).Pairwise ((· ≠ ·) on radical) | case refine_2.refine_1
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
s : Finset (Ideal R)
hs : s.inf id = I
hs' : ∀ ⦃J : Ideal R⦄, J ∈ s → J.IsPrimary
J : Ideal R
hJ : J ∈ s
⊢ J ∈ Finset.filter (fun I => I.radical = J.radical) s | simp [hJ] | no goals | a2c3bb9fdf20ac9b |
Real.eq_Gamma_of_log_convex | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem eq_Gamma_of_log_convex {f : ℝ → ℝ} (hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f))
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y) (hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y)
(hf_one : f 1 = 1) : EqOn f Gamma (Ioi (0 : ℝ)) | f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f)
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y
hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y
hf_one : f 1 = 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (log ∘ f) x = (log ∘ Gamma) x | have e1 := BohrMollerup.tendsto_logGammaSeq hf_conv ?_ hx | case refine_2
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f)
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y
hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y
hf_one : f 1 = 1
x : ℝ
hx : x ∈ Ioi 0
e1 : Tendsto (BohrMollerup.logGammaSeq x) atTop (𝓝 ((log ∘ f) x - (log ∘ f) 1))
⊢ (log ∘ f) x = (log ∘ Gamma) x
case refine_1
f : ℝ →... | a24445e3d457766a |
WeierstrassCurve.exists_variableChange_of_char_three_of_j_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_three_of_j_eq_zero
[E.IsShortNF] [E'.IsShortNF] :
∃ C : VariableChange F, E.variableChange C = E' | F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
⊢ ∃ C, E.variableChange C = E' | have ha₄ := E.Δ'.ne_zero | F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : ↑E.Δ' ≠ 0
⊢ ∃ C, E.variableChange C = E' | c130cdcad82e2167 |
Vector.foldlM_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Monadic.lean | theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β) (l : Vector α n) (init : β) :
(l.filter p).foldlM g init =
l.foldlM (fun x y => if p y then g x y else pure x) init | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
n : Nat
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → Bool
g : β → α → m β
l : Vector α n
init : β
⊢ Array.foldlM g init (Array.filter p l.toArray) = foldlM (fun x y => if p y = true then g x y else pure x) init l | rcases l with ⟨l, rfl⟩ | case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → Bool
g : β → α → m β
init : β
l : Array α
⊢ Array.foldlM g init (Array.filter p { toArray := l, size_toArray := ⋯ }.toArray) =
foldlM (fun x y => if p y = true then g x y else pure x) init { toArray := l, size_t... | 8746996535e968c0 |
Subalgebra.centralizer_coe_image_includeLeft_eq_center_tensorProduct | Mathlib/Algebra/Algebra/Subalgebra/Centralizer.lean | /--
Let `R` be a commutative ring and `A, B` be `R`-algebras where `B` is free as `R`-module.
For any subset `S ⊆ A`, the centralizer of `S ⊗ 1 ⊆ A ⊗ B` is `C_A(S) ⊗ B` where `C_A(S)` is the
centralizer of `S` in `A`.
-/
lemma centralizer_coe_image_includeLeft_eq_center_tensorProduct
(S : Set A) [Module.Free R B] :... | case h.mp.intro
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
hw ... | specialize hw (x ⊗ₜ[R] 1) ⟨x, hx, rfl⟩ | case h.mp.intro
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
j :... | 104f26fc78558199 |
Nat.Partrec.Code.evaln_complete | Mathlib/Computability/PartrecCode.lean | theorem evaln_complete {c n x} : x ∈ eval c n ↔ ∃ k, x ∈ evaln k c n | case right
n x : ℕ
h : x = (unpair n).2
⊢ (∃ x, n ≤ x) ∧ (unpair n).2 = x | exact ⟨⟨_, le_rfl⟩, h.symm⟩ | no goals | 2157576702b10f5b |
Std.DHashMap.Internal.List.containsKey_alterKey | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem containsKey_alterKey {k k' : α} {f : Option (β k) → Option (β k)}
{l : List ((a : α) × β a)} (hl : DistinctKeys l) :
containsKey k' (alterKey k f l) =
