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 |
|---|---|---|---|---|---|---|
CommApplicative.commutative_map | Mathlib/Control/Basic.lean | theorem CommApplicative.commutative_map {m : Type u → Type v} [h : Applicative m]
[CommApplicative m] {α β γ} (a : m α) (b : m β) {f : α → β → γ} :
f <$> a <*> b = flip f <$> b <*> a :=
calc
f <$> a <*> b = (fun p : α × β => f p.1 p.2) <$> (Prod.mk <$> a <*> b) | m : Type u → Type v
h : Applicative m
inst✝ : CommApplicative m
α β γ : Type u
a : m α
b : m β
f : α → β → γ
⊢ (fun a => (fun p => f p.fst p.snd) ∘ fun a_1 => (a_1, a)) <$> b <*> a = (fun b a => f a b) <$> b <*> a | rfl | no goals | 7b94d6ca6a1d855a |
Nat.factorial_lt | Mathlib/Data/Nat/Factorial/Basic.lean | theorem factorial_lt (hn : 0 < n) : n ! < m ! ↔ n < m | m n : ℕ
hn : 0 < n
h : n < m
⊢ ∀ {n : ℕ}, 0 < n → n ! < (n + 1)! | intro k hk | m n : ℕ
hn : 0 < n
h : n < m
k : ℕ
hk : 0 < k
⊢ k ! < (k + 1)! | a491a6261b8d2e38 |
Function.Injective.lieAlgebra_isNilpotent | Mathlib/Algebra/Lie/Nilpotent.lean | theorem Function.Injective.lieAlgebra_isNilpotent [h₁ : IsNilpotent L'] {f : L →ₗ⁅R⁆ L'}
(h₂ : Function.Injective f) : IsNilpotent L | R : Type u
L : Type v
L' : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
h₁ : ∃ k, lowerCentralSeries R L' L' k = ⊥
f : L →ₗ⁅R⁆ L'
h₂ : Injective ⇑f
⊢ ∃ k, lowerCentralSeries R L L k = ⊥ | peel h₁ with k hk | case h
R : Type u
L : Type v
L' : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
h₁ : ∃ k, lowerCentralSeries R L' L' k = ⊥
f : L →ₗ⁅R⁆ L'
h₂ : Injective ⇑f
k : ℕ
hk : lowerCentralSeries R L' L' k = ⊥
⊢ lowerCentralSeries R L L k = ⊥ | 01446f64e52ea22e |
ONote.cmp_compares | Mathlib/SetTheory/Ordinal/Notation.lean | theorem cmp_compares : ∀ (a b : ONote) [NF a] [NF b], (cmp a b).Compares a b
| 0, 0, _, _ => rfl
| oadd _ _ _, 0, _, _ => oadd_pos _ _ _
| 0, oadd _ _ _, _, _ => oadd_pos _ _ _
| o₁@(oadd e₁ n₁ a₁), o₂@(oadd e₂ n₂ a₂), h₁, h₂ => by -- TODO: golf
rw [cmp]
have IHe := @cmp_compares _ _ h₁.fst h₂.fst
s... | case inr.intro
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
left✝ : ↑n₂ ≤ ↑n₁
nh : (if ↑n₂ < ↑n₁ then Ordering.gt else Ordering.eq) = Ordering.gt
⊢ (Ordering.eq.t... | rw [ite_eq_iff] at nh | case inr.intro
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
left✝ : ↑n₂ ≤ ↑n₁
nh : ↑n₂ < ↑n₁ ∧ Ordering.gt = Ordering.gt ∨ ¬↑n₂ < ↑n₁ ∧ Ordering.eq = Ordering.gt
... | 52fdbaf9e931cd5c |
isZGroup_of_coprime | Mathlib/GroupTheory/SpecificGroups/ZGroup.lean | theorem isZGroup_of_coprime [Finite G] [IsZGroup G] [IsZGroup G'']
(h_le : f'.ker ≤ f.range) (h_cop : (Nat.card G).Coprime (Nat.card G'')) :
IsZGroup G' | case inr
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
f : G →* G'
f' : G' →* G''
inst✝² : Finite G
inst✝¹ : IsZGroup G
inst✝ : IsZGroup G''
h_le : f'.ker ≤ f.range
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
this : Fact (Nat.Prime p)
h_cop : (Nat.card ↥f'.ker).Coprime f'.ker... | have := (P.2.map f').isCyclic_of_isZGroup | case inr
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
f : G →* G'
f' : G' →* G''
inst✝² : Finite G
inst✝¹ : IsZGroup G
inst✝ : IsZGroup G''
h_le : f'.ker ≤ f.range
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
this✝ : Fact (Nat.Prime p)
h_cop : (Nat.card ↥f'.ker).Coprime f'.ke... | 8e6ebf384c14685f |
CategoryTheory.IsIso.of_isIso_fac_right | Mathlib/CategoryTheory/Iso.lean | theorem of_isIso_fac_right {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [IsIso g]
[hh : IsIso h] (w : f ≫ g = h) : IsIso f | C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h : X ⟶ Z
inst✝ : IsIso g
hh : IsIso (f ≫ g)
w : f ≫ g = h
this : IsIso (f ≫ g)
⊢ IsIso f | exact of_isIso_comp_right f g | no goals | 9dbbbbf52ab93b3e |
coeSubmodule_differentIdeal_fractionRing | Mathlib/RingTheory/DedekindDomain/Different.lean | lemma coeSubmodule_differentIdeal_fractionRing
[NoZeroSMulDivisors A B] [Algebra.IsIntegral A B]
[Algebra.IsSeparable (FractionRing A) (FractionRing B)]
[FiniteDimensional (FractionRing A) (FractionRing B)] :
coeSubmodule (FractionRing B) (differentIdeal A B) =
1 / Submodule.traceDual A (FractionR... | A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (F... | have := FractionalIdeal.dual_inv_le (A := A) (K := FractionRing A)
(1 : FractionalIdeal B⁰ (FractionRing B)) | A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (F... | 1468f2eabd5818af |
CompHausLike.LocallyConstant.presheaf_ext | Mathlib/Condensed/Discrete/LocallyConstant.lean | /--
To check equality of two elements of `X(S)`, it suffices to check equality after composing with
each `X(S) → X(Sᵢ)`.
