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
|
|---|---|---|---|---|---|---|
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x
|
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonpos : f u ≤ 0
hfg₃ : ‖g u‖ ≤ -1 / 2 * f u
⊢ f u + g u ≥ f u - ‖g u‖
|
rw [sub_eq_add_neg, norm_eq_abs]
|
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonpos : f u ≤ 0
hfg₃ : ‖g u‖ ≤ -1 / 2 * f u
⊢ f u + g u ≥ f u + -|g u|
|
dd533a6f58fe3819
|
Turing.ToPartrec.Code.comp_eval
|
Mathlib/Computability/TMConfig.lean
|
theorem comp_eval (f g) : (comp f g).eval = fun v => g.eval v >>= f.eval
|
f g : Code
⊢ (f.comp g).eval = fun v => g.eval v >>= f.eval
|
simp [eval]
|
no goals
|
e02b885a8a523280
|
MeasureTheory.Measure.exists_isOpen_everywherePosSubset_eq_diff
|
Mathlib/MeasureTheory/Measure/EverywherePos.lean
|
/-- The everywhere positive subset of a set is obtained by removing an open set. -/
lemma exists_isOpen_everywherePosSubset_eq_diff (μ : Measure α) (s : Set α) :
∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
x : α
n : Set α
ns : n ∈ 𝓝[s] x
hx : μ n = 0
v : Set α
vx : v ∈ 𝓝 x
hv : v ∩ s ⊆ n
w : Set α
wv : w ⊆ v
w_open : IsOpen w
xw : x ∈ w
y : α
yw : y ∈ w
⊢ s ∩ w ⊆ n
|
rw [inter_comm]
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
x : α
n : Set α
ns : n ∈ 𝓝[s] x
hx : μ n = 0
v : Set α
vx : v ∈ 𝓝 x
hv : v ∩ s ⊆ n
w : Set α
wv : w ⊆ v
w_open : IsOpen w
xw : x ∈ w
y : α
yw : y ∈ w
⊢ w ∩ s ⊆ n
|
7270b311930e87cb
|
Equiv.Perm.support_extend_domain
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
support (g.extendDomain f) = g.support.map f.asEmbedding
|
case pos.mpr.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Fintype α
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Fintype β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
g : Perm α
b : β
pb : p b
ha : ↑(f (g (f.symm ⟨b, pb⟩))) = b
hb : f.asEmbedding (f.symm ⟨b, pb⟩) = b
⊢ g (f.symm ⟨b, pb⟩) = f.symm ⟨b, pb⟩
|
rw [eq_symm_apply]
|
case pos.mpr.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Fintype α
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Fintype β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
g : Perm α
b : β
pb : p b
ha : ↑(f (g (f.symm ⟨b, pb⟩))) = b
hb : f.asEmbedding (f.symm ⟨b, pb⟩) = b
⊢ f (g (f.symm ⟨b, pb⟩)) = ⟨b, pb⟩
|
18d8756ce068490b
|
WithTop.coe_sInf'
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithTop α)
|
case intro
α : Type u_1
inst✝¹ : Preorder α
inst✝ : InfSet α
s : Set α
h's : BddBelow s
x : α
hx : x ∈ s
⊢ ↑(sInf s) = sInf ((fun a => ↑a) '' s)
|
change _ = ite _ _ _
|
case intro
α : Type u_1
inst✝¹ : Preorder α
inst✝ : InfSet α
s : Set α
h's : BddBelow s
x : α
hx : x ∈ s
⊢ ↑(sInf s) =
if (fun a => ↑a) '' s ⊆ {⊤} ∨ ¬BddBelow ((fun a => ↑a) '' s) then ⊤
else ↑(sInf ((fun a => ↑a) ⁻¹' ((fun a => ↑a) '' s)))
|
7fbc8ff5f1da276e
|
MeasureTheory.tendsto_of_integral_tendsto_of_monotone
|
Mathlib/MeasureTheory/Integral/Bochner.lean
|
/-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
(hf_int : ∀ n, Integrable (f n) μ) (hF_int : Integrable F μ)
(hf_tendsto : Tendsto (fun i ↦ ∫ a, f i a ∂μ) atTop (𝓝 (∫ a, F a ∂μ)))
(hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
(hf_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) :
∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i => ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
|
let F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i => ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
|
a54f15f3a6cf0f39
|
CategoryTheory.BinaryCofan.isVanKampen_iff
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
IsVanKampenColimit c ↔
∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
(_ : αX ≫ c.inl = c'.inl ≫ f) (_ : αY ≫ c.inr = c'.inr ≫ f),
Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr
|
case mp.mp
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
H : IsVanKampenColimit c
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hαX : αX ≫ c.inl = c'.inl ≫ f
hαY : αY ≫ c.inr = c'.inr ≫ f
⊢ (∀ (j : Discrete WalkingPair), IsPullback (c'.ι.app j) ((mapPair αX αY).app j) f (c.ι.app j)) →
IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr
|
intro H
|
case mp.mp
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
H✝ : IsVanKampenColimit c
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hαX : αX ≫ c.inl = c'.inl ≫ f
hαY : αY ≫ c.inr = c'.inr ≫ f
H : ∀ (j : Discrete WalkingPair), IsPullback (c'.ι.app j) ((mapPair αX αY).app j) f (c.ι.app j)
⊢ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr
|
7e7d3e488062777b
|
MeasureTheory.VectorMeasure.restrict_add
|
Mathlib/MeasureTheory/VectorMeasure/Basic.lean
|
theorem restrict_add (v w : VectorMeasure α M) (i : Set α) :
(v + w).restrict i = v.restrict i + w.restrict i
|
case pos
α : Type u_1
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousAdd M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
⊢ (v + w).restrict i = v.restrict i + w.restrict i
|
ext j hj
|
case pos.h
α : Type u_1
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousAdd M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
j : Set α
hj : MeasurableSet j
⊢ ↑((v + w).restrict i) j = ↑(v.restrict i + w.restrict i) j
|
d7e1efed09c0f5be
|
MeasureTheory.mul_le_addHaar_image_of_lt_det
|
Mathlib/MeasureTheory/Function/Jacobian.lean
|
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
(hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s)
|
case a.inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * μ s ≤ μ (f '' s)) x} ∈ 𝓝 0
|
have hA : A.det ≠ 0 := by
intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
|
case a.inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * μ s ≤ μ (f '' s)) x} ∈ 𝓝 0
|
22f8afc7492fca52
|
CategoryTheory.Localization.Preadditive.neg'_eq
|
Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean
|
lemma neg'_eq (f : L.obj X ⟶ L.obj Y) (φ : W.LeftFraction X Y)
(hφ : f = φ.map L (inverts L W)) :
neg' W f = φ.neg.map L (inverts L W)
|
case intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ neg' W (φ₀.map L ⋯) ≫ L.map (φ₀.s ≫ t₁) = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)
|
nth_rw 1 [L.map_comp]
|
case intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ neg' W (φ₀.map L ⋯) ≫ L.map φ₀.s ≫ L.map t₁ = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)
|
8a41e48563a45b5e
|
differentiableWithinAt_localInvariantProp
|
Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
|
theorem differentiableWithinAt_localInvariantProp :
(contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
(DifferentiableWithinAtProp I I') :=
{ is_local
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
u : Set H
f : H → H'
u_open : IsOpen u
xu : x ∈ u
⊢ DifferentiableWithinAtProp I I' f s x ↔ DifferentiableWithinAtProp I I' f (s ∩ u) x
|
have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
simp only [Set.inter_right_comm, Set.preimage_inter]
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
u : Set H
f : H → H'
u_open : IsOpen u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAtProp I I' f s x ↔ DifferentiableWithinAtProp I I' f (s ∩ u) x
|
3d1405d37db22f10
|
LieAlgebra.IsKilling.eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
|
Mathlib/Algebra/Lie/Weights/RootSystem.lean
|
lemma eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
(hα : α.IsNonZero) (k : K) (h : (β : H → K) = k • α) :
k = -1 ∨ k = 0 ∨ k = 1
|
case inr.inr.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ⇑β = k • ⇑α
h✝ : Nontrivial L
n : ℕ
hn : chainLength α β = 2 * n + 1
H : k • 2 / 2 = ↑(↑(chainLength α β) - 2 * ↑(chainTopCoeff (⇑α) β)) / 2
⊢ k = -1 ∨ k = 0 ∨ k = 1
|
rw [hn, smul_eq_mul] at H
|
case inr.inr.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ⇑β = k • ⇑α
h✝ : Nontrivial L
n : ℕ
hn : chainLength α β = 2 * n + 1
H : k * 2 / 2 = ↑(↑(2 * n + 1) - 2 * ↑(chainTopCoeff (⇑α) β)) / 2
⊢ k = -1 ∨ k = 0 ∨ k = 1
|
1279a8c2c7f69b11
|
isLocalStructomorphOn_contDiffGroupoid_iff
|
Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean
|
theorem isLocalStructomorphOn_contDiffGroupoid_iff (f : PartialHomeomorph M M') :
LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt f f.source ↔
ContMDiffOn I I n f f.source ∧ ContMDiffOn I I n f.symm f.target
|
case hx.h
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : WithTop ℕ∞
inst✝² : IsManifold I n M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : PartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : PartialHomeomorph M H := chartAt H x
c' : PartialHomeomorph M' H := chartAt H X
hxf :
(contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(c.symm ≫ₕ f ≫ₕ c')) (↑(chartAt H x).symm ⁻¹' f.source)
(↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
h1 : c' = chartAt H (↑f x)
h2 : ↑c' ∘ ↑f ∘ ↑c.symm = ↑(c.symm ≫ₕ f ≫ₕ c')
hcx : ↑c x ∈ ↑c.symm ⁻¹' f.source
⊢ ↑(chartAt H x) x ∈ (c.symm ≫ₕ f ≫ₕ c').source
|
simp only [c, hx, h1, mfld_simps]
|
no goals
|
52ce8090c899c270
|
LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
lemma inf_orthogonal_self_le_ker_restrict {W : Submodule R M} (b₁ : B.IsRefl) :
W ⊓ B.orthogonal W ≤ (LinearMap.ker <| B.restrict W).map W.subtype
|
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
B : BilinForm R M
W : Submodule R M
b₁ : B.IsRefl
⊢ W ⊓ B.orthogonal W ≤ Submodule.map W.subtype (ker (B.restrict W))
|
rintro v ⟨hv : v ∈ W, hv' : v ∈ B.orthogonal W⟩
|
case intro
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
B : BilinForm R M
W : Submodule R M
b₁ : B.IsRefl
v : M
hv : v ∈ W
hv' : v ∈ B.orthogonal W
⊢ v ∈ Submodule.map W.subtype (ker (B.restrict W))
|
50c09860c3ac2a16
|
Ordinal.isOpen_singleton_iff
|
Mathlib/SetTheory/Ordinal/Topology.lean
|
theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) ↔ ¬IsLimit a
|
case refine_2.inr.inl.intro
b : Ordinal.{u}
ha : ¬(succ b).IsLimit
⊢ IsOpen (Set.Ioo b (succ (succ b)))
|
exact isOpen_Ioo
|
no goals
|
c638dc4be42bbf41
|
LinearMap.flip_bijective_iff₁
|
Mathlib/LinearAlgebra/Dual.lean
|
theorem flip_bijective_iff₁ [FiniteDimensional K V₁] : Bijective B.flip ↔ Bijective B
|
K : Type uK
inst✝⁵ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K V₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
B : V₁ →ₗ[K] V₂ →ₗ[K] K
inst✝ : FiniteDimensional K V₁
⊢ Bijective ⇑B.flip ↔ Bijective ⇑B
|
simp_rw [Bijective, flip_injective_iff₁, flip_surjective_iff₁, and_comm]
|
no goals
|
68aada2bbc3b9612
|
StrictMonoOn.Iic_id_le
|
Mathlib/Order/Interval/Set/Monotone.lean
|
theorem StrictMonoOn.Iic_id_le [SuccOrder α] [IsSuccArchimedean α] [OrderBot α] {n : α} {φ : α → α}
(hφ : StrictMonoOn φ (Set.Iic n)) : ∀ m ≤ n, m ≤ φ m
|
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : OrderBot α
n : α
φ : α → α
k : α
ih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m
hφ : StrictMonoOn φ (Iic (succ k))
m : α
hm : m ≤ succ k
⊢ m ≤ φ m
|
by_cases hk : IsMax k
|
case pos
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : OrderBot α
n : α
φ : α → α
k : α
ih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m
hφ : StrictMonoOn φ (Iic (succ k))
m : α
hm : m ≤ succ k
hk : IsMax k
⊢ m ≤ φ m
case neg
α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : OrderBot α
n : α
φ : α → α
k : α
ih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m
hφ : StrictMonoOn φ (Iic (succ k))
m : α
hm : m ≤ succ k
hk : ¬IsMax k
⊢ m ≤ φ m
|
3368b27a353274fb
|
Matrix.linfty_opNNNorm_eq_opNNNorm
|
Mathlib/Analysis/Matrix.lean
|
lemma linfty_opNNNorm_eq_opNNNorm (A : Matrix m n α) :
‖A‖₊ = ‖ContinuousLinearMap.mk (Matrix.mulVecLin A)‖₊
|
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j => Matrix.unitOf (A i j)
hxn : ‖x‖₊ = 1
hN : ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
⊢ ∑ j : n, ‖A i j‖₊ ≤ N
|
rw [hxn, mul_one, Pi.nnnorm_def, Finset.sup_le_iff] at hN
|
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j => Matrix.unitOf (A i j)
hxn : ‖x‖₊ = 1
hN : ∀ b ∈ Finset.univ, ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x b‖₊ ≤ N
⊢ ∑ j : n, ‖A i j‖₊ ≤ N
|
e5f792842b2dab2d
|
List.isSuffixOf?_eq_some_iff_append_eq
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem isSuffixOf?_eq_some_iff_append_eq [BEq α] [LawfulBEq α] {xs ys zs : List α} :
xs.isSuffixOf? ys = some zs ↔ zs ++ xs = ys
|
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
xs ys zs : List α
x✝ : ∃ a, xs.reverse ++ a = ys.reverse ∧ a.reverse = zs
w✝ : List α
h : xs.reverse ++ zs.reverse = ys.reverse
heq : w✝ = zs.reverse
⊢ zs ++ xs = ys
|
