name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
Real.tendsto_eulerMascheroniSeq' | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | lemma tendsto_eulerMascheroniSeq' :
Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant) | ⊢ Tendsto (fun n => eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0) | suffices Tendsto (fun x : ℝ ↦ log (x + 1) - log x) atTop (𝓝 0) by
apply (this.comp tendsto_natCast_atTop_atTop).congr'
filter_upwards [eventually_ne_atTop 0] with n hn
simp [eulerMascheroniSeq, eulerMascheroniSeq', eq_false_intro hn] | ⊢ Tendsto (fun x => log (x + 1) - log x) atTop (𝓝 0) | cc6480414e009cf7 |
String.Iterator.ValidFor.remainingToString | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem remainingToString {it} (h : ValidFor l r it) : it.remainingToString = ⟨r⟩ | l r : List Char
it : Iterator
h : ValidFor l r it
⊢ it.remainingToString = { data := r } | cases h.out | case refl
l r : List Char
h : ValidFor l r { s := { data := l.reverseAux r }, i := { byteIdx := utf8Len l } }
⊢ { s := { data := l.reverseAux r }, i := { byteIdx := utf8Len l } }.remainingToString = { data := r } | 782b70dd556d5667 |
RootPairing.polarization_apply_eq_zero_iff | Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean | lemma polarization_apply_eq_zero_iff (m : M) :
P.Polarization m = 0 ↔ P.RootForm m = 0 | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
m : M
⊢ P.Polarization m = 0 ↔ P.RootForm m = 0 | rw [← flip_comp_polarization_eq_rootForm] | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
m : M
⊢ P.Polarization m = 0 ↔ (P.flip.toLin ∘ₗ P.Polarization) m = 0 | 8405c462392e53d5 |
BddAbove.continuous_convolution_right_of_integrable | Mathlib/Analysis/Convolution.lean | theorem _root_.BddAbove.continuous_convolution_right_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E']
(hbg : BddAbove (range fun x => ‖g x‖)) (hf : Integrable f μ) (hg : Continuous g) :
Continuous (f ⋆[L, μ] g) | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t : G ∂μ, ‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖ := by
filter_upwards with x; filter_upwards with t
apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl, le_ciSup hbg (x - t)] | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | 7cef45832d5d6aeb |
MeasureTheory.Measure.Regular.restrict_of_measure_ne_top | Mathlib/MeasureTheory/Measure/Regular.lean | theorem restrict_of_measure_ne_top [R1Space α] [BorelSpace α] [Regular μ]
{A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A) | case innerRegular
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).InnerRegularWRT IsCompact IsOpen | intro V hV r hr | case innerRegular
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A... | 99942bfc62b2d596 |
HolorIndex.drop_drop | Mathlib/Data/Holor.lean | theorem drop_drop : ∀ t : HolorIndex (ds₁ ++ ds₂ ++ ds₃), t.assocRight.drop.drop = t.drop
| ⟨is, h⟩ => Subtype.eq (by simp [add_comm, assocRight, drop, cast_type, List.drop_drop])
| ds₁ ds₂ ds₃ is : List ℕ
h : Forall₂ (fun x1 x2 => x1 < x2) is (ds₁ ++ ds₂ ++ ds₃)
⊢ ↑(assocRight ⟨is, h⟩).drop.drop = ↑(drop ⟨is, h⟩) | simp [add_comm, assocRight, drop, cast_type, List.drop_drop] | no goals | 82d0a6c432db2ef0 |
smul_finprod' | Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean | theorem smul_finprod' {ι : Sort*} [Finite ι] {f : ι → β} (r : α) :
r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x) | α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : CommMonoid β
inst✝¹ : MulDistribMulAction α β
ι : Sort u_4
inst✝ : Finite ι
f : ι → β
r : α
⊢ r • ∏ᶠ (x : ι), f x = ∏ᶠ (x : ι), r • f x | cases nonempty_fintype (PLift ι) | case intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : CommMonoid β
inst✝¹ : MulDistribMulAction α β
ι : Sort u_4
inst✝ : Finite ι
f : ι → β
r : α
val✝ : Fintype (PLift ι)
⊢ r • ∏ᶠ (x : ι), f x = ∏ᶠ (x : ι), r • f x | c39a4be152c4f848 |
Cycle.support_formPerm | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | theorem support_formPerm [Fintype α] (s : Cycle α) (h : Nodup s) (hn : Nontrivial s) :
support (formPerm s h) = s.toFinset | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : List α
h : Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)
hn : Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)
⊢ ∀ (x : α), s ≠ [x] | rintro _ rfl | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x✝ : α
h : Nodup (Quot.mk ⇑(IsRotated.setoid α) [x✝])
hn : Nontrivial (Quot.mk ⇑(IsRotated.setoid α) [x✝])
⊢ False | c975caac8b3ab4b4 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments... | simp [hasAssignment, hl, getElem!, l_in_bounds, h2, hasNegAssignment, decidableGetElem?] at h5 | no goals | 4e9908103f3f6d5f |
InfiniteGalois.fixingSubgroup_fixedField | Mathlib/FieldTheory/Galois/Infinite.lean | lemma fixingSubgroup_fixedField (H : ClosedSubgroup (K ≃ₐ[k] K)) [IsGalois k K] :
(IntermediateField.fixedField H).fixingSubgroup = H.1 | k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup (K ≃ₐ[k] K)
inst✝ : IsGalois k K
σ : K ≃ₐ[k] K
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
⊢ σ ∈ ↑H | by_contra h | k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup (K ≃ₐ[k] K)
inst✝ : IsGalois k K
σ : K ≃ₐ[k] K
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
⊢ False | 4195a2a4cd22a820 |
ProbabilityTheory.sum_variance_truncation_le | Mathlib/Probability/StrongLaw.lean | theorem sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
∑ j ∈ range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation X j ^ 2] ≤ 2 * 𝔼[X] | Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx :... | convert (div_le_one _).2 hx.2 | case h.e'_3.h.e'_6
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k ∈ range K
Ik : ↑k ≤ ... | 4c37cf41985268cf |
MeasureTheory.VectorMeasure.trim_eq_self | Mathlib/MeasureTheory/VectorMeasure/Basic.lean | theorem trim_eq_self : v.trim le_rfl = v | case h
α : Type u_1
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
i : Set α
hi : MeasurableSet i
⊢ ↑(v.trim ⋯) i = ↑v i | exact if_pos hi | no goals | e5d0658cec430037 |
Equiv.Perm.IsCycleOn.pow_apply_eq | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycleOn.pow_apply_eq {s : Finset α} (hf : f.IsCycleOn s) (ha : a ∈ s) {n : ℕ} :
(f ^ n) a = a ↔ #s ∣ n | case inr
α : Type u_2
f : Perm α
a : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