-/
lemma presheaf_ext (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w)
[PreservesFiniteProducts X] (x y : X.obj ⟨S⟩)
[HasExplicitFiniteCoproducts.{u} P]
(h : ∀ (a : Fiber f), X.map (sigmaIncl ... | case a.a.h
P : TopCat → Prop
inst✝³ : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝² : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑S.toTop) (Y.obj (op (of P PUnit.{u + 1})))
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝¹ : Preserve... | exact h | no goals | 4aad38c3a12edfe7 |
Valued.continuous_extension | Mathlib/Topology/Algebra/Valued/ValuedField.lean | theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) | K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Func... | intro x | K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Func... | af62a79ad478b7b6 |
Ideal.IsHomogeneous.iff_exists | Mathlib/RingTheory/GradedAlgebra/Homogeneous/Ideal.lean | theorem Ideal.IsHomogeneous.iff_exists :
I.IsHomogeneous 𝒜 ↔ ∃ S : Set (homogeneousSubmonoid 𝒜), I = Ideal.span ((↑) '' S) | ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : GradedRing 𝒜
I : Ideal A
⊢ I = (homogeneousCore 𝒜 I).toIdeal ↔ ∃ S, I = span (Subtype.val '' S) | exact ((Set.image_preimage.compose (Submodule.gi _ _).gc).exists_eq_l _).symm | no goals | f8de24b792153c27 |
Set.image_inter_preimage | Mathlib/Data/Set/Image.lean | theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t | case h₂
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
⊢ f '' s ∩ t ⊆ f '' (s ∩ f ⁻¹' t) | rintro _ ⟨⟨x, h', rfl⟩, h⟩ | case h₂.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
h' : x ∈ s
h : f x ∈ t
⊢ f x ∈ f '' (s ∩ f ⁻¹' t) | f5a73b4bafb61e59 |
Bool.ofNat_toNat | Mathlib/Data/Bool/Basic.lean | theorem ofNat_toNat (b : Bool) : ofNat (toNat b) = b | b : Bool
⊢ ofNat b.toNat = b | cases b <;> rfl | no goals | 2b2549dbdc8fbe3c |
Polynomial.orderOf_root_cyclotomic_dvd | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.Prime] {a : ℕ}
(hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) :
orderOf (ZMod.unitOfCoprime a (coprime_of_root_cyclotomic hpos hroot)) ∣ n | case h
n : ℕ
hpos : 0 < n
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
⊢ eval ((Nat.castRingHom (ZMod p)) a) (X ^ n - 1) = 0 | rw [IsRoot.def] at hroot | case h
n : ℕ
hpos : 0 < n
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
hroot : eval ((Nat.castRingHom (ZMod p)) a) (cyclotomic n (ZMod p)) = 0
⊢ eval ((Nat.castRingHom (ZMod p)) a) (X ^ n - 1) = 0 | 0bbad10037176716 |
MeasureTheory.le_integral_rnDeriv_of_ac | Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean | /-- For a convex continuous function `f` on `[0, ∞)`, if `μ` is absolutely continuous
with respect to a probability measure `ν`, then
`f (μ univ).toReal ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`. -/
lemma le_integral_rnDeriv_of_ac [IsFiniteMeasure μ] [IsProbabilityMeasure ν]
(hf_cvx : ConvexOn ℝ (Ici 0) f) (hf_cont : Con... | case inr
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : x ∈ Ici 0
hx_pos : 0 < x
h : x ∈ interi... | simp only [nonempty_Iio, interior_Ici', mem_Ioi] at h | case inr
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : x ∈ Ici 0
hx_pos : 0 < x
h : 0 < x → Co... | 661fab8a26d078bf |
blimsup_cthickening_mul_ae_eq | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
(r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) | case neg
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → ... | positivity | no goals | 8749d72cf710712d |
HomotopicalAlgebra.nonempty_attachCells_iff | Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.lean | lemma nonempty_attachCells_iff :
Nonempty (AttachCells.{w} g f) ↔ (coproducts.{w} (ofHoms g)).pushouts f | case mpr.intro.intro.intro.intro.intro.intro.intro.mk
C : Type u
inst✝ : Category.{v, u} C
α : Type t
A B : α → C
g : (a : α) → A a ⟶ B a
X₁ X₂ : C
f : X₁ ⟶ X₂
ι : Type w
X Y : C
F₁ F₂ : Discrete ι ⥤ C
c₁ : Cocone F₁
c₂ : Cocone F₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
φ : F₁ ⟶ F₂
hφ : (ofHoms g).functorCategory (Discret... | let e₂ (i : ι) : F₂.obj ⟨i⟩ ≅ B (π i) := Arrow.rightFunc.mapIso (e i) | case mpr.intro.intro.intro.intro.intro.intro.intro.mk
C : Type u
inst✝ : Category.{v, u} C
α : Type t
A B : α → C
g : (a : α) → A a ⟶ B a
X₁ X₂ : C
f : X₁ ⟶ X₂
ι : Type w
X Y : C
F₁ F₂ : Discrete ι ⥤ C
c₁ : Cocone F₁
c₂ : Cocone F₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
φ : F₁ ⟶ F₂
hφ : (ofHoms g).functorCategory (Discret... | a02d1f6a7c721f88 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastConst.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Const.lean | theorem go_get_aux (aig : AIG α) (c : BitVec w) (curr : Nat) (hcurr : curr ≤ w)
(s : AIG.RefVec aig curr) :
-- `hfoo` makes it possible to `generalize` below. With a concrete proof term this
-- `generalize` would produce a type incorrect term as the proof term would talk about
-- a `go` application inst... | case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
h✝ : ¬curr < w
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (hfoo : aig.decls.size ≤ { aig := aig, vec := ⋯ ▸ s }.aig.decls.s... | simp only [Nat.le_refl, get, Ref.gate_cast, Ref.mk.injEq, true_implies] | case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
h✝ : ¬curr < w
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (hfoo : True), (⋯ ▸ s).get idx ⋯ = (s.get idx hidx).cast ⋯ | ba3ced8118779694 |
Real.not_continuousAt_deriv_qaryEntropy_one | Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean | lemma not_continuousAt_deriv_qaryEntropy_one :
¬ContinuousAt (deriv (qaryEntropy q)) 1 | case h
q : ℕ
tendstoBot : Tendsto (fun p => log (↑q - 1) + log (1 - p) - log p) (𝓝[<] 1) atBot
a✝¹ : ℝ
a✝ : a✝¹ ∈ Ioo (1 - 2⁻¹) 1
⊢ log (↑q - 1) + log (1 - a✝¹) - log a✝¹ = deriv (qaryEntropy q) a✝¹ | apply (deriv_qaryEntropy _ _).symm | q : ℕ
tendstoBot : Tendsto (fun p => log (↑q - 1) + log (1 - p) - log p) (𝓝[<] 1) atBot
a✝¹ : ℝ
a✝ : a✝¹ ∈ Ioo (1 - 2⁻¹) 1
⊢ a✝¹ ≠ 0
q : ℕ
tendstoBot : Tendsto (fun p => log (↑q - 1) + log (1 - p) - log p) (𝓝[<] 1) atBot
a✝¹ : ℝ
a✝ : a✝¹ ∈ Ioo (1 - 2⁻¹) 1
⊢ a✝¹ ≠ 1 | 75b19cc4897418b6 |
ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_atTop_one | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma IsRatCondKernelCDFAux.tendsto_atTop_one (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
(a : α) :
∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atTop (𝓝 1) | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n => f (a, t) ↑n) atTop (𝓝 1)
t : β
ht : Tendsto (fun n => f (a, t) ↑n) atTop (𝓝 1)
... | rw [tendsto_iff_tendsto_subseq_of_monotone h_mono tendsto_natCast_atTop_atTop] | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n => f (a, t) ↑n) atTop (𝓝 1)
t : β
ht : Tendsto (fun n => f (a, t) ↑n) atTop (𝓝 1)
... | 8dafd0eb3b713696 |
QuaternionGroup.orderOf_xa | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | theorem orderOf_xa [NeZero n] (i : ZMod (2 * n)) : orderOf (xa i) = 4 | n : ℕ
inst✝ : NeZero n
i : ZMod (2 * n)
⊢ orderOf (xa i) = 2 ^ 2 | haveI : Fact (Nat.Prime 2) := Fact.mk Nat.prime_two | n : ℕ
inst✝ : NeZero n
i : ZMod (2 * n)
this : Fact (Nat.Prime 2)
⊢ orderOf (xa i) = 2 ^ 2 | 53aed3d24b3f681b |
CategoryTheory.MonoidalCategory.rightUnitor_monoidal | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem rightUnitor_monoidal (X₁ X₂ : C) :
(ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
tensorμ X₁ (𝟙_ C) X₂ (𝟙_ C) ≫ ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom =
((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
