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 |
|---|---|---|---|---|---|---|
Mathlib.Tactic.Ring.add_pf_add_lt | Mathlib/Tactic/Ring/Basic.lean | theorem add_pf_add_lt (a₁ : R) (_ : a₂ + b = c) : (a₁ + a₂) + b = a₁ + c | R : Type u_1
inst✝ : CommSemiring R
a₂ b c a₁ : R
x✝ : a₂ + b = c
⊢ a₁ + a₂ + b = a₁ + c | simp [*, add_assoc] | no goals | 901ae75805bb27f1 |
Polynomial.sumIDeriv_apply_of_lt | Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean | theorem sumIDeriv_apply_of_lt {p : R[X]} {n : ℕ} (hn : p.natDegree < n) :
sumIDeriv p = ∑ i ∈ range n, derivative^[i] p | R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
hn : p.natDegree < n
⊢ (derivativeFinsupp p).support ⊆ range n | simp [hn] | no goals | 382c86a4c222b5bf |
integral_sin_sq_mul_cos_sq | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_sin_sq_mul_cos_sq :
∫ x in a..b, sin x ^ 2 * cos x ^ 2 = (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 | a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
⊢ ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2 | intro | a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = sin (2 * x✝) / 2 | 91a2e24802ea5818 |
Equiv.Perm.IsCycle.pow_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.pow_iff [Finite β] {f : Perm β} (hf : IsCycle f) {n : ℕ} :
IsCycle (f ^ n) ↔ n.Coprime (orderOf f) | case intro.mpr.intro
β : Type u_3
inst✝ : Finite β
f : Perm β
hf : f.IsCycle
n : ℕ
val✝ : Fintype β
h : n.Coprime (orderOf f)
m : ℕ
hm : (f ^ n) ^ m = f
hf' : ((f ^ n) ^ m).IsCycle
x : β
hx : x ∈ (f ^ n).support
⊢ x ∈ ((f ^ n) ^ m).support | rw [hm] | case intro.mpr.intro
β : Type u_3
inst✝ : Finite β
f : Perm β
hf : f.IsCycle
n : ℕ
val✝ : Fintype β
h : n.Coprime (orderOf f)
m : ℕ
hm : (f ^ n) ^ m = f
hf' : ((f ^ n) ^ m).IsCycle
x : β
hx : x ∈ (f ^ n).support
⊢ x ∈ f.support | 66cd6cfb46d266c4 |
CompletelyDistribLattice.MinimalAxioms.iSup_iInf_eq | Mathlib/Order/CompleteBooleanAlgebra.lean | lemma iSup_iInf_eq (f : ∀ i, κ i → α) :
let _ := minAx.toCompleteLattice
⨆ i, ⨅ j, f i j = ⨅ g : ∀ i, κ i, ⨆ i, f i (g i) | α : Type u
ι : Sort w
κ : ι → Sort w'
minAx : MinimalAxioms α
f : (i : ι) → κ i → α
x✝ : CompleteLattice α := minAx.toCompleteLattice
g : ((i : ι) → κ i) → ι
a : ι
ha : ∀ (b : κ a), ∃ f, ∃ (h : a = g f), h ▸ b = f (g f)
⊢ ⨅ i, f (g i) (i (g i)) ≤ ⨆ i, ⨅ j, f i j | refine le_trans ?_ (le_iSup _ a) | α : Type u
ι : Sort w
κ : ι → Sort w'
minAx : MinimalAxioms α
f : (i : ι) → κ i → α
x✝ : CompleteLattice α := minAx.toCompleteLattice
g : ((i : ι) → κ i) → ι
a : ι
ha : ∀ (b : κ a), ∃ f, ∃ (h : a = g f), h ▸ b = f (g f)
⊢ ⨅ i, f (g i) (i (g i)) ≤ ⨅ j, f a j | be0a16e095a6ee29 |
Filter.isBoundedUnder_le_mul_of_nonneg | Mathlib/Order/LiminfLimsup.lean | lemma isBoundedUnder_le_mul_of_nonneg [Mul α] [Zero α] [Preorder α] [PosMulMono α]
[MulPosMono α] {f : Filter ι} {u v : ι → α} (h₁ : 0 ≤ᶠ[f] u)
(h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v)
(h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v) :
IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u... | case intro
α : Type u_1
ι : Type u_4
inst✝⁴ : Mul α
inst✝³ : Zero α
inst✝² : Preorder α
inst✝¹ : PosMulMono α
inst✝ : MulPosMono α
f : Filter ι
u v : ι → α
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
U : α
hU : ∀ᶠ (x : ι) in f, u x ≤ U
⊢ Is... | obtain ⟨V, hV⟩ := h₄.eventually_le | case intro.intro
α : Type u_1
ι : Type u_4
inst✝⁴ : Mul α
inst✝³ : Zero α
inst✝² : Preorder α
inst✝¹ : PosMulMono α
inst✝ : MulPosMono α
f : Filter ι
u v : ι → α
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
U : α
hU : ∀ᶠ (x : ι) in f, u x ≤ ... | 5d150f01f7a3ba19 |
IsSl2Triple.symm | Mathlib/Algebra/Lie/Sl2.lean | lemma symm (ht : IsSl2Triple h e f) : IsSl2Triple (-h) f e where
h_ne_zero | L : Type u_2
inst✝ : LieRing L
h e f : L
ht : IsSl2Triple h e f
⊢ ⁅f, e⁆ = -h | rw [← neg_eq_iff_eq_neg, lie_skew, ht.lie_e_f] | no goals | 8d56a793b514175c |
Real.fderiv_fourierChar_neg_bilinear_left_apply | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | lemma fderiv_fourierChar_neg_bilinear_left_apply (v y : V) (w : W) :
fderiv ℝ (fun v ↦ (𝐞 (-L v w) : ℂ)) v y = -2 * π * I * L y w * 𝐞 (-L v w) | V : Type u_1
W : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
v y : V
w : W
⊢ 2 * ↑π * I * ↑(𝐞 (-(L v) w)) * ↑((L y) w) = 2 * ↑π * I * ↑((L y) w) * ↑(𝐞 (-(L v) w)) | ring | no goals | 4c67b50984595267 |
FermatLastTheoremForThreeGen.Solution.lambda_dvd_a_add_eta_sq_mul_b | Mathlib/NumberTheory/FLT/Three.lean | /-- Given `(S : Solution)`, we have that `λ ∣ (S.a + η ^ 2 * S.b)`. -/
lemma lambda_dvd_a_add_eta_sq_mul_b : λ ∣ (S.a + η ^ 2 * S.b) | K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
⊢ λ ∣ S.a + ↑η ^ 2 * S.b | rw [show S.a + η ^ 2 * S.b = (S.a + S.b) + λ ^ 2 * S.b + 2 * λ * S.b by rw [coe_eta]; ring] | K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
⊢ λ ∣ S.a + S.b + λ ^ 2 * S.b + 2 * λ * S.b | f2c6a6b69d1d3fef |
Finsupp.some_single_some | Mathlib/Data/Finsupp/Basic.lean | theorem some_single_some [Zero M] (a : α) (m : M) :
(single (Option.some a) m : Option α →₀ M).some = single a m | α : Type u_1
M : Type u_5
inst✝ : Zero M
a : α
m : M
⊢ (single (Option.some a) m).some = single a m | ext b | case h
α : Type u_1
M : Type u_5
inst✝ : Zero M
a : α
m : M
b : α
⊢ (single (Option.some a) m).some b = (single a m) b | 6bf60aa718869f43 |
ProbabilityTheory.gaussianPDFReal_inv_mul | Mathlib/Probability/Distributions/Gaussian.lean | lemma gaussianPDFReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPDFReal μ v (c⁻¹ * x) = |c| * gaussianPDFReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | case refine_2.e_x
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ -(x - c * μ) ^ 2 / (c ^ 2 * (2 * ↑v)) = -(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)) | congr 1 | case refine_2.e_x.e_a
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ c ^ 2 * (2 * ↑v) = 2 * (c ^ 2 * ↑v) | 3d362dec61aca464 |
MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | theorem TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTop g) :
∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) | case intro.intro
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : MetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns ... | exact le_trans hNx.le h_inv_n_le_k | no goals | 125226675bee6dd7 |
finprod_eq_mulIndicator_apply | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_eq_mulIndicator_apply (s : Set α) (f : α → M) (a : α) :
∏ᶠ _ : a ∈ s, f a = mulIndicator s f a | α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
s : Set α
f : α → M
a : α
⊢ ∏ᶠ (_ : a ∈ s), f a = s.mulIndicator f a | convert finprod_eq_if (M := M) (p := a ∈ s) (x := f a) | no goals | 6f1b0d2d634b0a0d |
smoothSheafCommRing.isUnit_stalk_iff | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
(f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) | 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoot... | convert (Set.range_inclusion hUV).symm | case h.e'_2.h
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.o... | f7ca29d724eff4f4 |