if k == k' then
f (getValueCast? k l) |>.isSome
else
containsKey k' l | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
k k' : α
f : Option (β k) → Option (β k)
l : List ((a : α) × β a)
hl : DistinctKeys l
h : ¬(k == k') = true
⊢ containsKey k' (alterKey k f l) = containsKey k' l | rw [alterKey] | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
k k' : α
f : Option (β k) → Option (β k)
l : List ((a : α) × β a)
hl : DistinctKeys l
h : ¬(k == k') = true
⊢ containsKey k'
(match f (getValueCast? k l) with
| none => eraseKey k l
| some v => insertEntry k v l) =
containsKey k' l | 20cdb3aa8a15d1ad |
fwdDiff_smul_const | Mathlib/Algebra/Group/ForwardDiff.lean | @[simp] lemma fwdDiff_smul_const {R : Type} [Ring R] [Module R G] (f : M → R) (g : G) :
Δ_[h] (fun y ↦ f y • g) = Δ_[h] f • fun _ ↦ g | case h
M : Type u_1
G : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommGroup G
h : M
R : Type
inst✝¹ : Ring R
inst✝ : Module R G
f : M → R
g : G
y : M
⊢ Δ_[h] (fun y => f y • g) y = (Δ_[h] f • fun x => g) y | simp only [fwdDiff, Pi.smul_apply', sub_smul] | no goals | b8f177ca1d724586 |
ContinuousLinearMap.opNNNorm_le_of_unit_nnnorm | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | theorem opNNNorm_le_of_unit_nnnorm [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ≥0}
(hf : ∀ x, ‖x‖₊ = 1 → ‖f x‖₊ ≤ C) : ‖f‖₊ ≤ C :=
opNorm_le_of_unit_norm C.coe_nonneg fun x hx => hf x <| by rwa [← NNReal.coe_eq_one]
| E : Type u_4
F : Type u_6
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
f : E →L[ℝ] F
C : ℝ≥0
hf : ∀ (x : E), ‖x‖₊ = 1 → ‖f x‖₊ ≤ C
x : E
hx : ‖x‖ = 1
⊢ ‖x‖₊ = 1 | rwa [← NNReal.coe_eq_one] | no goals | 26662b433add620c |
Set.pow_eq_empty | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | @[to_additive (attr := simp)] lemma pow_eq_empty : s ^ n = ∅ ↔ s = ∅ ∧ n ≠ 0 | case mpr
α : Type u_2
inst✝ : Monoid α
s : Set α
n : ℕ
⊢ s = ∅ ∧ n ≠ 0 → s ^ n = ∅ | rintro ⟨rfl, hn⟩ | case mpr.intro
α : Type u_2
inst✝ : Monoid α
n : ℕ
hn : n ≠ 0
⊢ ∅ ^ n = ∅ | 98972d1f4b607a9b |
NumberField.FinitePlace.prod_eq_inv_abs_norm | Mathlib/NumberTheory/NumberField/ProductFormula.lean | theorem FinitePlace.prod_eq_inv_abs_norm {x : K} (h_x_nezero : x ≠ 0) :
∏ᶠ w : FinitePlace K, w x = |(Algebra.norm ℚ) x|⁻¹ | case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : 𝓞 K
h_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0
hb : b ≠ 0
⊢ ∏ᶠ (w : FinitePlace K), w ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b) =
↑|(Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b)... | have ha : a ≠ 0 := by
rintro rfl
simp at h_x_nezero | case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : 𝓞 K
h_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0
hb : b ≠ 0
ha : a ≠ 0
⊢ ∏ᶠ (w : FinitePlace K), w ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b) =
↑|(Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (... | bc2653900b49a69c |
Turing.mem_eval | Mathlib/Computability/PostTuringMachine.lean | theorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none | case refine_1.some.inr.intro.intro.refl.intro
σ : Type u_1
f : σ → Option σ
a✝ b : σ
h✝¹ : b ∈ eval f a✝
a : σ
h✝ : b ∈ eval f a
IH : ∀ (a' : σ), f a = some a' → Reaches f a' b ∧ f b = none
a' : σ
e : f a = some a'
h : Sum.inr a' ∈ Part.some ((some a').elim (Sum.inl a) Sum.inr)
right✝ : b ∈ PFun.fix (fun s => Part.some... | exact ⟨ReflTransGen.head e h₁, h₂⟩ | no goals | f40442f1b60c5272 |
ConvexOn.lipschitzOnWith_of_abs_le | Mathlib/Analysis/Convex/Continuous.lean | lemma ConvexOn.lipschitzOnWith_of_abs_le (hf : ConvexOn ℝ (ball x₀ r) f) (hε : 0 < ε)
(hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε)) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ba... | calc
_ = ‖(x - x₀) + (ε / ‖x - y‖) • (x - y)‖ := by simp only [z, add_sub_right_comm]
_ ≤ ‖x - x₀‖ + ‖(ε / ‖x - y‖) • (x - y)‖ := norm_add_le ..