X₁... | rw [← braiding_rightUnitor] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ ((β_ (𝟙_ C) X₂).hom ≫ (ρ_ X₂).hom) =
((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
X₁ ◁ (α_ (𝟙_ C) ... | 58daa7c29a4abff2 |
Real.logb_eq_iff_rpow_eq | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem logb_eq_iff_rpow_eq (hy : 0 < y) : logb b y = x ↔ b ^ x = y | b x y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hy : 0 < y
⊢ logb b y = x ↔ b ^ x = y | constructor <;> rintro rfl | case mp
b y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hy : 0 < y
⊢ b ^ logb b y = y
case mpr
b x : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hy : 0 < b ^ x
⊢ logb b (b ^ x) = x | 91501ddfc687ca4e |
ProbabilityTheory.IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici (a : α) (x : ℝ) :
ContinuousWithinAt (IsMeasurableRatCDF.stieltjesFunctionAux f a) (Ici x) x | case a
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
h' : ⨅ r, stieltjesFunctionAux f a ↑r = ⨅ r, stieltjesFunctionAux f a ↑↑r
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (⨅ a_1, stieltjesFunctionAux f a ↑a... | have h'' :
⨅ r : { r' : ℚ // x < r' }, stieltjesFunctionAux f a r =
⨅ r : { r' : ℚ // x < r' }, f a r := by
congr with r
exact stieltjesFunctionAux_eq hf a r | case a
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
h' : ⨅ r, stieltjesFunctionAux f a ↑r = ⨅ r, stieltjesFunctionAux f a ↑↑r
h'' : ⨅ r, stieltjesFunctionAux f a ↑↑r = ⨅ r, f a ↑r
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
Tendsto (stieltjesFunctionAux ... | 61ea09981116b634 |
IsIntegralCurveOn.hasDerivAt | Mathlib/Geometry/Manifold/IntegralCurve/Basic.lean | /-- If `γ` is an integral curve of a vector field `v`, then `γ t` is tangent to `v (γ t)` when
expressed in the local chart around the initial point `γ t₀`. -/
lemma IsIntegralCurveOn.hasDerivAt (hγ : IsIntegralCurveOn γ v s) {t : ℝ} (ht : t ∈ s)
(hsrc : γ t ∈ (extChartAt I (γ t₀)).source) :
HasDerivAt ((extCha... | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveOn γ v s
t : ℝ... | intro a | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveOn γ v s
t : ℝ... | e2bdac18aea0b848 |
OrderEmbedding.covBy_of_apply | Mathlib/Order/Cover.lean | theorem OrderEmbedding.covBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
(f : α ↪o β) {x y : α} (h : f x ⋖ f y) : x ⋖ y | case right
α : Type u_3
β : Type u_4
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ↪o β
x y : α
h : f x ⋖ f y
a : α
⊢ f x < f a → ¬f a < f y | apply h.2 | no goals | 331e966d90af3623 |
MeasureTheory.lintegral_mul_const' | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r | α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r | simp_rw [mul_comm, lintegral_const_mul' r f hr] | no goals | e96c65f6b507ac55 |
Int.ofNat_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem ofNat_sub (h : m ≤ n) : ((n - m : Nat) : Int) = n - m | m n : Nat
h : 0 ≤ n
⊢ ↑(n - 0) = ↑n - ↑0 | rfl | no goals | 0b3f339ee83c08f6 |
Equiv.pointReflection_midpoint_right | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | theorem Equiv.pointReflection_midpoint_right (x y : P) :
(Equiv.pointReflection (midpoint R x y)) y = x | R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁴ : Ring R
inst✝³ : Invertible 2
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y : P
⊢ (pointReflection (midpoint R x y)) y = x | rw [midpoint_comm, Equiv.pointReflection_midpoint_left] | no goals | 6ee0e9757cb4e18d |
OnePoint.continuousAt_coe | Mathlib/Topology/Compactification/OnePoint.lean | theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x | X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x | rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt] | X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x)) | e3a19212773164f5 |
ModuleCat.Tilde.localizationToStalk_mk | Mathlib/AlgebraicGeometry/Modules/Tilde.lean | theorem localizationToStalk_mk (x : PrimeSpectrum.Top R) (f : M) (s : x.asIdeal.primeCompl) :
(localizationToStalk M x).hom (LocalizedModule.mk f s) =
(tildeInModuleCat M).germ (PrimeSpectrum.basicOpen (s : R)) x s.2
(const M f s (PrimeSpectrum.basicOpen s) fun _ => id) :=
(Module.End_isUnit_iff _ |... | case iWU
R : Type u
inst✝ : CommRing R
M : ModuleCat R
x : ↑(PrimeSpectrum.Top R)
f : ↑M
s : ↥x.asIdeal.primeCompl
⊢ PrimeSpectrum.basicOpen ↑s ⟶ ⊤ | exact homOfLE le_top | no goals | 729d47d1450c6db4 |
eHolderNorm_eq_zero | Mathlib/Topology/MetricSpace/HolderNorm.lean | lemma eHolderNorm_eq_zero {r : ℝ≥0} {f : X → Y} :
eHolderNorm r f = 0 ↔ ∀ x₁ x₂, f x₁ = f x₂ | case neg
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
h : ∀ b > ⊥, ∃ i, ∃ (_ : HolderWith i r f), ↑i < b
x₁ x₂ : X
hx : ¬x₁ = x₂
⊢ edist (f x₁) (f x₂) ≤ 0 | refine le_of_forall_lt' fun b hb => ?_ | case neg
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
h : ∀ b > ⊥, ∃ i, ∃ (_ : HolderWith i r f), ↑i < b
x₁ x₂ : X
hx : ¬x₁ = x₂
b : ℝ≥0∞
hb : 0 < b
⊢ edist (f x₁) (f x₂) < b | 61029cfd55718559 |
AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_inv_snd | Mathlib/AlgebraicGeometry/Pullbacks.lean | theorem pullbackP1Iso_inv_snd (i : 𝒰.J) :
(pullbackP1Iso 𝒰 f g i).inv ≫ pullback.snd _ _ = pullback.fst _ _ | X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i : 𝒰.J
⊢ (pullbackP1Iso 𝒰 f g i).inv ≫ pullback.snd (p1 𝒰 f g) (𝒰.map i) = pullback.fst (𝒰.map i ≫ f) g | simp_rw [pullbackP1Iso, pullback.lift_snd] | no goals | 012093e10894fc44 |
Complex.HadamardThreeLines.norm_mul_invInterpStrip_le_one_of_mem_verticalClosedStrip | Mathlib/Analysis/Complex/Hadamard.lean | theorem norm_mul_invInterpStrip_le_one_of_mem_verticalClosedStrip (f : ℂ → E) (ε : ℝ) (hε : 0 < ε)
(z : ℂ) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
(hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1)) (hz : z ∈ verticalClosedStrip 0 1) :
‖F f ε z‖ ≤ 1 | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : 0 < ε
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1
⊢ ∃ B,
(fun x => invInterpStrip f x ε • f x) =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹... | obtain ⟨BF, hBF⟩ := F_BddAbove f ε hε hB | case h.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : 0 < ε
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1
BF : ℝ
hBF : BF ∈ upperBounds (norm ∘ F f ε '' verticalClosedStrip 0 1)
⊢ ∃ B,
... | 98bef02cbf34a7ba |
LinearMap.span_singleton_sup_orthogonal_eq_top | Mathlib/LinearAlgebra/SesquilinearForm.lean | theorem span_singleton_sup_orthogonal_eq_top {B : V →ₗ[K] V →ₗ[K] K} {x : V} (hx : ¬B.IsOrtho x x) :
(K ∙ x) ⊔ Submodule.orthogonalBilin (N := K ∙ x) (B := B) = ⊤ | K : Type u_13
V : Type u_16
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B : V →ₗ[K] V →ₗ[K] K
x : V
hx : ¬B.IsOrtho x x
⊢ Submodule.span K {x} ⊔ ker (B x) = ⊤ | exact (B x).span_singleton_sup_ker_eq_top hx | no goals | b8e64ce8cd021479 |
Choose.choose_modEq_choose_mod_mul_choose_div | Mathlib/Data/Nat/Choose/Lucas.lean | theorem choose_modEq_choose_mod_mul_choose_div :
choose n k ≡ choose (n % p) (k % p) * choose (n / p) (k / p) [ZMOD p] | case mp
n k p : ℕ
inst✝ : Fact (Nat.Prime p)
decompose : (X + 1) ^ n = (X + 1) ^ (n % p) * (X ^ p + 1) ^ (n / p)
x₁ x₂ : ℕ
hx : (x₁, x₂) ∈ range (n % p + 1) ×ˢ range (n / p + 1)
h : k = (x₁, x₂).1 + p * (x₁, x₂).2
⊢ k % p = x₁ ∧ k / p = x₂ | simp only [mem_product, mem_range] at hx | case mp
n k p : ℕ
inst✝ : Fact (Nat.Prime p)
decompose : (X + 1) ^ n = (X + 1) ^ (n % p) * (X ^ p + 1) ^ (n / p)
x₁ x₂ : ℕ
h : k = (x₁, x₂).1 + p * (x₁, x₂).2