n : ℕ
hs : s.Nontrivial
h : ∀ (x : { x // x ∈ s }), ¬f ↑x = ↑x
this : orderOf (f.subtypePerm ⋯) = #(f.subtypePerm ⋯).support
⊢ (f ^ n) a = a ↔ #s ∣ n | simp only [coe_sort_coe, support_subtype_perm, ne_eq, h, not_false_eq_true, univ_eq_attach,
mem_attach, imp_self, implies_true, filter_true_of_mem, card_attach] at this | case inr
α : Type u_2
f : Perm α
a : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
n : ℕ
hs : s.Nontrivial
h : ∀ (x : { x // x ∈ s }), ¬f ↑x = ↑x
this : orderOf (f.subtypePerm ⋯) = #s
⊢ (f ^ n) a = a ↔ #s ∣ n | 86354cb7edac4dca |
AntitoneOn.integral_le_sum | Mathlib/Analysis/SumIntegralComparisons.lean | theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + i) | x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
hint : ∀ k < a, IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))
i : ℕ
hi : i ∈ Finset.range a
ia : i < a
⊢ IntervalIntegrable (fun x => f (x₀ + ↑i)) volume (x₀ + ↑i) (x₀ + ↑(i + 1)) | simp | no goals | 709ac30a4a15e5af |
Real.exists_extension_norm_eq | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ | case intro.intro
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
p : Subspace ℝ E
f : ↥p →L[ℝ] ℝ
g : E →ₗ[ℝ] ℝ
g_eq : ∀ (x : ↥{ domain := p, toFun := ↑f }.domain), g ↑x = ↑{ domain := p, toFun := ↑f } x
g_le : ∀ (x : E), g x ≤ ‖f‖ * ‖x‖
g' : E →L[ℝ] ℝ := g.mkContinuous ‖f‖ ⋯
⊢ ‖f‖ ≤ ‖g.mkContinuo... | refine f.opNorm_le_bound (norm_nonneg _) fun x => ?_ | case intro.intro
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
p : Subspace ℝ E
f : ↥p →L[ℝ] ℝ
g : E →ₗ[ℝ] ℝ
g_eq : ∀ (x : ↥{ domain := p, toFun := ↑f }.domain), g ↑x = ↑{ domain := p, toFun := ↑f } x
g_le : ∀ (x : E), g x ≤ ‖f‖ * ‖x‖
g' : E →L[ℝ] ℝ := g.mkContinuous ‖f‖ ⋯
x : ↥p
⊢ ‖f x‖ ≤ ‖g.m... | 0f306ef1f16e3415 |
Array.anyM_loop_iff_exists | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem anyM_loop_iff_exists {p : α → Bool} {as : Array α} {start stop} (h : stop ≤ as.size) :
anyM.loop (m := Id) p as stop h start = true ↔
∃ (i : Nat) (_ : i < as.size), start ≤ i ∧ i < stop ∧ p as[i] = true | α : Type u_1
p : α → Bool
as : Array α
start stop : Nat
h✝ : stop ≤ as.size
h₁ : start < stop
h₂ : ¬p as[start] = true
i : Nat
hi : i < as.size
ge : start + 1 ≤ i
lt : i < stop
h : p as[i] = true
this : start ≠ i
⊢ start ≤ i | omega | no goals | ffc0dbb5549e1171 |
Finset.pi_insert | Mathlib/Data/Finset/Pi.lean | theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
(ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) | case a
α : Type u_1
β : α → Type u
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : a ∉ s
⊢ ((insert a s).pi t).val = ((t a).biUnion fun b => image (Pi.cons s a b) (s.pi t)).val | rw [← (pi (insert a s) t).2.dedup] | case a
α : Type u_1
β : α → Type u
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : a ∉ s
⊢ ((insert a s).pi t).val.dedup = ((t a).biUnion fun b => image (Pi.cons s a b) (s.pi t)).val | 6b4ab4a69f59cfdf |
isComplete_iUnion_separated | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
IsComplete (⋃ i, s i) | case intro.intro
α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : Set (α × α)
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
hls : S ∈ l
hl_ne : l.NeBot
hl' : ∀ s ∈ 𝓤 α, ∃ t ∈ l, t ×ˢ t ⊆ ... | rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩ | case intro.intro.intro
α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : Set (α × α)
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
hls : S ∈ l
hl_ne : l.NeBot
hl' : ∀ s ∈ 𝓤 α, ∃ t ∈ l, t ×... | bb147cb415fdbcec |
inv_le_inv_iff | Mathlib/Algebra/Order/Group/Unbundled/Basic.lean | theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a | α : Type u
inst✝³ : Group α
inst✝² : LE α
inst✝¹ : MulLeftMono α
a b : α
inst✝ : MulRightMono α
⊢ a * a⁻¹ * b ≤ a * b⁻¹ * b ↔ b ≤ a | simp | no goals | 85f86bc66c862d56 |
Order.sub_one_covBy | Mathlib/Algebra/Order/SuccPred.lean | theorem sub_one_covBy [NoMinOrder α] (x : α) : x - 1 ⋖ x | α : Type u_1
inst✝⁴ : Preorder α
inst✝³ : Sub α
inst✝² : One α
inst✝¹ : PredSubOrder α
inst✝ : NoMinOrder α
x : α
⊢ x - 1 ⋖ x | rw [← pred_eq_sub_one] | α : Type u_1
inst✝⁴ : Preorder α
inst✝³ : Sub α
inst✝² : One α
inst✝¹ : PredSubOrder α
inst✝ : NoMinOrder α
x : α
⊢ pred x ⋖ x | 46680f86c4bcf2a0 |
FDerivMeasurableAux.D_subset_differentiable_set | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L ... | ring | no goals | 28c357cacce7671e |
AEMeasurable.sum_measure | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | theorem sum_measure [Countable ι] {μ : ι → Measure α} (h : ∀ i, AEMeasurable f (μ i)) :
AEMeasurable f (sum μ) | case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) ... | intro t ht | case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) ... | df9a789791001abe |
EuclideanGeometry.tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.tan (∡ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂ | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ (∡ p₂ p₃ p₁).tan * dist p₃ p₂ = dist p₁ p₂ | have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₂ p₃ p₁).sign = 1
⊢ (∡ p₂ p₃ p₁).tan * dist p₃ p₂ = dist p₁ p₂ | b67c0a9de406c5b9 |
ContinuousMap.induction_on | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem ContinuousMap.induction_on {𝕜 : Type*} [RCLike 𝕜] {s : Set 𝕜}
{p : C(s, 𝕜) → Prop} (const : ∀ r, p (.const s r)) (id : p (.restrict s <| .id 𝕜))
(star_id : p (star (.restrict s <| .id 𝕜)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(closure : (∀ f ∈ (polynomi... | case mem.inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g... | simpa only [toContinuousMapOnAlgHom_apply, toContinuousMapOn_X_eq_restrict_id] | no goals | 01b202580110cbda |
Polynomial.degree_pow_le | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem degree_pow_le (p : R[X]) : ∀ n : ℕ, degree (p ^ n) ≤ n • degree p
| 0 => by rw [pow_zero, zero_nsmul]; exact degree_one_le
| n + 1 =>
calc
degree (p ^ (n + 1)) ≤ degree (p ^ n) + degree p | R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ (p ^ (n + 1)).degree ≤ (p ^ n).degree + p.degree | rw [pow_succ] | R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ (p ^ n * p).degree ≤ (p ^ n).degree + p.degree | 1a032331b3bf632a |
sUnion_memPartition | Mathlib/Data/Set/MemPartition.lean | @[simp]
lemma sUnion_memPartition (f : ℕ → Set α) (n : ℕ) : ⋃₀ memPartition f n = univ | case succ.h