LaurentPolynomial.isLocalization | Mathlib/Algebra/Polynomial/Laurent.lean | theorem isLocalization : IsLocalization (Submonoid.powers (X : R[X])) R[T;T⁻¹] :=
{ map_units' := fun ⟨t, ht⟩ => by
obtain ⟨n, rfl⟩ := ht
rw [algebraMap_eq_toLaurent, toLaurent_X_pow]
exact isUnit_T ↑n
surj' := fun f => by
induction' f using LaurentPolynomial.induction_on_mul_T with f n
... | R : Type u_1
inst✝ : CommSemiring R
x✝ : ↥(Submonoid.powers X)
t : R[X]
ht : t ∈ Submonoid.powers X
⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨t, ht⟩) | obtain ⟨n, rfl⟩ := ht | case intro
R : Type u_1
inst✝ : CommSemiring R
x✝ : ↥(Submonoid.powers X)
n : ℕ
⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨(fun x => X ^ x) n, ⋯⟩) | 142317cb84f0217d |
lt_inv_comm₀ | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | /-- See also `lt_inv_of_lt_inv₀` for a one-sided implication with one fewer assumption. -/
lemma lt_inv_comm₀ (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ | G₀ : Type u_2
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : PartialOrder G₀
inst✝³ : ZeroLEOneClass G₀
inst✝² : PosMulReflectLT G₀
a b : G₀
inst✝¹ : MulPosStrictMono G₀
inst✝ : PosMulStrictMono G₀
ha : 0 < a
hb : 0 < b
⊢ a < b⁻¹ ↔ b < a⁻¹ | rw [← inv_lt_inv₀ (inv_pos.2 hb) ha, inv_inv] | no goals | 47d89a3268cf1fcb |
AlgebraicGeometry.sourceAffineLocally_isLocal | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | theorem sourceAffineLocally_isLocal (h₁ : RingHom.RespectsIso P)
(h₂ : RingHom.LocalizationAwayPreserves P) (h₃ : RingHom.OfLocalizationSpan P) :
(sourceAffineLocally P).IsLocal | case to_basicOpen.a
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] =>... | exact H U | no goals | 222ac1262a013627 |
RingSubgroupsBasis.hasBasis_nhds | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | theorem hasBasis_nhds (a : A) :
HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
⟨by
intro s
rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
simp only [true_and]
constructor
· rintro ⟨-, ⟨i, rfl⟩, hi⟩
use i
suffices h : { b ... | A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ (fun y => a + y) '' ↑(B i) ⊆ s | assumption | no goals | 9189cb0f666030cd |
Associates.factors_eq_some_iff_ne_zero | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem factors_eq_some_iff_ne_zero {a : Associates α} :
(∃ s : Multiset { p : Associates α // Irreducible p }, a.factors = s) ↔ a ≠ 0 | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ (∃ s, a.factors = ↑s) ↔ a ≠ 0 | simp_rw [@eq_comm _ a.factors, ← WithTop.ne_top_iff_exists] | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ a.factors ≠ ⊤ ↔ a ≠ 0 | b75a386d076c5824 |
CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle | Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.lean | lemma distinguished_cocone_triangle {X Y : Cᵒᵖ} (f : X ⟶ Y) :
∃ (Z : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ X⟦(1 : ℤ)⟧),
Triangle.mk f g h ∈ distinguishedTriangles C | case intro.intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X Y : Cᵒᵖ
f : X ⟶ Y
Z : C
g : Z ⟶ Opposite.unop Y
h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z
H : Triangle... | exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) (by simp) (by simp)
(Quiver.Hom.op_inj (by simp [shift_unop_opShiftFunctorEquivalence_counitIso_inv_app])) | no goals | 272fd5c34afae8cc |
continuous_of_uniform_approx_of_continuous | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem continuous_of_uniform_approx_of_continuous
(L : ∀ u ∈ 𝓤 β, ∃ F, Continuous F ∧ ∀ y, (f y, F y) ∈ u) : Continuous f :=
continuous_iff_continuousOn_univ.mpr <|
continuousOn_of_uniform_approx_of_continuousOn <| by
simpa [continuous_iff_continuousOn_univ] using L
| α : Type u
β : Type v
inst✝¹ : UniformSpace β
f : α → β
inst✝ : TopologicalSpace α
L : ∀ u ∈ 𝓤 β, ∃ F, Continuous F ∧ ∀ (y : α), (f y, F y) ∈ u
⊢ ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F univ ∧ ∀ y ∈ univ, (f y, F y) ∈ u | simpa [continuous_iff_continuousOn_univ] using L | no goals | f0dfef3ce1b3096c |
SimpleGraph.Subgraph.coe_deleteEdges_eq | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
(G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) | V : Type u
G : SimpleGraph V
G' : G.Subgraph
s : Set (Sym2 V)
⊢ (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map Subtype.val ⁻¹' s) | ext ⟨v, hv⟩ ⟨w, hw⟩ | case Adj.h.mk.h.mk.a
V : Type u
G : SimpleGraph V
G' : G.Subgraph
s : Set (Sym2 V)
v : V
hv : v ∈ (G'.deleteEdges s).verts
w : V
hw : w ∈ (G'.deleteEdges s).verts
⊢ (G'.deleteEdges s).coe.Adj ⟨v, hv⟩ ⟨w, hw⟩ ↔ (G'.coe.deleteEdges (Sym2.map Subtype.val ⁻¹' s)).Adj ⟨v, hv⟩ ⟨w, hw⟩ | 4a500f910601c6eb |
Nat.ascFactorial_eq_factorial_mul_choose' | Mathlib/Data/Nat/Choose/Basic.lean | theorem ascFactorial_eq_factorial_mul_choose' (n k : ℕ) :
n.ascFactorial k = k ! * (n + k - 1).choose k | n k : ℕ
⊢ n.ascFactorial k = k ! * (n + k - 1).choose k | cases n | case zero
k : ℕ
⊢ ascFactorial 0 k = k ! * (0 + k - 1).choose k
case succ
k n✝ : ℕ
⊢ (n✝ + 1).ascFactorial k = k ! * (n✝ + 1 + k - 1).choose k | 007644bc8cc9572c |
MeasureTheory.integral_fintype_prod_eq_prod | Mathlib/MeasureTheory/Integral/Pi.lean | theorem integral_fintype_prod_eq_prod (ι : Type*) [Fintype ι] {E : ι → Type*}
(f : (i : ι) → E i → 𝕜) [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))] :
∫ x : (i : ι) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
ι : Type u_2
inst✝² : Fintype ι
E : ι → Type u_3
f : (i : ι) → E i → 𝕜
inst✝¹ : (i : ι) → MeasureSpace (E i)
inst✝ : ∀ (i : ι), SigmaFinite volume
e : Fin (card ι) ≃ ι := (equivFin ι).symm
⊢ ∫ (x : (b : Fin (card ι)) → E (e b)), ∏ i : ι, f i ((MeasurableEquiv.piCongrLeft E e) x i) =
... | simp_rw [← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply_apply,
MeasureTheory.integral_fin_nat_prod_eq_prod] | no goals | ab006eba2a45fcbb |
Complex.hasSum_taylorSeries_log | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | /-- The Taylor series of the complex logarithm at `1` converges to the logarithm in the
open unit disk. -/
lemma hasSum_taylorSeries_log {z : ℂ} (hz : ‖z‖ < 1) :
HasSum (fun n : ℕ ↦ (-1) ^ (n + 1) * z ^ n / n) (log (1 + z)) | case refine_2
z : ℂ
hz : ‖z‖ < 1
⊢ (fun x => logTaylor x z - log (1 + z)) =o[atTop] fun x => 1 | refine IsLittleO.trans_isBigO ?_ <| isBigO_const_one ℂ (1 : ℝ) atTop | case refine_2
z : ℂ
hz : ‖z‖ < 1
⊢ (fun x => logTaylor x z - log (1 + z)) =o[atTop] fun _x => 1 | c6ad5f4f2717ef3d |
SimpleGraph.IsTuranMaximal.not_adj_trans | Mathlib/Combinatorics/SimpleGraph/Turan.lean | /-- In a Turán-maximal graph, non-adjacency is transitive. -/
lemma not_adj_trans (h : G.IsTuranMaximal r) (hts : ¬G.Adj t s) (hsu : ¬G.Adj s u) :
¬G.Adj t u | case right
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
s t u : V
hts : ¬G.Adj t s
hsu : ¬G.Adj s u
hst : ¬G.Adj s t
dst : G.degree s = G.degree t
dsu : G.degree s = G.degree u
h : G.Adj t u
cf : G.CliqueFree (r + 1)
nst : s ≠ t
ntu : t ≠ u
this : ¬(G.replaceVertex s t).Adj s u
l1 ... | have l2 : (G.replaceVertex s t).degree u = G.degree u - 1 := by
rw [degree, degree, ← card_singleton t, ← card_sdiff (by simp [h.symm])]
congr 1; ext v