hx : x₁ < n % p + 1 ∧ x₂ < n / p + 1
⊢ k % p = x₁ ∧ k / p = x₂ | 33a40e7a05a91053 |
irrational_nrt_of_n_not_dvd_multiplicity | Mathlib/Data/Real/Irrational.lean | theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)
[hp : Fact p.Prime] (hxr : x ^ n = m)
(hv : multiplicity (p : ℤ) m % n ≠ 0) :
Irrational x | x : ℝ
n : ℕ
m : ℤ
hm : m ≠ 0
p : ℕ
hp : Fact (Nat.Prime p)
hxr : x ^ n = ↑m
hv : multiplicity (↑p) m % n ≠ 0
⊢ Irrational x | rcases Nat.eq_zero_or_pos n with (rfl | hnpos) | case inl
x : ℝ
m : ℤ
hm : m ≠ 0
p : ℕ
hp : Fact (Nat.Prime p)
hxr : x ^ 0 = ↑m
hv : multiplicity (↑p) m % 0 ≠ 0
⊢ Irrational x
case inr
x : ℝ
n : ℕ
m : ℤ
hm : m ≠ 0
p : ℕ
hp : Fact (Nat.Prime p)
hxr : x ^ n = ↑m
hv : multiplicity (↑p) m % n ≠ 0
hnpos : n > 0
⊢ Irrational x | e5116bdccdf5b584 |
MeasureTheory.hitting_mono | Mathlib/Probability/Process/HittingTime.lean | theorem hitting_mono {m₁ m₂ : ι} (hm : m₁ ≤ m₂) : hitting u s n m₁ ω ≤ hitting u s n m₂ ω | case pos.intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
hm : m₁ ≤ m₂
h : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s
j : ι
hj₁ : j ∈ Set.Icc n m₂
hj₂ : u j ω ∈ s
⊢ m₁ ≤ sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) | refine le_csInf ⟨j, hj₁, hj₂⟩ ?_ | case pos.intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
hm : m₁ ≤ m₂
h : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s
j : ι
hj₁ : j ∈ Set.Icc n m₂
hj₂ : u j ω ∈ s
⊢ ∀ b ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}, m₁ ≤ b | 4352e36fcea59599 |
FreeCommRing.map_subtype_val_restriction | Mathlib/RingTheory/FreeCommRing.lean | theorem map_subtype_val_restriction {x} (s : Set α) [DecidablePred (· ∈ s)]
(hxs : IsSupported x s) : map (↑) (restriction s x) = x | case refine_3
α : Type u
x : FreeCommRing α
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
hxs : x.IsSupported s
⊢ ∀ z ∈ of '' s,
∀ (n : FreeCommRing α),
(map Subtype.val) ((restriction s) n) = n → (map Subtype.val) ((restriction s) (z * n)) = z * n | rintro _ ⟨p, hps, rfl⟩ n ih | case refine_3.intro.intro
α : Type u
x : FreeCommRing α
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
hxs : x.IsSupported s
p : α
hps : p ∈ s
n : FreeCommRing α
ih : (map Subtype.val) ((restriction s) n) = n
⊢ (map Subtype.val) ((restriction s) (of p * n)) = of p * n | 8af8f3ca1b5a0bde |
TensorAlgebra.ι_eq_algebraMap_iff | Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 | case refine_1
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
r : R
h : (ι R) x = (algebraMap R (TensorAlgebra R M)) r
this✝ : Module Rᵐᵒᵖ M := Module.compHom M ((RingHom.id R).fromOpposite ⋯)
this : IsCentralScalar R M
hf0 : toTrivSqZeroExt ((ι R) x) = (0, x)
⊢ x = 0... | rw [h, AlgHom.commutes] at hf0 | case refine_1
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
r : R
h : (ι R) x = (algebraMap R (TensorAlgebra R M)) r
this✝ : Module Rᵐᵒᵖ M := Module.compHom M ((RingHom.id R).fromOpposite ⋯)
this : IsCentralScalar R M
hf0 : (algebraMap R (TrivSqZeroExt R M)) r = (0,... | a75fb78d09e9680b |
SimpleGraph.Walk.count_support_takeUntil_eq_one | Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean | theorem count_support_takeUntil_eq_one {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) :
(p.takeUntil u h).support.count u = 1 | case cons.tail
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (h : u ∈ p✝.support), List.count u (p✝.takeUntil u h).support = 1
a✝ : List.Mem u p✝.support
⊢ List.count u ((cons h✝ p✝).takeUntil u ⋯).support = 1 | simp! only | case cons.tail
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (h : u ∈ p✝.support), List.count u (p✝.takeUntil u h).support = 1
a✝ : List.Mem u p✝.support
⊢ List.count u (if hx : u✝ = u then hx ▸ nil else cons h✝ (p✝.takeUntil u ⋯)).support = 1 | f16bba71e73c8e80 |
ProbabilityTheory.Kernel.densityProcess_fst_univ_ae | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma densityProcess_fst_univ_ae (κ : Kernel α (γ × β)) [IsFiniteKernel κ] (n : ℕ) (a : α) :
∀ᵐ x ∂(fst κ a), densityProcess κ (fst κ) n a x univ = 1 | case hd
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this✝ : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}
this : {x | (... | exact fun _ _ ↦ disjoint_countablePartition hs ht hst | no goals | 3cc629ecd9ee70bb |
Mathlib.Tactic.Ring.neg_one_mul | Mathlib/Tactic/Ring/Basic.lean | theorem neg_one_mul {R} [Ring R] {a b : R} (_ : (Int.negOfNat (nat_lit 1)).rawCast * a = b) :
-a = b | R : Type u_2
inst✝ : Ring R
a : R
⊢ -a = (Int.negOfNat 1).rawCast * a | simp [Int.negOfNat] | no goals | 8cf03d70e225cd71 |
LinearMap.split_surjective_of_localization_maximal | Mathlib/RingTheory/LocalProperties/Projective.lean | theorem LinearMap.split_surjective_of_localization_maximal
(f : M →ₗ[R] N) [Module.FinitePresentation R N]
(H : ∀ (I : Ideal R) (_ : I.IsMaximal),
∃ (g : _ →ₗ[Localization.AtPrime I] _),
(LocalizedModule.map I.primeCompl f).comp g = LinearMap.id) :
∃ (g : N →ₗ[R] M), f.comp g = LinearMap.id | R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = id
⊢ ∃ g, f ∘ₗ g = id | show LinearMap.id ∈ LinearMap.range (LinearMap.llcomp R N M N f) | R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = id
⊢ id ∈ range ((llcomp R N M N) ... | d80210efccbecc76 |
CategoryTheory.NonPreadditiveAbelian.add_comm | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | theorem add_comm {X Y : C} (a b : X ⟶ Y) : a + b = b + a | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - 0 - (0 - a - b) = b + a | conv_lhs =>
congr
next => skip
rw [← neg_def, neg_sub] | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - 0 - (-b - a) = b + a | 93ff39b7d8cf110a |
Reflexive.rel_of_ne_imp | Mathlib/Logic/Relation.lean | theorem Reflexive.rel_of_ne_imp (h : Reflexive r) {x y : α} (hr : x ≠ y → r x y) : r x y | case neg
α : Type u_1
r : α → α → Prop
h : Reflexive r
x y : α
hr : x ≠ y → r x y
hxy : ¬x = y
⊢ r x y | exact hr hxy | no goals | 706c15bcf7e0832c |
Module.finite_of_finrank_eq_succ | Mathlib/LinearAlgebra/Dimension/Free.lean | theorem finite_of_finrank_eq_succ {n : ℕ} (hn : finrank R M = n.succ) : Module.Finite R M :=
finite_of_finrank_pos <| by rw [hn]; exact n.succ_pos
| R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : StrongRankCondition R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
n : ℕ
hn : finrank R M = n.succ
⊢ 0 < n.succ | exact n.succ_pos | no goals | 07916223ed73054e |
AlgebraicGeometry.Scheme.ker_of_isAffine | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma ker_of_isAffine {X Y : Scheme} (f : X ⟶ Y) [IsAffine Y] :
f.ker = ofIdealTop (RingHom.ker f.appTop.hom) | X Y : Scheme
f : X ⟶ Y
inst✝ : IsAffine Y
⊢ RingHom.ker (CommRingCat.Hom.hom (Hom.app f ↑⟨⊤, ⋯⟩)) ≤
(ofIdealTop (RingHom.ker (CommRingCat.Hom.hom (Hom.appTop f)))).ideal ⟨⊤, ⋯⟩ | simp | no goals | f7b91b3482dc0e75 |
eq_of_powMul_faithful | Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean | theorem eq_of_powMul_faithful (f₁ : AlgebraNorm R S) (hf₁_pm : IsPowMul f₁) (f₂ : AlgebraNorm R S)
(hf₂_pm : IsPowMul f₂)
(h_eq : ∀ y : S, ∃ (C₁ C₂ : ℝ) (_ : 0 < C₁) (_ : 0 < C₂),
∀ x : Algebra.adjoin R {y}, f₁ x.val ≤ C₁ * f₂ x.val ∧ f₂ x.val ≤ C₂ * f₁ x.val) :
f₁ = f₂ | case a.intro.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f₁ : AlgebraNorm R S
hf₁_pm : IsPowMul ⇑f₁
f₂ : AlgebraNorm R S
hf₂_pm : IsPowMul ⇑f₂
h_eq :
∀ (y : S),
∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥(Algebra.adjoin R {y})), f₁ ↑x ≤ C₁... | rw [hy, h1, h2, eq_seminorms hg₁_pm hg₂_pm ⟨C₁, hC₁_pos, hC₁⟩ ⟨C₂, hC₂_pos, hC₂⟩] | no goals | f86a56fc3f1b670d |
