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continuous_zpow | Mathlib/Topology/Algebra/Group/Basic.lean | theorem continuous_zpow : ∀ z : ℤ, Continuous fun a : G => a ^ z
| Int.ofNat n => by simpa using continuous_pow n
| Int.negSucc n => by simpa using (continuous_pow (n + 1)).inv
| G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
n : ℕ
⊢ Continuous fun a => a ^ Int.ofNat n | simpa using continuous_pow n | no goals | c3ebc7d6b2302a3b |
Topology.IsLower.isTopologicalSpace_basis | Mathlib/Topology/Order/LowerUpperTopology.lean | lemma isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U | case h.a
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen U
S : Set (Set α)
hS1 : S ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}
hS2 : U = ⋃₀ S
hUS : univ ∉ S
b : α
hb : b ∈ ⋂₀ (compl '' S)
c : α
hc : c ∈ {a | (Ici a)ᶜ ∈ S}
⊢ c ≤ b | simp only [sInter_image, mem_iInter] at hb | case h.a
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen U
S : Set (Set α)
hS1 : S ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}
hS2 : U = ⋃₀ S
hUS : univ ∉ S
b c : α
hc : c ∈ {a | (Ici a)ᶜ ∈ S}
hb : ∀ i ∈ S, b ∈ iᶜ
⊢ c ≤ b | f1c47d20246a5fbc |
LinearMap.continuousAt_zero_of_locally_bounded | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : UniformSpace F
inst✝⁶ : FirstCountableTopology E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : RCLike 𝕜'
inst✝¹ : Module 𝕜' F
inst✝ : ContinuousSMul 𝕜' F
σ : 𝕜 →+* 𝕜'
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i => b i
bE' : (𝓝 0).HasBasis (fun x => x ≠ 0) fun n => (↑n)⁻¹ • b n
V : Set F
hV : V ∈ 𝓝 0
u : ℕ → E
hu : ∀ (ia : ℕ), ia ≠ 0 → u ia ∈ (↑ia)⁻¹ • b ia
hu' : ∀ (ia : ℕ), ia ≠ 0 → f (u ia) ∉ V
h_tendsto : Tendsto (fun n => ↑n • u n) atTop (𝓝 0)
h_bounded : IsVonNBounded 𝕜 (Set.range fun n => ↑n • u n)
r : ℝ
hr : r > 0
h' : ∀ (c : 𝕜'), r ≤ ‖c‖ → (⇑f '' Set.range fun n => ↑n • u n) ⊆ c • V
n : ℕ
hn : r < ↑n
⊢ False | have h1 : r ≤ ‖(n : 𝕜')‖ := by
rw [RCLike.norm_natCast]
exact hn.le | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : UniformSpace F
inst✝⁶ : FirstCountableTopology E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : RCLike 𝕜'
inst✝¹ : Module 𝕜' F
inst✝ : ContinuousSMul 𝕜' F
σ : 𝕜 →+* 𝕜'
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i => b i
bE' : (𝓝 0).HasBasis (fun x => x ≠ 0) fun n => (↑n)⁻¹ • b n
V : Set F
hV : V ∈ 𝓝 0
u : ℕ → E
hu : ∀ (ia : ℕ), ia ≠ 0 → u ia ∈ (↑ia)⁻¹ • b ia
hu' : ∀ (ia : ℕ), ia ≠ 0 → f (u ia) ∉ V
h_tendsto : Tendsto (fun n => ↑n • u n) atTop (𝓝 0)
h_bounded : IsVonNBounded 𝕜 (Set.range fun n => ↑n • u n)
r : ℝ
hr : r > 0
h' : ∀ (c : 𝕜'), r ≤ ‖c‖ → (⇑f '' Set.range fun n => ↑n • u n) ⊆ c • V
n : ℕ
hn : r < ↑n
h1 : r ≤ ‖↑n‖
⊢ False | eb01391c7362c5dd |
WeierstrassCurve.Jacobian.neg_of_Z_eq_zero' | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma neg_of_Z_eq_zero' {P : Fin 3 → R} (hPz : P z = 0) : W'.neg P = ![P x, -P y, 0] | R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
hPz : P z = 0
⊢ W'.neg P = ![P x, -P y, 0] | rw [neg, negY_of_Z_eq_zero hPz, hPz] | no goals | 2cd2c918df78da3e |
Grp_.isPullback | Mathlib/CategoryTheory/Monoidal/Grp_.lean | theorem isPullback (A : Grp_ C) :
IsPullback (A.mul ▷ A.X) ((α_ A.X A.X A.X).hom ≫ (A.X ◁ A.mul)) A.mul A.mul where
w | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : ChosenFiniteProducts C
A : Grp_ C
s : PullbackCone A.mul A.mul
⊢ lift (lift (s.snd ≫ fst A.X A.X ≫ A.inv) (s.fst ≫ fst A.X A.X) ≫ A.mul) (s.fst ≫ snd A.X A.X) ≫ A.mul =
s.snd ≫ snd A.X A.X | have : lift (s.snd ≫ fst _ _ ≫ A.inv) (s.fst ≫ fst _ _) ≫ A.mul =
lift (s.snd ≫ snd _ _) (s.fst ≫ snd _ _ ≫ A.inv) ≫ A.mul := by
rw [← assoc s.fst, eq_lift_inv_right, lift_lift_assoc, ← assoc s.snd, lift_inv_left_eq,
lift_comp_fst_snd, lift_comp_fst_snd, s.condition] | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : ChosenFiniteProducts C
A : Grp_ C
s : PullbackCone A.mul A.mul
this :
lift (s.snd ≫ fst A.X A.X ≫ A.inv) (s.fst ≫ fst A.X A.X) ≫ A.mul =
lift (s.snd ≫ snd A.X A.X) (s.fst ≫ snd A.X A.X ≫ A.inv) ≫ A.mul
⊢ lift (lift (s.snd ≫ fst A.X A.X ≫ A.inv) (s.fst ≫ fst A.X A.X) ≫ A.mul) (s.fst ≫ snd A.X A.X) ≫ A.mul =
s.snd ≫ snd A.X A.X | d4cb31ef2ed58598 |
NumberField.mixedEmbedding.iUnion_negAt_plusPart_union | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem iUnion_negAt_plusPart_union :
(⋃ s, negAt s '' (plusPart A)) ∪ (A ∩ (⋃ w, {x | x.1 w = 0})) = A | case h.refine_1.inr
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w => |x.1 w|, x.2) ∈ A
x : mixedSpace K
h : x ∈ A ∧ ∃ i, x ∈ {x | x.1 i = 0}
⊢ x ∈ A | exact h.left | no goals | 618707fcf4bf60a6 |
Algebra.FormallySmooth.of_pi | Mathlib/RingTheory/Smooth/Pi.lean | theorem of_pi [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✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ FormallySmooth R (A i) | fapply FormallySmooth.of_split (Pi.evalAlgHom R A i) | case g
R : Type (max u v)
I : Type u
A : I → Type (max u v)
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ A i →ₐ[R] ((i : I) → A i) ⧸ RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2
case hg
R : Type (max u v)
I : Type u
A : I → Type (max u v)
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ (Pi.evalAlgHom R A i).kerSquareLift.comp ?g = AlgHom.id R (A i) | 05006a1a091ecde0 |
ascPochhammer_eval_zero | Mathlib/RingTheory/Polynomial/Pochhammer.lean | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 | case zero
S : Type u
inst✝ : Semiring S
⊢ eval 0 (ascPochhammer S 0) = if 0 = 0 then 1 else 0 | simp | no goals | 6def024cbe9e9070 |
ConvexOn.slope_mono_adjacent | Mathlib/Analysis/Convex/Slope.lean | theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) | 𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
a : 𝕜 := (z - y) / (z - x)
b : 𝕜 := (y - x) / (z - x)
hy : a • x + b • z = y
⊢ a + b = 1 | field_simp [a, b] | no goals | bc57f356e81e7bde |
Batteries.RBNode.upperBound?_eq_find? | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem upperBound?_eq_find? {t : RBNode α} {cut} (ub) (H : t.find? cut = some x) :
t.upperBound? cut ub = some x | case node.h_1
α : Type u_1
x : α
cut : α → Ordering
c : RBColor
a : RBNode α
y : α
b : RBNode α
iha : ∀ (ub : Option α), find? cut a = some x → upperBound? cut a ub = some x
ihb : ∀ (ub : Option α), find? cut b = some x → upperBound? cut b ub = some x
ub : Option α
x✝ : Ordering
heq✝ : cut y = Ordering.lt
H : find? cut a = some x
⊢ upperBound? cut a (some y) = some x | apply iha _ H | no goals | 4f0bd97d16fa95f5 |
Equiv.Perm.ofSubtype_eq_iff | Mathlib/GroupTheory/Perm/Support.lean | /-- A permutation c is the extension of a restriction of g to s
iff its support is contained in s and its restriction is that of g -/
lemma ofSubtype_eq_iff {g c : Equiv.Perm α} {s : Finset α}
(hg : ∀ x, x ∈ s ↔ g x ∈ s) :
ofSubtype (g.subtypePerm hg) = c ↔
c.support ≤ s ∧
∀ (hc' : ∀ x, x ∈ s ↔ c x ∈ s), c.subtypePerm hc' = g.subtypePerm hg | case mpr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
s : Finset α
hg : ∀ (x : α), x ∈ s ↔ g x ∈ s
hc : c.support ≤ s
h : (∀ (x : α), x ∈ s ↔ c x ∈ s) → ∀ a ∈ s, c a = g a
a : α
⊢ (ofSubtype (g.subtypePerm hg)) a = c a | specialize h (isInvariant_of_support_le hc) | case mpr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
s : Finset α
hg : ∀ (x : α), x ∈ s ↔ g x ∈ s
hc : c.support ≤ s
a : α
h : ∀ a ∈ s, c a = g a
⊢ (ofSubtype (g.subtypePerm hg)) a = c a | ab1e7a47010d0039 |
Algebra.FormallySmooth.localization_base | Mathlib/RingTheory/Smooth/Basic.lean | theorem localization_base [FormallySmooth R Sₘ] : FormallySmooth Rₘ Sₘ | case comp_surjective
R Rₘ Sₘ : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₘ
inst✝⁸ : CommRing Sₘ
M : Submonoid R
inst✝⁷ : Algebra R Sₘ
inst✝⁶ : Algebra R Rₘ
inst✝⁵ : Algebra Rₘ Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f✝ : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
f : Sₘ →ₐ[Rₘ] Q :=
let __src := lift I ⋯ (AlgHom.restrictScalars R f✝);
{ toRingHom := __src.toRingHom, commutes' := ⋯ }
⊢ ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f✝ | use f | case h
R Rₘ Sₘ : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₘ
inst✝⁸ : CommRing Sₘ
M : Submonoid R
inst✝⁷ : Algebra R Sₘ
inst✝⁶ : Algebra R Rₘ
inst✝⁵ : Algebra Rₘ Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f✝ : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
f : Sₘ →ₐ[Rₘ] Q :=
let __src := lift I ⋯ (AlgHom.restrictScalars R f✝);
{ toRingHom := __src.toRingHom, commutes' := ⋯ }
⊢ (Ideal.Quotient.mkₐ Rₘ I).comp f = f✝ | 03f56deeca80d586 |
Compactum.cl_cl | Mathlib/Topology/Category/Compactum.lean | theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A | X : Compactum
A : Set X.A
⊢ Compactum.cl (Compactum.cl A) ⊆ Compactum.cl A | rintro _ ⟨F, hF, rfl⟩ | case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
⊢ X.str F ∈ Compactum.cl A | 1c30fb7238f2120d |