simp only [mem_neighborFinset, mem_sdiff, mem_singleton, replaceVertex]
split_ifs <;> simp_all [adj_comm] | case right
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
s t u : V
hts : ¬G.Adj t s
hsu : ¬G.Adj s u
hst : ¬G.Adj s t
dst : G.degree s = G.degree t
dsu : G.degree s = G.degree u
h : G.Adj t u
cf : G.CliqueFree (r + 1)
nst : s ≠ t
ntu : t ≠ u
this : ¬(G.replaceVertex s t).Adj s u
l1 ... | 320247f515c2bcdc |
Array.swap_swap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem swap_swap (a : Array α) {i j : Nat} (hi hj) :
(a.swap i j hi hj).swap i j ((a.size_swap ..).symm ▸ hi) ((a.size_swap ..).symm ▸ hj) = a | case h₁
α : Type u_1
a : Array α
i j : Nat
hi : i < a.size
hj : j < a.size
⊢ ((a.swap i j hi hj).swap i j ⋯ ⋯).size = a.size | simp only [size_swap] | no goals | 8af4d36a7bf83b1e |
Set.MapsTo.restrict_surjective_iff | Mathlib/Data/Set/Function.lean | theorem MapsTo.restrict_surjective_iff (h : MapsTo f s t) :
Surjective (MapsTo.restrict _ _ _ h) ↔ SurjOn f s t | α : Type u_1
β : Type u_2
s : Set α
t : Set β
f : α → β
h : MapsTo f s t
h' : Surjective (restrict f s t h)
b : β
hb : b ∈ t
a : α
ha : a ∈ s
ha' : restrict f s t h ⟨a, ha⟩ = ⟨b, hb⟩
⊢ f a = b | simpa [Subtype.ext_iff] using ha' | no goals | 3f017aab25bc5663 |
Dynamics.log_coverMincard_le_add | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | lemma log_coverMincard_le_add {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
{U : Set (X × X)} (U_symm : SymmetricRel U) {m n : ℕ} (m_pos : 0 < m) (n_pos : 0 < n) :
log (coverMincard T F (U ○ U) n) / n
≤ log (coverMincard T F U m) / m + log (coverMincard T F U m) / n | case inr
X : Type u_1
T : X → X
F : Set X
F_inv : MapsTo T F F
U : Set (X × X)
U_symm : SymmetricRel U
m n : ℕ
m_pos : 0 < m
n_pos : 0 < n
F_nemp : F.Nonempty
h_nm : 0 ≤ ↑(n / m)
h_log : 0 ≤ (↑(coverMincard T F U m)).log
⊢ (↑(coverMincard T F (U ○ U) n)).log / ↑n ≤ (↑(coverMincard T F U m)).log / ↑m + (↑(coverMincard T... | have n_div_n := EReal.div_self (natCast_ne_bot n) (natCast_ne_top n)
(Nat.cast_pos'.2 n_pos).ne.symm | case inr
X : Type u_1
T : X → X
F : Set X
F_inv : MapsTo T F F
U : Set (X × X)
U_symm : SymmetricRel U
m n : ℕ
m_pos : 0 < m
n_pos : 0 < n
F_nemp : F.Nonempty
h_nm : 0 ≤ ↑(n / m)
h_log : 0 ≤ (↑(coverMincard T F U m)).log
n_div_n : ↑n / ↑n = 1
⊢ (↑(coverMincard T F (U ○ U) n)).log / ↑n ≤ (↑(coverMincard T F U m)).log / ... | bfe34c9eb0da6d6f |
Finset.pairwiseDisjoint_fibers | Mathlib/Data/Finset/Union.lean | private lemma pairwiseDisjoint_fibers : Set.PairwiseDisjoint ↑t fun a ↦ s.filter (f · = a) :=
fun x' hx y' hy hne ↦ by
simp_rw [disjoint_left, mem_filter]; rintro i ⟨_, rfl⟩ ⟨_, rfl⟩; exact hne rfl
| case intro.intro
α : Type u_1
β : Type u_2
inst✝ : DecidableEq β
s : Finset α
t : Finset β
f : α → β
i : α
left✝¹ : i ∈ s
hx : f i ∈ ↑t
left✝ : i ∈ s
hy : f i ∈ ↑t
hne : f i ≠ f i
⊢ False | exact hne rfl | no goals | b21a107221de7caf |
exists_continuous_one_zero_of_isCompact_of_isGδ | Mathlib/Topology/UrysohnsLemma.lean | theorem exists_continuous_one_zero_of_isCompact_of_isGδ [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), s = f ⁻¹' {1} ∧ EqOn f 0 t ∧ HasCompactSupport f
∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsComp... | let g : X → ℝ := fun x ↦ ∑' n, u n * f n x | case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsComp... | 06fdcc97382f69bb |
Complex.ofReal_arctan | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | theorem ofReal_arctan (x : ℝ) : (Real.arctan x : ℂ) = arctan x | x : ℝ
⊢ ↑(Real.arctan x) = (↑x).arctan | conv_rhs => rw [← Real.tan_arctan x] | x : ℝ
⊢ ↑(Real.arctan x) = (↑(Real.tan (Real.arctan x))).arctan | d799d1afd10876e7 |
Nat.bisect_lt_stop | Mathlib/.lake/packages/batteries/Batteries/Data/Nat/Bisect.lean | theorem bisect_lt_stop {p : Nat → Bool} (h : start < stop) (hstart : p start = true)
(hstop : p stop = false) : bisect h hstart hstop < stop | case isTrue
start stop : Nat
p : Nat → Bool
h : start < stop
hstart : p start = true
hstop : p stop = false
h✝ : start < start.avg stop
⊢ (match hmid : p (start.avg stop) with
| false => bisect h✝ hstart hmid
| true => bisect ⋯ hmid hstop) <
stop | split | case isTrue.h_1
start stop : Nat
p : Nat → Bool
h : start < stop
hstart : p start = true
hstop : p stop = false
h✝ : start < start.avg stop
heq✝ : p (start.avg stop) = false
⊢ bisect h✝ hstart ⋯ < stop
case isTrue.h_2
start stop : Nat
p : Nat → Bool
h : start < stop
hstart : p start = true
hstop : p stop = false
h✝ : ... | fc35ee5a21c978ac |
Int.two_pow_two_pow_add_two_pow_two_pow | Mathlib/NumberTheory/Multiplicity.lean | theorem Int.two_pow_two_pow_add_two_pow_two_pow {x y : ℤ} (hx : ¬2 ∣ x) (hxy : 4 ∣ x - y) (i : ℕ) :
emultiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑(1 : ℕ) | x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
⊢ Odd x | rwa [← Int.not_even_iff_odd, even_iff_two_dvd] | no goals | c2c4f247c9f39fe5 |
Sigma.isConnected_iff | Mathlib/Topology/Connected/Clopen.lean | theorem Sigma.isConnected_iff [∀ i, TopologicalSpace (π i)] {s : Set (Σi, π i)} :
IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t | case refine_2.intro.intro.intro
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
i : ι
t : Set (π i)
ht : IsConnected t
⊢ IsConnected (mk i '' t) | exact ht.image _ continuous_sigmaMk.continuousOn | no goals | 1455e050525a61d2 |
MeasureTheory.SignedMeasure.toJordanDecomposition_zero | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem toJordanDecomposition_zero : (0 : SignedMeasure α).toJordanDecomposition = 0 | case a
α : Type u_1
inst✝ : MeasurableSpace α
⊢ (toJordanDecomposition 0).toSignedMeasure = toSignedMeasure 0 | simp [toSignedMeasure_zero] | no goals | 00269c9e6b672618 |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | ... | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in... | rw [ENNReal.mul_top] | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in... | 6098eac6b53bede3 |
OrthonormalBasis.volume_parallelepiped | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
volume (parallelepiped b) = 1 | ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
b : OrthonormalBasis ι ℝ F
this : Fact (finrank ℝ F = finrank ℝ F)
⊢ volume (parallelepiped ⇑b) = 1 | let o := (stdOrthonormalBasis ℝ F).toBasis.orientation | ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
b : OrthonormalBasis ι ℝ F
this : Fact (finrank ℝ F = finrank ℝ F)
o : Orientation ℝ F (Fin (finrank ℝ F)) := (stdOrthonormalBasis ℝ F)... | 6ace9de66174bbdf |
Ordnode.Raised.dist_le' | Mathlib/Data/Ordmap/Ordset.lean | theorem Raised.dist_le' {n m} (H : Raised n m) : Nat.dist m n ≤ 1 | n m : ℕ
H : Raised n m
⊢ m.dist n ≤ 1 | rw [Nat.dist_comm] | n m : ℕ
H : Raised n m
⊢ n.dist m ≤ 1 | 914217f208d8a07a |
MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem range_eq_empty_of_isEmpty {β} [hα : IsEmpty α] (f : α →ₛ β) : f.range = ∅ | case intro
α : Type u_1
inst✝ : MeasurableSpace α
β : Type u_5
hα : IsEmpty α
f : α →ₛ β
y : β
hy_mem : y ∈ f.range
⊢ False | rw [SimpleFunc.mem_range, Set.mem_range] at hy_mem | case intro
α : Type u_1
inst✝ : MeasurableSpace α