Nat.le_log2 | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem le_log2 (h : n ≠ 0) : k ≤ n.log2 ↔ 2 ^ k ≤ n | n k✝ : Nat
h : n ≠ 0
k : Nat
h✝ : n ≥ 2
n0 : 0 < n / 2
⊢ 0 < 2 | decide | no goals | 61e6f5635f3912bb |
Multiset.count_finset_sup | Mathlib/Data/Finset/Lattice/Fold.lean | theorem count_finset_sup [DecidableEq β] (s : Finset α) (f : α → Multiset β) (b : β) :
count b (s.sup f) = s.sup fun a => count b (f a) | case refine_2
α : Type u_2
β : Type u_3
inst✝ : DecidableEq β
s✝ : Finset α
f : α → Multiset β
b : β
this : DecidableEq α := Classical.decEq α
i : α
s : Finset α
a✝ : i ∉ s
ih : count b (s.sup f) = s.sup fun a => count b (f a)
⊢ count b ((insert i s).sup f) = (insert i s).sup fun a => count b (f a) | rw [Finset.sup_insert, sup_eq_union, count_union, Finset.sup_insert, ih] | no goals | 63d14a7b80528253 |
QuaternionGroup.orderOf_a_one | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | theorem orderOf_a_one : orderOf (a 1 : QuaternionGroup n) = 2 * n | case inl
n : ℕ
hn : n = 0
⊢ orderOf (a 1) = 2 * n | subst hn | case inl
⊢ orderOf (a 1) = 2 * 0 | 6a0b04ee75963fd8 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.safe_insert_of_performRupCheck_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem safe_insert_of_performRupCheck_insertRup {n : Nat} (f : DefaultFormula n)
(f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n) (rupHints : Array Nat) :
(performRupCheck (insertRupUnits f (negate c)).1 rupHints).2.2.1 = true
→
Limplies (PosFin n) f (f.insert c) | case inl
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
rupHints : Array Nat
performRupCheck_success :
(Array.foldl (confirmRupHint (f.insertRupUnits c.negate).1.clauses)
((f.insertRupUnits c.negate).1.assignments, [], false, false) rupHints).2.2.fst =
true
p : Po... | rw [c'_eq_c] | case inl
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
rupHints : Array Nat
performRupCheck_success :
(Array.foldl (confirmRupHint (f.insertRupUnits c.negate).1.clauses)
((f.insertRupUnits c.negate).1.assignments, [], false, false) rupHints).2.2.fst =
true
p : Po... | 56eb1fda3331659a |
Finset.image₂_right_comm | Mathlib/Data/Finset/NAry.lean | theorem image₂_right_comm {γ : Type*} {u : Finset γ} {f : δ → γ → ε} {g : α → β → δ}
{f' : α → γ → δ'} {g' : δ' → β → ε} (h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
image₂ f (image₂ g s t) u = image₂ g' (image₂ f' s u) t :=
coe_injective <| by
push_cast
exact image2_right_comm h_right_comm... | α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : δ → γ → ε
g : α → β → δ
f' : α → γ → δ'
g' : δ' → β → ε
h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b
⊢ ... | exact image2_right_comm h_right_comm | no goals | aca330c91814883d |
Order.mem_range_pred_of_not_isPredPrelimit | Mathlib/Order/SuccPred/Limit.lean | theorem mem_range_pred_of_not_isPredPrelimit (h : ¬ IsPredPrelimit a) :
a ∈ range (pred : α → α) | case intro
α : Type u_1
a : α
inst✝¹ : PartialOrder α
inst✝ : PredOrder α
h : ¬IsPredPrelimit a
b : α
hb : ¬IsMin b ∧ pred b = a
⊢ a ∈ range pred | exact ⟨b, hb.2⟩ | no goals | b031daf673792464 |
Multiset.isDershowitzMannaLT_singleton_insert | Mathlib/Data/Multiset/DershowitzManna.lean | private lemma isDershowitzMannaLT_singleton_insert (h : OneStep N (a ::ₘ M)) :
∃ M', N = a ::ₘ M' ∧ OneStep M' M ∨ N = M + M' ∧ ∀ x ∈ M', x < a | case intro.intro.intro.intro.intro.inr.refine_1
α : Type u_1
inst✝ : Preorder α
M : Multiset α
a : α
X Y : Multiset α
b : α
h0 : a ::ₘ M = X + {b}
h2 : ∀ y ∈ Y, y < b
hab : a ≠ b
⊢ X + Y = a ::ₘ (Y + (M - {b})) | rw [← singleton_add, add_comm] at h0 | case intro.intro.intro.intro.intro.inr.refine_1
α : Type u_1
inst✝ : Preorder α
M : Multiset α
a : α
X Y : Multiset α
b : α
h0 : M + {a} = X + {b}
h2 : ∀ y ∈ Y, y < b
hab : a ≠ b
⊢ X + Y = a ::ₘ (Y + (M - {b})) | fe1657f5397146bc |
Matrix.det_blockDiagonal | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det | case refine_3.refine_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preserving_snd : ∀ {σ : Perm (n × o)},... | intro x | case refine_3.refine_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := filter (fun σ => ∀ (x : n × o), (σ x).2 = x.2) univ
mem_preserving_snd : ∀ {σ : Perm (n × o)},... | 85a593e300a16f88 |
exists_affineIndependent | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem exists_affineIndependent (s : Set P) :
∃ t ⊆ s, affineSpan k t = affineSpan k s ∧ AffineIndependent k ((↑) : t → P) | case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p : P
hp : p ∈ s
b : Set V
hb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s
hb₂ : Submodule.span k b = vectorSpan k s
hb₃ : AffineIndependent k fun p_1 => ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0... | exact ⟨mem_spanPoints k _ _ (Set.mem_insert p _), mem_spanPoints k _ _ hp⟩ | no goals | eed2f1fe5d5f5b68 |
List.mapIdxMGo_eq_mapIdxMAuxSpec | Mathlib/Data/List/Indexes.lean | theorem mapIdxMGo_eq_mapIdxMAuxSpec
[LawfulMonad m] {β} (f : ℕ → α → m β) (arr : Array β) (as : List α) :
mapIdxM.go f as arr = (arr.toList ++ ·) <$> mapIdxMAuxSpec f arr.size as | α : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
β : Type u
f : ℕ → α → m β
len : ℕ
ih :
∀ (arr : Array β) (as : List α),
as.length = len → mapIdxM.go f as arr = (fun x => arr.toList ++ x) <$> mapIdxMAuxSpec f arr.size as
arr : Array β
as : List α
head : α
tail : List α
h : tail.length = len
... | congr | case e_a
α : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
β : Type u
f : ℕ → α → m β
len : ℕ
ih :
∀ (arr : Array β) (as : List α),
as.length = len → mapIdxM.go f as arr = (fun x => arr.toList ++ x) <$> mapIdxMAuxSpec f arr.size as
arr : Array β
as : List α
head : α
tail : List α
h : tail.leng... | 484508954c433f55 |
piecewise_ae_eq_restrict | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem piecewise_ae_eq_restrict [DecidablePred (· ∈ s)] (hs : MeasurableSet s) :
piecewise s f g =ᵐ[μ.restrict s] f | α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 s] f | exact (piecewise_eqOn s f g).eventuallyEq.filter_mono inf_le_right | no goals | 952c1c111ae423b3 |
LaurentSeries.valuation_LaurentSeries_equal_extension | Mathlib/RingTheory/LaurentSeries.lean | theorem valuation_LaurentSeries_equal_extension :
(LaurentSeriesPkg K).isDenseInducing.extend Valued.v = (Valued.v : K⸨X⸩ → ℤₘ₀) | case hg
K : Type u_2
inst✝ : Field K
⊢ Continuous ⇑Valued.v | exact Valued.continuous_valuation (K := K⸨X⸩) | no goals | 8df9f023f0e8c820 |
ProbabilityTheory.Kernel.iIndepSets.iIndep | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepSets.iIndep (m : ι → MeasurableSpace Ω)
(h_le : ∀ i, m i ≤ _mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
(h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π κ μ) :
iIndep m κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
m : ι → MeasurableSpace Ω
h_le : ∀ (i : ι), m i ≤ _mΩ
π : ι → Set (Set Ω)
h_pi : ∀ (n : ι), IsPiSystem (π n)
h_generate : ∀ (i : ι), m i = generateFrom (π i)
h_ind : iIndepSets π κ μ
hμ : μ ≠ 0
η : Kernel... | have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
m : ι → MeasurableSpace Ω
h_le : ∀ (i : ι), m i ≤ _mΩ
π : ι → Set (Set Ω)
h_pi : ∀ (n : ι), IsPiSystem (π n)
h_generate : ∀ (i : ι), m i = generateFrom (π i)
h_ind : iIndepSets π κ μ
hμ : μ ≠ 0
η : Kernel... | 839768f1641edae0 |
CategoryTheory.OplaxNatTrans.whiskerRight_naturality_comp | Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.lean | theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