Subalgebra.isSimpleOrder_of_finrank | Mathlib/LinearAlgebra/FiniteDimensional.lean | theorem Subalgebra.isSimpleOrder_of_finrank (hr : finrank F E = 2) :
IsSimpleOrder (Subalgebra F E) :=
let i := nontrivial_of_finrank_pos (zero_lt_two.trans_eq hr.symm)
{ toNontrivial :=
⟨⟨⊥, ⊤, fun h => by cases hr.symm.trans (Subalgebra.bot_eq_top_iff_finrank_eq_one.1 h)⟩⟩
eq_bot_or_eq_top | case «2».h
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
S : Subalgebra F E
this✝² : FiniteDimensional F E
this✝¹ : FiniteDimensional F ↥S
h : finrank F ↥S = finrank F E
this✝ : 2 ≤ 2
this : 0 < 2
⊢ S = ⊤ | rw [← Algebra.toSubmodule_eq_top] | case «2».h
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
S : Subalgebra F E
this✝² : FiniteDimensional F E
this✝¹ : FiniteDimensional F ↥S
h : finrank F ↥S = finrank F E
this✝ : 2 ≤ 2
this : 0 < 2
⊢ toSubmodule S = ⊤ | c7e4894f93ed9891 |
Real.vector_fourierIntegral_eq_integral_exp_smul | Mathlib/Analysis/Fourier/FourierTransform.lean | theorem vector_fourierIntegral_eq_integral_exp_smul {V : Type*} [AddCommGroup V] [Module ℝ V]
[MeasurableSpace V] {W : Type*} [AddCommGroup W] [Module ℝ W] (L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ)
(μ : Measure V) (f : V → E) (w : W) :
VectorFourier.fourierIntegral fourierChar μ L f w =
∫ v : V, Complex.exp (↑(-2 * π * L v w) * Complex.I) • f v ∂μ | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
inst✝² : MeasurableSpace V
W : Type u_3
inst✝¹ : AddCommGroup W
inst✝ : Module ℝ W
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
μ : Measure V
f : V → E
w : W
⊢ VectorFourier.fourierIntegral 𝐞 μ L f w = ∫ (v : V), Complex.exp (↑(-2 * π * (L v) w) * Complex.I) • f v ∂μ | simp_rw [VectorFourier.fourierIntegral, Circle.smul_def, Real.fourierChar_apply, mul_neg,
neg_mul] | no goals | 9df2ea4a25cbef20 |
ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
(A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ | 𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
A₀ : Submodule ℝ C(X, ℝ) := Submodule.comap I (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))
SW : A₀.topologicalClosure = ⊤
h₁ :
Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealCLM)) A₀.topologicalClosure ≤
(Submodule.map (↑(ContinuousLinearMap.compLeftContinuousCompact X ofRealCLM)) A₀).topologicalClosure
h₂ :
Submodule.map I (Submodule.comap I (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra))) ≤
Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)
⊢ Submodule.map I A₀.topologicalClosure ≤
(Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure | exact h₁.trans (Submodule.topologicalClosure_mono h₂) | no goals | 587f29aadba1393d |
Set.image_update_uIcc_right | Mathlib/Order/Interval/Set/Pi.lean | theorem image_update_uIcc_right (f : ∀ i, α i) (i : ι) (b : α i) :
update f i '' uIcc (f i) b = uIcc f (update f i b) | ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Lattice (α i)
inst✝ : DecidableEq ι
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' uIcc (f i) b = uIcc f (update f i b) | simpa using image_update_uIcc f i (f i) b | no goals | b3851f77be639435 |
MeasureTheory.continuousWithinAt_of_dominated | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
(hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
(h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
(h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ | case pos
α : Type u_1
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
s : Set X
hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀
hG : CompleteSpace G
⊢ ContinuousWithinAt
(fun x =>
if h : True then
if hf : Integrable (fun a => F x a) μ then L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0
else 0)
s x₀ | exact continuousWithinAt_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
hF_meas h_bound bound_integrable h_cont | no goals | d77cd827192ec8fd |
Polynomial.X_pow_mem_lifts | Mathlib/Algebra/Polynomial/Lifts.lean | theorem X_pow_mem_lifts (f : R →+* S) (n : ℕ) : (X ^ n : S[X]) ∈ lifts f :=
⟨X ^ n, by
simp only [coe_mapRingHom, map_pow, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true,
map_X, and_self_iff]⟩
| R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
⊢ (mapRingHom f) (X ^ n) = X ^ n | simp only [coe_mapRingHom, map_pow, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true,
map_X, and_self_iff] | no goals | a8ad5606ae23bf76 |
Metric.cobounded_eq_cocompact | Mathlib/Topology/MetricSpace/Bounded.lean | theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α | α : Type u
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
a✝ : Nontrivial α
inhabited_h : Inhabited α
⊢ cobounded α = cocompact α | exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact | no goals | 2cf5377ba4d7d105 |
CategoryTheory.ShortComplex.RightHomologyMapData.quasiIso_iff | Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | lemma RightHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}
{h₂ : S₂.RightHomologyData} (γ : RightHomologyMapData φ h₁ h₂) :
QuasiIso φ ↔ IsIso γ.φH | case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) → IsIso γ.φH | intro h | case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH | dcb9895c8bb79cf1 |
Ideal.sum_ramification_inertia | Mathlib/NumberTheory/RamificationInertia/Basic.lean | theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDomain R]
[Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L]
[Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S]
[p.IsMaximal] (hp0 : p ≠ ⊥) :
(∑ P ∈ (factors (map (algebraMap R S) p)).toFinset,
ramificationIdx (algebraMap R S) p P * inertiaDeg p P) =
finrank K L | case calc_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
p : Ideal R
inst✝¹⁴ : IsDedekindDomain S
inst✝¹³ : Algebra R S
K : Type u_1
L : Type u_2
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Algebra R K
inst✝⁸ : IsFractionRing R K
inst✝⁷ : Algebra S L
inst✝⁶ : IsFractionRing S L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Module.Finite R S
inst✝ : p.IsMaximal
hp0 : p ≠ ⊥
e : Ideal S → ℕ := ramificationIdx (algebraMap R S) p
f : Ideal S → [inst : Algebra R S] → ℕ := p.inertiaDeg
⊢ map (algebraMap R S) p ≠ ⊥ | rwa [Ne, Ideal.map_eq_bot_iff_le_ker, (RingHom.injective_iff_ker_eq_bot _).mp <|
algebraMap_injective_of_field_isFractionRing R S K L, le_bot_iff] | no goals | 93ea3cefba21e7c4 |
inv_eq_of_root_of_coeff_zero_ne_zero | Mathlib/RingTheory/Algebraic/Basic.lean | theorem inv_eq_of_root_of_coeff_zero_ne_zero {x : L} {p : K[X]} (aeval_eq : aeval x p = 0)
(coeff_zero_ne : p.coeff 0 ≠ 0) : x⁻¹ = -(aeval x (divX p) / algebraMap _ _ (p.coeff 0)) | case convert_2
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
p : K[X]
aeval_eq : (aeval x) p = 0
coeff_zero_ne : p.coeff 0 ≠ 0
h : (aeval x) p.divX = 0
⊢ (algebraMap K L) (p.coeff 0) = (algebraMap K L) 0 | rw [RingHom.map_zero] | case convert_2
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
p : K[X]
aeval_eq : (aeval x) p = 0
coeff_zero_ne : p.coeff 0 ≠ 0
h : (aeval x) p.divX = 0
⊢ (algebraMap K L) (p.coeff 0) = 0 | 7dd9a2f842fde806 |
nonempty_linearEquiv_of_lift_rank_eq | Mathlib/LinearAlgebra/Dimension/Free.lean | theorem nonempty_linearEquiv_of_lift_rank_eq
(cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
Nonempty (M ≃ₗ[R] M') | R : Type u
M : Type v
M' : Type v'
inst✝⁷ : Semiring R
inst✝⁶ : StrongRankCondition R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Free R M
inst✝² : AddCommMonoid M'
inst✝¹ : Module R M'
inst✝ : Free R M'
cnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')
α : Type v
B : Basis α R M
β : Type v'
B' : Basis β R M'
⊢ lift.{v', v} #α = lift.{v, v'} #β | rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank''] | no goals | 21fe368c1b53de28 |
Module.free_of_maximalIdeal_rTensor_injective | Mathlib/RingTheory/LocalRing/Module.lean | theorem free_of_maximalIdeal_rTensor_injective [Module.FinitePresentation R M]
(H : Function.Injective ((𝔪).subtype.rTensor M)) :
Module.Free R M | case intro.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
w✝¹ : Type u_2
w✝ : w✝¹ → M
b : Basis w✝¹ R M
h✝ : ∀ (i : w✝¹), b i = id (w✝ i)
⊢ Free R M | exact Free.of_basis b | no goals | 51ea42859f002e98 |
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' | 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
h_cop : (Nat.card G).Coprime (Nat.card G'')
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
⊢ IsCyclic ↥↑P | have := Fact.mk hp | 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
h_cop : (Nat.card G).Coprime (Nat.card G'')
p : ℕ
hp : Nat.Prime p
P : Sylow p G'
this : Fact (Nat.Prime p)
⊢ IsCyclic ↥↑P | 8e6ebf384c14685f |
Monoid.CoprodI.lift_word_prod_nontrivial_of_not_empty | Mathlib/GroupTheory/CoprodI.lean | theorem lift_word_prod_nontrivial_of_not_empty {i j} (w : NeWord H i j) :
lift f w.prod ≠ 1 | case pos
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
hcard : 3 ≤ #(H j)