β : Type u_5
hα : IsEmpty α
f : α →ₛ β
y : β
hy_mem : ∃ y_1, f y_1 = y
⊢ False | ba9f32d2d6566746 |
PowerSeries.binomialSeries_nat | Mathlib/RingTheory/PowerSeries/Binomial.lean | @[simp]
lemma binomialSeries_nat [CommRing A] (d : ℕ) :
binomialSeries A (d : ℤ) = (1 + X) ^ d | case pos
A : Type u_2
inst✝ : CommRing A
d n : ℕ
h : d < n
k : ℕ
hk : k ∈ range (d + 1)
hkd : k ≤ d
⊢ (coeff A (n - k)) ↑(d.choose k) = 0 | rw [← map_natCast (C A), coeff_ne_zero_C (by omega)] | no goals | 41f98537d51fbeb7 |
PowerBasis.dim_le_natDegree_of_root | Mathlib/RingTheory/PowerBasis.lean | theorem dim_le_natDegree_of_root (pb : PowerBasis A S) {p : A[X]} (ne_zero : p ≠ 0)
(root : aeval pb.gen p = 0) : pb.dim ≤ p.natDegree | S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
p : A[X]
ne_zero : p ≠ 0
root : (aeval pb.gen) p = 0
hlt : p.natDegree < pb.dim
⊢ ∑ i : Fin pb.dim, (monomial ↑i) (p.coeff ↑i) = 0 | refine Fintype.sum_eq_zero _ fun i => ?_ | S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
p : A[X]
ne_zero : p ≠ 0
root : (aeval pb.gen) p = 0
hlt : p.natDegree < pb.dim
i : Fin pb.dim
⊢ (monomial ↑i) (p.coeff ↑i) = 0 | ce294df16a59e170 |
CategoryTheory.Limits.IsBilimit.total | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | theorem IsBilimit.total {f : J → C} {b : Bicone f} (i : b.IsBilimit) :
∑ j : J, b.π j ≫ b.ι j = 𝟙 b.pt :=
i.isLimit.hom_ext fun j => by
classical
cases j
simp [sum_comp, b.ι_π, comp_dite]
| C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type
inst✝ : Fintype J
f : J → C
b : Bicone f
i : b.IsBilimit
j : Discrete J
⊢ (∑ j : J, b.π j ≫ b.ι j) ≫ b.toCone.π.app j = 𝟙 b.pt ≫ b.toCone.π.app j | cases j | case mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type
inst✝ : Fintype J
f : J → C
b : Bicone f
i : b.IsBilimit
as✝ : J
⊢ (∑ j : J, b.π j ≫ b.ι j) ≫ b.toCone.π.app { as := as✝ } = 𝟙 b.pt ≫ b.toCone.π.app { as := as✝ } | 2d1f111e47cc7625 |
Subgroup.SchurZassenhausInduction.step2 | Mathlib/GroupTheory/SchurZassenhaus.lean | theorem step2 (K : Subgroup G) [K.Normal] (hK : K ≤ N) : K = ⊥ ∨ K = N | case intro.refine_2
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
∀ (G' : Type u) [inst : Group G'] [inst_1 : Finite G'],
Nat.card G' < Nat.card G →
∀ {N' : Subgroup G'} [inst_2 : N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h... | rw [← comap_top (QuotientGroup.mk' K)] | case intro.refine_2
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
∀ (G' : Type u) [inst : Group G'] [inst_1 : Finite G'],
Nat.card G' < Nat.card G →
∀ {N' : Subgroup G'} [inst_2 : N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h... | 1296e7e9af1910d3 |
CategoryTheory.Paths.morphismProperty_eq_top | Mathlib/CategoryTheory/PathCategory/MorphismProperty.lean | /-- A reformulation of `CategoryTheory.Paths.induction` in terms of `MorphismProperty`. -/
lemma morphismProperty_eq_top
(P : MorphismProperty (Paths V))
(id : ∀ {v : V}, P (𝟙 (of.obj v)))
(comp : ∀ {u v w : V} (p : of.obj u ⟶ of.obj v) (q : v ⟶ w), P p → P (p ≫ of.map q)) :
P = ⊤ | V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 (of.obj v))
comp : ∀ {u v w : V} (p : of.obj u ⟶ of.obj v) (q : v ⟶ w), P p → P (p ≫ of.map q)
⊢ P = ⊤ | ext | case h
V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 (of.obj v))
comp : ∀ {u v w : V} (p : of.obj u ⟶ of.obj v) (q : v ⟶ w), P p → P (p ≫ of.map q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝ | d99e50c08e2889c2 |
CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app | Mathlib/CategoryTheory/Triangulated/Opposite/Basic.lean | lemma shiftFunctorAdd'_op_hom_app (X : Cᵒᵖ) (a₁ a₂ a₃ : ℤ) (h : a₁ + a₂ = a₃)
(b₁ b₂ b₃ : ℤ) (h₁ : a₁ + b₁ = 0) (h₂ : a₂ + b₂ = 0) (h₃ : a₃ + b₃ = 0) :
(shiftFunctorAdd' Cᵒᵖ a₁ a₂ a₃ h).hom.app X =
(shiftFunctorOpIso C _ _ h₃).hom.app X ≫
((shiftFunctorAdd' C b₁ b₂ b₃ (by omega)).inv.app X.unop).o... | C : Type u_1
inst✝¹ : Category.{?u.8759, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₁ + b₂ = b₃ | omega | no goals | b022e71c424a408a |
Matrix.zero_vecMul | Mathlib/Data/Matrix/Mul.lean | theorem zero_vecMul [Fintype m] (A : Matrix m n α) : 0 ᵥ* A = 0 | m : Type u_2
n : Type u_3
α : Type v
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : Fintype m
A : Matrix m n α
⊢ 0 ᵥ* A = 0 | ext | case h
m : Type u_2
n : Type u_3
α : Type v
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : Fintype m
A : Matrix m n α
x✝ : n
⊢ (0 ᵥ* A) x✝ = 0 x✝ | 6f71e2a75f0cc832 |
hasSum_one_div_nat_pow_mul_sin | Mathlib/NumberTheory/ZetaValues.lean | theorem hasSum_one_div_nat_pow_mul_sin {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k + 1) * Real.sin (2 * π * n * x))
((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
(Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1))... | case h.e'_6
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x) =
-(2 * ↑π * I) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x) | congr | case h.e'_6.e_a.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
⊢ (-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) = -(2 * ↑π * I) ^ (2 * k + 1) | 4ffa37bf4994910e |
iSup_ge_eq_iSup_nat_add | Mathlib/Order/CompleteLattice.lean | theorem iSup_ge_eq_iSup_nat_add (u : ℕ → α) (n : ℕ) : ⨆ i ≥ n, u i = ⨆ i, u (i + n) | case a
α : Type u_1
inst✝ : CompleteLattice α
u : ℕ → α
n : ℕ
⊢ ∀ (i : ℕ), u (i + n) ≤ ⨆ i, ⨆ (_ : i ≥ n), u i | exact fun i => le_sSup ⟨i + n, iSup_pos (Nat.le_add_left _ _)⟩ | no goals | bc70894f4edb606a |
Perfect.exists_nat_bool_injection | Mathlib/Topology/MetricSpace/Perfect.lean | theorem Perfect.exists_nat_bool_injection
(hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f | case property
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.None... | exact ⟨(h1 _ _ _).1, (h1 _ _ _).2.1⟩ | no goals | ef9f9f33b845189a |
Ordinal.isNormal_preOmega | Mathlib/SetTheory/Cardinal/Aleph.lean | theorem isNormal_preOmega : IsNormal preOmega | o : Ordinal.{u_1}
ho : o.IsLimit
a : Ordinal.{u_1}
ha : ∀ b < o, preOmega b ≤ a
b : Ordinal.{u_1}
hb : b < o
⊢ (preOmega b).card < a.card | apply lt_of_lt_of_le _ (card_le_card <| ha _ (ho.succ_lt hb)) | o : Ordinal.{u_1}
ho : o.IsLimit
a : Ordinal.{u_1}
ha : ∀ b < o, preOmega b ≤ a
b : Ordinal.{u_1}
hb : b < o
⊢ (preOmega b).card < (preOmega (succ b)).card | a6ce36b2d6f95c46 |
MvPolynomial.isWeightedHomogeneous_monomial | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | theorem isWeightedHomogeneous_monomial (w : σ → M) (d : σ →₀ ℕ) (r : R) {m : M}
(hm : weight w d = m) : IsWeightedHomogeneous w (monomial d r) m | R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
d : σ →₀ ℕ
r : R
m : M
hm : (weight w) d = m
c : σ →₀ ℕ
hc : coeff c ((monomial d) r) ≠ 0
⊢ (weight w) c = m | rw [coeff_monomial] at hc | R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
d : σ →₀ ℕ
r : R
m : M
hm : (weight w) d = m
c : σ →₀ ℕ
hc : (if d = c then r else 0) ≠ 0
⊢ (weight w) c = m | 8e3ee3ec219eecc4 |
csSup_mem_of_not_isSuccPrelimit | Mathlib/Order/SuccPred/CompleteLinearOrder.lean | lemma csSup_mem_of_not_isSuccPrelimit
(hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) : sSup s ∈ s | α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hne : s.Nonempty