_ < r - ε + ε :=
add_lt_add_of_lt_of_le (mem_ball_iff_norm.1 hx) <| by
simp [norm_smul, abs_of_nonneg, hε.le, hxy.ne']
_ = r := by simp | no goals | f794e80453a6298e |
CategoryTheory.Idempotents.whiskeringLeft_obj_preimage_app | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | theorem whiskeringLeft_obj_preimage_app {F G : Karoubi C ⥤ D}
(τ : toKaroubi _ ⋙ F ⟶ toKaroubi _ ⋙ G) (P : Karoubi C) :
(((whiskeringLeft _ _ _).obj (toKaroubi _)).preimage τ).app P =
F.map P.decompId_i ≫ τ.app P.X ≫ G.map P.decompId_p | case e_a.e_a
C : Type u_1
D : Type u_2
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : IsIdempotentComplete D
F G : Karoubi C ⥤ D
τ : toKaroubi C ⋙ F ⟶ toKaroubi C ⋙ G
P : Karoubi C
⊢ (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage τ).app { X := P.X, p := 𝟙 P.X, idem := ⋯ } = τ.app... | rw [← congr_app (((whiskeringLeft _ _ _).obj (toKaroubi _)).map_preimage τ) P.X] | case e_a.e_a
C : Type u_1
D : Type u_2
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : IsIdempotentComplete D
F G : Karoubi C ⥤ D
τ : toKaroubi C ⋙ F ⟶ toKaroubi C ⋙ G
P : Karoubi C
⊢ (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage τ).app { X := P.X, p := 𝟙 P.X, idem := ⋯ } =
(... | e20f3b913aa0bca7 |
Vector.getElem_drop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem getElem_drop (a : Vector α n) (m : Nat) (hi : i < n - m) :
(a.drop m)[i] = a[m + i] | α : Type u_1
n i : Nat
a : Vector α n
m : Nat
hi : i < n - m
⊢ (a.drop m)[i] = a[m + i] | cases a | case mk
α : Type u_1
n i m : Nat
hi : i < n - m
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
⊢ ({ toArray := toArray✝, size_toArray := size_toArray✝ }.drop m)[i] =
{ toArray := toArray✝, size_toArray := size_toArray✝ }[m + i] | 1b0144d787a02925 |
left_inv_eq_right_inv | Mathlib/Algebra/Group/Defs.lean | @[to_additive] lemma left_inv_eq_right_inv (hba : b * a = 1) (hac : a * c = 1) : b = c | M : Type u_2
inst✝ : Monoid M
a b c : M
hba : b * a = 1
hac : a * c = 1
⊢ b = c | rw [← one_mul c, ← hba, mul_assoc, hac, mul_one b] | no goals | eebfe6c939c7636b |
Orientation.eq_zero_or_eq_zero_of_kahler_eq_zero | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | theorem eq_zero_or_eq_zero_of_kahler_eq_zero {x y : E} (hx : o.kahler x y = 0) : x = 0 ∨ y = 0 | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
⊢ x = 0 ∨ y = 0 | rcases eq_zero_or_eq_zero_of_mul_eq_zero this with h | h | case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖x‖ = 0
⊢ x = 0 ∨ y = 0
case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact ... | 6defa13177029bb6 |
StructureGroupoid.isLocalStructomorphWithinAt_localInvariantProp | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local | case mpr.intro.intro.intro.refine_2
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
s : Set H
x : H
u : Set H
f : H → H
hu : IsOpen u
hux : x ∈ u
h : G.IsLocalStructomorphWithinAt f (s ∩ u) x
hx : x ∈ s
e : PartialHomeomorph H H
heG : e ∈ G
hef : EqOn f (↑e.toPartialEqu... | simpa only [this, interior_interior, hu.interior_eq, mfld_simps] using hef | no goals | 546fb444522c1b0b |
MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : MeasurableSet s)
(x : E) (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1))
(t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x... | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