hh : i ≠ j
this : (lift f) w.inv.prod ≠ 1
heq : (lift f) w.prod = 1
⊢ (lift f) w.inv.prod = 1 | simpa using heq | no goals | 3e8798a67334cdb3 |
SimpleGraph.isClique_map_iff_of_nontrivial | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem isClique_map_iff_of_nontrivial {f : α ↪ β} {t : Set β} (ht : t.Nontrivial) :
(G.map f).IsClique t ↔ ∃ (s : Set α), G.IsClique s ∧ f '' s = t | case refine_2
α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
t : Set β
ht : t.Nontrivial
h : (SimpleGraph.map f G).IsClique t
⊢ ⇑f '' (⇑f ⁻¹' t) = t | rw [Set.image_preimage_eq_iff] | case refine_2
α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
t : Set β
ht : t.Nontrivial
h : (SimpleGraph.map f G).IsClique t
⊢ t ⊆ Set.range ⇑f | 67c4bf83d155ae44 |
ProbabilityTheory.IndepFun.mgf_add' | Mathlib/Probability/Moments/Basic.lean | theorem IndepFun.mgf_add' {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : mgf (X + Y) μ t = mgf X μ t * mgf Y μ t | Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
A : Continuous fun x => rexp (t * x)
h'X : AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ
h'Y : AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ
⊢ mgf (X + Y) μ t = mgf X μ t * mgf Y μ t | exact h_indep.mgf_add h'X h'Y | no goals | 34796f9be20b1f78 |
Polynomial.existsUnique_hilbertPoly | Mathlib/RingTheory/Polynomial/HilbertPoly.lean | theorem existsUnique_hilbertPoly (p : F[X]) (d : ℕ) :
∃! h : F[X], ∃ N : ℕ, ∀ n > N,
PowerSeries.coeff F n (p * invOneSubPow F d) = h.eval (n : F) | case h.right.intro
F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
p : F[X]
d : ℕ
h : F[X]
N : ℕ
hhN : ∀ n > N, (PowerSeries.coeff F n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h
⊢ h = p.hilbertPoly d | apply eq_of_infinite_eval_eq h (hilbertPoly p d) | case h.right.intro
F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
p : F[X]
d : ℕ
h : F[X]
N : ℕ
hhN : ∀ n > N, (PowerSeries.coeff F n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h
⊢ {x | eval x h = eval x (p.hilbertPoly d)}.Infinite | 45a6eaf17bc668dd |
List.range'_eq_range'TR | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean | theorem range'_eq_range'TR : @range' = @range'TR | ⊢ @range' = @range'TR | apply funext | case h
⊢ ∀ (x : Nat), @range' x = @range'TR x | e9265b974e40b86d |
ZLattice.isAddFundamentalDomain | Mathlib/Algebra/Module/ZLattice/Basic.lean | theorem ZLattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] {L : Submodule ℤ E} [DiscreteTopology L] [IsZLattice ℝ L] [Finite ι]
(b : Basis ι ℤ L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : Measure E) :
IsAddFundamentalDomain L (fundamentalDomain (b.ofZLatticeBasis ℝ)) μ | case h.e'_1.h.e'_2.h.h.e'_4
ι : Type u_3
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
inst✝² : Finite ι
b : Basis ι ℤ ↥L
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
x✝ : E
⊢ L = span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L b))
case h.e'_3.e'_6
ι : Type u_3
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
inst✝² : Finite ι
b : Basis ι ℤ ↥L
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
e_1✝ : ↥L = ↥(span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L b)))
⊢ L = span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L b))
case h.e'_4.e'_6
ι : Type u_3
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
inst✝² : Finite ι
b : Basis ι ℤ ↥L
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
e_1✝ : ↥L = ↥(span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L b)))
⊢ L = span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L b)) | all_goals exact (b.ofZLatticeBasis_span ℝ).symm | no goals | b519d41af37f923f |
CategoryTheory.Functor.preservesFiniteColimits_of_preservesHomology | Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean | /-- An additive which preserves homology preserves finite colimits. -/
lemma preservesFiniteColimits_of_preservesHomology
[HasFiniteCoproducts C] [HasCokernels C] : PreservesFiniteColimits F | C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasCokernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasCoequalizers C
⊢ IsZero (F.obj 0) | rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero] | no goals | eba7eed2b0d3086e |
Finite.exists_infinite_fiber | Mathlib/Data/Fintype/Pigeonhole.lean | theorem Finite.exists_infinite_fiber [Infinite α] [Finite β] (f : α → β) :
∃ y : β, Infinite (f ⁻¹' {y}) | α : Type u_1
β : Type u_2
inst✝¹ : Infinite α
inst✝ : Finite β
f : α → β
hf : ∀ (y : β), ¬Infinite ↑(f ⁻¹' {y})
⊢ False | cases nonempty_fintype β | case intro
α : Type u_1
β : Type u_2
inst✝¹ : Infinite α
inst✝ : Finite β
f : α → β
hf : ∀ (y : β), ¬Infinite ↑(f ⁻¹' {y})
val✝ : Fintype β
⊢ False | 1fb76a52f781b215 |
MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToL1S_indicatorConst {T : Set α → E →L[ℝ] F} {s : Set α}
(h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T)
(hs : MeasurableSet s) (hμs : μ s < ∞) (x : E) :
setToL1S T (simpleFunc.indicatorConst 1 hs hμs.ne x) = T s x | α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : μ s < ⊤
x : E
⊢ setToL1S T (indicatorConst 1 hs ⋯ x) = (T s) x | have h_empty : T ∅ = 0 := h_zero _ MeasurableSet.empty measure_empty | α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : μ s < ⊤
x : E
h_empty : T ∅ = 0
⊢ setToL1S T (indicatorConst 1 hs ⋯ x) = (T s) x | 49eca8056f95cd51 |
ZLattice.covolume.tendsto_card_le_div'' | Mathlib/Algebra/Module/ZLattice/Covolume.lean | theorem tendsto_card_le_div'' [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
[Nonempty ι] {X : Set E} (hX : ∀ ⦃x⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X)
{F : E → ℝ} (h₁ : ∀ x ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * (F x))
(h₂ : IsBounded {x ∈ X | F x ≤ 1}) (h₃ : MeasurableSet {x ∈ X | F x ≤ 1})
(h₄ : volume (frontier ((b.ofZLatticeBasis ℝ L).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0) :
Tendsto (fun c : ℝ ↦
Nat.card ({x ∈ X | F x ≤ c} ∩ L : Set E) / (c : ℝ))
atTop (𝓝 (volume ((b.ofZLatticeBasis ℝ).equivFun '' {x ∈ X | F x ≤ 1})).toReal) | case h.intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
c : ℝ
hc : 0 < c
aux₁ : ↑(card ι) ≠ 0
aux₂ : 0 < c ^ (↑(card ι))⁻¹
aux₃ : (c ^ (↑(card ι))⁻¹)⁻¹ ≠ 0
aux₄ : c ^ (-(↑(card ι))⁻¹) ≠ 0
hc₁ : 0 ≤ c
hc₂ : 0 ≠ c
⊢ ↑(Nat.card ↑({x | x ∈ X ∧ F x ≤ c} ∩ ↑L)) / c =
↑(Nat.card
↑(⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∩
(c ^ (↑(card ι))⁻¹)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) /
c | congr | case h.intro.e_a.e_a
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
c : ℝ
hc : 0 < c
aux₁ : ↑(card ι) ≠ 0
aux₂ : 0 < c ^ (↑(card ι))⁻¹
aux₃ : (c ^ (↑(card ι))⁻¹)⁻¹ ≠ 0
aux₄ : c ^ (-(↑(card ι))⁻¹) ≠ 0
hc₁ : 0 ≤ c
hc₂ : 0 ≠ c
⊢ Nat.card ↑({x | x ∈ X ∧ F x ≤ c} ∩ ↑L) =
Nat.card
↑(⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∩
(c ^ (↑(card ι))⁻¹)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) | 1e1817e88f98e881 |
Zsqrtd.nonnegg_pos_neg | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem nonnegg_pos_neg {c d} {a b : ℕ} : Nonnegg c d a (-b) ↔ SqLe b c a d | c d a b : ℕ
⊢ Nonnegg c d (↑a) (-↑b) ↔ SqLe b c a d | rw [nonnegg_comm] | c d a b : ℕ
⊢ Nonnegg d c (-↑b) ↑a ↔ SqLe b c a d | a55b09f60e8769a3 |
NumberField.mixedEmbedding.exists_primitive_element_lt_of_isComplex | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀ : IsComplex w₀)
{B : ℝ≥0} (hB : minkowskiBound K ↑1 < convexBodyLT'Factor K * B) :
∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
∀ w : InfinitePlace K, w a < Real.sqrt (1 + B ^ 2) | case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
|((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
|((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2 | norm_num | no goals | b1817ae1b2a5d1e1 |
linearIndependent_of_ne_zero_of_inner_eq_zero | Mathlib/Analysis/InnerProductSpace/Basic.lean | theorem linearIndependent_of_ne_zero_of_inner_eq_zero {ι : Type*} {v : ι → E} (hz : ∀ i, v i ≠ 0)
(ho : Pairwise fun i j => ⟪v i, v j⟫ = 0) : LinearIndependent 𝕜 v | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j => inner (v i) (v j) = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
h' : g i * inner (v i) (v i) = inner (v i) (∑ j ∈ s, g j • v j)
⊢ g i = 0 | simpa [hg, hz] using h' | no goals | 8e5b7920a8b38e39 |
List.length_dropLast_cons | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean | theorem length_dropLast_cons (a : α) (as : List α) : (a :: as).dropLast.length = as.length | α : Type u
a : α
as : List α
b : α
bs : List α
ih : (b :: bs).dropLast.length = bs.length
⊢ (a :: b :: bs).dropLast.length = (b :: bs).length | simp [dropLast, ih] | no goals | db7033c17c610e4e |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id) | n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_eq_i : l.fst.val = i.val
l_ne_b : l.snd = !b
hb✝ : hasAssignment b (removeAssignment (!b) f.assignments[i.val]) = true