hbdd : BddAbove s
hlim : ¬IsSuccPrelimit (sSup s)
⊢ sSup s ∈ s | obtain ⟨y, hy⟩ := not_forall_not.mp hlim | case intro
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hne : s.Nonempty
hbdd : BddAbove s
hlim : ¬IsSuccPrelimit (sSup s)
y : α
hy : y ⋖ sSup s
⊢ sSup s ∈ s | f419e4f7b2c0f097 |
Real.b_ne_one' | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem b_ne_one' : b ≠ 1 | b : ℝ
hb : 1 < b
⊢ b ≠ 1 | linarith | no goals | f91dfa1112df941b |
List.Vector.eraseIdx_insertIdx' | Mathlib/Data/Vector/Basic.lean | theorem eraseIdx_insertIdx' {v : Vector α (n + 1)} :
∀ {i : Fin (n + 1)} {j : Fin (n + 2)},
eraseIdx (j.succAbove i) (insertIdx a j v) = insertIdx a (i.predAbove j) (eraseIdx i v)
| ⟨i, hi⟩, ⟨j, hj⟩ => by
dsimp [insertIdx, eraseIdx, Fin.succAbove, Fin.predAbove]
rw [Subtype.mk_eq_mk]
simp only [... | α : Type u_1
n : ℕ
a : α
v : Vector α (n + 1)
i : ℕ
hi : i < n + 1
j : ℕ
hj : j < n + 2
⊢ (List.insertIdx j a ↑v).eraseIdx ↑(if i < j then ⟨i, ⋯⟩ else ⟨i + 1, ⋯⟩) =
List.insertIdx (↑(if h : i < j then ⟨j, hj⟩.pred ⋯ else ⟨j, hj⟩.castPred ⋯)) a
↑(match v with
| ⟨l, p⟩ => ⟨l.eraseIdx i, ⋯⟩) | split_ifs with hij | case pos
α : Type u_1
n : ℕ
a : α
v : Vector α (n + 1)
i : ℕ
hi : i < n + 1
j : ℕ
hj : j < n + 2
hij : i < j
⊢ (List.insertIdx j a ↑v).eraseIdx ↑⟨i, ⋯⟩ =
List.insertIdx (↑(⟨j, hj⟩.pred ⋯)) a
↑(match v with
| ⟨l, p⟩ => ⟨l.eraseIdx i, ⋯⟩)
case neg
α : Type u_1
n : ℕ
a : α
v : Vector α (n + 1)
i : ℕ
hi ... | 8db247d22bea6750 |
MeasureTheory.setToFun_neg | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
setToFun μ T hT (-f) = -setToFun μ T hT f | case neg
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 : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : ¬Integrable f μ
⊢ ¬Inte... | rwa [← integrable_neg_iff] at hf | no goals | ac93434dec24c688 |
RingHom.OfLocalizationSpanTarget.ofIsLocalization | Mathlib/RingTheory/LocalProperties/Basic.lean | lemma RingHom.OfLocalizationSpanTarget.ofIsLocalization
(hP : RingHom.OfLocalizationSpanTarget P) (hP' : RingHom.RespectsIso P)
{R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (hs : Ideal.span s = ⊤)
(hT : ∀ r : s, ∃ (T : Type u) (_ : CommRing T) (_ : Algebra S T)
(_ : IsLocalizatio... | case intro.intro.intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] => P
hP' : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs ... | convert hP'.1 _
(Localization.algEquiv (R := S) (Submonoid.powers (r : S)) T).symm.toRingEquiv hT | case h.e'_5
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] => P
hP' : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ... | e05ee18a3e87f750 |
ENNReal.mul_div_cancel_right | Mathlib/Data/ENNReal/Inv.lean | /-- See `ENNReal.mul_div_cancel_right'` for a stronger version. -/
protected lemma mul_div_cancel_right (hb₀ : b ≠ 0) (hb : b ≠ ∞) : a * b / b = a :=
ENNReal.mul_div_cancel_right' (by simp [hb₀]) (by simp [hb])
| a b : ℝ≥0∞
hb₀ : b ≠ 0
hb : b ≠ ⊤
⊢ b = ⊤ → a = 0 | simp [hb] | no goals | 39621360c887d130 |
GaussianInt.mod_four_eq_three_of_nat_prime_of_prime | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : Fact p.Prime]
(hpi : Prime (p : ℤ[i])) : p % 4 = 3 :=
hp.1.eq_two_or_odd.elim
(fun hp2 => by
have := hpi.irreducible.isUnit_or_isUnit (a := ⟨1, 1⟩) (b := ⟨1, -1⟩)
simp [hp2, Zsqrtd.ext_iff, ← norm_eq_one_iff, norm_def] at this)
fun ... | p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp1 : p % 2 = 1
hp3 : p % 4 ≠ 3
hp41 : p % 4 = 1
k : ZMod p
hk : -1 = k * k
⊢ False | obtain ⟨k, k_lt_p, rfl⟩ : ∃ (k' : ℕ) (_ : k' < p), (k' : ZMod p) = k := by
exact ⟨k.val, k.val_lt, ZMod.natCast_zmod_val k⟩ | case intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp1 : p % 2 = 1
hp3 : p % 4 ≠ 3
hp41 : p % 4 = 1
k : ℕ
k_lt_p : k < p
hk : -1 = ↑k * ↑k
⊢ False | 9ab49eaa8ecd58e2 |
MvQPF.wEquiv.symm | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | theorem wEquiv.symm {α : TypeVec n} (x y : q.P.W α) : WEquiv x y → WEquiv y x | case trans
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ y✝ x y z : (P F).W α
_e₁ : WEquiv x y
_e₂ : WEquiv y z
ih₁ : WEquiv y x
ih₂ : WEquiv z y
⊢ WEquiv z x | exact MvQPF.WEquiv.trans _ _ _ ih₂ ih₁ | no goals | 1c556c738b614841 |
RegularExpression.star_rmatch_iff | Mathlib/Computability/RegularExpressions.lean | theorem star_rmatch_iff (P : RegularExpression α) :
∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
= S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
fun x => by
have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
clear star_rmatch_iff
constructor
· rcases x... | α : Type u_1
inst✝ : DecidableEq α
P : RegularExpression α
a : α
x : List α
IH :
∀ (t : List α),
t.length < (a :: x).length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ P.rmatch t_1 = true
hsum : ... | rw [hsum.1, hsum.2] | α : Type u_1
inst✝ : DecidableEq α
P : RegularExpression α
a : α
x : List α
IH :
∀ (t : List α),
t.length < (a :: x).length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ P.rmatch t_1 = true
hsum : ... | 8215ce2ddbd6bd2a |
tendsto_integral_mulExpNegMulSq_comp | Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.lean | theorem tendsto_integral_mulExpNegMulSq_comp (g : E →ᵇ ℝ) :
Tendsto (fun ε => ∫ x, mulExpNegMulSq ε (g x) ∂P) (𝓝[>] 0) (𝓝 (∫ x, g x ∂P)) | case a.intro
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
g : E →ᵇ ℝ
u : ℕ → ℝ
hu : Tendsto u atTop (𝓝[>] 0)
N : ℕ
hupos : ∀ b ≥ N, u b ∈ Set.Ioi 0
⊢ Tendsto ((fun ε => ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P) ∘ u) atTop (𝓝 (∫ (x : E), ... | apply tendsto_integral_filter_of_norm_le_const ?h_meas ?h_bound ?h_lim | case h_meas
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
g : E →ᵇ ℝ
u : ℕ → ℝ
hu : Tendsto u atTop (𝓝[>] 0)
N : ℕ
hupos : ∀ b ≥ N, u b ∈ Set.Ioi 0
⊢ ∀ᶠ (n : ℕ) in atTop, AEStronglyMeasurable (fun ω => (u n).mulExpNegMulSq (g ω)) P
ca... | 9677e28f00b7be5e |
MeasureTheory.lintegral_lintegral_mul_inv | Mathlib/MeasureTheory/Group/Prod.lean | theorem lintegral_lintegral_mul_inv [IsMulLeftInvariant ν] (f : G → G → ℝ≥0∞)
(hf : AEMeasurable (uncurry f) (μ.prod ν)) :
(∫⁻ x, ∫⁻ y, f (y * x) x⁻¹ ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ | G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹... | conv_rhs => rw [← (measurePreserving_mul_prod_inv μ ν).map_eq] | G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹... | 5030442f8dec594f |
AffineSubspace.direction_affineSpan_insert | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean | theorem direction_affineSpan_insert {s : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) :
(affineSpan k (insert p₂ (s : Set P))).direction =
Submodule.span k {p₂ -ᵥ p₁} ⊔ s.direction | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p₁ p₂ : P
hp₁ : p₁ ∈ s
⊢ (affineSpan k (↑s ∪ ↑(affineSpan k {p₂}))).direction = s.direction ⊔ Submodule.span k {p₂ -ᵥ p₁} | change (s ⊔ affineSpan k {p₂}).direction = _ | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p₁ p₂ : P
hp₁ : p₁ ∈ s
⊢ (s ⊔ affineSpan k {p₂}).direction = s.direction ⊔ Submodule.span k {p₂ -ᵥ p₁} | 35d3596013473b81 |