η.naturality (f ≫ g) ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapComp f g ▷ h =
F.mapComp f g ▷ η.app c ▷ h ≫
(α_ _ _ _).hom ▷ h ≫
(α_ _ _ _).hom ≫
F.map f ◁ η.naturality g ▷ h ≫
(α_ _ _ _).... | B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : OplaxFunctor B C
η : OplaxNatTrans F G
a b c : B
a' : C
f : a ⟶ b
g : b ⟶ c
h : G.obj c ⟶ a'
⊢ η.naturality (f ≫ g) ▷ h ≫ (α_ (η.app a) (G.map (f ≫ g)) h).hom ≫ η.app a ◁ G.mapComp f g ▷ h =
F.mapComp f g ▷ η.app c ▷ h ≫
(α_ (F.map f) (F... | rw [← associator_naturality_middle, ← comp_whiskerRight_assoc, naturality_comp] | B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : OplaxFunctor B C
η : OplaxNatTrans F G
a b c : B
a' : C
f : a ⟶ b
g : b ⟶ c
h : G.obj c ⟶ a'
⊢ (F.mapComp f g ▷ η.app c ≫
(α_ (F.map f) (F.map g) (η.app c)).hom ≫
F.map f ◁ η.naturality g ≫
(α_ (F.map f) (η.app ... | 43d5d0dc1a09d30b |
SimpleGraph.ediam_eq_one | Mathlib/Combinatorics/SimpleGraph/Diam.lean | @[simp]
lemma ediam_eq_one [Nontrivial α] : G.ediam = 1 ↔ G = ⊤ | case Adj.h.h.a
α : Type u_1
G : SimpleGraph α
inst✝ : Nontrivial α
h₁ : G.ediam = 1
u v : α
h₂ : 0 < G.edist u v
⊢ G.Adj u v | apply le_of_eq at h₁ | case Adj.h.h.a
α : Type u_1
G : SimpleGraph α
inst✝ : Nontrivial α
u v : α
h₂ : 0 < G.edist u v
h₁ : G.ediam ≤ 1
⊢ G.Adj u v | f79273a120dd6be5 |
Associates.dvd_of_mem_factors | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem dvd_of_mem_factors {a p : Associates α} (hm : p ∈ factors a) :
p ∣ a | case inr.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a | rw [← Associates.factors_prod a] | case inr.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a.factors.prod | 5299e19f6694fb2c |
tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (f... | case hmg
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : Measurable... | simp only [integrable_indicator_iff ht, integrableOn_const, ht, Measure.restrict_apply] | case hmg
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : Measurable... | fcf9a03d4108054c |
ProbabilityTheory.integrable_rpow_mul_exp_of_mem_interior_integrableExpSet | Mathlib/Probability/Moments/IntegrableExpMul.lean | /-- If `v` belongs to the interior of the interval `integrableExpSet X μ`,
then `X ^ p * exp (v * X)` is integrable for all nonnegative `p : ℝ`. -/
lemma integrable_rpow_mul_exp_of_mem_interior_integrableExpSet
(hv : v ∈ interior (integrableExpSet X μ)) {p : ℝ} (hp : 0 ≤ p) :
Integrable (fun ω ↦ X ω ^ p * exp (... | case intro.intro.intro.refine_3
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v p : ℝ
hp : 0 ≤ p
l u : ℝ
hvlu : v ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
h_pos : 0 < (v - l) ⊓ (u - v)
⊢ Integrable (fun ω => rexp ((v - ((v - l) ⊓ (u - v)) / 2) * X ω)) μ | exact h_subset (sub_half_inf_sub_mem_Ioo hvlu) | no goals | 0e622ce8317f663e |
Finset.mul_inv_eq_inv_mul_of_doubling_lt_two_aux | Mathlib/Combinatorics/Additive/VerySmallDoubling.lean | private lemma mul_inv_eq_inv_mul_of_doubling_lt_two_aux (h : #(A * A) < 2 * #A) :
A⁻¹ * A ⊆ A * A⁻¹ | G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
h : #(A * A) < 2 * #A
z : G
⊢ z ∈ A⁻¹ * A → z ∈ A * A⁻¹ | simp only [mem_mul, forall_exists_index, exists_and_left, and_imp, mem_inv,
exists_exists_and_eq_and] | G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
h : #(A * A) < 2 * #A
z : G
⊢ ∀ x ∈ A, ∀ x_1 ∈ A, x⁻¹ * x_1 = z → ∃ y ∈ A, ∃ a ∈ A, y * a⁻¹ = z | b8d9a308629ad83d |
Polynomial.Monic.geom_sum | Mathlib/RingTheory/Polynomial/Basic.lean | theorem Monic.geom_sum {P : R[X]} (hP : P.Monic) (hdeg : 0 < P.natDegree) {n : ℕ} (hn : n ≠ 0) :
(∑ i ∈ range n, P ^ i).Monic | case intro
R : Type u
inst✝ : Semiring R
P : R[X]
hP : P.Monic
hdeg : 0 < P.natDegree
a✝ : Nontrivial R
n : ℕ
hn : n.succ ≠ 0
k : ℕ
⊢ k < n → k * P.natDegree < n * P.natDegree | exact nsmul_lt_nsmul_left hdeg | no goals | 88dcf173b62f3103 |
MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_enorm | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem tendsto_approxOn_range_L1_enorm [OpensMeasurableSpace E] {f : β → E} {μ : Measure β}
[SeparableSpace (range f ∪ {0} : Set E)] (fmeas : Measurable f) (hf : Integrable f μ) :
Tendsto (fun n => ∫⁻ x, ‖approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖ₑ ∂μ) atTop
(𝓝 0) | β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f fmeas (Set.range f ∪ {0}... | apply tendsto_approxOn_L1_enorm fmeas | case hμ
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure (Set.range f ∪ {0})
case hi
β... | 9089b3887e889536 |
Subadditive.tendsto_lim | Mathlib/Analysis/Subadditive.lean | theorem tendsto_lim (hbdd : BddBelow (range fun n => u n / n)) :
Tendsto (fun n => u n / n) atTop (𝓝 h.lim) | u : ℕ → ℝ
h : Subadditive u
hbdd : BddBelow (range fun n => u n / ↑n)
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 h.lim) | refine tendsto_order.2 ⟨fun l hl => ?_, fun L hL => ?_⟩ | case refine_1
u : ℕ → ℝ
h : Subadditive u
hbdd : BddBelow (range fun n => u n / ↑n)
l : ℝ
hl : l < h.lim
⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b
case refine_2
u : ℕ → ℝ
h : Subadditive u
hbdd : BddBelow (range fun n => u n / ↑n)
L : ℝ
hL : L > h.lim
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < L | 4ac4c5b856326f16 |
Finset.insert_compl_self | Mathlib/Data/Finset/BooleanAlgebra.lean | theorem insert_compl_self (x : α) : insert x ({x}ᶜ : Finset α) = univ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
x : α
⊢ insert x {x}ᶜ = univ | rw [← compl_erase, erase_singleton, compl_empty] | no goals | bac433c196f0ec9d |
MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) :
DenseRange (coeSimpleFuncNonnegToLpNonneg p μ G) := fun g ↦ by
borelize G
rw [mem_closure_iff_seq_limit]
have hg_memLp : MemLp (g : α → G) p μ := Lp.memLp (g : Lp G p μ)
have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ ... | α : Type u_1
inst✝¹ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝ : NormedLatticeAddCommGroup G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : Separab... | have g_meas : Measurable (g : α → G) := (Lp.stronglyMeasurable (g : Lp G p μ)).measurable | α : Type u_1
inst✝¹ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝ : NormedLatticeAddCommGroup G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : Separab... | 1fc70ad144f2653f |
MeasureTheory.exists_seq_tendstoInMeasure_atTop_iff | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | theorem exists_seq_tendstoInMeasure_atTop_iff [IsFiniteMeasure μ]
{f : ℕ → α → E} (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) {g : α → E} :
TendstoInMeasure μ f atTop g ↔
∀ ns : ℕ → ℕ, StrictMono ns → ∃ ns' : ℕ → ℕ, StrictMono ns' ∧
∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f (ns (ns' i)) ω) atTop (𝓝 (g... | case intro.intro
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
ε : ℝ
hε : 0 < ε
h2 : ¬Tendsto (fun i => (μ {x | ε ≤ dist (f i x) (g x)}).toNNReal) atTop (𝓝 0)
s : Set ℝ≥0
hs : s ∈ 𝓝 0
... | obtain ⟨δ, hδ, h5⟩ := NNReal.nhds_zero_basis.mem_iff.1 hs | case intro.intro.intro.intro
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
ε : ℝ
hε : 0 < ε
h2 : ¬Tendsto (fun i => (μ {x | ε ≤ dist (f i x) (g x)}).toNNReal) atTop (𝓝 0)
s : Set ℝ≥0
hs... | f829873f3e8498ff |
MeasureTheory.Measure.exists_positive_of_not_mutuallySingular | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem exists_positive_of_not_mutuallySingular (μ ν : Measure α) [IsFiniteMeasure μ]
[IsFiniteMeasure ν] (h : ¬ μ ⟂ₘ ν) :
∃ ε : ℝ≥0, 0 < ε ∧
∃ E : Set α, MeasurableSet E ∧ 0 < ν E