hb : hasAssignment b f.assignments[i.val] = true
⊢ unit (i, b) ∈
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList | specialize hf i b hb | n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_eq_i : l.fst.val = i.val
l_ne_b : l.snd = !b
hb✝ : hasAssignment b (removeAssignment (!b) f.assignments[i.val]) = true
hb : hasAssignment b f.assignments[i.val] = true
hf : unit (i, b) ∈ f.toList
⊢ unit (i, b) ∈
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList | 0ab20a88767ce69e |
WithTop.strictMono_map_iff | Mathlib/Order/WithBot.lean | theorem strictMono_map_iff {f : α → β} : StrictMono (WithTop.map f) ↔ StrictMono f :=
strictMono_iff.trans <| by simp [StrictMono, coe_lt_top]
| α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α → β
⊢ ((StrictMono fun a => map f ↑a) ∧ ∀ (x : α), map f ↑x < map f ⊤) ↔ StrictMono f | simp [StrictMono, coe_lt_top] | no goals | 7765458217a159e9 |
Ideal.iUnion_minimalPrimes | Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | theorem Ideal.iUnion_minimalPrimes :
⋃ p ∈ I.minimalPrimes, p = { x | ∃ y ∉ I.radical, x * y ∈ I.radical } | R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x y : R
hy : y ∉ I.radical
hx : x * y ∈ I.radical
⊢ ∃ p ∈ I.minimalPrimes, y ∉ p | simpa [← Ideal.sInf_minimalPrimes] using hy | no goals | c2bb79363c5fb210 |
IsBaseChange.comp | Mathlib/RingTheory/IsTensorProduct.lean | theorem IsBaseChange.comp {f : M →ₗ[R] N} (hf : IsBaseChange S f) {g : N →ₗ[S] O}
(hg : IsBaseChange T g) : IsBaseChange T ((g.restrictScalars R).comp f) | case h
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝²³ : AddCommMonoid M
inst✝²² : AddCommMonoid N
inst✝²¹ : CommSemiring R
inst✝²⁰ : CommSemiring S
inst✝¹⁹ : Algebra R S
inst✝¹⁸ : Module R M
inst✝¹⁷ : Module R N
inst✝¹⁶ : Module S N
inst✝¹⁵ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁴ : CommSemiring T
inst✝¹³ : Algebra R T
inst✝¹² : Algebra S T
inst✝¹¹ : IsScalarTower R S T
inst✝¹⁰ : AddCommMonoid O
inst✝⁹ : Module R O
inst✝⁸ : Module S O
inst✝⁷ : Module T O
inst✝⁶ : IsScalarTower S T O
inst✝⁵ : IsScalarTower R S O
inst✝⁴ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
g : N →ₗ[S] O
hg : IsBaseChange T g
Q : Type (max v₁ u_5 u_4)
inst✝³ : AddCommMonoid Q
inst✝² : Module R Q
inst✝¹ : Module T Q
inst✝ : IsScalarTower R T Q
i : M →ₗ[R] Q
this✝¹ : Module S Q := Module.compHom Q (algebraMap S T)
this✝ : IsScalarTower S T Q
this : IsScalarTower R S Q
g' : O →ₗ[T] Q
e : ↑R g' ∘ₗ ↑R g ∘ₗ f = i
⊢ ↑R (↑S g' ∘ₗ g) ∘ₗ f = ↑R g' ∘ₗ ↑R g ∘ₗ f | ext | case h.h
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝²³ : AddCommMonoid M
inst✝²² : AddCommMonoid N
inst✝²¹ : CommSemiring R
inst✝²⁰ : CommSemiring S
inst✝¹⁹ : Algebra R S
inst✝¹⁸ : Module R M
inst✝¹⁷ : Module R N
inst✝¹⁶ : Module S N
inst✝¹⁵ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁴ : CommSemiring T
inst✝¹³ : Algebra R T
inst✝¹² : Algebra S T
inst✝¹¹ : IsScalarTower R S T
inst✝¹⁰ : AddCommMonoid O
inst✝⁹ : Module R O
inst✝⁸ : Module S O
inst✝⁷ : Module T O
inst✝⁶ : IsScalarTower S T O
inst✝⁵ : IsScalarTower R S O
inst✝⁴ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
g : N →ₗ[S] O
hg : IsBaseChange T g
Q : Type (max v₁ u_5 u_4)
inst✝³ : AddCommMonoid Q
inst✝² : Module R Q
inst✝¹ : Module T Q
inst✝ : IsScalarTower R T Q
i : M →ₗ[R] Q
this✝¹ : Module S Q := Module.compHom Q (algebraMap S T)
this✝ : IsScalarTower S T Q
this : IsScalarTower R S Q
g' : O →ₗ[T] Q
e : ↑R g' ∘ₗ ↑R g ∘ₗ f = i
x✝ : M
⊢ (↑R (↑S g' ∘ₗ g) ∘ₗ f) x✝ = (↑R g' ∘ₗ ↑R g ∘ₗ f) x✝ | 5f083a0c88695e0c |
VitaliFamily.ae_tendsto_average_norm_sub | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ⨍ y in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0) | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a => (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ⊤
h''a : IntegrableOn f a μ
⊢ IntegrableOn (fun y => ↑‖f y - f x‖₊) a μ | simp_rw [coe_nnnorm] | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a => (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ⊤
h''a : IntegrableOn f a μ
⊢ IntegrableOn (fun y => ‖f y - f x‖) a μ | 58e38ef8cb10ccfc |
SatisfiesM_ExceptT_eq | Mathlib/.lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | theorem SatisfiesM_ExceptT_eq [Monad m] [LawfulMonad m] :
SatisfiesM (m := ExceptT ρ m) (α := α) p x ↔
SatisfiesM (m := m) (∀ a, · = .ok a → p a) x.run | case refine_2.e_a
m : Type u_1 → Type u_2
α ρ : Type u_1
p : α → Prop
x : ExceptT ρ m α
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x✝ : SatisfiesM (fun x => ∀ (a : α), x = Except.ok a → p a) x
f : m { a // (fun x => ∀ (a : α), x = Except.ok a → p a) a }
eq : Subtype.val <$> f = x
⊢ (fun x =>
match
match x with
| ⟨Except.ok a, h⟩ => Except.ok ⟨a, ⋯⟩
| ⟨Except.error e, property⟩ => Except.error e with
| Except.ok a => pure (Except.ok a.val)
| Except.error e => pure (Except.error e)) =
fun a => pure a.val | funext ⟨a, h⟩ | case refine_2.e_a.h
m : Type u_1 → Type u_2
α ρ : Type u_1
p : α → Prop
x : ExceptT ρ m α
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x✝ : SatisfiesM (fun x => ∀ (a : α), x = Except.ok a → p a) x
f : m { a // (fun x => ∀ (a : α), x = Except.ok a → p a) a }
eq : Subtype.val <$> f = x
a : Except ρ α
h : ∀ (a_1 : α), a = Except.ok a_1 → p a_1
⊢ (match
match ⟨a, h⟩ with
| ⟨Except.ok a, h⟩ => Except.ok ⟨a, ⋯⟩
| ⟨Except.error e, property⟩ => Except.error e with
| Except.ok a => pure (Except.ok a.val)
| Except.error e => pure (Except.error e)) =
pure ⟨a, h⟩.val | 0d5f8bbc1987cbbe |
Polynomial.coeff_bdd_of_roots_le | Mathlib/Topology/Algebra/Polynomial.lean | theorem coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1 : p.Monic)
(h2 : Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) (i : ℕ) :
‖(map f p).coeff i‖ ≤ max B 1 ^ d * d.choose (d / 2) | case h.hmn
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
B : ℝ
d : ℕ
f : F →+* K
p : F[X]
h1 : p.Monic
h2 : Splits f p
h3 : p.natDegree ≤ d
h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
i : ℕ
hB : 0 ≤ B
⊢ p.natDegree - i ≤ d | exact le_trans (Nat.sub_le _ _) h3 | no goals | 49af0bf207486cbc |
CochainComplex.HomComplex.Cochain.leftShift_v | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | lemma leftShift_v (a n' : ℤ) (hn' : n + a = n') (p q : ℤ) (hpq : p + n' = q)
(p' : ℤ) (hp' : p' + n = q) :
(γ.leftShift a n' hn').v p q hpq = (a * n' + ((a * (a - 1))/2)).negOnePow •
(K.shiftFunctorObjXIso a p p'
(by rw [← add_left_inj n, hp', add_assoc, add_comm a, hn', hpq])).hom ≫ γ.v p' q hp' | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n' = q
p' : ℤ
hp' : p' + n = q
⊢ p' = p + a | omega | no goals | 7e18951062272244 |
WeierstrassCurve.Projective.addMap_of_Z_eq_zero_left | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma addMap_of_Z_eq_zero_left {P : Fin 3 → F} {Q : PointClass F} (hP : W.Nonsingular P)
(hQ : W.NonsingularLift Q) (hPz : P z = 0) : W.addMap ⟦P⟧ Q = Q | case pos
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
Q✝ : PointClass F
hP : W.Nonsingular P
hPz : P z = 0
Q : Fin 3 → F
hQ : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) Q)
hQz : Q z = 0
⊢ W.addMap ⟦P⟧ (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) Q) = Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) Q | erw [addMap_eq, add_of_Z_eq_zero hP hQ hPz hQz,
smul_eq _ <| (isUnit_Y_of_Z_eq_zero hP hPz).pow 4, Quotient.eq] | case pos
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
Q✝ : PointClass F
hP : W.Nonsingular P
hPz : P z = 0
Q : Fin 3 → F
hQ : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) Q)
hQz : Q z = 0
⊢ (MulAction.orbitRel Fˣ (Fin 3 → F)) ![0, 1, 0] Q | 6da18fa7da391a26 |
FiberBundleCore.open_source' | Mathlib/Topology/FiberBundle/Basic.lean | theorem open_source' (i : ι) : IsOpen (Z.localTrivAsPartialEquiv i).source | case a
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
⊢ (Z.localTrivAsPartialEquiv i).source =
(Z.localTrivAsPartialEquiv i).source ∩ ↑(Z.localTrivAsPartialEquiv i) ⁻¹' Z.baseSet i ×ˢ univ | ext p | case a.h
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
p : Z.TotalSpace
⊢ p ∈ (Z.localTrivAsPartialEquiv i).source ↔
p ∈ (Z.localTrivAsPartialEquiv i).source ∩ ↑(Z.localTrivAsPartialEquiv i) ⁻¹' Z.baseSet i ×ˢ univ | 437cc1867a06723f |
mfderiv_neg | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | theorem mfderiv_neg (f : M → E') (x : M) :
(mfderiv I 𝓘(𝕜, E') (-f) x : TangentSpace I x →L[𝕜] E') =
(-mfderiv I 𝓘(𝕜, E') f x : TangentSpace I x →L[𝕜] E') | 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : M → E'
x : M
⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)
else 0) =
-if MDifferentiableAt I 𝓘(𝕜, E') f x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)
else 0 | by_cases hf : MDifferentiableAt I 𝓘(𝕜, E') f x | case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : M → E'
x : M
hf : MDifferentiableAt I 𝓘(𝕜, E') f x
⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)
else 0) =
-if MDifferentiableAt I 𝓘(𝕜, E') f x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)
else 0
case neg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : M → E'
x : M
hf : ¬MDifferentiableAt I 𝓘(𝕜, E') f x
⊢ (if MDifferentiableAt I 𝓘(𝕜, E') (-f) x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x (-f)) (range ↑I) (↑(extChartAt I x) x)
else 0) =
-if MDifferentiableAt I 𝓘(𝕜, E') f x then
fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)
else 0 | bb6d984738c96be5 |
WeierstrassCurve.Affine.map_polynomial | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | lemma map_polynomial : (W'.map f).toAffine.polynomial = W'.polynomial.map (mapRingHom f) | R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Affine R
f : R →+* S
⊢ Y ^ 2 + C (C (map W' f).toAffine.a₁ * X + C (map W' f).toAffine.a₃) * Y -
C (X ^ 3 + C (map W' f).toAffine.a₂ * X ^ 2 + C (map W' f).toAffine.a₄ * X + C (map W' f).toAffine.a₆) =
Polynomial.map (mapRingHom f)
(Y ^ 2 + C (C W'.a₁ * X + C W'.a₃) * Y - C (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)) | map_simp | no goals | 6c994053a676fe74 |
CategoryTheory.Functor.isRightKanExtension_iff_isIso | Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean | lemma isRightKanExtension_iff_isIso {F' : D ⥤ H} {F'' : D ⥤ H} (φ : F'' ⟶ F')
{L : C ⥤ D} {F : C ⥤ H} (α : L ⋙ F' ⟶ F) (α' : L ⋙ F'' ⟶ F)
(comm : whiskerLeft L φ ≫ α = α') [F'.IsRightKanExtension α] :
F''.IsRightKanExtension α' ↔ IsIso φ | C : Type u_1
H : Type u_3
D : Type u_4
inst✝³ : Category.{u_8, u_1} C
inst✝² : Category.{u_7, u_3} H
inst✝¹ : Category.{u_6, u_4} D
F' F'' : D ⥤ H
φ : F'' ⟶ F'
L : C ⥤ D
F : C ⥤ H
α : L ⋙ F' ⟶ F
α' : L ⋙ F'' ⟶ F
comm : whiskerLeft L φ ≫ α = α'
inst✝ : F'.IsRightKanExtension α
⊢ F''.IsRightKanExtension α' ↔ IsIso φ | constructor | case mp
C : Type u_1
H : Type u_3
D : Type u_4
inst✝³ : Category.{u_8, u_1} C
inst✝² : Category.{u_7, u_3} H
inst✝¹ : Category.{u_6, u_4} D
F' F'' : D ⥤ H
φ : F'' ⟶ F'
L : C ⥤ D
F : C ⥤ H
α : L ⋙ F' ⟶ F
α' : L ⋙ F'' ⟶ F
comm : whiskerLeft L φ ≫ α = α'
inst✝ : F'.IsRightKanExtension α
⊢ F''.IsRightKanExtension α' → IsIso φ
case mpr
C : Type u_1
H : Type u_3
D : Type u_4
inst✝³ : Category.{u_8, u_1} C
inst✝² : Category.{u_7, u_3} H
inst✝¹ : Category.{u_6, u_4} D
F' F'' : D ⥤ H
φ : F'' ⟶ F'
L : C ⥤ D
F : C ⥤ H
α : L ⋙ F' ⟶ F
α' : L ⋙ F'' ⟶ F
comm : whiskerLeft L φ ≫ α = α'
inst✝ : F'.IsRightKanExtension α
⊢ IsIso φ → F''.IsRightKanExtension α' | 14f3fa1f3712e7b2 |
Padic.AddValuation.map_mul | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem AddValuation.map_mul (x y : ℚ_[p]) :
addValuationDef (x * y : ℚ_[p]) = addValuationDef x + addValuationDef y | case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hx : ¬x = 0
⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) =
(if x = 0 then ⊤ else ↑x.valuation) + if y = 0 then ⊤ else ↑y.valuation | by_cases hy : y = 0 | case pos
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hx : ¬x = 0
hy : y = 0
⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) =
(if x = 0 then ⊤ else ↑x.valuation) + if y = 0 then ⊤ else ↑y.valuation
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hx : ¬x = 0
hy : ¬y = 0
⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) =
(if x = 0 then ⊤ else ↑x.valuation) + if y = 0 then ⊤ else ↑y.valuation | c60ad8ab6f4d7496 |
CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_pushouts | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/EnoughInjectives.lean | /-- If `p : X ⟶ Y` is a monomorphism that is not an isomorphism, there exists
a subobject `X'` of `Y` containing `X` (but different from `X`) such that
the inclusion `X ⟶ X'` is a pushout of a monomorphism in the family
`generatingMonomorphisms G`. -/
lemma exists_pushouts
{X Y : C} (p : X ⟶ Y) [Mono p] (hp : ¬ IsIso p) :
∃ (X' : C) (i : X ⟶ X') (p' : X' ⟶ Y) (_ : (generatingMonomorphisms G).pushouts i)
(_ : ¬ IsIso i) (_ : Mono p'), i ≫ p' = p | case intro
C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X Y : C
p : X ⟶ Y
inst✝ : Mono p
f : G ⟶ Y
hf : ∀ (x : G ⟶ X), ¬x ≫ p = f
⊢ ∃ X' i p', ∃ (_ : (generatingMonomorphisms G).pushouts i) (_ : ¬IsIso i) (_ : Mono p'), i ≫ p' = p | refine ⟨pushout (pullback.fst p f) (pullback.snd p f), pushout.inl _ _,
pushout.desc p f pullback.condition,
⟨_, _, _, (Subobject.underlyingIso _).hom ≫ pullback.fst _ _,
pushout.inr _ _, ⟨Subobject.mk (pullback.snd p f)⟩,
(IsPushout.of_hasPushout (pullback.fst p f) (pullback.snd p f)).of_iso
((Subobject.underlyingIso _).symm) (Iso.refl _) (Iso.refl _)
(Iso.refl _) (by simp) (by simp) (by simp) (by simp)⟩, ?_, ?_, by simp⟩ | case intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X Y : C
p : X ⟶ Y
inst✝ : Mono p
f : G ⟶ Y
hf : ∀ (x : G ⟶ X), ¬x ≫ p = f
⊢ ¬IsIso (pushout.inl (pullback.fst p f) (pullback.snd p f))
case intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X Y : C
p : X ⟶ Y
inst✝ : Mono p
f : G ⟶ Y
hf : ∀ (x : G ⟶ X), ¬x ≫ p = f
⊢ Mono (pushout.desc p f ⋯) | a7ac65ea27f73093 |
CategoryTheory.le_topology_of_closedSieves_isSheaf | Mathlib/CategoryTheory/Sites/Closed.lean | theorem le_topology_of_closedSieves_isSheaf {J₁ J₂ : GrothendieckTopology C}
(h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)) : J₁ ≤ J₂ | C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)
X : C
S : Sieve X
hS : S ∈ J₁ X
this : J₂.IsClosed ⊤
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
S.arrows f → (Functor.closedSieves J₂).map f.op ⟨J₂.close S, ⋯⟩ = (Functor.closedSieves J₂).map f.op ⟨⊤, this⟩ | intro Y f hf | C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)
X : C
S : Sieve X
hS : S ∈ J₁ X
this : J₂.IsClosed ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ (Functor.closedSieves J₂).map f.op ⟨J₂.close S, ⋯⟩ = (Functor.closedSieves J₂).map f.op ⟨⊤, this⟩ | 13067aa834d1fd87 |
separate_convex_open_set | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | theorem separate_convex_open_set [TopologicalSpace E] [AddCommGroup E] [IsTopologicalAddGroup E]
[Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
(hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s) : ∃ f : E →L[ℝ] ℝ, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1 | case refine_2.intro.intro
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
φ : E →ₗ[ℝ] ℝ
hφ₁ : ∀ (x : ↥f.domain), φ ↑x = ↑f x
hφ₂ : ∀ (x : E), φ x ≤ gauge s x
hφ₃ : φ x₀ = 1
hφ₄ : ∀ x ∈ s, φ x < 1
⊢ ∃ f, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1 | refine ⟨⟨φ, ?_⟩, hφ₃, hφ₄⟩ | case refine_2.intro.intro
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
φ : E →ₗ[ℝ] ℝ
hφ₁ : ∀ (x : ↥f.domain), φ ↑x = ↑f x
hφ₂ : ∀ (x : E), φ x ≤ gauge s x
hφ₃ : φ x₀ = 1
hφ₄ : ∀ x ∈ s, φ x < 1
⊢ Continuous φ.toFun | 5059de53d562f5c3 |
TopCat.GlueData.preimage_image_eq_image | Mathlib/Topology/Gluing.lean | theorem preimage_image_eq_image (i j : D.J) (U : Set (𝖣.U i)) :
𝖣.ι j ⁻¹' (𝖣.ι i '' U) = D.f _ _ '' ((D.t j i ≫ D.f _ _) ⁻¹' U) | case a
D : GlueData
i j : D.J
U : Set ↑(D.U i)
this :
⇑(ConcreteCategory.hom (D.f j i)) ⁻¹' (⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' (⇑(ConcreteCategory.hom (D.ι i)) '' U)) =
⇑(ConcreteCategory.hom (D.t j i ≫ D.f i j)) ⁻¹' U
⊢ ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' (⇑(ConcreteCategory.hom (D.ι i)) '' U) ⊆
⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' Set.range ⇑(ConcreteCategory.hom (D.ι i)) | exact Set.preimage_mono (Set.image_subset_range _ _) | no goals | 4ed43fa86c2baa31 |
List.scanl_cons | Mathlib/Data/List/Scan.lean | theorem scanl_cons : scanl f b (a :: l) = [b] ++ scanl f (f b a) l | α : Type u_1
β : Type u_2
f : β → α → β
b : β
a : α
l : List α
⊢ scanl f b (a :: l) = [b] ++ scanl f (f b a) l | simp only [scanl, eq_self_iff_true, singleton_append, and_self_iff] | no goals | 5e91e7dc3c7f9a36 |
Finset.left_mem_Ico | Mathlib/Order/Interval/Finset/Basic.lean | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b | α : Type u_2
a b : α
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
⊢ a ∈ Ico a b ↔ a < b | simp only [mem_Ico, true_and, le_refl] | no goals | a850e3e6a2a797e6 |
AnalyticOnNhd.iteratedFDeriv | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem AnalyticOnNhd.iteratedFDeriv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) (n : ℕ) :
AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
n : ℕ
IH : AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s
⊢ (E →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F) →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F | exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F).symm | no goals | 30a2541ac18d8023 |
IsCoprime.prod_left | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x | R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