Nat.Partrec.Code.evaln_sound | Mathlib/Computability/PartrecCode.lean | theorem evaln_sound : ∀ {k c n x}, x ∈ evaln k c n → x ∈ eval c n
| 0, _, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
induction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generalizing x n <;>
simp [eval, evaln, Option.bind_eq_some, Seq.seq] at h ⊢ <;>
obtain ⟨_, h⟩ := h
it... | k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
y : ℕ
hy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))
hy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpair n).2 + 1))), ¬a = 0
h₂ : ev... | simpa [add_comm, add_left_comm] using hz | no goals | 0a43fa3ebdc9ae7d |
Asymptotics.IsBigOWith.prod_left_same | Mathlib/Analysis/Asymptotics/Defs.lean | theorem IsBigOWith.prod_left_same (hf : IsBigOWith c l f' k') (hg : IsBigOWith c l g' k') :
IsBigOWith c l (fun x => (f' x, g' x)) k' | α : Type u_1
E' : Type u_6
F' : Type u_7
G' : Type u_8
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : SeminormedAddCommGroup F'
inst✝ : SeminormedAddCommGroup G'
c : ℝ
f' : α → E'
g' : α → F'
k' : α → G'
l : Filter α
hf : IsBigOWith c l f' k'
hg : IsBigOWith c l g' k'
⊢ IsBigOWith c l (fun x => (f' x, g' x)) k' | rw [isBigOWith_iff] at * | α : Type u_1
E' : Type u_6
F' : Type u_7
G' : Type u_8
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : SeminormedAddCommGroup F'
inst✝ : SeminormedAddCommGroup G'
c : ℝ
f' : α → E'
g' : α → F'
k' : α → G'
l : Filter α
hf : ∀ᶠ (x : α) in l, ‖f' x‖ ≤ c * ‖k' x‖
hg : ∀ᶠ (x : α) in l, ‖g' x‖ ≤ c * ‖k' x‖
⊢ ∀ᶠ (x : α) in l, ‖(f... | 4d78b2c090b9c6ef |
MvPolynomial.IsWeightedHomogeneous.weightedHomogeneousComponent_same | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | theorem IsWeightedHomogeneous.weightedHomogeneousComponent_same {m : M} {p : MvPolynomial σ R}
(hp : IsWeightedHomogeneous w p m) :
weightedHomogeneousComponent w m p = p | case a
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x p | by_cases zero_coeff : coeff x p = 0 | case pos
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m : M
p : MvPolynomial σ R
hp : IsWeightedHomogeneous w p m
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x p
case neg
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring ... | 61924f9884b810f2 |
Algebra.FormallyEtale.iff_exists_algEquiv_prod | Mathlib/RingTheory/Etale/Field.lean | theorem iff_exists_algEquiv_prod [EssFiniteType K A] :
FormallyEtale K A ↔
∃ (I : Type u) (_ : Finite I) (Ai : I → Type u) (_ : ∀ i, Field (Ai i))
(_ : ∀ i, Algebra K (Ai i)) (_ : A ≃ₐ[K] Π i, Ai i),
∀ i, Algebra.IsSeparable K (Ai i) | K A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
I : Type u
w✝² : Finite I
Ai : I → Type u
w✝¹ : (i : I) → Field (Ai i)
w✝ : (i : I) → Algebra K (Ai i)
e : A ≃ₐ[K] (i : I) → Ai i
h✝ : ∀ (i : I), Algebra.IsSeparable K (Ai i)
i : I
this✝¹ : Algebra A (Ai i) := ((Pi.evalRing... | exact EssFiniteType.comp K A (Ai i) | no goals | bed71e21534c428f |
DualNumber.isMaximal_span_singleton_eps | Mathlib/RingTheory/DualNumber.lean | lemma isMaximal_span_singleton_eps [DivisionRing K] :
(Ideal.span {ε} : Ideal K[ε]).IsMaximal | case refine_2.inr.inr
K : Type u_2
inst✝ : DivisionRing K
hI : Ideal.span {ε} < ⊤
⊢ ⊤ = ⊤ | simp | no goals | 4856da4c4e023ce4 |
GromovHausdorff.toGHSpace_eq_toGHSpace_iff_isometryEquiv | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem toGHSpace_eq_toGHSpace_iff_isometryEquiv {X : Type u} [MetricSpace X] [CompactSpace X]
[Nonempty X] {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
toGHSpace X = toGHSpace Y ↔ Nonempty (X ≃ᵢ Y) :=
⟨by
simp only [toGHSpace]
rw [Quotient.eq]
rintro ⟨e⟩
have I :
(Nonemp... | X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ ⟦NonemptyCompacts.kuratowskiEmbedding X⟧ = ⟦NonemptyCompacts.kuratowskiEmbedding Y⟧ → Nonempty (X ≃ᵢ Y) | rw [Quotient.eq] | X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ IsometryRel.setoid (NonemptyCompacts.kuratowskiEmbedding X) (NonemptyCompacts.kuratowskiEmbedding Y) →
Nonempty (X ≃ᵢ Y) | 458e5e4306906052 |
Turing.PartrecToTM2.tr_ret_respects | Mathlib/Computability/TMToPartrec.lean | theorem tr_ret_respects (k v s) : ∃ b₂,
TrCfg (stepRet k v) b₂ ∧
Reaches₁ (TM2.step tr)
⟨some (Λ'.ret (trCont k)), s, K'.elim (trList v) [] [] (trContStack k)⟩ b₂ | case fix
f : Code
k : Cont
IH :
∀ (v : List ℕ) (s : Option Γ'),
∃ b₂,
TrCfg (stepRet k v) b₂ ∧
Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
b₂
v : List ℕ
s : Option Γ'
this :
if v.headI = 0 then natEnd (trList v).head?.i... | by_cases h : v.headI = 0 <;> simp only [h, ite_true, ite_false] at this ⊢ | case pos
f : Code
k : Cont
IH :
∀ (v : List ℕ) (s : Option Γ'),
∃ b₂,
TrCfg (stepRet k v) b₂ ∧
Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
b₂
v : List ℕ
s : Option Γ'
h : v.headI = 0
this : natEnd (trList v).head?.iget = ... | 5079d8447f8a95d7 |
hasFDerivWithinAt_closure_of_tendsto_fderiv | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | theorem hasFDerivWithinAt_closure_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
(f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
(h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
HasFDerivWithinA... | case mk.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y
h : Tends... | have diff : DifferentiableOn ℝ f (B ∩ s) := f_diff.mono inter_subset_right | case mk.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y
h : Tends... | b73104c22d482296 |
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed | Mathlib/Topology/Compactness/Compact.lean | theorem IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
{ι : Type v} [hι : Nonempty ι] (t : ι → Set X) (htd : Directed (· ⊇ ·) t)
(htn : ∀ i, (t i).Nonempty) (htc : ∀ i, IsCompact (t i)) (htcl : ∀ i, IsClosed (t i)) :
(⋂ i, t i).Nonempty | case intro.intro
X : Type u
inst✝ : TopologicalSpace X
ι : Type v
hι : Nonempty ι
t : ι → Set X
htd : Directed (fun x1 x2 => x1 ⊇ x2) t
htc : ∀ (i : ι), IsCompact (t i)
htcl : ∀ (i : ι), IsClosed (t i)
i₀ : ι := hι.some
htn : ∀ (i : ι), (t i).Nonempty
i j : ι
hji₀ : t i₀ ⊇ t j
hji : t i ⊇ t j
⊢ (t i₀ ∩ t i).Nonempty | exact (htn j).mono (subset_inter hji₀ hji) | no goals | 6ce44587c65756d8 |
bernoulliPowerSeries_mul_exp_sub_one | Mathlib/NumberTheory/Bernoulli.lean | theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X | case h.succ.succ.h
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if n.succ = 0 then 1 else 0) = 0
x : ℕ × ℕ
h : x.1 + x.2 = n.succ
⊢ ↑x.2! * (↑x.2 + 1) = (↑x.2 + 1) * ↑x.2! ∨ (x.1 + x.2)! = 0 | left | case h.succ.succ.h.h
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if n.succ = 0 then 1 else 0) = 0
x : ℕ × ℕ
h : x.1 + x.2 = n.succ
⊢ ↑x.2! * (↑x.2 + 1) = (↑x.2 + 1) * ↑x.2! | 0c6cdd2eaa278054 |
EReal.nhds_bot_basis | Mathlib/Topology/Instances/EReal/Lemmas.lean | theorem nhds_bot_basis : (𝓝 (⊥ : EReal)).HasBasis (fun _ : ℝ ↦ True) (Iio ·) | case intro.intro
x : EReal
hx : ⊥ < x
y : ℚ
hxy : ↑↑y < x
⊢ ∃ i', True ∧ Iio ↑i' ⊆ Iio x | exact ⟨_, trivial, Iio_subset_Iio hxy.le⟩ | no goals | ed8f653a28830301 |