... | filter_upwards [self_mem_nhdsWithin] | case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t :... | a092845449b3addf |
InfIrred.isPrimary | Mathlib/RingTheory/Lasker.lean | lemma _root_.InfIrred.isPrimary {I : Ideal R} (h : InfIrred I) : I.IsPrimary | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
a b : R
hab : a * b ∈ I
f : ℕ → Ideal R := fun n => Submodule.colon I (span {b ^ n})
hf : Monotone f
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → { toFun := f, monotone' := hf } n = { toFun := f, monotone' := hf } m
h : Submodule.colon I (span {b ^ n}) = I
⊢ I = ... | rcases eq_or_ne n 0 with rfl|hn' | case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
a b : R
hab : a * b ∈ I
f : ℕ → Ideal R := fun n => Submodule.colon I (span {b ^ n})
hf : Monotone f
hn : ∀ (m : ℕ), 0 ≤ m → { toFun := f, monotone' := hf } 0 = { toFun := f, monotone' := hf } m
h : Submodule.colon I (span {b ^ 0}) = I
⊢ I... | 48120bc89c7755a4 |
AddSubmonoid.smul_iSup | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | theorem smul_iSup (T : AddSubmonoid R) (S : ι → AddSubmonoid A) : (T • ⨆ i, S i) = ⨆ i, T • S i :=
le_antisymm (smul_le.mpr fun t ht s hs ↦ iSup_induction _ (C := (t • · ∈ _)) hs
(fun i s hs ↦ mem_iSup_of_mem i <| smul_mem_smul ht hs)
(by simp_rw [smul_zero]; apply zero_mem) fun x y ↦ by simp_rw [smul_add]; a... | R : Type u_4
A : Type u_5
inst✝² : AddMonoid A
inst✝¹ : AddMonoid R
inst✝ : DistribSMul R A
ι : Sort u_7
T : AddSubmonoid R
S : ι → AddSubmonoid A
t : R
ht : t ∈ T
s : A
hs : s ∈ ⨆ i, S i
x y : A
⊢ (fun x => t • x ∈ ⨆ i, T • S i) x → (fun x => t • x ∈ ⨆ i, T • S i) y → (fun x => t • x ∈ ⨆ i, T • S i) (x + y) | simp_rw [smul_add] | R : Type u_4
A : Type u_5
inst✝² : AddMonoid A
inst✝¹ : AddMonoid R
inst✝ : DistribSMul R A
ι : Sort u_7
T : AddSubmonoid R
S : ι → AddSubmonoid A
t : R
ht : t ∈ T
s : A
hs : s ∈ ⨆ i, S i
x y : A
⊢ t • x ∈ ⨆ i, T • S i → t • y ∈ ⨆ i, T • S i → t • x + t • y ∈ ⨆ i, T • S i | 28baf8adc81dba6a |
geom_sum_alternating_of_lt_neg_one | Mathlib/Algebra/GeomSum.lean | theorem geom_sum_alternating_of_lt_neg_one [StrictOrderedRing α] (hx : x + 1 < 0) (hn : 1 < n) :
if Even n then (∑ i ∈ range n, x ^ i) < 0 else 1 < ∑ i ∈ range n, x ^ i | case refine_2
α : Type u
n : ℕ
x : α
inst✝ : StrictOrderedRing α
hx : x + 1 < 0
hn : 1 < n
hx0 : x < 0
⊢ ∀ (n : ℕ),
2 ≤ n →
(if Even n then ∑ i ∈ range n, x ^ i < 0 else 1 < ∑ i ∈ range n, x ^ i) →
if Even (n + 1) then ∑ i ∈ range (n + 1), x ^ i < 0 else 1 < ∑ i ∈ range (n + 1), x ^ i | clear hn | case refine_2
α : Type u
n : ℕ
x : α
inst✝ : StrictOrderedRing α
hx : x + 1 < 0
hx0 : x < 0
⊢ ∀ (n : ℕ),
2 ≤ n →
(if Even n then ∑ i ∈ range n, x ^ i < 0 else 1 < ∑ i ∈ range n, x ^ i) →
if Even (n + 1) then ∑ i ∈ range (n + 1), x ^ i < 0 else 1 < ∑ i ∈ range (n + 1), x ^ i | 068770c1157e84dd |
VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem measure_le_mul_of_subset_limRatioMeas_lt {p : ℝ≥0} {s : Set α}
(h : s ⊆ {x | v.limRatioMeas hρ x < p}) : ρ s ≤ p * μ s | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
⊢ ρ s ≤ ↑p * μ s | let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))} | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x ... | a284bae1d86d6c33 |