α : Type u_1
f : ℕ → Set α
n : ℕ
ih : ⋃₀ memPartition f n = univ
x : α
this : ∃ t ∈ memPartition f n, x ∈ t
⊢ ∃ t ∈ {s | ∃ u ∈ memPartition f n, s = u ∩ f n ∨ s = u \ f n}, x ∈ t | obtain ⟨t, ht, hxt⟩ := this | case succ.h.intro.intro
α : Type u_1
f : ℕ → Set α
n : ℕ
ih : ⋃₀ memPartition f n = univ
x : α
t : Set α
ht : t ∈ memPartition f n
hxt : x ∈ t
⊢ ∃ t ∈ {s | ∃ u ∈ memPartition f n, s = u ∩ f n ∨ s = u \ f n}, x ∈ t | 92b61a089b6c0707 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRatUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem assignmentsInvariant_insertRatUnits {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRatUnits f units).1 | case left
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitInvariant f.assignments ⋯ ratUnits assignments... | have h2 : (insertRatUnits f units).fst.ratUnits[j2] = (i, false) := by
rw [h2]
simp only [Prod.mk.injEq, and_true]
rfl | case left
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitInvariant f.assignments ⋯ ratUnits assignments... | b6d274376b14ee0b |
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_of_disjoint | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepSet.indep_generateFrom_of_disjoint {s : ι → Set Ω}
(hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (S T : Set ι) (hST : Disjoint S T) :
Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) κ μ | case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
⊢ Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t... | obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
exists_ae_eq_isMarkovKernel hs.ae_isProbabilityMeasure hμ | case inr.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᶠ[ae μ] ⇑η
hη : IsMarkovKernel η
⊢ Indep (g... | a2cf118cd07d37b1 |
Filter.eventually_lt_of_lt_liminf | Mathlib/Order/LiminfLimsup.lean | theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
{b : β} (h : b < liminf u f)
(hu : f.IsBoundedUnder (· ≥ ·) u | α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
⊢ ∃ c, ∃ (_ : c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}), b < c | simp_rw [exists_prop] | α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
⊢ ∃ c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}, b < c | 82ef5ffe328c509b |
MonomialOrder.leadingCoeff_monomial | Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean | theorem leadingCoeff_monomial {d : σ →₀ ℕ} (c : R) :
m.leadingCoeff (monomial d c) = c | σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
d : σ →₀ ℕ
c : R
⊢ m.leadingCoeff ((monomial d) c) = c | classical
simp only [leadingCoeff, degree_monomial]
split_ifs with hc <;> simp [hc] | no goals | f8562f63559ed44e |
StrictMono.not_bddAbove_range_of_wellFoundedLT | Mathlib/Order/WellFounded.lean | theorem StrictMono.not_bddAbove_range_of_wellFoundedLT {f : β → β} [WellFoundedLT β] [NoMaxOrder β]
(hf : StrictMono f) : ¬ BddAbove (Set.range f) | case intro.intro
β : Type u_2
inst✝² : LinearOrder β
f : β → β
inst✝¹ : WellFoundedLT β
inst✝ : NoMaxOrder β
hf : StrictMono f
a : β
ha : a ∈ upperBounds (Set.range f)
b : β
hb : a < b
⊢ False | exact ((hf.le_apply.trans_lt (hf hb)).trans_le <| ha (Set.mem_range_self _)).false | no goals | 5c12033566e5c122 |
MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_memLp | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem measure_preimage_lt_top_of_memLp (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) (f : α →ₛ E)
(hf : MemLp f p μ) (y : E) (hy_ne : y ≠ 0) : μ (f ⁻¹' {y}) < ∞ | case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
hp_pos_real : 0 < p.toReal
hf_eLpNorm : ∀ a ∈ f.range, ‖a‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {a}) < ⊤
hyf : y ∈ f.range
⊢ μ (⇑f ⁻¹' ... | specialize hf_eLpNorm y hyf | case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
hp_pos_real : 0 < p.toReal
hyf : y ∈ f.range
hf_eLpNorm : ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y}) < ⊤
⊢ μ (⇑f ⁻¹' {y}) < ⊤ | 33a6699fa8a77822 |
Matrix.PosSemidef.conjTranspose_mul_mul_same | Mathlib/LinearAlgebra/Matrix/PosDef.lean | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) | case right
n : Type u_2
R : Type u_3
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
A : Matrix n n R
hA : A.PosSemidef
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
⊢ ∀ (x : m → R), 0 ≤ star x ⬝ᵥ (Bᴴ * A * B) *ᵥ x | intro x | case right
n : Type u_2
R : Type u_3
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
A : Matrix n n R
hA : A.PosSemidef
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
x : m → R
⊢ 0 ≤ star x ⬝ᵥ (Bᴴ * A * B) *ᵥ x | 6cfd56aa47280d8c |
gramSchmidt_ne_zero_coe | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
(h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∑ i ∈ Finset.Iio n, ↑((orthogo... | apply Submodule.sum_mem _ _ | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∀ c ∈ Finset.Iio n,
↑((ort... | 69b931e205d84775 |
Nat.log_eq_iff | Mathlib/Data/Nat/Log.lean | theorem log_eq_iff {b m n : ℕ} (h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0) :
log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1) | case inr
b m n : ℕ
h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0
hbn : ¬(1 < b ∧ n ≠ 0)
hm : m ≠ 0
⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1) | rw [not_and_or, not_lt, Ne, not_not] at hbn | case inr
b m n : ℕ
h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0
hbn : b ≤ 1 ∨ n = 0
hm : m ≠ 0
⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1) | d0a2456541327695 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.safe_insert_of_performRupCheck_insertRat | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem safe_insert_of_performRupCheck_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (rupHints : Array Nat) :
(performRupCheck (insertRatUnits f (negate c)).1 rupHints).2.2.1 = true
→
Limplies (PosFin n) f (f.insert c) | case inr
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
rupHints : Array Nat
performRupCheck_success :
(Array.foldl (confirmRupHint (f.insertRatUnits c.negate).1.clauses)
((f.insertRatUnits c.negate).1.assignments, [], false, false) rupHints).2.2.fst =
... | exact pf c' c'_in_f | no goals | 7b923cb96f92e5d9 |
Associates.prod_le_prod_iff_le | Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean | theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
(hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q | case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q | rw [← prod_eq_zero_iff, eqc, hc, mul_zero] | no goals | 9197e5d1051792cf |
MeasureTheory.measure_univ_of_isMulLeftInvariant | Mathlib/MeasureTheory/Group/Measure.lean | theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
(μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ | G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed... | exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _) | no goals | d58a29d70ac786c1 |
Multiset.bind_bind | Mathlib/Data/Multiset/Bind.lean | theorem bind_bind (m : Multiset α) (n : Multiset β) {f : α → β → Multiset γ} :
((bind m) fun a => (bind n) fun b => f a b) = (bind n) fun b => (bind m) fun a => f a b :=
Multiset.induction_on m (by simp) (by simp +contextual)
| α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
n : Multiset β
f : α → β → Multiset γ
⊢ ∀ (a : α) (s : Multiset α),
((s.bind fun a => n.bind fun b => f a b) = n.bind fun b => s.bind fun a => f a b) →
((a ::ₘ s).bind fun a => n.bind fun b => f a b) = n.bind fun b => (a ::ₘ s).bind fun a => f a b | simp +contextual | no goals | abdc90d1a3f89e14 |
HomologicalComplex.pOpcycles_extendOpcyclesIso_inv | Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean | @[reassoc (attr := simp)]
lemma pOpcycles_extendOpcyclesIso_inv :
K.pOpcycles j ≫ (K.extendOpcyclesIso e hj').inv =
(K.extendXIso e hj').inv ≫ (K.extend e).pOpcycles j' | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j : ι
j' : ι'
hj' : e.f j = j'
inst✝¹ : K.HasHomology j
inst✝ : (K.extend e).HasHomology j'
⊢ (K.sc j).pOpcycl... | rw [ShortComplex.RightHomologyData.pOpcycles_comp_opcyclesIso_hom_assoc] | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j : ι
j' : ι'
hj' : e.f j = j'
inst✝¹ : K.HasHomology j
inst✝ : (K.extend e).HasHomology j'
⊢ (K.sc j).pOpcycl... | d598f615c159cc2f |
PrimeSpectrum.iSup_basicOpen_eq_top_iff' | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | lemma iSup_basicOpen_eq_top_iff' {s : Set R} :
(⨆ i ∈ s, PrimeSpectrum.basicOpen i) = ⊤ ↔ Ideal.span s = ⊤ | R : Type u
inst✝ : CommSemiring R
s : Set R
⊢ ⨆ i ∈ s, basicOpen i = ⊤ ↔ ⨆ i, basicOpen ↑i = ⊤ | simp | no goals | c483e22b2c8fc038 |
HasFiniteFPowerSeriesOnBall.fderiv | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem HasFiniteFPowerSeriesOnBall.fderiv
(h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r) :
HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
n : ℕ
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r
⊢ HasFiniteFPowerSeries... | refine .congr (f := fun z ↦ continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1)) ?_
fun z hz ↦ ?_ | case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
n : ℕ
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r
⊢ HasFini... | b335a221088906cf |
List.take_of_length_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean | theorem take_of_length_le {l : List α} (h : l.length ≤ i) : take i l = l | α : Type u_1
i : Nat
l : List α
h : l.length ≤ i
this : take i l ++ drop i l = l
⊢ take i l = l | rw [drop_of_length_le h, append_nil] at this | α : Type u_1
i : Nat
l : List α
h : l.length ≤ i
this : take i l = l
⊢ take i l = l | 6c0eac07c538a813 |
ComplexShape.π_symm | Mathlib/Algebra/Homology/ComplexShapeSigns.lean | lemma π_symm (i₁ : I₁) (i₂ : I₂) :
π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ | I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : TotalComplexShape c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c₁₂
i₁ : I₁
i₂ : I₂
⊢ c₂.π c₁ c₁₂ (i₂, i₁) = c₁.π c₂ c₁₂ (i₁, i₂) | apply TotalComplexShapeSymmetry.symm | no goals | aefa0261e4cfe0e7 |
finiteMultiplicity_mul_aux | Mathlib/RingTheory/Multiplicity.lean | theorem finiteMultiplicity_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fu... | α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this✝ : p ∣ a * b
x✝ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x * b * p = p ^ ... | simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add] | no goals | e3775fe83a9cfc31 |
MeasureTheory.integrableOn_univ | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ | α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
μ : Measure α
⊢ IntegrableOn f univ μ ↔ Integrable f μ | rw [IntegrableOn, Measure.restrict_univ] | no goals | dc45be82ebcf739a |
Equiv.Perm.ofSign_disjoint | Mathlib/GroupTheory/Perm/Sign.lean | lemma ofSign_disjoint : _root_.Disjoint (ofSign 1 : Finset (Perm α)) (ofSign (-1)) | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
hσ : σ ∈ ofSign 1
hτ : σ ∈ ofSign (-1)
⊢ False | rw [mem_ofSign] at hσ hτ | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
hσ : sign σ = 1
hτ : sign σ = -1
⊢ False | e53c5c58b6af3a43 |
ProbabilityTheory.integrable_exp_mul_of_abs_le | Mathlib/Probability/Moments/IntegrableExpMul.lean | /-- If `ω ↦ exp (u * X ω)` is integrable at `u` and `-u`, then it is integrable on `[-u, u]`. -/
lemma integrable_exp_mul_of_abs_le
(hu_int_pos : Integrable (fun ω ↦ exp (u * X ω)) μ)
(hu_int_neg : Integrable (fun ω ↦ exp (- u * X ω)) μ)
(htu : |t| ≤ |u|) :
Integrable (fun ω ↦ exp (t * X ω)) μ | case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω => rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω => rexp (-u * X ω)) μ
htu : |t| ≤ |u|
⊢ Integrable (fun ω => rexp (|u| * X ω)) μ | rcases le_total 0 u with hu | hu | case refine_2.inl
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω => rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω => rexp (-u * X ω)) μ
htu : |t| ≤ |u|
hu : 0 ≤ u
⊢ Integrable (fun ω => rexp (|u| * X ω)) μ
case refine_2.inr
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω ... | c59caf165b150fde |
BitVec.zero_sshiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem zero_sshiftRight {n : Nat} : (0#w).sshiftRight n = 0#w | w n : Nat
⊢ (0#w).sshiftRight n = 0#w | ext i h | case pred
w n i : Nat
h : i < w
⊢ ((0#w).sshiftRight n).getLsbD i = (0#w).getLsbD i | d91b7635f218cbff |
Trivialization.apply_eq_prod_continuousLinearEquivAt | Mathlib/Topology/VectorBundle/Basic.lean | theorem apply_eq_prod_continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
(hb : b ∈ e.baseSet) (z : E b) : e ⟨b, z⟩ = (b, e.continuousLinearEquivAt R b hb z) | case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → To... | exact hb | no goals | 5868769150d7cbc0 |