EuclideanDomain.mod_eq_zero | Mathlib/Algebra/EuclideanDomain/Basic.lean | theorem mod_eq_zero {a b : R} : a % b = 0 ↔ b ∣ a :=
⟨fun h => by
rw [← div_add_mod a b, h, add_zero]
exact dvd_mul_right _ _, fun ⟨c, e⟩ => by
rw [e, ← add_left_cancel_iff, div_add_mod, add_zero]
haveI := Classical.dec
by_cases b0 : b = 0
· simp only [b0, zero_mul]
· rw [mul_div_cancel_le... | R : Type u
inst✝ : EuclideanDomain R
a b : R
x✝ : b ∣ a
c : R
e : a = b * c
this : (p : Prop) → Decidable p
⊢ b * c = b * (b * c / b) | by_cases b0 : b = 0 | case pos
R : Type u
inst✝ : EuclideanDomain R
a b : R
x✝ : b ∣ a
c : R
e : a = b * c
this : (p : Prop) → Decidable p
b0 : b = 0
⊢ b * c = b * (b * c / b)
case neg
R : Type u
inst✝ : EuclideanDomain R
a b : R
x✝ : b ∣ a
c : R
e : a = b * c
this : (p : Prop) → Decidable p
b0 : ¬b = 0
⊢ b * c = b * (b * c / b) | b42d0fe6e6a8faaf |
ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | theorem apply_norm_sq_eq_inner_adjoint_left (A : E →L[𝕜] F) (x : E) :
‖A x‖ ^ 2 = re ⟪(A† ∘L A) x, x⟫ | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : E
⊢ ⟪((adjoint A).comp A) x, x⟫_𝕜 = ⟪(adjoint A) (A x), x⟫_𝕜 | rfl | no goals | 777303bf55f47c11 |
ENNReal.funMulInvSnorm_rpow | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem funMulInvSnorm_rpow {p : ℝ} (hp0 : 0 < p) {f : α → ℝ≥0∞} {a : α} :
funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ c, f c ^ p ∂μ)⁻¹ | α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0 : 0 < p
f : α → ℝ≥0∞
a : α
⊢ ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ | rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one] | no goals | 221200e8b4b5a745 |
MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant | Mathlib/MeasureTheory/Measure/EverywherePos.lean | /-- If a compact closed set is everywhere positive with respect to a left-invariant measure on a
topological group, then it is a Gδ set. This is nontrivial, as there is no second-countability or
metrizability assumption in the statement, so a general compact closed set has no reason to be
a countable intersection of op... | case intro.intro.intro.intro.intro
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k... | have B : μ (((x * z ⁻¹) • k) \ k) = 0 :=
le_antisymm (ge_of_tendsto u_lim (Eventually.of_forall A)) bot_le | case intro.intro.intro.intro.intro
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k... | 5dba91f75a485a07 |
MeasureTheory.Measure.mkMetric_le_liminf_tsum | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem mkMetric_le_liminf_tsum {β : Type*} {ι : β → Type*} [∀ n, Countable (ι n)] (s : Set X)
{l : Filter β} (r : β → ℝ≥0∞) (hr : Tendsto r l (𝓝 0)) (t : ∀ n : β, ι n → Set X)
(ht : ∀ᶠ n in l, ∀ i, diam (t n i) ≤ r n) (hst : ∀ᶠ n in l, s ⊆ ⋃ i, t n i) (m : ℝ≥0∞ → ℝ≥0∞) :
mkMetric m s ≤ liminf (fun n => ∑'... | X : Type u_2
inst✝³ : EMetricSpace X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst✝ : ∀ (n : β), Countable (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), diam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ ... | refine le_of_forall_gt_imp_ge_of_dense fun c hc => ?_ | X : Type u_2
inst✝³ : EMetricSpace X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst✝ : ∀ (n : β), Countable (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), diam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ ... | 45914e8e7bc6bf9d |
CategoryTheory.PresheafHom.IsSheafFor.exists_app | Mathlib/CategoryTheory/Sites/SheafHom.lean | lemma exists_app (hx : x.Compatible) (g : Y ⟶ X) :
∃ (φ : F.obj (op Y) ⟶ G.obj (op Y)),
∀ {Z : C} (p : Z ⟶ Y) (hp : S (p ≫ g)), φ ≫ G.map p.op =
F.map p.op ≫ (x (p ≫ g) hp).app ⟨Over.mk (𝟙 Z)⟩ | C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
Y : C
hx : x.Compatible
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).a... | use (hG g).lift c | case h
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
Y : C
hx : x.Compatible
g : Y ⟶ X
c : Cone ((Sieve.pullback... | 8103034d905afbca |
HasStrictDerivAt.clm_comp | Mathlib/Analysis/Calculus/Deriv/Mul.lean | theorem HasStrictDerivAt.clm_comp (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x | 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
d' : E ... | have := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt | 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
d' : E ... | ac2874cca97d8015 |
Nat.one_ascFactorial | Mathlib/Data/Nat/Factorial/Basic.lean | theorem one_ascFactorial : ∀ (k : ℕ), (1 : ℕ).ascFactorial k = k.factorial
| 0 => ascFactorial_zero 1
| (k+1) => by
rw [ascFactorial_succ, one_ascFactorial k, Nat.add_comm, factorial_succ]
| k : ℕ
⊢ ascFactorial 1 (k + 1) = (k + 1)! | rw [ascFactorial_succ, one_ascFactorial k, Nat.add_comm, factorial_succ] | no goals | 087e1a0544fd70ba |
NumberField.hermiteTheorem.finite_of_discr_bdd_of_isReal | Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean | theorem finite_of_discr_bdd_of_isReal :
{K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
haveI : NumberField K := @NumberField.mk _ _ inferInstance K.prop
{w : InfinitePlace K | IsReal w}.Nonempty ∧ |discr K| ≤ N }.Finite | case intro.intro.refine_4
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈(B ⊔ 1) ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧... | rw [← (IntermediateField.lift_injective _).eq_iff, eq_comm] at hx₁ | case intro.intro.refine_4
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈(B ⊔ 1) ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧... | 258b240e119b72fe |
MeasureTheory.limsup_trim | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | theorem limsup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
limsup f (ae (μ.trim hm)) = limsup f (ae μ) | case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞
⊢ a ∈ {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} ↔ a ∈ {a | ∀ᵐ (n : α) ∂μ, f n ≤ a} | suffices h_meas_eq : μ { x | ¬f x ≤ a } = μ.trim hm { x | ¬f x ≤ a } by
simp_rw [Set.mem_setOf_eq, ae_iff, h_meas_eq] | case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞
⊢ μ {x | ¬f x ≤ a} = (μ.trim hm) {x | ¬f x ≤ a} | 77906aab397d62a7 |
isPreconnected_of_forall | Mathlib/Topology/Connected/Basic.lean | theorem isPreconnected_of_forall {s : Set α} (x : α)
(H : ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) : IsPreconnected s | case intro.intro.intro.intro.inl
α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen u
hv : IsOpen v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
xs : x ∈ s
xu : x ∈ u
⊢ (s ∩ (u ∩ v)).Nonempty | rcases H y ys with ⟨t, ts, xt, yt, ht⟩ | case intro.intro.intro.intro.inl.intro.intro.intro.intro
α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen u
hv : IsOpen v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
xs : x ∈ s
xu : x ∈ u
t : Set α
ts : t ⊆ s
... | 72217f788baa5fcd |
ONote.repr_opow | Mathlib/SetTheory/Ordinal/Notation.lean | theorem repr_opow (o₁ o₂) [NF o₁] [NF o₂] : repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ | case mk.intro.oadd.intro.mk.intro.succ
o₁ o₂ : ONote
inst✝¹ : o₁.NF
inst✝ : o₂.NF
m : ℕ
a0 : ONote
n : ℕ+
a' : ONote
e₁ : o₁.split = (a0.oadd n a', m)
N₁ : (a0.oadd n a').NF
r₁ : o₁.repr = (a0.oadd n a').repr + ↑m