H : ∀ i ∈ insert b t, IsCoprime (s i) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x | rw [Finset.forall_mem_insert] at H | R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
H : IsCoprime (s b) x ∧ ∀ x_1 ∈ t, IsCoprime (s x_1) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x | c624860008ac4be4 |
MvPolynomial.IsWeightedHomogeneous.pderiv | Mathlib/RingTheory/MvPolynomial/EulerIdentity.lean | protected lemma IsWeightedHomogeneous.pderiv [AddCancelCommMonoid M] {w : σ → M} {n n' : M} {i : σ}
(h : φ.IsWeightedHomogeneous w n) (h' : n' + w i = n) :
(pderiv i φ).IsWeightedHomogeneous w n' | case refine_1
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun i => single i 1) '' {d | (weight w) d = n})
h' : n' + w i = n
⊢ ∀ x ∈ (fun i => single i 1) '' {d | (weight w) d = n}, IsWeightedHomogeneous w ((pderiv i) x) n' | rintro _ ⟨m, hm, rfl⟩ | case refine_1.intro.intro
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun i => single i 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
⊢ IsWeightedHomogeneous w ((pderiv i) ((fun i => single i 1) m)) n' | 8ec031de5f3c8a44 |
CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj (P : C) :
Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms | case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
P : C
a✝ : (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms
⊢ (coyoneda.obj (op P)).PreservesEpimorphisms | exact (inferInstance : (preadditiveCoyoneda.obj (op P) ⋙ forget _).PreservesEpimorphisms) | no goals | df0da8b1d81c31aa |
List.min?_eq_some_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean | theorem min?_eq_some_iff [Min α] [LE α] [anti : Std.Antisymm ((· : α) ≤ ·)]
(le_refl : ∀ a : α, a ≤ a)
(min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b)
(le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} :
xs.min? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b | case cons
α : Type u_1
a : α
inst✝¹ : Min α
inst✝ : LE α
anti : Std.Antisymm fun x1 x2 => x1 ≤ x2
le_refl : ∀ (a : α), a ≤ a
min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b
le_min_iff : ∀ (a b c : α), a ≤ min b c ↔ a ≤ b ∧ a ≤ c
x : α
xs : List α
h₁ : a ∈ x :: xs
h₂ : ∀ (b : α), b ∈ x :: xs → a ≤ b
⊢ (x :: xs).min? = some a | exact congrArg some <| anti.1 _ _
((le_min?_iff le_min_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
(h₂ _ (min?_mem min_eq_or (xs := x::xs) rfl)) | no goals | 20feb9ca0ad00dc0 |
Std.DHashMap.Internal.Raw₀.Const.toListModel_alterₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem toListModel_alterₘ [BEq α] [EquivBEq α] [Hashable α] [LawfulHashable α]
{m : Raw₀ α (fun _ => β)} (h : Raw.WFImp m.1) {a : α} {f : Option β → Option β} :
Perm (toListModel (Const.alterₘ m a f).1.2) (Const.alterKey a f (toListModel m.1.2)) | α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
hc : containsKey a (toListModel m.val.buckets) = false
⊢ toListModel
(match f none with
| none => m
| some b => (m.consₘ a b).expandIfNecessary).val.buckets ~
Const.alterKey a f (toListModel m.val.buckets) | rw [Const.alterKey, getValue?_eq_none.mpr hc] | α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
hc : containsKey a (toListModel m.val.buckets) = false
⊢ toListModel
(match f none with
| none => m
| some b => (m.consₘ a b).expandIfNecessary).val.buckets ~
match f none with
| none => eraseKey a (toListModel m.val.buckets)
| some v => insertEntry a v (toListModel m.val.buckets) | 46325448e4f73ee5 |
Finset.map_traverse | Mathlib/Data/Finset/Functor.lean | theorem map_traverse (g : α → G β) (h : β → γ) (s : Finset α) :
Functor.map h <$> traverse g s = traverse (Functor.map h ∘ g) s | α β γ : Type u
G : Type u → Type u
inst✝¹ : Applicative G
inst✝ : CommApplicative G
g : α → G β
h : β → γ
s : Finset α
⊢ Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =
Multiset.toFinset <$> Multiset.traverse (Functor.map h ∘ g) s.val | simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←
Multiset.map_traverse] | no goals | aea3dce0394b1832 |
CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | theorem Presieve.isSheaf_iff_preservesFiniteProducts (F : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F | case refine_1.refine_2
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type w
hF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).covering X, IsSheafFor F R
n : ℕ
K : Discrete (Fin n) ⥤ Cᵒᵖ
Z : Fin n → C := fun i => unop (K.obj { as := i })
this✝ : (ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (∐ Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun i => op (Z i) := Discrete.natIsoFunctor
⊢ (Sigma.desc fun i => Sigma.ι Z i) = 𝟙 (∐ Z) | ext | case refine_1.refine_2.h
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type w
hF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).covering X, IsSheafFor F R
n : ℕ
K : Discrete (Fin n) ⥤ Cᵒᵖ
Z : Fin n → C := fun i => unop (K.obj { as := i })
this✝ : (ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (∐ Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun i => op (Z i) := Discrete.natIsoFunctor
b✝ : Fin n
⊢ (Sigma.ι Z b✝ ≫ Sigma.desc fun i => Sigma.ι Z i) = Sigma.ι Z b✝ ≫ 𝟙 (∐ Z) | 6216cf6cf2959d2c |
exteriorPower.pairingDual_apply_apply_eq_one | Mathlib/LinearAlgebra/ExteriorPower/Pairing.lean | lemma pairingDual_apply_apply_eq_one (a : Fin n ↪o ι) :
pairingDual R M n (ιMulti _ _ (f ∘ a)) (ιMulti _ _ (x ∘ a)) = 1 | R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₁ : ∀ (i : ι), (f i) (x i) = 1
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a : Fin n ↪o ι
⊢ (Matrix.of fun i j => (f (a j)) (x (a i))).det = Matrix.det 1 | congr | case e_M
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₁ : ∀ (i : ι), (f i) (x i) = 1
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a : Fin n ↪o ι
⊢ (Matrix.of fun i j => (f (a j)) (x (a i))) = 1 | c301cc6cf4cbc2cf |
List.Nat.nodup_antidiagonalTuple | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | theorem nodup_antidiagonalTuple (k n : ℕ) : List.Nodup (antidiagonalTuple k n) | case succ.right.succ
k : ℕ
ih : ∀ (n : ℕ), (antidiagonalTuple k n).Nodup
n : ℕ
n_ih :
Pairwise (Function.onFun Disjoint fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2))
(antidiagonal n)
⊢ Pairwise (Function.onFun Disjoint fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2))
((0, n + 1) :: map (Prod.map Nat.succ id) (antidiagonal n)) | refine List.Pairwise.cons (fun a ha x hx₁ hx₂ => ?_) (n_ih.map _ fun a b h x hx₁ hx₂ => ?_) | case succ.right.succ.refine_1
k : ℕ
ih : ∀ (n : ℕ), (antidiagonalTuple k n).Nodup
n : ℕ
n_ih :
Pairwise (Function.onFun Disjoint fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2))
(antidiagonal n)
a : ℕ × ℕ
ha : a ∈ map (Prod.map Nat.succ id) (antidiagonal n)
x : Fin (k + 1) → ℕ
hx₁ : x ∈ (fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2)) (0, n + 1)
hx₂ : x ∈ (fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2)) a
⊢ False
case succ.right.succ.refine_2
k : ℕ
ih : ∀ (n : ℕ), (antidiagonalTuple k n).Nodup
n : ℕ
n_ih :
Pairwise (Function.onFun Disjoint fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2))
(antidiagonal n)
a b : ℕ × ℕ
h : Function.onFun Disjoint (fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2)) a b
x : Fin (k + 1) → ℕ
hx₁ : x ∈ (fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2)) (Prod.map Nat.succ id a)
hx₂ : x ∈ (fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple k ni.2)) (Prod.map Nat.succ id b)
⊢ False | 4c17a19ec5bf0ba2 |
ProbabilityTheory.gaussianReal_map_add_const | Mathlib/Probability/Distributions/Gaussian.lean | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | case pos
μ : ℝ
v : ℝ≥0
y : ℝ
hv : v = 0
⊢ Measure.map (fun x => x + y) (Measure.dirac μ) = Measure.dirac (μ + y) | exact Measure.map_dirac (measurable_id'.add_const _) _ | no goals | c255a30ef6e9b8ed |
Sum.liftRel_swap_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Sum/Lemmas.lean | theorem liftRel_swap_iff : LiftRel s r x.swap y.swap ↔ LiftRel r s x y :=
⟨fun h => by rw [← swap_swap x, ← swap_swap y]; exact h.swap, LiftRel.swap⟩
| β✝¹ : Type u_1
β✝ : Type u_2
s : β✝¹ → β✝ → Prop
α✝¹ : Type u_3
α✝ : Type u_4
r : α✝¹ → α✝ → Prop
x : α✝¹ ⊕ β✝¹
y : α✝ ⊕ β✝
h : LiftRel s r x.swap y.swap
⊢ LiftRel r s x.swap.swap y.swap.swap | exact h.swap | no goals | b9260286946e2ea5 |
bernsteinPolynomial.sum_mul_smul | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem sum_mul_smul (n : ℕ) :
(∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν) =
(n * (n - 1)) • X ^ 2 | case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
k : ℕ
⊢ (↑k + 1 + 1) * (↑k + 1) * (↑(n.choose (k + 1 + 1)) * (X ^ k * X * X) * (1 - X) ^ (n - (k + 1 + 1))) =
↑(n.choose (k + 1 + 1)) * ((1 - X) ^ (n - (k + 1 + 1)) * ((↑k + 1 + 1) * ((↑k + 1) * X ^ k))) * (X ^ 0 * X * X) | ring | no goals | 5a1ea5f98d0a949e |
gradient_eq_deriv | Mathlib/Analysis/Calculus/Gradient/Basic.lean | theorem gradient_eq_deriv : ∇ g u = starRingEnd 𝕜 (deriv g u) | case neg