PythagoreanTriple.even_odd_of_coprime | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :
x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0 | case inr.inr
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hx : x % 2 = 1
hy : y % 2 = 1
⊢ False | obtain ⟨x0, y0, rfl, rfl⟩ : ∃ x0 y0, x = x0 * 2 + 1 ∧ y = y0 * 2 + 1 := by
obtain ⟨x0, hx2⟩ := exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hx)
obtain ⟨y0, hy2⟩ := exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hy)
rw [sub_eq_iff_eq_add] at hx2 hy2
exact ⟨x0, y0, hx2, hy2⟩ | case inr.inr.intro.intro.intro
z x0 y0 : ℤ
hx : (x0 * 2 + 1) % 2 = 1
hy : (y0 * 2 + 1) % 2 = 1
h : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z
hc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1
⊢ False | 48f64ea05953c676 |
Batteries.RBNode.foldr_reverse | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem foldr_reverse {α β : Type _} {t : RBNode α} {f : α → β → β} {init : β} :
t.reverse.foldr f init = t.foldl (flip f) init :=
foldl_reverse.symm.trans (by simp; rfl)
| α : Type u_1
β : Type u_2
t : RBNode α
f : α → β → β
init : β
⊢ foldl (fun a b => f b a) init t.reverse.reverse = foldl (flip f) init t | simp | α : Type u_1
β : Type u_2
t : RBNode α
f : α → β → β
init : β
⊢ foldl (fun a b => f b a) init t = foldl (flip f) init t | 50b680d84eae0c3b |
PartENat.lt_find | Mathlib/Data/Nat/PartENat.lean | theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : PartENat) < find P | P : ℕ → Prop
inst✝ : DecidablePred P
n : ℕ
h : ∀ m ≤ n, ¬P m
⊢ ∀ (h : (find P).Dom), n < (find P).get h | intro h₁ | P : ℕ → Prop
inst✝ : DecidablePred P
n : ℕ
h : ∀ m ≤ n, ¬P m
h₁ : (find P).Dom
⊢ n < (find P).get h₁ | 33f16598e16dc969 |
one_div_pow_le_one_div_pow_of_le | Mathlib/Algebra/Order/Field/Basic.lean | theorem one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
1 / a ^ n ≤ 1 / a ^ m | α : Type u_2
inst✝ : LinearOrderedSemifield α
a : α
a1 : 1 ≤ a
m n : ℕ
mn : m ≤ n
⊢ 1 / a ^ n ≤ 1 / a ^ m | refine (one_div_le_one_div ?_ ?_).mpr (pow_right_mono₀ a1 mn) <;>
exact pow_pos (zero_lt_one.trans_le a1) _ | no goals | 30efe5e517cf3a6d |
Subgroup.fg_iff_submonoid_fg | Mathlib/GroupTheory/Finiteness.lean | theorem Subgroup.fg_iff_submonoid_fg (P : Subgroup G) : P.FG ↔ P.toSubmonoid.FG | case mpr.intro.refine_1
G : Type u_3
inst✝ : Group G
P : Subgroup G
S : Finset G
hS : Submonoid.closure ↑S = P.toSubmonoid
⊢ closure ↑S ≤ P | rw [Subgroup.closure_le, ← Subgroup.coe_toSubmonoid, ← hS] | case mpr.intro.refine_1
G : Type u_3
inst✝ : Group G
P : Subgroup G
S : Finset G
hS : Submonoid.closure ↑S = P.toSubmonoid
⊢ ↑S ⊆ ↑(Submonoid.closure ↑S) | 11ccabe6f8ec3164 |
MvPolynomial.vars_sub_subset | Mathlib/Algebra/MvPolynomial/CommRing.lean | theorem vars_sub_subset [DecidableEq σ] : (p - q).vars ⊆ p.vars ∪ q.vars | R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p q : MvPolynomial σ R
inst✝ : DecidableEq σ
⊢ (p - q).vars ⊆ p.vars ∪ q.vars | convert vars_add_subset p (-q) using 2 <;> simp [sub_eq_add_neg] | no goals | 2d80017a45e04164 |
IsCompact.exists_forall_le' | Mathlib/Topology/Order/Compact.lean | theorem IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
{s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b | case inr.intro.intro
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
hs' : s.Nonempty
x : β
hx : x ∈ s
hx' : IsMinOn f s x
⊢ ... | exact ⟨f x, hf' x hx, hx'⟩ | no goals | 0928e6d3df8c75af |
HahnSeries.addOppositeEquiv_orderTop | Mathlib/RingTheory/HahnSeries/Addition.lean | @[simp]
lemma addOppositeEquiv_orderTop (x : HahnSeries Γ (Rᵃᵒᵖ)) :
(addOppositeEquiv x).unop.orderTop = x.orderTop | Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddMonoid R
x : HahnSeries Γ Rᵃᵒᵖ
⊢ (AddOpposite.unop (addOppositeEquiv x)).orderTop = x.orderTop | classical
simp only [orderTop, AddOpposite.unop_op, mk_eq_zero, EmbeddingLike.map_eq_zero_iff,
addOppositeEquiv_support, ne_eq]
simp only [addOppositeEquiv_apply, AddOpposite.unop_op, mk_eq_zero, coeff_zero]
simp_rw [HahnSeries.ext_iff, funext_iff]
simp only [Pi.zero_apply, AddOpposite.unop_eq_zero_iff, coeff_zero] | no goals | 66cf7b891a048377 |
ContDiffWithinAt.fderivWithin'' | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem ContDiffWithinAt.fderivWithin'' {f : E → F → G} {g : E → F} {t : Set F}
(hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : ContDiffWithinAt 𝕜 m g s x₀)
(ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)) (hmn : m + 1 ≤ n)
(hgt : t ∈ 𝓝[g '' s] g x... | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m : WithTop ℕ∞
f : E → F → G
g : E → F
t : Set F
h... | obtain ⟨v, hv, -, f', hvf', hf'⟩ := hf.hasFDerivWithinAt_nhds (by simp) hg hgt | case intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m : WithTop ℕ∞
... | ce8ad388b9bbe46b |
Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd | Mathlib/Data/Set/MulAntidiagonal.lean | theorem fst_eq_fst_iff_snd_eq_snd : (x : α × α).1 = (y : α × α).1 ↔ (x : α × α).2 = (y : α × α).2 :=
⟨fun h =>
mul_left_cancel
(y.2.2.2.trans <| by
rw [← h]
exact x.2.2.2.symm).symm,
fun h =>
mul_right_cancel
(y.2.2.2.trans <| by
rw [← h]
exact x.2.2.2.s... | α : Type u_1
inst✝ : CancelCommMonoid α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑y).1 * (↑x).2 | rw [← h] | α : Type u_1
inst✝ : CancelCommMonoid α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑x).1 * (↑x).2 | c2b68b98fd393ff0 |
contMDiff_of_contMDiff_inl | Mathlib/Geometry/Manifold/ContMDiff/Constructions.lean | lemma contMDiff_of_contMDiff_inl {f : M → N}
(h : ContMDiff I J n ((@Sum.inl N N') ∘ f)) : ContMDiff I J n f | 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
i... | apply (contMDiff_id.sumElim contMDiff_const).contMDiffOn (s := @Sum.inl N N' '' univ).comp h | 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
i... | 9dafec4728754145 |
IsPGroup.iff_card | Mathlib/GroupTheory/PGroup.lean | theorem iff_card [Fact p.Prime] [Finite G] : IsPGroup p G ↔ ∃ n : ℕ, Nat.card G = p ^ n | case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite G
hG : Nat.card G ≠ 0
h : IsPGroup p G
q : ℕ
hq : q ∈ (Nat.card G).primeFactorsList
hq1 : Nat.Prime q
hq2 : q ∣ Nat.card G
this : Fact (Nat.Prime q)
g : G
hg : orderOf g = q