∧ ∀ A, MeasurableSet A → ε * ν (A ∩ E) ≤ μ (A ∩ E) | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ)... | exact hb | no goals | 48437d57402f42cd |
swap_mul_swap_mul_swap | Mathlib/Algebra/Group/End.lean | theorem swap_mul_swap_mul_swap {x y z : α} (hxy : x ≠ y) (hxz : x ≠ z) :
swap y z * swap x y * swap y z = swap z x | α : Type u_4
inst✝ : DecidableEq α
x y z : α
hxy : x ≠ y
hxz : x ≠ z
⊢ swap y z * swap x y * swap y z = swap z x | nth_rewrite 3 [← swap_inv] | α : Type u_4
inst✝ : DecidableEq α
x y z : α
hxy : x ≠ y
hxz : x ≠ z
⊢ swap y z * swap x y * (swap y z)⁻¹ = swap z x | a6fbb5a9b4adfbc4 |
Submodule.FG.stabilizes_of_iSup_eq | Mathlib/RingTheory/Finiteness/Basic.lean | theorem FG.stabilizes_of_iSup_eq {M' : Submodule R M} (hM' : M'.FG) (N : ℕ →o Submodule R M)
(H : iSup N = M') : ∃ n, M' = N n | case h.a
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
M' : Submodule R M
N : ℕ →o Submodule R M
H : iSup ⇑N = M'
S : Finset M
hS : span R ↑S = M'
f : { x // x ∈ S } → ℕ
hf : ∀ (s : { x // x ∈ S }), ↑s ∈ N (f s)
s : M
hs : s ∈ ↑S
⊢ s ∈ ↑(N (S.attach.sup f)) | exact N.2 (Finset.le_sup <| S.mem_attach ⟨s, hs⟩) (hf _) | no goals | 8a1a767d6cea695a |
BitVec.msb_abs | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem msb_abs {w : Nat} {x : BitVec w} :
x.abs.msb = (decide (x = intMin w) && decide (0 < w)) | case neg
w : Nat
x : BitVec w
h₀ : 0 < w
h₁ : ¬x = intMin w
h₂ : ¬x.msb = true
⊢ (if x.msb = true then -x else x).msb = (false && decide (0 < w)) | simp [h₂] | no goals | b96e82b9c2fda510 |
exists_dist_eq | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z | E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z | use a • x + b • z | case h
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ dist x (a • x + b • z) = b * dist x z ∧ dist (a • x + b • z) z = a * dist x z | b95a636671d023aa |
Besicovitch.SatelliteConfig.exists_normalized_aux3 | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | theorem exists_normalized_aux3 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
(lastc : a.c (last N) = 0) (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4)
(i j : Fin N.succ) (inej : i ≠ j) (hi : 2 < ‖a.c i‖) (hij : ‖a.c i‖ ≤ ‖a.c j‖) :
1 - δ ≤ ‖(2 / ‖a.c i‖) • a.c i - (2 / ‖a.c j‖) • a.c j‖... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
i j : Fin N.succ
inej : i ≠ j
hi : 2 < ‖a.c i‖
hij : ‖a.c i‖ ≤ ‖a.c j‖
ah : Pairwise fun i j => a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ... | have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1] | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
i j : Fin N.succ
inej : i ≠ j
hi : 2 < ‖a.c i‖
hij : ‖a.c i‖ ≤ ‖a.c j‖
ah : Pairwise fun i j => a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ... | 03f5b92dee305e5f |
CategoryTheory.Pretriangulated.productTriangle_distinguished | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | /-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
(hT : ∀ j, T j ∈ distTriang C)
[HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
[HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)... | C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun ... | simpa using this | no goals | ce9c86c420201fe7 |
CStarRing.norm_coe_unitary_mul | Mathlib/Analysis/CStarAlgebra/Basic.lean | theorem norm_coe_unitary_mul (U : unitary E) (A : E) : ‖(U : E) * A‖ = ‖A‖ | E : Type u_2
inst✝² : NormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
U : ↥(unitary E)
A : E
⊢ ‖↑U * A‖ = ‖A‖ | nontriviality E | E : Type u_2
inst✝² : NormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
U : ↥(unitary E)
A : E
a✝ : Nontrivial E
⊢ ‖↑U * A‖ = ‖A‖ | f09abe48df8a1a8c |
List.head?_pmap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Attach.lean | theorem head?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
(H : ∀ (a : α), a ∈ xs → P a) :
(xs.pmap f H).head? = xs.attach.head?.map fun ⟨a, m⟩ => f a (H a m) | case nil
α : Type u_1
β : Type u_2
P : α → Prop
f : (a : α) → P a → β
H : ∀ (a : α), a ∈ [] → P a
⊢ (pmap f [] H).head? =
Option.map
(fun x =>
match x with
| ⟨a, m⟩ => f a ⋯)
[].attach.head? | simp | no goals | 3831ebbe35343d5a |
CategoryTheory.IsGrothendieckAbelian.subobjectMk_of_isColimit_eq_iSup | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Subobject.lean | /-- If `C` is a Grothendieck abelian category, `X : C`, if `F : J ⥤ MonoOver X` is a
functor from a filtered category `J`, the colimit of `F` (computed in `C`) gives
a subobject of `F` which is a supremum of the subobjects corresponding to
the objects in the image of the functor `F`. -/
lemma subobjectMk_of_isColimit_e... | C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : IsGrothendieckAbelian.{w, v, u} C
X : C
J : Type w
inst✝¹ : SmallCategory J
F : J ⥤ MonoOver X
inst✝ : IsFiltered J
c : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)
hc : IsColimit c
f : c.pt ⟶ X
hf : ∀ (j : J), c.ι.app j ≫ f = (F.obj j).obj.hom
this : ... | apply le_antisymm | case a
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : IsGrothendieckAbelian.{w, v, u} C
X : C
J : Type w
inst✝¹ : SmallCategory J
F : J ⥤ MonoOver X
inst✝ : IsFiltered J
c : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)
hc : IsColimit c
f : c.pt ⟶ X
hf : ∀ (j : J), c.ι.app j ≫ f = (F.obj j).obj.hom
... | 23cc3e98d4dca6a2 |
adjoin_le_integralClosure | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
Algebra.adjoin R {x} ≤ integralClosure R A | R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A
hx : IsIntegral R x
⊢ Algebra.adjoin R {x} ≤ integralClosure R A | rw [Algebra.adjoin_le_iff] | R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A
hx : IsIntegral R x
⊢ {x} ⊆ ↑(integralClosure R A) | d466a6f9b3a6fd44 |
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 } | case neg.intro.intro
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... | rw [km] at hk h'k | case neg.intro.intro
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... | 28c357cacce7671e |
Std.DHashMap.get!_insertMany_list_of_mem | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Lemmas.lean | theorem get!_insertMany_list_of_mem [LawfulBEq α]
{l : List ((a : α) × β a)} {k k' : α} (k_beq : k == k') {v : β k} [Inhabited (β k')]
(distinct : l.Pairwise (fun a b => (a.1 == b.1) = false))
(mem : ⟨k, v⟩ ∈ l) :
(m.insertMany l).get! k' = cast (by congr; apply LawfulBEq.eq_of_beq k_beq) v :=
Raw₀.ge... | case e_a
α : Type u
β : α → Type v
x✝¹ : BEq α
x✝ : Hashable α
m : DHashMap α β
inst✝¹ : LawfulBEq α
l : List ((a : α) × β a)
k k' : α
k_beq : (k == k') = true
v : β k
inst✝ : Inhabited (β k')
distinct : List.Pairwise (fun a b => (a.fst == b.fst) = false) l
mem : ⟨k, v⟩ ∈ l
⊢ k = k' | apply LawfulBEq.eq_of_beq k_beq | no goals | bed2cb906f5e044b |
LinearLocallyFiniteOrder.succFn_le_of_lt | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | theorem succFn_le_of_lt (i j : ι) (hij : i < j) : succFn i ≤ j | ι : Type u_1
inst✝ : LinearOrder ι
i j : ι
hij : i < j
h : (∀ x ∈ Set.Ioi i, succFn i ≤ x) ∧ succFn i ∈ upperBounds (lowerBounds (Set.Ioi i))
⊢ succFn i ≤ j | exact h.1 j hij | no goals | 22c67ce05b62e111 |
Field.isAlgebraic_of_adjoin_eq_adjoin | Mathlib/FieldTheory/PrimitiveElement.lean | theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α | case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
r s : F[X]
h✝ : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hm :... | simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this | no goals | ce2babdd1545fb9d |
Algebra.FormallySmooth.pi_iff | Mathlib/RingTheory/Smooth/Pi.lean | theorem pi_iff [Finite I] :