rw [← reverse_inj, reverse_append, h]
|
no goals
|
5e5a9da994bdcc5a
|
wittStructureInt_rename
|
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
theorem wittStructureInt_rename {σ : Type*} (Φ : MvPolynomial idx ℤ) (f : idx → σ) (n : ℕ) :
wittStructureInt p (rename f Φ) n = rename (Prod.map f id) (wittStructureInt p Φ n)
|
case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
σ : Type u_3
Φ : MvPolynomial idx ℤ
f : idx → σ
n : ℕ
⊢ (map (Int.castRingHom ℚ)) (wittStructureInt p ((rename f) Φ) n) =
(map (Int.castRingHom ℚ)) ((rename (Prod.map f id)) (wittStructureInt p Φ n))
|
simp only [map_rename, map_wittStructureInt, wittStructureRat, rename_bind₁, rename_rename,
bind₁_rename]
|
case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
σ : Type u_3
Φ : MvPolynomial idx ℤ
f : idx → σ
n : ℕ
⊢ (bind₁ fun k => (bind₁ ((fun i => (rename (Prod.mk i)) (W_ ℚ k)) ∘ f)) ((map (Int.castRingHom ℚ)) Φ))
(xInTermsOfW p ℚ n) =
(bind₁ fun i => (bind₁ fun i_1 => (rename (Prod.map f id ∘ Prod.mk i_1)) (W_ ℚ i)) ((map (Int.castRingHom ℚ)) Φ))
(xInTermsOfW p ℚ n)
|
ca26d0c99e320043
|
CategoryTheory.Limits.hasZeroObject_of_hasInitial_object
|
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
theorem hasZeroObject_of_hasInitial_object [HasZeroMorphisms C] [HasInitial C] :
HasZeroObject C
|
case h.e'_7
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasInitial C
X : C
f : X ⟶ ⊥_ C
⊢ 𝟙 (⊥_ C) = 0
|
subsingleton
|
no goals
|
aacc01f350069db8
|
BitVec.toInt_abs_eq_natAbs
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem toInt_abs_eq_natAbs {x : BitVec w} : x.abs.toInt =
if x = intMin w then (intMin w).toInt else x.toInt.natAbs
|
case pos
w : Nat
x : BitVec w
hx : ¬x = intMin w
h : x.msb = true
⊢ (if x.msb = true then -x.toInt else x.toInt) = ↑x.toInt.natAbs
|
simp only [h, ↓reduceIte]
|
case pos
w : Nat
x : BitVec w
hx : ¬x = intMin w
h : x.msb = true
⊢ -x.toInt = ↑x.toInt.natAbs
|
49362e766b77f6e6
|
Ordnode.Valid'.merge_aux₁
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.merge_aux₁ {o₁ o₂ ls ll lx lr rs rl rx rr t}
(hl : Valid' o₁ (@Ordnode.node α ls ll lx lr) o₂) (hr : Valid' o₁ (.node rs rl rx rr) o₂)
(h : delta * ls < rs) (v : Valid' o₁ t rx) (e : size t = ls + size rl) :
Valid' o₁ (.balanceL t rx rr) o₂ ∧ size (.balanceL t rx rr) = ls + rs
|
case inr.intro
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
⊢ 1 ≤ t.size → 1 ≤ rr.size → rr.size ≤ delta * t.size
|
intro _ _
|
case inr.intro
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Valid' o₁ t ↑rx
e : t.size = ll.size + lr.size + 1 + rl.size
hr₁ : rl.size ≤ delta * rr.size
hr₂ : rr.size ≤ delta * rl.size
a✝¹ : 1 ≤ t.size
a✝ : 1 ≤ rr.size
⊢ rr.size ≤ delta * t.size
|
dde6393e47c80664
|
IntermediateField.restrictScalars_sup
|
Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean
|
theorem restrictScalars_sup :
L.restrictScalars K ⊔ L'.restrictScalars K = (L ⊔ L').restrictScalars K :=
toSubfield_injective (by simp)
|
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra K E
inst✝¹ : Algebra K F
inst✝ : IsScalarTower K F E
L L' : IntermediateField F E
⊢ (restrictScalars K L ⊔ restrictScalars K L').toSubfield = (restrictScalars K (L ⊔ L')).toSubfield
|
simp
|
no goals
|
2969b9cc382dda5e
|
Filter.HasBasis.restrict
|
Mathlib/Order/Filter/Bases.lean
|
theorem HasBasis.restrict (h : l.HasBasis p s) {q : ι → Prop}
(hq : ∀ i, p i → ∃ j, p j ∧ q j ∧ s j ⊆ s i) : l.HasBasis (fun i => p i ∧ q i) s
|
case intro.intro.intro.intro.intro
α : Type u_1
ι : Sort u_4
l : Filter α
p : ι → Prop
s : ι → Set α
h : l.HasBasis p s
q : ι → Prop
hq : ∀ (i : ι), p i → ∃ j, p j ∧ q j ∧ s j ⊆ s i
t : Set α
ht : t ∈ l
i : ι
hpi : p i
hti : s i ⊆ t
j : ι
hpj : p j
hqj : q j
hji : s j ⊆ s i
⊢ ∃ i, (p i ∧ q i) ∧ s i ⊆ t
|
exact ⟨j, ⟨hpj, hqj⟩, hji.trans hti⟩
|
no goals
|
e8349a680250bf41
|
Std.DHashMap.Internal.Raw₀.Const.getKey!_insertManyIfNewUnit_list_of_contains
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem getKey!_insertManyIfNewUnit_list_of_contains [EquivBEq α] [LawfulHashable α]
[Inhabited α] (h : m.1.WF) {l : List α} {k : α} :
m.contains k → getKey! (insertManyIfNewUnit m l).1 k = getKey! m k
|
α : Type u
inst✝⁴ : BEq α
inst✝³ : Hashable α
m : Raw₀ α fun x => Unit
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
h : m.val.WF
l : List α
k : α
⊢ m.contains k = true → (insertManyIfNewUnit m l).val.getKey! k = m.getKey! k
|
simp_to_model [Const.insertManyIfNewUnit] using List.getKey!_insertListIfNewUnit_of_contains
|
no goals
|
8150323122ca71b3
|
Set.ncard_eq_toFinset_card
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_toFinset_card (s : Set α) (hs : s.Finite
|
α : Type u_1
s : Set α
hs : autoParam s.Finite _auto✝
⊢ s.ncard = (Finite.toFinset hs).card
|
rw [← Nat.card_coe_set_eq, @Nat.card_eq_fintype_card _ hs.fintype,
@Finite.card_toFinset _ _ hs.fintype hs]
|
no goals
|
ff059e4b1009ae59
|
Ideal.finrank_quotient_map
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
theorem finrank_quotient_map [IsDomain S] [IsDedekindDomain R] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L]
[hp : p.IsMaximal] [Module.Finite R S] :
finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L
|
case refine_2.refine_1.intro.intro
R : Type u
inst✝¹⁴ : CommRing R
S : Type v
inst✝¹³ : CommRing S
p : Ideal R
inst✝¹² : Algebra R S
K : Type u_1
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
L : Type u_2
inst✝⁹ : Field L
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
hRK : IsFractionRing R K
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Algebra K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsScalarTower R S L
hp : p.IsMaximal
inst✝ : Module.Finite R S
ι : Type v := Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Free.chooseBasis (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b' : ι → S := fun i => ⋯.choose
b_eq_b' : ⇑b = ⇑(↑R (Submodule.mkQ (map (algebraMap R S) p))) ∘ b'
b'' : ι → L := ⇑(algebraMap S L) ∘ b'
b''_li : LinearIndependent K b''
x : S
a✝ : x ∈ ⊤
mem_span_b :
(Submodule.mkQ (map (algebraMap R S) p)) x ∈
Submodule.map (↑R (Submodule.mkQ (map (algebraMap R S) p))) (Submodule.span R (Set.range b'))
y : S
y_mem : y ∈ Submodule.span R (Set.range b')