Std.Tactic.BVDecide.BVPred.eval_getLsbD | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Basic.lean | theorem eval_getLsbD : eval assign (.getLsbD expr idx) = (expr.eval assign).getLsbD idx | assign : BVExpr.Assignment
a✝ : Nat
expr : BVExpr a✝
idx : Nat
⊢ eval assign (getLsbD expr idx) = (BVExpr.eval assign expr).getLsbD idx | rfl | no goals | 0a0cc8977cc62ff7 |
Batteries.RBNode.Path.Ordered.fill | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | theorem Ordered.fill : ∀ {path : Path α} {t},
(path.fill t).Ordered cmp ↔ path.Ordered cmp ∧ t.Ordered cmp ∧ t.All (path.RootOrdered cmp)
| .root, _ => ⟨fun H => ⟨⟨⟩, H, .trivial ⟨⟩⟩, (·.2.1)⟩
| .left .., _ => by
simp [Ordered.fill, RBNode.Ordered, Ordered, RootOrdered, All_and]
exact ⟨
fun ⟨hp, ⟨... | α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
parent✝ : Path α
x✝ : RBNode α
⊢ RBNode.Ordered cmp ((right c✝ l✝ v✝ parent✝).fill x✝) ↔
Ordered cmp (right c✝ l✝ v✝ parent✝) ∧ RBNode.Ordered cmp x✝ ∧ All (RootOrdered cmp (right c✝ l✝ v✝ parent✝)) x✝ | simp [Ordered.fill, RBNode.Ordered, Ordered, RootOrdered, All_and] | α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
parent✝ : Path α
x✝ : RBNode α
⊢ Ordered cmp parent✝ ∧
(All (fun x => cmpLT cmp x v✝) l✝ ∧
All (fun x => cmpLT cmp v✝ x) x✝ ∧ RBNode.Ordered cmp l✝ ∧ RBNode.Ordered cmp x✝) ∧
RootOrdered cmp parent✝ v✝ ∧ All (RootOrdered cmp p... | 1c2c1b95f6b08d32 |
partialSups_eq_biUnion_range | Mathlib/Order/PartialSups.lean | lemma partialSups_eq_biUnion_range (s : ℕ → Set α) (n : ℕ) :
partialSups s n = ⋃ i ∈ Finset.range (n + 1), s i | α : Type u_1
s : ℕ → Set α
n : ℕ
⊢ (partialSups s) n = ⋃ i ∈ range (n + 1), s i | ext | case h
α : Type u_1
s : ℕ → Set α
n : ℕ
x✝ : α
⊢ x✝ ∈ (partialSups s) n ↔ x✝ ∈ ⋃ i ∈ range (n + 1), s i | 4b944f186e62eb6e |
Language.leftQuotient_append | Mathlib/Computability/MyhillNerode.lean | theorem leftQuotient_append (x y : List α) :
L.leftQuotient (x ++ y) = (L.leftQuotient x).leftQuotient y | α : Type u
L : Language α
x y : List α
⊢ L.leftQuotient (x ++ y) = (L.leftQuotient x).leftQuotient y | simp [leftQuotient, Language] | no goals | 8941c51609492971 |
EuclideanGeometry.Sphere.secondInter_eq_self_iff | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | theorem Sphere.secondInter_eq_self_iff {s : Sphere P} {p : P} {v : V} :
s.secondInter p v = p ↔ ⟪v, p -ᵥ s.center⟫ = 0 | 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 : Sphere P
p : P
v : V
hp : -2 = 0 ∨ inner v (p -ᵥ s.center) = 0
hv : ¬v = 0
⊢ -2 ≠ 0 | norm_num | no goals | 765c9fa1cdc26c9d |
List.sublistForall₂_iff | Mathlib/Data/List/Forall2.lean | theorem sublistForall₂_iff {l₁ : List α} {l₂ : List β} :
SublistForall₂ R l₁ l₂ ↔ ∃ l, Forall₂ R l₁ l ∧ l <+ l₂ | case mpr
α : Type u_1
β : Type u_2
R : α → β → Prop
l₁ : List α
l₂ : List β
h : ∃ l, Forall₂ R l₁ l ∧ l <+ l₂
⊢ SublistForall₂ R l₁ l₂ | obtain ⟨l, hl1, hl2⟩ := h | case mpr.intro.intro
α : Type u_1
β : Type u_2
R : α → β → Prop
l₁ : List α
l₂ l : List β
hl1 : Forall₂ R l₁ l
hl2 : l <+ l₂
⊢ SublistForall₂ R l₁ l₂ | dd5b799edf73dec1 |
Set.preimage_singleton_false | Mathlib/Data/Set/Image.lean | theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} | α : Type u_1
p : α → Prop
⊢ p ⁻¹' {False} = {a | ¬p a} | ext | case h
α : Type u_1
p : α → Prop
x✝ : α