FormallySmooth R (Π i, A i) ↔ ∀ i, FormallySmooth R (A i) | R : Type (max u v)
I : Type u
A : I → Type (max u v)
inst✝⁵ : CommRing R
inst✝⁴ : (i : I) → CommRing (A i)
inst✝³ : (i : I) → Algebra R (A i)
inst✝² : Finite I
val✝ : Fintype I
H : ∀ (i : I), FormallySmooth R (A i)
B : Type (max u v)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
g : ((i : I) → A i)... | rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton] | R : Type (max u v)
I : Type u
A : I → Type (max u v)
inst✝⁵ : CommRing R
inst✝⁴ : (i : I) → CommRing (A i)
inst✝³ : (i : I) → Algebra R (A i)
inst✝² : Finite I
val✝ : Fintype I
H : ∀ (i : I), FormallySmooth R (A i)
B : Type (max u v)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
g : ((i : I) → A i)... | 0128f0207720570c |
UV.compress_injOn | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compress_injOn : Set.InjOn (compress u v) ↑{a ∈ s | compress u v a ∉ s} | case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v : α
inst✝ : DecidableEq α
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : Disjoint u b ∧ v ≤ b
hb : b ∈ s ∧ (b ⊔ u) \ v ∉ s
hab : (a ⊔ u) \ v = (b ⊔... | exact sup_sdiff_injOn u v has hbs hab | no goals | 4c8b28ba0d63c5ee |
Array.getElem_insertIdx_loop_gt | Mathlib/.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | theorem getElem_insertIdx_loop_gt {as : Array α} {i : Nat} {j : Nat} {hj : j < as.size}
{k : Nat} {h} (w : i < k) :
(insertIdx.loop i as ⟨j, hj⟩)[k] =
if k ≤ j then as[k-1]'(by simp at h; omega) else as[k]'(by simpa using h) | α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
w : i < k
⊢ (insertIdx.loop i as ⟨j, hj⟩)[k] = if k ≤ j then as[k - 1] else as[k] | unfold insertIdx.loop | α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
w : i < k
⊢ (if i < ↑⟨j, hj⟩ then
let j' := ⟨↑⟨j, hj⟩ - 1, ⋯⟩;
let as_1 := as.swap ↑j' ↑⟨j, hj⟩ ⋯ ⋯;
insertIdx.loop i as_1 ⟨↑j', ⋯⟩
else as)[k] =
if k ≤ j then as[k - 1] else as[k] | ab494e4c94494354 |
isIntegral_localization | Mathlib/RingTheory/Localization/Integral.lean | theorem isIntegral_localization [Algebra.IsIntegral R S] :
(map Sₘ (algebraMap R S)
(show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) :
Rₘ →+* _).IsIntegral | R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁶ : CommRing Rₘ
inst✝⁵ : CommRing Sₘ
inst✝⁴ : Algebra R Rₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : Algebra S Sₘ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝ : Algeb... | intro x | R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁶ : CommRing Rₘ
inst✝⁵ : CommRing Sₘ
inst✝⁴ : Algebra R Rₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : Algebra S Sₘ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝ : Algeb... | 8b8cef4c34437a28 |
Stream'.drop_drop | Mathlib/Data/Stream/Init.lean | theorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s | case a
α : Type u
n m : ℕ
s : Stream' α
n✝ : ℕ
⊢ (drop n (drop m s)).get n✝ = (drop (m + n) s).get n✝ | simp [Nat.add_assoc] | no goals | 6fb978b75adf3eda |
PFunctor.M.bisim | Mathlib/Data/PFunctor/Univariate/M.lean | theorem bisim (R : M P → M P → Prop)
(h : ∀ x y, R x y → ∃ a f f', M.dest x = ⟨a, f⟩ ∧ M.dest y = ⟨a, f'⟩ ∧ ∀ i, R (f i) (f' i)) :
∀ x y, R x y → x = y | case head.intro.intro.intro.intro.intro
P : PFunctor.{u}
R : P.M → P.M → Prop
this : Inhabited P.A
a a' : P.A
f : P.B a → P.M
f' : P.B a' → P.M
ih : R (M.mk ⟨a, f⟩) (M.mk ⟨a', f'⟩)
a'' : P.A
g g' : P.B a'' → P.M
h₂ : ∀ (i : P.B a''), R (g i) (g' i)
h₀ : (M.mk ⟨a, f⟩).dest.fst = ⟨a'', g⟩.fst
h₁ : (M.mk ⟨a', f'⟩).dest.fs... | simp only [dest_mk] at h₀ h₁ | case head.intro.intro.intro.intro.intro
P : PFunctor.{u}
R : P.M → P.M → Prop
this : Inhabited P.A
a a' : P.A
f : P.B a → P.M
f' : P.B a' → P.M
ih : R (M.mk ⟨a, f⟩) (M.mk ⟨a', f'⟩)
a'' : P.A
g g' : P.B a'' → P.M
h₂ : ∀ (i : P.B a''), R (g i) (g' i)
h₀ : a = a''
h₁ : a' = a''
⊢ a = a' | ec0e4121dc7f2f06 |
MvPolynomial.rank_R | Mathlib/FieldTheory/Finite/Polynomial.lean | theorem rank_R [Fintype σ] : Module.rank K (R σ K) = Fintype.card (σ → K) :=
calc
Module.rank K (R σ K) =
Module.rank K (↥{ s : σ →₀ ℕ | ∀ n : σ, s n ≤ Fintype.card K - 1 } →₀ K) :=
LinearEquiv.rank_eq
(Finsupp.supportedEquivFinsupp { s : σ →₀ ℕ | ∀ n : σ, s n ≤ Fintype.card K - 1 })
_ =... | σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Fintype σ
f : σ →₀ ℕ
⊢ f ∈ {s | ∀ (n : σ), s n ≤ Fintype.card K - 1} ↔ Finsupp.equivFunOnFinite f ∈ {s | ∀ (n : σ), s n < Fintype.card K} | refine forall_congr' fun n => le_tsub_iff_right ?_ | σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Fintype σ
f : σ →₀ ℕ
n : σ
⊢ 1 ≤ Fintype.card K | 9d3af127cf4d8525 |
AkraBazziRecurrence.one_mem_range_sumCoeffsExp | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma one_mem_range_sumCoeffsExp : 1 ∈ Set.range (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) | case le_one
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∃ a_1, ∑ i : α, a i * b i ^ a_1 ≤ 1
case ge_one
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a... | case le_one =>
exact R.tendsto_zero_sumCoeffsExp.eventually_le_const zero_lt_one |>.exists | case ge_one
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∃ b_1, 1 ≤ ∑ i : α, a i * b i ^ b_1 | 682141ea82f75568 |
Lean.Data.AC.Context.evalList_insert | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.evalList_insert
(ctx : Context α)
(h : Commutative ctx.op)
(x : Nat)
(xs : List Nat)
: evalList α ctx (insert x xs) = evalList α ctx (x::xs) | case isFalse
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
x y z : Nat
zs : List Nat
ih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)
h✝¹ : ¬x < y
h✝ : ¬x < z
⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs) | next => simp_all [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)] | no goals | 37aa11cde795cd0c |
AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | theorem toΓSpecCApp_iff
(f :
(structureSheaf <| Γ.obj <| op X).val.obj (op <| basicOpen r) ⟶
X.presheaf.obj (op <| X.toΓSpecMapBasicOpen r)) :
toOpen _ (basicOpen r) ≫ f = X.toToΓSpecMapBasicOpen r ↔ f = X.toΓSpecCApp r | case mp.hf
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
f : (structureSheaf ↑(Γ.obj (op X))).val.obj (op (basicOpen r)) ⟶ X.presheaf.obj (op (X.toΓSpecMapBasicOpen r))
loc_inst : IsLocalization.Away r ↑((structureSheaf ↑(Γ.obj (op X))).val.obj (op (basicOpen r)))
h :
ConcreteCategory.hom (toOpen (↑(Γ.obj (op X))) (basi... | exact IsLocalization.ringHom_ext (Submonoid.powers r) h | no goals | e699f685bb0ed6c2 |
isLocalHom_of_le_jacobson_bot | Mathlib/RingTheory/Henselian.lean | theorem isLocalHom_of_le_jacobson_bot {R : Type*} [CommRing R] (I : Ideal R)
(h : I ≤ Ideal.jacobson ⊥) : IsLocalHom (Ideal.Quotient.mk I) | R : Type u_1
inst✝ : CommRing R
I : Ideal R
h✝ : I ≤ ⊥.jacobson
a : R
h : ∃ b, (Ideal.Quotient.mk I) a * b = 1
⊢ ∃ b, (Ideal.Quotient.mk ⊥.jacobson) a * b = 1 | obtain ⟨b, hb⟩ := h | case intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R
h : I ≤ ⊥.jacobson
a : R
b : R ⧸ I
hb : (Ideal.Quotient.mk I) a * b = 1
⊢ ∃ b, (Ideal.Quotient.mk ⊥.jacobson) a * b = 1 | a961c7da2da42bc3 |
Besicovitch.SatelliteConfig.exists_normalized | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
(lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ Pairwise fun i j => 1 - δ ≤ ‖c' i - c' j‖ | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ)... | rcases le_or_lt ‖a.c j‖ 2 with (Hj | Hj) | case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fi... | ef3e2825a59b79a7 |
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