y_eq : (↑R (Submodule.mkQ (map (algebraMap R S) p))) y = (Submodule.mkQ (map (algebraMap R S) p)) x
⊢ y + -(y - x) ∈ Submodule.span R (Set.range b') ⊔ Submodule.restrictScalars R (map (algebraMap R S) p)
|
rw [LinearMap.restrictScalars_apply, Submodule.mkQ_apply, Submodule.mkQ_apply,
Submodule.Quotient.eq] at y_eq
|
case refine_2.refine_1.intro.intro
R : Type u
inst✝¹⁴ : CommRing R
S : Type v
inst✝¹³ : CommRing S
p : Ideal R
inst✝¹² : Algebra R S
K : Type u_1
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
L : Type u_2
inst✝⁹ : Field L
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
hRK : IsFractionRing R K
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Algebra K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsScalarTower R S L
hp : p.IsMaximal
inst✝ : Module.Finite R S
ι : Type v := Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Free.chooseBasis (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b' : ι → S := fun i => ⋯.choose
b_eq_b' : ⇑b = ⇑(↑R (Submodule.mkQ (map (algebraMap R S) p))) ∘ b'
b'' : ι → L := ⇑(algebraMap S L) ∘ b'
b''_li : LinearIndependent K b''
x : S
a✝ : x ∈ ⊤
mem_span_b :
(Submodule.mkQ (map (algebraMap R S) p)) x ∈
Submodule.map (↑R (Submodule.mkQ (map (algebraMap R S) p))) (Submodule.span R (Set.range b'))
y : S
y_mem : y ∈ Submodule.span R (Set.range b')
y_eq : y - x ∈ map (algebraMap R S) p
⊢ y + -(y - x) ∈ Submodule.span R (Set.range b') ⊔ Submodule.restrictScalars R (map (algebraMap R S) p)
|
c261b0ceb1fb92db
|
Computation.terminates_parallel.aux
|
Mathlib/Data/Seq/Parallel.lean
|
theorem terminates_parallel.aux :
∀ {l : List (Computation α)} {S c},
c ∈ l → Terminates c → Terminates (corec parallel.aux1 (l, S))
|
α : Type u
lem1 :
∀ (l : List (Computation α)) (S : WSeq (Computation α)),
(∃ a, parallel.aux2 l = Sum.inl a) → (corec parallel.aux1 (l, S)).Terminates
c : Computation α
T : c.Terminates
⊢ ∀ (s : Computation α),
(∀ {l : List (Computation α)} {S : WSeq (Computation α)}, s ∈ l → (corec parallel.aux1 (l, S)).Terminates) →
∀ {l : List (Computation α)} {S : WSeq (Computation α)}, s.think ∈ l → (corec parallel.aux1 (l, S)).Terminates
|
intro s IH l S m
|
α : Type u
lem1 :
∀ (l : List (Computation α)) (S : WSeq (Computation α)),
(∃ a, parallel.aux2 l = Sum.inl a) → (corec parallel.aux1 (l, S)).Terminates
c : Computation α
T : c.Terminates
s : Computation α
IH : ∀ {l : List (Computation α)} {S : WSeq (Computation α)}, s ∈ l → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : WSeq (Computation α)
m : s.think ∈ l
⊢ (corec parallel.aux1 (l, S)).Terminates
|
933ed16bf89826a9
|
List.erase_cons_head
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean
|
theorem erase_cons_head [LawfulBEq α] (a : α) (l : List α) : (a :: l).erase a = l
|
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
l : List α
⊢ (a :: l).erase a = l
|
simp [erase_cons]
|
no goals
|
0f41b2c0e7bd0ead
|
Order.coheight_add_one_le
|
Mathlib/Order/KrullDimension.lean
|
lemma coheight_add_one_le {a b : α} (hab : b < a) : coheight a + 1 ≤ coheight b
|
α : Type u_1
inst✝ : Preorder α
a b : α
hab : b < a
⊢ coheight a + 1 ≤ coheight b
|
cases hfin : coheight a with
| top =>
have : ⊤ ≤ coheight b := by
rw [← hfin]
gcongr
simp [this]
| coe n =>
apply Order.add_one_le_of_lt
rw [← hfin]
gcongr
simp [hfin]
|
no goals
|
d04e3a2735fe3bd1
|
Homotopy.comp_nullHomotopicMap
|
Mathlib/Algebra/Homology/Homotopy.lean
|
theorem comp_nullHomotopicMap (f : C ⟶ D) (hom : ∀ i j, D.X i ⟶ E.X j) :
f ≫ nullHomotopicMap hom = nullHomotopicMap fun i j => f.f i ≫ hom i j
|
case h
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → D.X i ⟶ E.X j
n : ι
⊢ f.f n ≫ (D.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ E.dTo n) =
C.dFrom n ≫ f.f (c.next n) ≫ hom (c.next n) n + (f.f n ≫ hom n (c.prev n)) ≫ E.dTo n
|
simp only [Preadditive.comp_add, assoc, f.comm_assoc]
|
no goals
|
180e3371759e376e
|
Finset.prod_dvd_of_isRelPrime
|
Mathlib/RingTheory/Coprime/Lemmas.lean
|
theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z
|
α : Type u_2
I : Type u_1
inst✝¹ : CommMonoid α
inst✝ : DecompositionMonoid α
z : α
s : I → α
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsRelPrime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsRelPrime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
aux1 : a ∈ ↑(insert a r)
hir : a ∈ r
⊢ False
|
exact har hir
|
no goals
|
83dc5ba78bc0ac42
|
Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
|
Mathlib/LinearAlgebra/Dimension/Finite.lean
|
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
{t : Finset M} (h : finrank R M + 1 < t.card) :
∃ f : M → R, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0
|
case intro.intro.intro.intro.intro.refine_1
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Finset M := Finset.map shift (t.erase x₀)
h' : finrank R M < #t'
g : M → R
gsum : ∑ x ∈ t.erase x₀, g (x - x₀) • (x - x₀) = 0
x₁ : M
x₁_mem : x₁ ∈ t'
nz : g x₁ ≠ 0
f : M → R := fun z => if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀)
⊢ ∑ x ∈ t.erase x₀, ((if x = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (x - x₀)) • x - g (x - x₀) • x₀) =
∑ x ∈ t.erase x₀, (g (x - x₀) • x - g (x - x₀) • x₀)
|
refine sum_congr rfl fun x x_mem ↦ ?_
|
case intro.intro.intro.intro.intro.refine_1
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Finset M := Finset.map shift (t.erase x₀)
h' : finrank R M < #t'
g : M → R
gsum : ∑ x ∈ t.erase x₀, g (x - x₀) • (x - x₀) = 0
x₁ : M
x₁_mem : x₁ ∈ t'
nz : g x₁ ≠ 0
f : M → R := fun z => if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀)
x : M
x_mem : x ∈ t.erase x₀
⊢ (if x = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (x - x₀)) • x - g (x - x₀) • x₀ =
g (x - x₀) • x - g (x - x₀) • x₀
|
9ccb45a24efab47f
|
Lean.Data.AC.Context.eval_norm
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean
|
theorem Context.eval_norm (ctx : Context α) (e : Expr) : evalList α ctx (norm ctx e) = eval α ctx e
|
α : Sort u_1
ctx : Context α
e : Expr
⊢ evalList α ctx
(if ContextInformation.isIdem ctx = true then
mergeIdem
(if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx e.toList)
else removeNeutrals ctx e.toList)
else
if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx e.toList)