this✝ : a0.NF
this : a'.NF
a00 : a0.repr ≠ 0
ad : ω ∣ a'.repr
al : a'.repr + ↑m < ω ^ a0.repr
aa : (a' + ↑... | simp [opow, opowAux2, r₂, opow_add, opow_mul, mul_assoc, add_assoc] | case mk.intro.oadd.intro.mk.intro.succ
o₁ o₂ : ONote
inst✝¹ : o₁.NF
inst✝ : o₂.NF
m : ℕ
a0 : ONote
n : ℕ+
a' : ONote
e₁ : o₁.split = (a0.oadd n a', m)
N₁ : (a0.oadd n a').NF
r₁ : o₁.repr = (a0.oadd n a').repr + ↑m
this✝ : a0.NF
this : a'.NF
a00 : a0.repr ≠ 0
ad : ω ∣ a'.repr
al : a'.repr + ↑m < ω ^ a0.repr
aa : (a' + ↑... | 778175cafc8055e7 |
SimpleGraph.colorable_iff_exists_bdd_nat_coloring | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem colorable_iff_exists_bdd_nat_coloring (n : ℕ) :
G.Colorable n ↔ ∃ C : G.Coloring ℕ, ∀ v, C v < n | case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : G.Coloring ℕ
Cf : ∀ (v : V), C v < n
⊢ G.Colorable n | refine ⟨Coloring.mk ?_ ?_⟩ | case mpr.intro.refine_1
V : Type u
G : SimpleGraph V
n : ℕ
C : G.Coloring ℕ
Cf : ∀ (v : V), C v < n
⊢ V → Fin n
case mpr.intro.refine_2
V : Type u
G : SimpleGraph V
n : ℕ
C : G.Coloring ℕ
Cf : ∀ (v : V), C v < n
⊢ ∀ {v w : V}, G.Adj v w → ?mpr.intro.refine_1 v ≠ ?mpr.intro.refine_1 w | d1efc174a6faab8a |
TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds | Mathlib/Topology/Bases.lean | theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
(h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) :
IsTopologicalBasis s :=
.of_hasBasis_nhds <| fun a ↦
(nhds_basis_opens a).to_hasBasis' (by simpa [and_assoc] using h_nhds a)
fun _... | α : Type u
t : TopologicalSpace α
s : Set (Set α)
h_open : ∀ u ∈ s, IsOpen u
h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u
a : α
⊢ ∀ (i : Set α), a ∈ i ∧ IsOpen i → ∃ i', (i' ∈ s ∧ a ∈ i') ∧ id i' ⊆ i | simpa [and_assoc] using h_nhds a | no goals | f15e782a19fbea2c |
root_X_pow_sub_C_ne_zero' | Mathlib/FieldTheory/KummerPolynomial.lean | lemma root_X_pow_sub_C_ne_zero' {n : ℕ} {a : K} (hn : 0 < n) (ha : a ≠ 0) :
(AdjoinRoot.root (X ^ n - C a)) ≠ 0 | case inl
K : Type u
inst✝ : Field K
a : K
ha : a ≠ 0
hn : 0 < Nat.succ 0
⊢ root (X ^ Nat.succ 0 - C a) ≠ 0 | rw [pow_one] | case inl
K : Type u
inst✝ : Field K
a : K
ha : a ≠ 0
hn : 0 < Nat.succ 0
⊢ root (X - C a) ≠ 0 | 153688240a8b7f53 |
Lean.Grind.Bool.not_eq_of_eq_true | Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Lemmas.lean | theorem Bool.not_eq_of_eq_true {a : Bool} (h : a = true) : (!a) = false | a : Bool
h : a = true
⊢ (!a) = false | simp [h] | no goals | 285707a31c5dceed |
powersEquivPowers_apply | Mathlib/GroupTheory/OrderOfElement.lean | theorem powersEquivPowers_apply (h : orderOf x = orderOf y) (n : ℕ) :
powersEquivPowers h ⟨x ^ n, n, rfl⟩ = ⟨y ^ n, n, rfl⟩ | G : Type u_1
inst✝¹ : LeftCancelMonoid G
inst✝ : Finite G
x y : G
h : orderOf x = orderOf y
n : ℕ
⊢ (finCongr h) ⟨n % orderOf x, ⋯⟩ = ⟨n % orderOf y, ⋯⟩ | simp [h] | no goals | 50b349098b756c8a |
padicNorm.values_discrete | Mathlib/NumberTheory/Padics/PadicNorm.lean | theorem values_discrete {q : ℚ} (hq : q ≠ 0) : ∃ z : ℤ, padicNorm p q = (p : ℚ) ^ (-z) :=
⟨padicValRat p q, by simp [padicNorm, hq]⟩
| p : ℕ
q : ℚ
hq : q ≠ 0
⊢ padicNorm p q = ↑p ^ (-padicValRat p q) | simp [padicNorm, hq] | no goals | 79f7011f2887507d |
Action.rightDual_ρ | Mathlib/CategoryTheory/Action/Monoidal.lean | theorem rightDual_ρ [RightRigidCategory V] (h : H) : Xᘁ.ρ h = (X.ρ (h⁻¹ : H))ᘁ | V : Type (u + 1)
inst✝³ : LargeCategory V
inst✝² : MonoidalCategory V
H : Type u
inst✝¹ : Group H
X : Action V H
inst✝ : RightRigidCategory V
h : H
⊢ Xᘁ.ρ h = X.ρ h⁻¹ᘁ | rw [← SingleObj.inv_as_inv] | V : Type (u + 1)
inst✝³ : LargeCategory V
inst✝² : MonoidalCategory V
H : Type u
inst✝¹ : Group H
X : Action V H
inst✝ : RightRigidCategory V
h : H
⊢ Xᘁ.ρ h = X.ρ (inv h)ᘁ
case x
V : Type (u + 1)
inst✝³ : LargeCategory V
inst✝² : MonoidalCategory V
H : Type u
inst✝¹ : Group H
X : Action V H
inst✝ : RightRigidCategory ... | 467c60d5375160b7 |
strictConcaveOn_log_Ioi | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log | x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < y - x
⊢ y⁻¹ < (log y - log x) / (y - x) | rw [lt_div_iff₀ h] | x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < y - x
⊢ y⁻¹ * (y - x) < log y - log x | b669e8d780cc761c |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_confirmRupHint_insertRup_fold | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem sat_of_confirmRupHint_insertRup_fold {n : Nat} (f : DefaultFormula n)
(f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n) (rupHints : Array Nat)
(p : PosFin n → Bool) (pf : p ⊨ f) :
let fc := insertRupUnits f (negate c)
let confirmRupHint_fold_res := rupHints.foldl (confirmRupHint fc.1.c... | case neg.intro.intro.inl.intro.inr.intro
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
rupHints : Array Nat
p : PosFin n → Bool
pf : p ⊨ f
fc : DefaultFormula n × Bool := f.insertRupUnits c.negate
confirmRupHint_fold_res : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool :=
... | simp only [negate_eq, List.mem_map, Prod.exists, Bool.exists_bool] at v_in_neg_c | case neg.intro.intro.inl.intro.inr.intro
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
rupHints : Array Nat
p : PosFin n → Bool
pf : p ⊨ f
fc : DefaultFormula n × Bool := f.insertRupUnits c.negate
confirmRupHint_fold_res : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool :=
... | 5ee4e43f082bad4a |
Padic.exi_rat_seq_conv_cauchy | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
let ⟨N2, hN2⟩ := f.cauchy₂ hε3
exists max N N2
intro j hj
suffices
padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f ... | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε | let ⟨N, hN⟩ := exi_rat_seq_conv f hε3 | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε | 5532e11826254be6 |
NNRat.cast_injective | Mathlib/Data/Rat/Cast/CharZero.lean | lemma cast_injective : Injective ((↑) : ℚ≥0 → α) | α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p = ↑q
⊢ p = q | rw [NNRat.cast_def, NNRat.cast_def, Commute.div_eq_div_iff] at hpq | α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ p = q
case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den
case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : ... | db8f302d56c944d4 |
HallMarriageTheorem.hall_cond_of_restrict | Mathlib/Combinatorics/Hall/Finite.lean | theorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (s' : Finset (s : Set ι)) :
#s' ≤ #(s'.biUnion fun a' => t a') | α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑↑s
⊢ #s' ≤ #(s'.biUnion fun a' => t ↑a') | rw [← card_image_of_injective s' Subtype.coe_injective] | α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑↑s
⊢ #(image (fun a => ↑a) s') ≤ #(s'.biUnion fun a' => t ↑a') | 3655e829fac3e08f |
ENat.add_one_natCast_le_withTop_of_lt | Mathlib/Data/ENat/Basic.lean | lemma add_one_natCast_le_withTop_of_lt {m : ℕ} {n : WithTop ℕ∞} (h : m < n) : (m + 1 : ℕ) ≤ n | m : ℕ
n✝ : WithTop ℕ∞
n : ℕ
h : ↑m < ↑n
⊢ ↑(m + 1) ≤ ↑n | simpa only [Nat.cast_le, ge_iff_le, Nat.cast_lt] using h | no goals | bb69bb09eeb550e5 |
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