⊢ q = p | obtain ⟨k, hk⟩ := (iff_orderOf.mp h) g | case intro.intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite G
hG : Nat.card G ≠ 0
h : IsPGroup p G
q : ℕ
hq : q ∈ (Nat.card G).primeFactorsList
hq1 : Nat.Prime q
hq2 : q ∣ Nat.card G
this : Fact (Nat.Prime q)
g : G
hg : orderOf g = q
k : ℕ
hk : orderOf g = p ^ k
⊢ q = p | cdf7a3b03afadbaa |
Cardinal.lt_wf | Mathlib/SetTheory/Cardinal/Basic.lean | theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
⟨fun a =>
by_contradiction fun h => by
let ι := { c : Cardinal // ¬Acc (· < ·) c }
let f : ι → Cardinal := Subtype.val
haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩
obtain ⟨⟨c : Cardinal, hc : ¬Acc (· < ·) c⟩, ⟨h_1 : ∀ j, (f ⟨c, hc⟩).out ↪ (f j).out... | a : Cardinal.{u}
h : ¬Acc (fun x1 x2 => x1 < x2) a
ι : Type (max 0 (?u.56390 + 1)) := { c // ¬Acc (fun x1 x2 => x1 < x2) c }
⊢ False | let f : ι → Cardinal := Subtype.val | a : Cardinal.{u}
h : ¬Acc (fun x1 x2 => x1 < x2) a
ι : Type (max 0 (?u.56390 + 1)) := { c // ¬Acc (fun x1 x2 => x1 < x2) c }
f : ι → Cardinal.{?u.56390} := Subtype.val
⊢ False | 2083f4a91b2b3d48 |
Complex.HadamardThreeLines.interpStrip_eq_of_pos | Mathlib/Analysis/Complex/Hadamard.lean | /-- Rewrite for `InterpStrip` when `0 < sSupNormIm f 0` and `0 < sSupNormIm f 1`. -/
lemma interpStrip_eq_of_pos (z : ℂ) (h0 : 0 < sSupNormIm f 0) (h1 : 0 < sSupNormIm f 1) :
interpStrip f z = sSupNormIm f 0 ^ (1 - z) * sSupNormIm f 1 ^ z | E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
h0 : 0 < sSupNormIm f 0
h1 : 0 < sSupNormIm f 1
⊢ interpStrip f z = ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z | simp only [ne_of_gt h0, ne_of_gt h1, interpStrip, if_false, or_false] | no goals | 8db93de84433c1a8 |
VitaliFamily.exists_measurable_supersets_limRatio | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0 | 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 q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a)... | refine le_antisymm ((measure_mono A).trans ?_) bot_le | 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 q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a)... | f2acb3dc8c3a0a55 |
SimpleGraph.card_commonNeighbors_top | Mathlib/Combinatorics/SimpleGraph/Finite.lean | theorem card_commonNeighbors_top [DecidableEq V] {v w : V} (h : v ≠ w) :
Fintype.card ((⊤ : SimpleGraph V).commonNeighbors v w) = Fintype.card V - 2 | V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ #(Set.univ.toFinset \ {v, w}.toFinset) = Fintype.card V - 2 | rw [Finset.card_sdiff] | V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ #Set.univ.toFinset - #{v, w}.toFinset = Fintype.card V - 2
V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ {v, w}.toFinset ⊆ Set.univ.toFinset | 3c759119a4b352a3 |
Array.getElem_swap' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_swap' (a : Array α) (i j : Nat) {hi hj} (k : Nat) (hk : k < a.size) :
(a.swap i j hi hj)[k]'(by simp_all) = if k = i then a[j] else if k = j then a[i] else a[k] | case isTrue
α : Type u_1
a : Array α
i j : Nat
hi : i < a.size
hj : j < a.size
k : Nat
hk : k < a.size
h✝ : k = i
⊢ (a.swap i j hi hj)[k] = a[j] | simp_all only [getElem_swap_left] | no goals | 57b3b7cba232ccbc |
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 | ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, (f i).IsPrime) →
s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn :... | simp only [Set.biUnion_insert] at h ⊢ | ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, (f i).IsPrime) →
s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn :... | fa41770a7d2d9fd9 |
AlgebraicGeometry.Scheme.GlueData.isOpen_iff | Mathlib/AlgebraicGeometry/Gluing.lean | theorem isOpen_iff (U : Set D.glued.carrier) : IsOpen U ↔ ∀ i, IsOpen ((D.ι i).base ⁻¹' U) | D : GlueData
U : Set ↑↑D.glued.toPresheafedSpace
⊢ (∀ (i : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J),
IsOpen
(⇑(ConcreteCategory.hom
(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i)) ⁻¹'... | apply forall_congr' | case h
D : GlueData
U : Set ↑↑D.glued.toPresheafedSpace
⊢ ∀ (a : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J),
IsOpen
(⇑(ConcreteCategory.hom
(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι a))... | a33f798209b44a35 |
RingHom.isStandardSmoothOfRelativeDimension_isStableUnderBaseChange | Mathlib/RingTheory/RingHom/StandardSmooth.lean | lemma isStandardSmoothOfRelativeDimension_isStableUnderBaseChange :
IsStableUnderBaseChange (@IsStandardSmoothOfRelativeDimension.{t, w} n) | case h.e'_6
n : ℕ
R S T : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardSmoothOfRelativeDimension n R T
this : Algebra.IsStandardSmoothOfRelativeDimension n S (S ⊗[R] T)
⊢ Algebra.TensorProduct.includeLeftRingHom.toAlgebra = Algebra.T... | ext | case h.e'_6.h
n : ℕ
R S T : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardSmoothOfRelativeDimension n R T
this : Algebra.IsStandardSmoothOfRelativeDimension n S (S ⊗[R] T)
r✝ : S
x✝ : S ⊗[R] T
⊢ (let_fun I := Algebra.TensorProduct.inc... | 14172c066b23aad2 |
Batteries.RBNode.Ordered.memP_iff_upperBound? | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem Ordered.memP_iff_upperBound? [@TransCmp α cmp] [IsCut cmp cut] (ht : Ordered cmp t) :
t.MemP cut ↔ ∃ x, t.upperBound? cut = some x ∧ cut x = .eq | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
⊢ cut x = Ordering.eq | cases ex : cut x | case lt
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
ex : cut x = Ordering.lt
⊢ Ordering.lt = Ordering.eq
case eq
α... | 34c79185e82ce46b |
Std.Tactic.BVDecide.LRAT.Internal.CNF.convertLRAT_readyForRatAdd | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Convert.lean | theorem CNF.convertLRAT_readyForRatAdd (cnf : CNF Nat) :
DefaultFormula.ReadyForRatAdd (CNF.convertLRAT cnf) | cnf : CNF Nat
⊢ (convertLRAT cnf).ReadyForRatAdd | unfold CNF.convertLRAT | cnf : CNF Nat
⊢ (let lifted := lift cnf;
let lratCnf := convertLRAT' lifted;
DefaultFormula.ofArray (none :: lratCnf).toArray).ReadyForRatAdd | 6721871e1b681c10 |
Continuous.continuousOn | Mathlib/Topology/ContinuousOn.lean | theorem Continuous.continuousOn (h : Continuous f) : ContinuousOn f s | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
h : Continuous f
⊢ ContinuousOn f s | rw [continuous_iff_continuousOn_univ] at h | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
h : ContinuousOn f univ