else removeNeutrals ctx e.toList) =
eval α ctx e
|
cases h₁ : ContextInformation.isIdem ctx <;> cases h₂ : ContextInformation.isComm ctx <;>
simp_all [evalList_removeNeutrals, eval_toList, toList_nonEmpty, evalList_mergeIdem, evalList_sort]
|
no goals
|
62f4ef397b8c946b
|
Std.Sat.AIG.RefVec.ite.go_get_aux
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/If.lean
|
theorem go_get_aux {w : Nat} (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (discr : Ref aig)
(lhs rhs : RefVec aig w) (s : RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig curr hcurr discr lhs rhs s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo
|
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
h✝ : curr < w
hgo :
go (aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get curr h✝ }).aig (curr + 1) ⋯ (discr.cast ⋯)
(lhs.cast ⋯) (rhs.cast ⋯)
((s.cast ⋯).push (aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get curr h✝ }).ref) =
res
hfoo✝ :
aig.decls.size ≤
(go (aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get curr h✝ }).aig (curr + 1) ⋯
(discr.cast ⋯) (lhs.cast ⋯) (rhs.cast ⋯)
((s.cast ⋯).push
(aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get curr h✝ }).ref)).aig.decls.size
⊢ ((s.get idx ?isTrue.hidx).cast ⋯).gate = (s.get idx hidx).gate
|
simp
|
no goals
|
c08e573c9b942d4a
|
Rat.num_den_mk
|
Mathlib/Data/Rat/Lemmas.lean
|
theorem num_den_mk {q : ℚ} {n d : ℤ} (hd : d ≠ 0) (qdf : q = n /. d) :
∃ c : ℤ, n = c * q.num ∧ d = c * q.den
|
q : ℚ
n d : ℤ
hd : d ≠ 0
qdf : q = n /. d
⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den
|
obtain rfl | hn := eq_or_ne n 0
|
case inl
q : ℚ
d : ℤ
hd : d ≠ 0
qdf : q = 0 /. d
⊢ ∃ c, 0 = c * q.num ∧ d = c * ↑q.den
case inr
q : ℚ
n d : ℤ
hd : d ≠ 0
qdf : q = n /. d
hn : n ≠ 0
⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den
|
6651e65d3ba1324c
|
Polynomial.rootMultiplicity_eq_natTrailingDegree'
|
Mathlib/Algebra/Polynomial/Div.lean
|
/-- See `Polynomial.rootMultiplicity_eq_natTrailingDegree` for the general case. -/
lemma rootMultiplicity_eq_natTrailingDegree' : p.rootMultiplicity 0 = p.natTrailingDegree
|
R : Type u
inst✝ : CommRing R
p : R[X]
⊢ rootMultiplicity 0 p = p.natTrailingDegree
|
by_cases h : p = 0
|
case pos
R : Type u
inst✝ : CommRing R
p : R[X]
h : p = 0
⊢ rootMultiplicity 0 p = p.natTrailingDegree
case neg
R : Type u
inst✝ : CommRing R
p : R[X]
h : ¬p = 0
⊢ rootMultiplicity 0 p = p.natTrailingDegree
|
fb3fe56f8f18fafe
|
Relation.cutExpand_double
|
Mathlib/Logic/Hydra.lean
|
lemma cutExpand_double {a a₁ a₂} (h₁ : r a₁ a) (h₂ : r a₂ a) : CutExpand r {a₁, a₂} {a} :=
cutExpand_singleton <| by
simp only [insert_eq_cons, mem_cons, mem_singleton, forall_eq_or_imp, forall_eq]
tauto
|
α : Type u_1
r : α → α → Prop
a a₁ a₂ : α
h₁ : r a₁ a
h₂ : r a₂ a
⊢ ∀ x' ∈ {a₁, a₂}, r x' a
|
simp only [insert_eq_cons, mem_cons, mem_singleton, forall_eq_or_imp, forall_eq]
|
α : Type u_1
r : α → α → Prop
a a₁ a₂ : α
h₁ : r a₁ a
h₂ : r a₂ a
⊢ r a₁ a ∧ r a₂ a
|
3db6640335ee3b12
|
FirstOrder.Language.Theory.models_formula_iff_onTheory_models_equivSentence
|
Mathlib/ModelTheory/Satisfiability.lean
|
theorem models_formula_iff_onTheory_models_equivSentence {φ : L.Formula α} :
T ⊨ᵇ φ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ Formula.equivSentence φ
|
case refine_2
L : Language
T : L.Theory
α : Type w
φ : L.Formula α
h : (L.lhomWithConstants α).onTheory T ⊨ᵇ Formula.equivSentence φ
M : T.ModelType
v : α → ↑M
this✝ : (constantsOn α).Structure ↑M := constantsOn.structure v
this : ↑M ⊨ (L.lhomWithConstants α).onTheory T
⊢ φ.Realize v
|
exact (Formula.realize_equivSentence _ _).1 (h.realize_sentence M)
|
no goals
|
68157140d9be2023
|
norm_cfcₙ_lt
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean
|
lemma norm_cfcₙ_lt {f : 𝕜 → 𝕜} {a : A} {c : ℝ} (h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c) :
‖cfcₙ f a‖ < c
|
case refine_2.intro.intro
𝕜 : Type u_1
A : Type u_2
p : outParam (A → Prop)
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
x : 𝕜
hx : x ∈ σₙ 𝕜 a
⊢ (fun x => ‖f x‖) x < c
|
exact h x hx
|
no goals
|
950e36292dc4fd4a
|
List.getElem_zipIdx
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Range.lean
|
theorem getElem_zipIdx (l : List α) (n) (i : Nat) (h : i < (l.zipIdx n).length) :
(l.zipIdx n)[i] = (l[i]'(by simpa [length_zipIdx] using h), n + i)
|
α : Type u_1
l : List α
n i : Nat
h✝ : i < (l.zipIdx n).length
h : i < l.length
⊢ (Option.map (fun a => (a, n + i)) (some l[i])).get ⋯ = (l[i], n + i)
|
simp
|
no goals
|
7de6a7b9a3350b01
|
List.foldl_ext
|
Mathlib/Data/List/Basic.lean
|
theorem foldl_ext (f g : α → β → α) (a : α) {l : List β} (H : ∀ a : α, ∀ b ∈ l, f a b = g a b) :
foldl f a l = foldl g a l
|
α : Type u
β : Type v
f g : α → β → α
a : α
l : List β
H : ∀ (a : α) (b : β), b ∈ l → f a b = g a b
⊢ foldl f a l = foldl g a l
|
induction l generalizing a with
| nil => rfl
| cons hd tl ih =>
unfold foldl
rw [ih _ fun a b bin => H a b <| mem_cons_of_mem _ bin, H a hd (mem_cons_self _ _)]
|
no goals
|
6ffe2cd5673f2e92
|
QuasispectrumRestricts.cfc
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
|
theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a
|
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
s : S
hs : s ∈ σₙ S ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) g)
⊢ (algebraMap R S) (f s) = s
|
rw [nonUnitalStarAlgHom_apply,
cfcₙHom_map_quasispectrum] at hs
|
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
s : S
hs :
s ∈
range
⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
(g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
⊢ (algebraMap R S) (f s) = s
|
3af5d542f22c6b37
|
aux1
|
Mathlib/Algebra/Jordan/Basic.lean
|
theorem aux1 {a b c : A} :
⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +
2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆
=
⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +
⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +
(⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +
⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +
(⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +
⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆)
|
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