𝕜 : Type u_1
inst✝ : RCLike 𝕜
g : 𝕜 → 𝕜
u : 𝕜
h : ¬DifferentiableAt 𝕜 g u
⊢ ∇ g u = (starRingEnd 𝕜) (deriv g u) | rw [gradient_eq_zero_of_not_differentiableAt h, deriv_zero_of_not_differentiableAt h, map_zero] | no goals | 28982b9536fea35f |
Ordinal.ord_cof_eq | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem ord_cof_eq (r : α → α → Prop) [IsWellOrder α r] :
∃ S, Unbounded r S ∧ type (Subrel r (· ∈ S)) = (cof (type r)).ord | α : Type u
r : α → α → Prop
inst✝ : IsWellOrder α r
S : Set α
hS : Unbounded r S
e : #↑S = (type r).cof
s : ↑S → ↑S → Prop
w✝ : IsWellOrder (↑S) s
e' : (#↑S).ord = type s
⊢ ∃ S, Unbounded r S ∧ type (Subrel r fun x => x ∈ S) = (type r).cof.ord | let T : Set α := { a | ∃ aS : a ∈ S, ∀ b : S, s b ⟨_, aS⟩ → r b a } | α : Type u
r : α → α → Prop
inst✝ : IsWellOrder α r
S : Set α
hS : Unbounded r S
e : #↑S = (type r).cof
s : ↑S → ↑S → Prop
w✝ : IsWellOrder (↑S) s
e' : (#↑S).ord = type s
T : Set α := {a | ∃ (aS : a ∈ S), ∀ (b : ↑S), s b ⟨a, aS⟩ → r (↑b) a}
⊢ ∃ S, Unbounded r S ∧ type (Subrel r fun x => x ∈ S) = (type r).cof.ord | 1935e0c9099cb4dd |
dvd_add_right | Mathlib/Algebra/Ring/Divisibility/Basic.lean | theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c | α : Type u_1
inst✝ : NonUnitalRing α
a b c : α
h : a ∣ b
⊢ a ∣ b + c ↔ a ∣ c | rw [add_comm] | α : Type u_1
inst✝ : NonUnitalRing α
a b c : α
h : a ∣ b
⊢ a ∣ c + b ↔ a ∣ c | 5260398ee4d792b4 |
Set.partiallyWellOrderedOn_iff_finite_antichains | Mathlib/Order/WellFoundedSet.lean | theorem partiallyWellOrderedOn_iff_finite_antichains [IsSymm α r] :
s.PartiallyWellOrderedOn r ↔ ∀ t, t ⊆ s → IsAntichain r t → t.Finite | case refine_1.inr.inr
α : Type u_2
r : α → α → Prop
s : Set α
inst✝¹ : IsRefl α r
inst✝ : IsSymm α r
hs : ∀ t ⊆ s, IsAntichain r t → t.Finite
f : ℕ → α
hf : ∀ (n : ℕ), f n ∈ s
H : ∀ (m n : ℕ), m < n → ¬r (f m) (f n)
m n : ℕ
hmn : f m = f n
h : n < m
⊢ r (f n) (f m) | rw [hmn] | case refine_1.inr.inr
α : Type u_2
r : α → α → Prop
s : Set α
inst✝¹ : IsRefl α r
inst✝ : IsSymm α r
hs : ∀ t ⊆ s, IsAntichain r t → t.Finite
f : ℕ → α
hf : ∀ (n : ℕ), f n ∈ s
H : ∀ (m n : ℕ), m < n → ¬r (f m) (f n)
m n : ℕ
hmn : f m = f n
h : n < m
⊢ r (f n) (f n) | 43bb422c139b758d |
LinearRecurrence.sol_eq_of_eq_init | Mathlib/Algebra/LinearRecurrence.lean | theorem sol_eq_of_eq_init (u v : ℕ → α) (hu : E.IsSolution u) (hv : E.IsSolution v) :
u = v ↔ Set.EqOn u v ↑(range E.order) | α : Type u_1
inst✝ : CommSemiring α
E : LinearRecurrence α
u v : ℕ → α
hu : E.IsSolution u
hv : E.IsSolution v
⊢ u = v ↔ Set.EqOn u v ↑(range E.order) | refine Iff.intro (fun h x _ ↦ h ▸ rfl) ?_ | α : Type u_1
inst✝ : CommSemiring α
E : LinearRecurrence α
u v : ℕ → α
hu : E.IsSolution u
hv : E.IsSolution v
⊢ Set.EqOn u v ↑(range E.order) → u = v | 5fb080a45c739da9 |
Option.filter_eq_some | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean | theorem filter_eq_some {o : Option α} {p : α → Bool} :
o.filter p = some a ↔ a ∈ o ∧ p a | case some
α : Type u_1
a✝ : α
p : α → Bool
a : α
⊢ (if p a = true then some a else none) = some a✝ ↔ a = a✝ ∧ p a✝ = true | split <;> rename_i h | case some.isTrue
α : Type u_1
a✝ : α
p : α → Bool
a : α
h : p a = true
⊢ some a = some a✝ ↔ a = a✝ ∧ p a✝ = true
case some.isFalse
α : Type u_1
a✝ : α
p : α → Bool
a : α
h : ¬p a = true
⊢ none = some a✝ ↔ a = a✝ ∧ p a✝ = true | 3260041bf741cef3 |
Ordnode.delta_lt_false | Mathlib/Data/Ordmap/Ordset.lean | theorem delta_lt_false {a b : ℕ} (h₁ : delta * a < b) (h₂ : delta * b < a) : False :=
not_le_of_lt (lt_trans ((mul_lt_mul_left (by decide)).2 h₁) h₂) <| by
simpa [mul_assoc] using Nat.mul_le_mul_right a (by decide : 1 ≤ delta * delta)
| a b : ℕ
h₁ : delta * a < b
h₂ : delta * b < a
⊢ 0 < delta | decide | no goals | 99f41f90e87d06fc |
AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover | Mathlib/AlgebraicGeometry/Noetherian.lean | theorem isLocallyNoetherian_iff_of_affine_openCover (𝒰 : Scheme.OpenCover.{v, u} X)
[∀ i, IsAffine (𝒰.obj i)] :
IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing Γ(𝒰.obj i, ⊤) | case mpr
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
hCNoeth : ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
fS : 𝒰.J → ↑X.affineOpens := fun i => ⟨Scheme.Hom.opensRange (𝒰.map i), ⋯⟩
⊢ IsLocallyNoetherian X | apply isLocallyNoetherian_of_affine_cover (S := fS) | case mpr.hS
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
hCNoeth : ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
fS : 𝒰.J → ↑X.affineOpens := fun i => ⟨Scheme.Hom.opensRange (𝒰.map i), ⋯⟩
⊢ ⨆ i, ↑(fS i) = ⊤
case mpr.hS'
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
hCNoeth : ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
fS : 𝒰.J → ↑X.affineOpens := fun i => ⟨Scheme.Hom.opensRange (𝒰.map i), ⋯⟩
⊢ ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(X, ↑(fS i)) | 1710a80ef0394cfb |
Ordnode.Valid'.rotateL | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
(H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ | case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size + 1) ≤ 9 * l.size + 5 ∨ rl.size + rr.size + 1 ≤ 3
⊢ Valid' o₁ (l.rotateL x (Ordnode.node rs rl rx rr)) o₂ | replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ | case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
⊢ Valid' o₁ (l.rotateL x (Ordnode.node rs rl rx rr)) o₂ | 0ebe7132c42c2866 |
Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
[DecidableEq n] (M : Matrix n n 𝕜) :
∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
(L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D | 𝕜 : Type u_3
inst✝² : Field 𝕜
r : ℕ
IH :
∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
Fintype.card n = r → ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = r + 1
⊢ r + 1 = Fintype.card (Fin r ⊕ Unit) | rw [@Fintype.card_sum (Fin r) Unit _ _] | 𝕜 : Type u_3
inst✝² : Field 𝕜
r : ℕ
IH :
∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
Fintype.card n = r → ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = r + 1
⊢ r + 1 = Fintype.card (Fin r) + Fintype.card Unit | 6fda99f6ca7c53ea |
Algebra.discr_zero_of_not_linearIndependent | Mathlib/RingTheory/Discriminant.lean | theorem discr_zero_of_not_linearIndependent [IsDomain A] {b : ι → B}
(hli : ¬LinearIndependent A b) : discr A b = 0 | A : Type u
B : Type v
ι : Type w
inst✝⁵ : DecidableEq ι
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : Fintype ι
inst✝ : IsDomain A
b : ι → B
hli : ¬LinearIndependent A b
g : ι → A
hg : ∑ i : ι, g i • b i = 0
i : ι
hi : g i ≠ 0
⊢ traceMatrix A b *ᵥ g = 0 | ext i | case h
A : Type u
B : Type v
ι : Type w
inst✝⁵ : DecidableEq ι
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : Fintype ι
inst✝ : IsDomain A
b : ι → B
hli : ¬LinearIndependent A b
g : ι → A
hg : ∑ i : ι, g i • b i = 0
i✝ : ι
hi : g i✝ ≠ 0
i : ι
⊢ (traceMatrix A b *ᵥ g) i = 0 i | f530ce30634a182a |
MvPowerSeries.one_mul | Mathlib/RingTheory/MvPowerSeries/Basic.lean | theorem one_mul : (1 : MvPowerSeries σ R) * φ = φ :=
ext fun n => by simpa using coeff_add_monomial_mul 0 n φ 1
| σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (1 * φ) = (coeff R n) φ | simpa using coeff_add_monomial_mul 0 n φ 1 | no goals | 051199c63a005a2b |
Equiv.swap_bijOn_self | Mathlib/Logic/Equiv/Set.lean | theorem Equiv.swap_bijOn_self (hs : a ∈ s ↔ b ∈ s) : BijOn (Equiv.swap a b) s s | case refine_2.inr
α : Type u_1
inst✝ : DecidableEq α
a b : α
s : Set α
hs : a ∈ s ↔ b ∈ s
x : α
hx : x ∈ s
hxa : x ≠ a
⊢ x ∈ ⇑(swap a b) '' s | obtain (rfl | hxb) := eq_or_ne x b | case refine_2.inr.inl
α : Type u_1
inst✝ : DecidableEq α
a : α
s : Set α
x : α
hx : x ∈ s
hxa : x ≠ a
hs : a ∈ s ↔ x ∈ s
⊢ x ∈ ⇑(swap a x) '' s
case refine_2.inr.inr
α : Type u_1
inst✝ : DecidableEq α
a b : α
s : Set α
hs : a ∈ s ↔ b ∈ s
x : α
hx : x ∈ s
hxa : x ≠ a
hxb : x ≠ b
⊢ x ∈ ⇑(swap a b) '' s | b9c2b4efbec77056 |
CategoryTheory.Functor.Elements.ext | Mathlib/CategoryTheory/Elements.lean | lemma Functor.Elements.ext {F : C ⥤ Type w} (x y : F.Elements) (h₁ : x.fst = y.fst)
(h₂ : F.map (eqToHom h₁) x.snd = y.snd) : x = y | case mk.mk.refl
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
fst✝ : C
snd✝¹ snd✝ : F.obj fst✝
h₂ : F.map (eqToHom ⋯) ⟨fst✝, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd
⊢ ⟨fst✝, snd✝¹⟩ = ⟨fst✝, snd✝⟩ | simp only [eqToHom_refl, FunctorToTypes.map_id_apply] at h₂ | case mk.mk.refl
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
fst✝ : C
snd✝¹ snd✝ : F.obj fst✝
h₂ : snd✝¹ = snd✝
⊢ ⟨fst✝, snd✝¹⟩ = ⟨fst✝, snd✝⟩ | 48952278ceaed9f9 |
Memℓp.const_smul | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem const_smul {f : ∀ i, E i} (hf : Memℓp f p) (c : 𝕜) : Memℓp (c • f) p | case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Memℓp (c • f) p | apply memℓp_gen | case inr.inr.hf
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Summable fun i => ‖(c • f) i‖ ^ p.toReal | c315b9f83470cdc9 |