⊢ ContinuousOn f s | f5a067e05d3992af |
CategoryTheory.Subgroupoid.mul_mem_cancel_left | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | theorem mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e | C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e | constructor | case mp
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ f ≫ g ∈ S.arrows c e → g ∈ S.arrows d e
case mpr
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ g ∈ S.arrows d e → f ≫ g ∈ S.arrows c e | 1dcdf4857cfeffc1 |
Complex.isConnected_of_lowerHalfPlane | Mathlib/Analysis/Complex/Convex.lean | lemma Complex.isConnected_of_lowerHalfPlane {r} {s : Set ℂ} (hs₁ : {z | z.im < r} ⊆ s)
(hs₂ : s ⊆ {z | z.im ≤ r}) : IsConnected s | r : ℝ
s : Set ℂ
hs₁ : {z | z.im < r} ⊆ s
hs₂ : s ⊆ {z | z.im ≤ r}
⊢ IsConnected {z | z.im < r} | exact (convex_halfSpace_im_lt r).isConnected ⟨(r - 1) * I, by simp⟩ | no goals | f2eeaafd90e9656f |
Group.ext | Mathlib/Algebra/Group/Ext.lean | theorem Group.ext {G : Type*} ⦃g₁ g₂ : Group G⦄ (h_mul : g₁.mul = g₂.mul) : g₁ = g₂ | G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
⊢ g₁ = g₂ | let f : @MonoidHom G G g₁.toMulOneClass g₂.toMulOneClass :=
@MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
(fun x y => congr_fun (congr_fun h_mul x) y) | G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
f : G →* G := { toFun := id, map_one' := h₁, map_mul' := ⋯ }
⊢ g₁ = g₂ | 40365b7f868018d4 |
Filter.HasBasis.equicontinuousAt_iff_right | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k | ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ EquicontinuousAt F x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k | rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff] | ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ (∀ (i : κ),
p i → ∀ᶠ (x : X) in 𝓝 x₀, (⇑ofFun ∘ swap F) x ∈ {g | ((⇑ofFun ∘ swap F) x₀, g) ∈ UniformFun.gen ι α (s i)}) ↔
∀ (k : κ),... | bb2ea194c9e25e81 |
Real.Angle.arg_toCircle | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | @[simp] lemma arg_toCircle (θ : Real.Angle) : (arg θ.toCircle : Angle) = θ | θ : Angle
⊢ ↑(↑θ.toCircle).arg = θ | induction θ using Real.Angle.induction_on | case h
x✝ : ℝ
⊢ ↑(↑(↑x✝).toCircle).arg = ↑x✝ | f3f2bde3cb20d916 |
sign_finRotate | Mathlib/GroupTheory/Perm/Fin.lean | theorem sign_finRotate (n : ℕ) : Perm.sign (finRotate (n + 1)) = (-1) ^ n | case succ
n : ℕ
ih : sign (finRotate (n + 1)) = (-1) ^ n
⊢ sign (finRotate (n + 1 + 1)) = (-1) ^ (n + 1) | rw [finRotate_succ_eq_decomposeFin] | case succ
n : ℕ
ih : sign (finRotate (n + 1)) = (-1) ^ n
⊢ sign (decomposeFin.symm (1, finRotate (n + 1))) = (-1) ^ (n + 1) | f9f2177ec4cba59d |
zeta_nat_eq_tsum_of_gt_one | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | theorem zeta_nat_eq_tsum_of_gt_one {k : ℕ} (hk : 1 < k) :
riemannZeta k = ∑' n : ℕ, 1 / (n : ℂ) ^ k | k : ℕ
hk : 1 < k
⊢ 1 < (↑k).re | rwa [← ofReal_natCast, ofReal_re, ← Nat.cast_one, Nat.cast_lt] | no goals | a3b6be3587306bf5 |
List.not_lex_antisymm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean | theorem not_lex_antisymm [DecidableEq α] {r : α → α → Prop} [DecidableRel r]
(antisymm : ∀ x y : α, ¬ r x y → ¬ r y x → x = y)
{as bs : List α} (h₁ : ¬ Lex r bs as) (h₂ : ¬ Lex r as bs) : as = bs :=
match as, bs with
| [], [] => rfl
| [], _::_ => False.elim <| h₂ (List.Lex.nil ..)
| _::_, [] ... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
r : α → α → Prop
inst✝ : DecidableRel r
antisymm : ∀ (x y : α), ¬r x y → ¬r y x → x = y
as✝ bs✝ : List α
a : α
as : List α
b : α
bs : List α
h₁ : ¬Lex r (b :: bs) (a :: as)
h₂ : ¬Lex r (a :: as) (b :: bs)
hab : ¬r a b
eq : ¬a = b
hba : r b a
⊢ False | exact h₁ (Lex.rel hba) | no goals | bc20d604ba52bc59 |
Set.Iic_mul_Iio_subset' | Mathlib/Algebra/Order/Group/Pointwise/Interval.lean | theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) | case intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b : α
this : MulRightMono α
y : α
hya : y ∈ Iic a
z : α
hzb : z ∈ Iio b
⊢ (fun x1 x2 => x1 * x2) y z ∈ Iio (a * b) | exact mul_lt_mul_of_le_of_lt hya hzb | no goals | d29ae3cc2f5221d0 |
SimpleGraph.IsAcyclic.path_unique | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q | case mk.mk.cons.inl.cons.inl.inl.refl
V : Type u
G : SimpleGraph V
v w u✝ v✝ w✝ : V
ph : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q
hp : p.IsPath ∧ u✝ ∉ p.support
h✝ : G.Adj u✝ v✝
q : G.Walk v✝ w✝
hq : q.IsPath ∧ u✝ ∉ q.support
⊢ cons ph p = cons h✝ q | cases ih hp.1 q hq.1 | case mk.mk.cons.inl.cons.inl.inl.refl.refl
V : Type u
G : SimpleGraph V
v w u✝ v✝ w✝ : V
ph : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q
hp : p.IsPath ∧ u✝ ∉ p.support
h✝ : G.Adj u✝ v✝
hq : p.IsPath ∧ u✝ ∉ p.support
⊢ cons ph p = cons h✝ p | c6bceaf8dd6ef474 |
Ordinal.card_iSup_Iio_le_card_mul_iSup | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem card_iSup_Iio_le_card_mul_iSup {o : Ordinal.{u}} (f : Iio o → Ordinal.{max u v}) :
(⨆ a : Iio o, f a).card ≤ Cardinal.lift.{v} o.card * ⨆ a : Iio o, (f a).card | case h.e'_4.h.e'_5.h.e'_1
o : Ordinal.{u}
f : ↑(Iio o) → Ordinal.{max u v}
⊢ #o.toType = o.card | exact mk_toType o | no goals | c3d702eab5714f39 |
RingHom.finiteType_ofLocalizationSpanTarget | Mathlib/RingTheory/RingHom/FiniteType.lean | theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteType | case h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l =... | apply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _ | R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
sf :... | 2dc9a560bb7745cb |
Normal.of_isSplittingField | Mathlib/FieldTheory/Normal/Basic.lean | theorem Normal.of_isSplittingField (p : F[X]) [hFEp : IsSplittingField F E p] : Normal F E | case inr.refine_1
F : Type u_1
inst✝² : Field F
E : Type u_3
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
hFEp : IsSplittingField F E p
hp : p ≠ 0
x : E
this : FiniteDimensional F E
hx : IsIntegral F x
L : Type u_1 := (p * minpoly F x).SplittingField
⊢ p ≠ 0 ∧ minpoly F x ≠ 0 | exact ⟨hp, minpoly.ne_zero hx⟩ | no goals | 0aed749abc5ecaab |
Commute.geom_sum₂ | Mathlib/Algebra/GeomSum.lean | theorem Commute.geom_sum₂ [DivisionRing α] {x y : α} (h' : Commute x y) (h : x ≠ y)
(n : ℕ) : ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y) | α : Type u
inst✝ : DivisionRing α
x y : α
h' : Commute x y
h : x ≠ y
n : ℕ