⁅L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =
⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
⁅L a, 2 • L (b * c)⁆ +
(⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
⁅L b, 2 • L (b * c)⁆) +
(⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
⁅L c, 2 • L (b * c)⁆)
|
iterate 15 rw [lie_add]
|
no goals
|
b5c23462450b58db
|
MulAction.IsPreprimitive.of_card_lt
|
Mathlib/GroupTheory/GroupAction/Primitive.lean
|
theorem of_card_lt [Finite Y] [IsPretransitive H Y] [IsPreprimitive G X]
(hf' : Nat.card Y < 2 * (Set.range f).ncard) :
IsPreprimitive H Y := by
refine ⟨fun {B} hB ↦ ?_⟩
rcases B.eq_empty_or_nonempty with hB' | hB'; · simp [IsTrivialBlock, hB']
rw [IsTrivialBlock, or_iff_not_imp_right]
intro hB_ne_top
-- we need Set.Subsingleton B ↔ Set.ncard B ≤ 1
suffices Set.ncard B < 2 by simpa [Nat.lt_succ] using this
-- We reduce to proving that (Set.range f).ncard ≤ (orbit N B).ncard
apply lt_of_mul_lt_mul_right (lt_of_le_of_lt _ hf') (zero_le _)
simp only [← Nat.card_eq_fintype_card, ← hB.ncard_block_mul_ncard_orbit_eq hB']
apply Nat.mul_le_mul_left
-- We reduce to proving that (Set.range f ∩ g • B).ncard ≤ 1 for every g
have hfin := Fintype.ofFinite (Set.range fun g : H ↦ g • B)
rw [(hB.isBlockSystem hB').left.ncard_eq_finsum, finsum_eq_sum_of_fintype]
apply le_trans (Finset.sum_le_card_nsmul _ _ 1 _)
· rw [nsmul_one, Finset.card_univ, ← Set.toFinset_card, ← Set.ncard_eq_toFinset_card',
orbit, Nat.cast_id]
· rintro ⟨x, ⟨g, rfl⟩⟩ -
suffices Set.Subsingleton (Set.range f ∩ g • B) by simpa
-- It suffices to prove that the preimage is subsingleton
rw [← Set.image_preimage_eq_range_inter]
apply Set.Subsingleton.image
-- Since the action of M on α is primitive, it suffices to prove that
-- the preimage is a block which is not ⊤
apply Or.resolve_right (isTrivialBlock_of_isBlock ((hB.translate g).preimage f))
intro h
simp only [Set.top_eq_univ, Set.preimage_eq_univ_iff] at h
-- We will prove that B is large, which will contradict the assumption that it is not ⊤
apply hB_ne_top
apply hB.eq_univ_of_card_lt
-- It remains to show that Nat.card β < Set.ncard B * 2
apply lt_of_lt_of_le hf'
rw [mul_comm, mul_le_mul_right Nat.succ_pos']
apply le_trans (Set.ncard_le_ncard h) (Set.ncard_image_le B.toFinite)
|
case mk.intro.hs
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
hB_ne_top : ¬B = Set.univ
hfin : Fintype ↑(Set.range fun g => g • B)
g : H
h : Set.range ⇑f ⊆ g • B
⊢ B = Set.univ
|
apply hB.eq_univ_of_card_lt
|
case mk.intro.hs
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
hB_ne_top : ¬B = Set.univ
hfin : Fintype ↑(Set.range fun g => g • B)
g : H
h : Set.range ⇑f ⊆ g • B
⊢ Nat.card Y < B.ncard * 2
|
ef6cb8f6bf6f5d36
|
SimpleGraph.Walk.IsPath.snd_of_toSubgraph_adj
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
|
lemma snd_of_toSubgraph_adj {u v v'} {p : G.Walk u v} (hp : p.IsPath)
(hadj : p.toSubgraph.Adj u v') : p.snd = v'
|
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
i : ℕ
hl1 : p.getVert i = u
hadj : p.toSubgraph.Adj u (p.getVert (i + 1))
hi :
(p.getVert i = u ∧ p.getVert (i + 1) = p.getVert (i + 1) ∨ p.getVert i = p.getVert (i + 1) ∧ p.getVert (i + 1) = u) ∧
i < p.length
⊢ 0 ≤ p.length
|
omega
|
no goals
|
6a4c9bd87e294aeb
|
Ideal.spanNorm_mul
|
Mathlib/RingTheory/Ideal/Norm/RelNorm.lean
|
theorem spanNorm_mul (I J : Ideal S) : spanNorm R (I * J) = spanNorm R I * spanNorm R J
|
case he.h.a
R : Type u_1
inst✝¹² : CommRing R
inst✝¹¹ : IsDomain R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : IsDomain S
inst✝⁸ : IsIntegrallyClosed R
inst✝⁷ : IsIntegrallyClosed S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Finite R S
inst✝⁴ : NoZeroSMulDivisors R S
inst✝³ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
I J : Ideal S
inst✝ : Nontrivial R
h✝ : Nontrivial S
P : Ideal R
hP : P.IsMaximal
hP0 : ¬P = ⊥
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_3 := Localization P'
x✝¹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁵ : IsDomain Sₚ
this✝⁴ : IsDedekindDomain Sₚ
this✝³ : IsPrincipalIdealRing Sₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
L : Type u_3 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
algInst✝ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap Rₚ (FractionRing Rₚ)) ((algebraMap R Rₚ) x✝)) =
(algebraMap Sₚ (FractionRing Sₚ)) ((algebraMap S Sₚ) ((algebraMap R S) x✝))
|
simp only [← IsScalarTower.algebraMap_apply]
|
case he.h.a
R : Type u_1
inst✝¹² : CommRing R
inst✝¹¹ : IsDomain R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : IsDomain S
inst✝⁸ : IsIntegrallyClosed R
inst✝⁷ : IsIntegrallyClosed S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Finite R S
inst✝⁴ : NoZeroSMulDivisors R S
inst✝³ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
I J : Ideal S
inst✝ : Nontrivial R
h✝ : Nontrivial S
P : Ideal R
hP : P.IsMaximal
hP0 : ¬P = ⊥
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
Sₚ : Type u_3 := Localization P'
x✝¹ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this✝⁶ : IsScalarTower R Rₚ Sₚ
h : P' ≤ S⁰
this✝⁵ : IsDomain Sₚ
this✝⁴ : IsDedekindDomain Sₚ
this✝³ : IsPrincipalIdealRing Sₚ
this✝² : NoZeroSMulDivisors Rₚ Sₚ
this✝¹ : Module.Finite Rₚ Sₚ
L : Type u_3 := FractionRing S
g : Sₚ →+* L := IsLocalization.map L (RingHom.id S) h
algInst✝ : Algebra Sₚ L := g.toAlgebra
this✝ : IsScalarTower S Sₚ (FractionRing S)
this : IsFractionRing Sₚ (FractionRing S)
x✝ : R
⊢ (algebraMap (FractionRing Rₚ) (FractionRing Sₚ)) ((algebraMap R (FractionRing Rₚ)) x✝) =
(algebraMap R (FractionRing Sₚ)) x✝
|
a44add16c102adb0
|
mellin_convergent_zero_of_isBigO
|
Mathlib/Analysis/MellinTransform.lean
|
theorem mellin_convergent_zero_of_isBigO {b : ℝ} {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0))
(hf : f =O[𝓝[>] 0] (· ^ (-b))) {s : ℝ} (hs : b < s) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioc 0 c)
|
case intro.intro.intro.intro.refine_2.h
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x => x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioo 0 ε), ‖a ^ (s - 1) * f a‖ ≤ d * a ^ (s - b - 1)
|
refine (ae_restrict_iff' measurableSet_Ioo).mpr (Eventually.of_forall fun t ht => ?_)
|
case intro.intro.intro.intro.refine_2.h