⊢ x✝ ∈ p ⁻¹' {False} ↔ x✝ ∈ {a | ¬p a} | b07aa977a919c602 |
Ordnode.Bounded.to_sep | Mathlib/Data/Ordmap/Ordset.lean | theorem Bounded.to_sep {t₁ t₂ o₁ o₂} {x : α}
(h₁ : Bounded t₁ o₁ (x : WithTop α)) (h₂ : Bounded t₂ (x : WithBot α) o₂) :
t₁.All fun y => t₂.All fun z : α => y < z | α : Type u_1
inst✝ : Preorder α
t₁ t₂ : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
x : α
h₁ : t₁.Bounded o₁ ↑x
h₂ : t₂.Bounded (↑x) o₂
⊢ All (fun y => All (fun z => y < z) t₂) t₁ | refine h₁.mem_lt.imp fun y yx => ?_ | α : Type u_1
inst✝ : Preorder α
t₁ t₂ : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
x : α
h₁ : t₁.Bounded o₁ ↑x
h₂ : t₂.Bounded (↑x) o₂
y : α
yx : y < x
⊢ All (fun z => y < z) t₂ | d57fcd55bb4a5d08 |
stereographic'_neg | Mathlib/Geometry/Manifold/Instances/Sphere.lean | private lemma stereographic'_neg {n : ℕ} [Fact (finrank ℝ E = n + 1)] (v : sphere (0 : E) 1) :
stereographic' n (-v) v = 0 | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr (↑(stereographic ⋯) v) = 0 | simp only [EmbeddingLike.map_eq_zero_iff] | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
⊢ ↑(stereographic ⋯) v = 0 | 20e68356a8b17c29 |
LeftCancelMonoid.mul_eq_one | Mathlib/Algebra/Group/Units/Basic.lean | theorem mul_eq_one : a * b = 1 ↔ a = 1 ∧ b = 1 :=
⟨fun h => ⟨LeftCancelMonoid.eq_one_of_mul_right h, LeftCancelMonoid.eq_one_of_mul_left h⟩, by
rintro ⟨rfl, rfl⟩
exact mul_one _⟩
| case intro
α : Type u
inst✝¹ : LeftCancelMonoid α
inst✝ : Subsingleton αˣ
⊢ 1 * 1 = 1 | exact mul_one _ | no goals | 0a4946293d6ba767 |
IsPrimitiveRoot.toInteger_sub_one_not_dvd_two | Mathlib/NumberTheory/Cyclotomic/Rat.lean | /-- We have that `hζ.toInteger - 1` does not divide `2`. -/
lemma toInteger_sub_one_not_dvd_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) : ¬ hζ.toInteger - 1 ∣ 2 := fun h ↦ by
have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
repl... | case hx
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K
h : hζ.toInteger - 1 ∣ ↑2
⊢ Prime ↑↑p | exact Nat.prime_iff_prime_int.1 hp.1 | no goals | 2a447481b82fe60a |
fourierIntegral_gaussian_innerProductSpace' | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :
𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =
(π / b) ^ (Module.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) | case h.e'_2.h.e'_7.h.e_z
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
x w v : V
⊢ -(2 * ↑π * ↑(inner w v) * I) + (-(b * ↑‖v‖ ^ 2) + 2 * ↑π * I * ↑(inner x v)) =
-(b * ↑‖v‖ ^ 2) + 2 * ↑π * ... | ring | no goals | 328850bb4ca1c803 |
summable_jacobiTheta₂_term_fderiv_iff | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | lemma summable_jacobiTheta₂_term_fderiv_iff (z τ : ℂ) :
Summable (jacobiTheta₂_term_fderiv · z τ) ↔ 0 < im τ | z τ : ℂ
hτ : 0 < τ.im
n : ℤ
⊢ 0 < 3 | norm_num | no goals | d5a065cb2c1034b1 |
Function.Surjective.pathConnectedSpace | Mathlib/Topology/Connected/PathConnected.lean | theorem Function.Surjective.pathConnectedSpace [PathConnectedSpace X]
{f : X → Y} (hf : Surjective f) (hf' : Continuous f) : PathConnectedSpace Y | X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
hf : Surjective f
hf' : Continuous f
⊢ IsPathConnected (range f) | exact isPathConnected_range hf' | no goals | 5b7a5066abab8e7e |
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