⊢ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y) | have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add] | α : Type u
inst✝ : DivisionRing α
x y : α
h' : Commute x y
h : x ≠ y
n : ℕ
this : x - y ≠ 0
⊢ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y) | 3c2f97cccd2e98b3 |
Polynomial.contract_mul_expand | Mathlib/Algebra/Polynomial/Expand.lean | theorem contract_mul_expand {p : ℕ} (hp : p ≠ 0) (f g : R[X]) :
contract p (f * expand R p g) = contract p f * g | case a.hf
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : p ≠ 0
f g : R[X]
n : ℕ
⊢ ∀ (x : ℕ × ℕ),
x.1 + x.2 = n * p → (¬∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = x) → f.coeff x.1 * ((expand R p) g).coeff x.2 = 0 | intro ⟨x, y⟩ eq nex | case a.hf
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : p ≠ 0
f g : R[X]
n x y : ℕ
eq : (x, y).1 + (x, y).2 = n * p
nex : ¬∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = (x, y)
⊢ f.coeff (x, y).1 * ((expand R p) g).coeff (x, y).2 = 0 | c12f546c03101c96 |
MulAction.mk' | Mathlib/GroupTheory/GroupAction/Primitive.lean | theorem mk' (Hnt : fixedPoints G X ≠ ⊤)
(H : ∀ {B : Set X} (_ : IsBlock G B), IsTrivialBlock B) :
IsPreprimitive G X | G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X | simp only [Set.top_eq_univ, Set.ne_univ_iff_exists_not_mem] at Hnt | G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
H : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
Hnt : ∃ a, a ∉ fixedPoints G X
⊢ IsPreprimitive G X | 93a1b4b478f7903a |
Polynomial.sum_bernoulli | Mathlib/NumberTheory/BernoulliPolynomials.lean | theorem sum_bernoulli (n : ℕ) :
(∑ k ∈ range (n + 1), ((n + 1).choose k : ℚ) • bernoulli k) = monomial n (n + 1 : ℚ) | n : ℕ
⊢ ∑ x ∈ range (n + 1),
∑ x_1 ∈ range (n + 1 - x), (↑((n + 1 - x).choose x_1) * _root_.bernoulli x_1) • (monomial x) ↑((n + 1).choose x) =
(monomial n) ↑n + (monomial n) 1 | simp_rw [← sum_smul] | n : ℕ
⊢ ∑ x ∈ range (n + 1),
(∑ i ∈ range (n + 1 - x), ↑((n + 1 - x).choose i) * _root_.bernoulli i) • (monomial x) ↑((n + 1).choose x) =
(monomial n) ↑n + (monomial n) 1 | b01e377955c96550 |
SimpleGraph.Subgraph.map_mono | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | @[gcongr] lemma map_mono {H₁ H₂ : G.Subgraph} (hH : H₁ ≤ H₂) : H₁.map f ≤ H₂.map f | V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ Subgraph.map f H₁ ≤ Subgraph.map f H₂ | constructor | case left
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ (Subgraph.map f H₁).verts ⊆ (Subgraph.map f H₂).verts
case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ ∀ ⦃v w : W⦄, (Subgraph.map f H₁).A... | 45e01797f0297c07 |
LSeriesSummable.congr' | Mathlib/NumberTheory/LSeries/Basic.lean | /-- If `f` and `g` agree on large `n : ℕ` and the `LSeries` of `f` converges at `s`,
then so does that of `g`. -/
lemma LSeriesSummable.congr' {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[atTop] g) (hf : LSeriesSummable f s) :
LSeriesSummable g s | case intro.intro
f g : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
S : Set ℕ
hS : S ∈ cofinite
hS' : Set.EqOn f g S
n : ℕ
hn : n ∈ S ∧ ¬n = 0
⊢ term f s n = term g s n | simp [hn.2, hS' hn.1] | no goals | 549909a247ffe2bd |
ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite | Mathlib/Probability/Kernel/MeasurableLIntegral.lean | theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) | case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun ... | rw [this] | case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun ... | 4c14e8ca6e5d1721 |
MeasureTheory.setToFun_congr_measure_of_integrable | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞)
(hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) :
setToFun μ T hT f = setToFun μ' T hT' f | case h_closed
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : Do... | exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' hμ'_le) | no goals | 65e35cd00eda3e5b |
PartENat.withTopEquiv_symm_coe | Mathlib/Data/Nat/PartENat.lean | theorem withTopEquiv_symm_coe (n : Nat) : withTopEquiv.symm n = n | n : ℕ
⊢ withTopEquiv.symm ↑n = ↑n | simp | no goals | 645a799cca9575eb |
MeromorphicAt.order_smul | Mathlib/Analysis/Meromorphic/Order.lean | theorem order_smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜}
(hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
(hf.smul hg).order = hf.order + hg.order | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → 𝕜
g : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : hf.order = ↑m
h₂g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, g z = 0
z : 𝕜
hz : g z = 0
⊢ (f • g) z = 0 | simp [hz] | no goals | 888980cd950630aa |
CategoryTheory.Iso.refl_conj | Mathlib/CategoryTheory/Conj.lean | theorem refl_conj (f : End X) : (Iso.refl X).conj f = f | C : Type u
inst✝ : Category.{v, u} C
X : C
f : End X
⊢ (refl X).conj f = f | rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id] | no goals | 9620a46c53a58c1f |
Module.finite_dual_iff | Mathlib/LinearAlgebra/Dual.lean | theorem finite_dual_iff [Free K V] : Module.Finite K (Dual K V) ↔ Module.Finite K V | case mk.intro.refine_1
K : Type uK
V : Type uV
inst✝³ : CommRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
ι : Type uV
b : Basis ι K V
a✝ : Nontrivial K
s : Finset (Dual K V)
span_s : span K ↑s = ⊤
⊢ Set.range (⇑b.toDual ∘ ⇑b) ≤ ↑(span K ↑s) | rw [span_s] | case mk.intro.refine_1
K : Type uK
V : Type uV
inst✝³ : CommRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
ι : Type uV
b : Basis ι K V
a✝ : Nontrivial K
s : Finset (Dual K V)
span_s : span K ↑s = ⊤
⊢ Set.range (⇑b.toDual ∘ ⇑b) ≤ ↑⊤ | 1a833564690fbd59 |
LiouvilleWith.add_rat | Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean | theorem add_rat (h : LiouvilleWith p x) (r : ℚ) : LiouvilleWith p (x + r) | p x : ℝ
h : LiouvilleWith p x
r : ℚ
⊢ LiouvilleWith p (x + ↑r) | rcases h.exists_pos with ⟨C, _hC₀, hC⟩ | case intro.intro
p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
⊢ LiouvilleWith p (x + ↑r) | 7652465b354d5b90 |
List.sym2_eq_sym_two | Mathlib/Data/List/Sym.lean | theorem sym2_eq_sym_two : xs.sym2.map (Sym2.equivSym α) = xs.sym 2 | case nil
α : Type u_1
xs : List α
⊢ map ⇑(Sym2.equivSym α) [].sym2 = List.sym 2 [] | simp only [List.sym, map_eq_nil_iff, sym2_eq_nil_iff] | no goals | 02f4fa6d82cb0101 |
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