Set.ncard_diff_singleton_add_one | Mathlib/Data/Set/Card.lean | theorem ncard_diff_singleton_add_one {a : α} (h : a ∈ s)
(hs : s.Finite | α : Type u_1
s : Set α
a : α
h : a ∈ s
hs : autoParam s.Finite _auto✝
⊢ (s \ {a}).ncard + 1 = s.ncard | to_encard_tac | α : Type u_1
s : Set α
a : α
h : a ∈ s
hs : autoParam s.Finite _auto✝
⊢ ↑(s \ {a}).ncard + 1 = ↑s.ncard | f16c1c8ed81c9874 |
IsSimpleGroup.prime_card | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | theorem prime_card [Finite α] : (Nat.card α).Prime | case intro.refine_1
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : IsSimpleGroup α
inst✝ : Finite α
h0 : 0 < Nat.card α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
n : ℕ
hn : n ∣ Nat.card α
h : Subgroup.zpowers (g ^ n) = ⊤
hgo : orderOf (g ^ n) = orderOf g / (orderOf g).gcd n
⊢ n = 1 | rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.gcd_eq_right_iff_dvd.1 hn,
orderOf_eq_card_of_forall_mem_zpowers, eq_comm,
Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo | case intro.refine_1
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : IsSimpleGroup α
inst✝ : Finite α
h0 : 0 < Nat.card α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
n : ℕ
hn : n ∣ Nat.card α
h : Subgroup.zpowers (g ^ n) = ⊤
hgo : Nat.card α = Nat.card α * n
⊢ n = 1
case intro.refine_1
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : IsSimpleGroup α
inst✝ : Finite α
h0 : 0 < Nat.card α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
n : ℕ
hn : n ∣ Nat.card α
h : Subgroup.zpowers (g ^ n) = ⊤
hgo : orderOf (g ^ n) = Nat.card α / n
⊢ ∀ (x : α), x ∈ Subgroup.zpowers (g ^ n) | 9a62ce0ba103643e |
Irrational.beattySeq'_pos_eq | Mathlib/NumberTheory/Rayleigh.lean | theorem Irrational.beattySeq'_pos_eq {r : ℝ} (hr : Irrational r) :
{beattySeq' r k | k > 0} = {beattySeq r k | k > 0} | case h.e'_2.h.h.e'_2.h.a.a.h.e'_2
r : ℝ
hr : Irrational r
x✝ k : ℤ
hk : k > 0
⊢ ↑⌊↑k * r⌋ < ↑k * r ∧ ↑k * r ≤ ↑⌊↑k * r⌋ + 1 | refine ⟨(Int.floor_le _).lt_of_ne fun h ↦ ?_, (Int.lt_floor_add_one _).le⟩ | case h.e'_2.h.h.e'_2.h.a.a.h.e'_2
r : ℝ
hr : Irrational r
x✝ k : ℤ
hk : k > 0
h : ↑⌊↑k * r⌋ = ↑k * r
⊢ False | 8a9525f89f636dc1 |
ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
{f : D → E} {g : D → F} {N : WithTop ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
(hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : n ≤ N) :
‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
‖B‖ * ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : WithTop ℕ∞
s : Set D
x✝ : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x✝ ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G :=
((B.comp ↑{ toLinearEquiv := isoE.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip.comp
↑{ toLinearEquiv := isoF.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu :=
((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
((compL 𝕜 Fu G Gu) ↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }))
Bu₀
hBu :
Bu =
((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
((compL 𝕜 Fu G Gu)
↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }))
Bu₀
Bu_eq : (fun y => (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y => (B (f y)) (g y)) ∘ ⇑isoD
y : Eu
x : Fu
⊢ ‖(Bu y) x‖ ≤ ‖B‖ * ‖y‖ * ‖x‖ | simp only [Du, Eu, Fu, Gu, hBu, Bu₀, compL_apply, coe_comp', Function.comp_apply,
ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_coe, flip_apply,
LinearIsometryEquiv.norm_map] | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : WithTop ℕ∞
s : Set D
x✝ : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x✝ ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G :=
((B.comp ↑{ toLinearEquiv := isoE.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip.comp
↑{ toLinearEquiv := isoF.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu :=
((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
((compL 𝕜 Fu G Gu) ↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }))
Bu₀
hBu :
Bu =
((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
((compL 𝕜 Fu G Gu)
↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }))
Bu₀
Bu_eq : (fun y => (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y => (B (f y)) (g y)) ∘ ⇑isoD
y : Eu
x : Fu
⊢ ‖(B (isoE y)) (isoF x)‖ ≤ ‖B‖ * ‖y‖ * ‖x‖ | c6fa542bdff00851 |
WittVector.coeff_p | Mathlib/RingTheory/WittVector/Identities.lean | theorem coeff_p [CharP R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
⊢ (↑p).coeff i = if i = 1 then 1 else 0 | split_ifs with hi | case pos
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hi : i = 1
⊢ (↑p).coeff i = 1
case neg
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hi : ¬i = 1
⊢ (↑p).coeff i = 0 | cb1d7d5ed1383bb1 |
HomologicalComplex.homotopyCofiber.inrX_d | Mathlib/Algebra/Homology/HomotopyCofiber.lean | @[reassoc (attr := simp)]
lemma inrX_d (i j : ι) :
inrX φ i ≫ d φ i j = G.d i j ≫ inrX φ j | case pos
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel i j
hj : c.Rel j (c.next j)
⊢ inrX φ i ≫ d φ i j = G.d i j ≫ inrX φ j | apply ext_to_X _ _ _ hj | case pos.h₁
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel i j
hj : c.Rel j (c.next j)
⊢ (inrX φ i ≫ d φ i j) ≫ fstX φ j (c.next j) hj = (G.d i j ≫ inrX φ j) ≫ fstX φ j (c.next j) hj
case pos.h₂
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel i j
hj : c.Rel j (c.next j)
⊢ (inrX φ i ≫ d φ i j) ≫ sndX φ j = (G.d i j ≫ inrX φ j) ≫ sndX φ j | a8c86133f9b3c635 |
exists_gt_t2space | Mathlib/Topology/ShrinkingLemma.lean | theorem exists_gt_t2space (v : PartialRefinement u s (fun w => IsCompact (closure w)))
(hs : IsCompact s) (i : ι) (hi : i ∉ v.carrier) :
∃ v' : PartialRefinement u s (fun w => IsCompact (closure w)),
v < v' ∧ IsCompact (closure (v' i)) | ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
this : ∀ (j : ι), j ≠ i → x ∉ v.toFun j
⊢ x ∈ v.toFun i | obtain ⟨j, hj⟩ := Set.mem_iUnion.mp
(v.subset_iUnion (Set.mem_of_mem_inter_left hx)) | case intro
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
this : ∀ (j : ι), j ≠ i → x ∉ v.toFun j
j : ι
hj : x ∈ v.toFun j
⊢ x ∈ v.toFun i | b61f3fede2ad34cb |
solvableByRad.induction2 | Mathlib/FieldTheory/AbelRuffini.lean | theorem induction2 {α β γ : solvableByRad F E} (hγ : γ ∈ F⟮α, β⟯) (hα : P α) (hβ : P β) : P γ | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
⊢ minpoly F γ = minpoly F (f ⟨γ, hγ⟩) | refine minpoly.eq_of_irreducible_of_monic
(minpoly.irreducible (isIntegral γ)) ?_ (minpoly.monic (isIntegral γ)) | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
⊢ (aeval (f ⟨γ, hγ⟩)) (minpoly F γ) = 0 | baad06864f8e9f42 |
MeasureTheory.measurable_stoppedValue | Mathlib/Probability/Process/Stopping.lean | theorem measurable_stoppedValue [MetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
(hf_prog : ProgMeasurable f u) (hτ : IsStoppingTime f τ) :
Measurable[hτ.measurableSpace] (stoppedValue u τ) | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝² : MetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : ProgMeasurable f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => τ ω ⊓ i)
t : Set β
ht : MeasurableSet t
i : ι
this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => τ ω ⊓ i) ⁻¹' t ∩ {ω | τ ω ≤ i}
⊢ MeasurableSet ((stoppedValue u fun ω => τ ω ⊓ i) ⁻¹' t ∩ {ω | τ ω ≤ i}) | exact ((h_str_meas i).measurable ht).inter (hτ.measurableSet_le i) | no goals | 1bd0ad42e2bec67b |
isOpen_iff_forall_mem_open | Mathlib/Topology/Basic.lean | theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t | X : Type u
s : Set X
inst✝ : TopologicalSpace X
⊢ s ⊆ interior s ↔ ∀ x ∈ s, ∃ t ⊆ s, IsOpen t ∧ x ∈ t | simp only [subset_def, mem_interior] | no goals | bbc12580404b6a0a |
List.foldr_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem foldr_filter (p : α → Bool) (f : α → β → β) (l : List α) (init : β) :
(l.filter p).foldr f init = l.foldr (fun x y => if p x then f x y else y) init | case nil
α : Type u_1
β : Type u_2
p : α → Bool
f : α → β → β
init : β
⊢ foldr f init (filter p []) = foldr (fun x y => if p x = true then f x y else y) init [] | rfl | no goals | 643547fd2eee8a3a |
BitVec.getMsbD_neg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem getMsbD_neg {i : Nat} {x : BitVec w} :
getMsbD (-x) i =
(getMsbD x i ^^ decide (∃ j < w, i < j ∧ getMsbD x j = true)) | w i : Nat
x : BitVec w
⊢ (-x).getMsbD i = (x.getMsbD i ^^ decide (∃ j, j < w ∧ i < j ∧ x.getMsbD j = true)) | simp only [getMsbD, getLsbD_neg, Bool.decide_and, Bool.and_eq_true, decide_eq_true_eq] | w i : Nat
x : BitVec w
⊢ (decide (i < w) &&
(x.getLsbD (w - 1 - i) ^^ decide (w - 1 - i < w) && decide (∃ j, j < w - 1 - i ∧ x.getLsbD j = true))) =
(decide (i < w) && x.getLsbD (w - 1 - i) ^^ decide (∃ j, j < w ∧ i < j ∧ j < w ∧ x.getLsbD (w - 1 - j) = true)) | a66804a595b3aedc |
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