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x => x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
t : ℝ
ht : t ∈ Ioo 0 ε
⊢ ‖t ^ (s - 1) * f t‖ ≤ d * t ^ (s - b - 1)
|
755d659e5dc10655
|
ProbabilityTheory.hasDerivAt_integral_pow_mul_exp
|
Mathlib/Probability/Moments/ComplexMGF.lean
|
/-- For `z : ℂ` with `z.re ∈ interior (integrableExpSet X μ)`, the derivative of the function
`z' ↦ μ[X ^ n * cexp (z' * X)]` at `z` is `μ[X ^ (n + 1) * cexp (z * X)]`. -/
lemma hasDerivAt_integral_pow_mul_exp (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
HasDerivAt (fun z ↦ μ[fun ω ↦ X ω ^ n * cexp (z * X ω)])
μ[fun ω ↦ X ω ^ (n + 1) * cexp (z * X ω)] z
|
case intro.intro.intro.refine_6
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := ((z.re - l) ⊓ (u - z.re)) / 2
h_pos : 0 < (z.re - l) ⊓ (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
⊢ HasDerivAt (fun x => cexp (x * ↑(X ω))) (↑(X ω) * cexp (ε * ↑(X ω))) ε
|
simp_rw [← smul_eq_mul, Complex.exp_eq_exp_ℂ]
|
case intro.intro.intro.refine_6
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := ((z.re - l) ⊓ (u - z.re)) / 2
h_pos : 0 < (z.re - l) ⊓ (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
⊢ HasDerivAt (fun x => NormedSpace.exp ℂ (x • ↑(X ω))) (↑(X ω) • NormedSpace.exp ℂ (ε • ↑(X ω))) ε
|
39ea945347d2d3e2
|
RegularExpression.star_rmatch_iff
|
Mathlib/Computability/RegularExpressions.lean
|
theorem star_rmatch_iff (P : RegularExpression α) :
∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
= S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
fun x => by
have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
clear star_rmatch_iff
constructor
· rcases x with - | ⟨a, x⟩
· intro _h
use []; dsimp; tauto
· rw [rmatch, deriv, mul_rmatch_iff]
rintro ⟨t, u, hs, ht, hu⟩
have hwf : u.length < (List.cons a x).length
|
case mp.nil
α : Type u_1
inst✝ : DecidableEq α
P : RegularExpression α
IH :
∀ (t : List α),
t.length < [].length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)
⊢ P.star.rmatch [] = true → ∃ S, [] = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true
|
intro _h
|
case mp.nil
α : Type u_1
inst✝ : DecidableEq α
P : RegularExpression α
IH :
∀ (t : List α),
t.length < [].length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)
_h : P.star.rmatch [] = true
⊢ ∃ S, [] = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true
|
8215ce2ddbd6bd2a
|
one_add_mul_self_lt_rpow_one_add
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
theorem one_add_mul_self_lt_rpow_one_add {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp : 1 < p) :
1 + p * s < (1 + s) ^ p
|
case inr.inr.a.inr
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s > 0
⊢ log (1 + p * s) < log (1 + s) * p
|
rw [← div_lt_iff₀ hp', ← div_lt_div_iff_of_pos_right hs']
|
case inr.inr.a.inr
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s > 0
⊢ log (1 + p * s) / p / s < log (1 + s) / s
|
05080339bfb00ac5
|
MeasurableEmbedding.comap_restrict
|
Mathlib/MeasureTheory/Measure/Restrict.lean
|
lemma comap_restrict (μ : Measure β) (s : Set β) :
(μ.restrict s).comap f = (μ.comap f).restrict (f ⁻¹' s)
|
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
⊢ Measure.comap f (μ.restrict s) = (Measure.comap f μ).restrict (f ⁻¹' s)
|
ext t ht
|
case h
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
t : Set α
ht : MeasurableSet t
⊢ (Measure.comap f (μ.restrict s)) t = ((Measure.comap f μ).restrict (f ⁻¹' s)) t
|
6c46b887e8c798e1
|
List.drop_zipWith
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean
|
theorem drop_zipWith : (zipWith f l l').drop n = zipWith f (l.drop n) (l'.drop n)
|
case cons.cons.succ
α✝² : Type u_1
α✝¹ : Type u_2
α✝ : Type u_3
f : α✝² → α✝¹ → α✝
hd : α✝²
tl : List α✝²
hl : ∀ {l' : List α✝¹} {n : Nat}, drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')
head✝ : α✝¹
tail✝ : List α✝¹
n✝ : Nat
⊢ drop (n✝ + 1) (zipWith f (hd :: tl) (head✝ :: tail✝)) =
zipWith f (drop (n✝ + 1) (hd :: tl)) (drop (n✝ + 1) (head✝ :: tail✝))
|
simp [hl]
|
no goals
|
85fad810981714d3
|
Std.Sat.CNF.relabel_congr
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Relabel.lean
|
theorem relabel_congr {f : CNF α} {r1 r2 : α → β} (hw : ∀ v, Mem v f → r1 v = r2 v) :
relabel r1 f = relabel r2 f
|
α : Type u_1
β : Type u_2
f : CNF α
r1 r2 : α → β
hw : ∀ (v : α), Mem v f → r1 v = r2 v
⊢ relabel r1 f = relabel r2 f
|
dsimp only [relabel]
|
α : Type u_1
β : Type u_2
f : CNF α
r1 r2 : α → β
hw : ∀ (v : α), Mem v f → r1 v = r2 v
⊢ List.map (Clause.relabel r1) f = List.map (Clause.relabel r2) f
|
9df4d829e86fbf4e
|
Polynomial.Monic.eq_X_pow_iff_natDegree_le_natTrailingDegree
|
Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
|
lemma eq_X_pow_iff_natDegree_le_natTrailingDegree (h₁ : p.Monic) :
p = X ^ p.natDegree ↔ p.natDegree ≤ p.natTrailingDegree
|
case refine_2.a.inl
R : Type u
inst✝ : Semiring R
p : R[X]
h₁ : p.Monic
h : p.natDegree ≤ p.natTrailingDegree
n : ℕ
hn : n < p.natDegree
⊢ p.coeff n = if n = p.natDegree then 1 else 0
|
rw [if_neg hn.ne, coeff_eq_zero_of_lt_natTrailingDegree (hn.trans_le h)]
|
no goals
|
d8897cf4497f59d4
|
Fin.findSome?_succ_of_isNone
|
Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean
|
theorem findSome?_succ_of_isNone {f : Fin (n+1) → Option α} (h : (f 0).isNone) :
findSome? f = findSome? fun i => f i.succ
|
n : Nat
α : Type u_1
f : Fin (n + 1) → Option α
h : (f 0).isNone = true
⊢ findSome? f = findSome? fun i => f i.succ
|
simp_all [findSome?_succ_of_none]
|
no goals
|
aef985eba48555fe
|
MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
(hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
(hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a)
|
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
y : ℝ
hy : y ∈ Ioc a (id x)
⊢ g' y = ‖g' y‖
|
dsimp
|
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
y : ℝ
hy : y ∈ Ioc a (id x)
⊢ g' y = |g' y|
|
1f10c323da5db2cd
|
MeasureTheory.LevyProkhorov.continuous_equiv_symm_probabilityMeasure
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm
|
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε
|
have third_ε_pos' : 0 < ENNReal.ofReal (ε / 3) := ofReal_pos.mpr third_ε_pos
|
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε
|
1196bd5a9632bf03
|
Polynomial.rootMultiplicity_X_sub_C
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
theorem rootMultiplicity_X_sub_C [Nontrivial R] [DecidableEq R] {x y : R} :
rootMultiplicity x (X - C y) = if x = y then 1 else 0
|
case neg
R : Type u
inst✝² : CommRing R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq R
x y : R
hxy : ¬x = y
⊢ rootMultiplicity x (X - C y) = 0
|
exact rootMultiplicity_eq_zero (mt root_X_sub_C.mp (Ne.symm hxy))
|
no goals
|
dbf0f6a53bd5c674
|
IsCyclotomicExtension.Rat.Three.Units.mem
|
Mathlib/NumberTheory/Cyclotomic/Three.lean
|
theorem Units.mem [NumberField K] [IsCyclotomicExtension {3} ℚ K] :
u ∈ [1, -1, η, -η, η ^ 2, -η ^ 2]
|
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = ↑(x, e).1 * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i
⊢ u = ↑x
|
rw [← mul_one x.1, hxu]
|
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = ↑(x, e).1 * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i
⊢ ↑(x, e).1 * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i = ↑x * 1
|
4597273329d964a3
|
MonotoneOn.csInf_eq_of_subset_of_forall_exists_le
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
lemma MonotoneOn.csInf_eq_of_subset_of_forall_exists_le
[Preorder α] [ConditionallyCompleteLattice β] {f : α → β}
{s t : Set α} (ht : BddBelow (f '' t)) (hf : MonotoneOn f t)
(hst : s ⊆ t) (h : ∀ y ∈ t, ∃ x ∈ s, x ≤ y) :
sInf (f '' s) = sInf (f '' t)
|
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : ConditionallyCompleteLattice β
f : α → β
s t : Set α
ht : BddBelow (f '' t)
hf : MonotoneOn f t
hst : s ⊆ t
h : ∀ y ∈ t, ∃ x ∈ s, x ≤ y
hs : s.Nonempty
⊢ sInf (f '' s) ≤ sInf (f '' t)
|
refine le_csInf ((hs.mono hst).image f) ?_
|
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : ConditionallyCompleteLattice β
f : α → β
s t : Set α
ht : BddBelow (f '' t)
hf : MonotoneOn f t
hst : s ⊆ t
h : ∀ y ∈ t, ∃ x ∈ s, x ≤ y
hs : s.Nonempty
⊢ ∀ b ∈ f '' t, sInf (f '' s) ≤ b
|
bebfbe43136d45af
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.rupAdd_sound
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem rupAdd_sound {n : Nat} (f : DefaultFormula n) (c : DefaultClause n) (rupHints : Array Nat)
(f' : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(rupAddSuccess : performRupAdd f c rupHints = (f', true)) :
Liff (PosFin n) f f'
|
n : Nat
f : DefaultFormula n
c : DefaultClause n
rupHints : Array Nat
f' : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
f'_def : f' = f.insert c
rupAddSuccess :
(if (f.insertRupUnits c.negate).snd = true then ((f.insertRupUnits c.negate).fst.clearRupUnits.insert c, true)
else
if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.snd = true then
(((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
else
if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.fst = false then
(((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
else
({ clauses := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.clauses,
rupUnits := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.rupUnits,
ratUnits := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.ratUnits,
assignments :=
restoreAssignments ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.assignments
((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.fst }.clearRupUnits.insert
c,
true)) =
(f', true)
⊢ Liff (PosFin n) f f'
|
split at rupAddSuccess
|
case isTrue
n : Nat
f : DefaultFormula n
c : DefaultClause n
rupHints : Array Nat
f' : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
f'_def : f' = f.insert c
h✝ : (f.insertRupUnits c.negate).snd = true
rupAddSuccess : ((f.insertRupUnits c.negate).fst.clearRupUnits.insert c, true) = (f', true)
⊢ Liff (PosFin n) f f'
case isFalse
n : Nat
f : DefaultFormula n
c : DefaultClause n
rupHints : Array Nat
f' : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
f'_def : f' = f.insert c
h✝ : ¬(f.insertRupUnits c.negate).snd = true
rupAddSuccess :
(if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.snd = true then
(((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
else
if ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.fst = false then
(((f.insertRupUnits c.negate).fst.performRupCheck rupHints).fst, false)
else
({ clauses := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.clauses,
rupUnits := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.rupUnits,
ratUnits := ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.ratUnits,
assignments :=
restoreAssignments ((f.insertRupUnits c.negate).fst.performRupCheck rupHints).1.assignments
((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.fst }.clearRupUnits.insert
c,
true)) =
(f', true)
⊢ Liff (PosFin n) f f'
|
54bbe5dd0a90a45f
|
Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
|
Mathlib/Analysis/Analytic/Uniqueness.lean
|
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0
|
case intro.intro.intro.intro.intro.intro.intro.zero
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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
p : ContinuousMultilinearMap 𝕜 (fun i => E) F
ht : ∀ y ∈ t, ‖p fun x => y‖ ≤ c * ‖‖y‖ ^ (0 + 1)‖
⊢ (p fun x => y) = 0
|
exact norm_eq_zero.mp (by
simpa only [fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
|
no goals
|
a2f8ef35a556aa68
|
LittleWedderburn.InductionHyp.center_eq_top
|
Mathlib/RingTheory/LittleWedderburn.lean
|
theorem center_eq_top [Finite D] (hD : InductionHyp D) : Subring.center D = ⊤
|
case intro.h.mk
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : LittleWedderburn.InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
⊢ eval (↑q) Φₙ ∣ ↑(ConjClasses.mk x).carrier.toFinset.card
|
set Zx := Subring.centralizer ({↑x} : Set D)
|
case intro.h.mk
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : LittleWedderburn.InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
Zx : Subring D := Subring.centralizer {↑x}
⊢ eval (↑q) Φₙ ∣ ↑(ConjClasses.mk x).carrier.toFinset.card
|
abd28bec54f62bc4
|
Set.Infinite.biUnion
|
Mathlib/Data/Set/Finite/Lattice.lean
|
theorem Infinite.biUnion {ι : Type*} {s : ι → Set α} {a : Set ι} (ha : a.Infinite)
(hs : a.InjOn s) : (⋃ i ∈ a, s i).Infinite
|
α : Type u
ι : Type u_1
s : ι → Set α
a : Set ι
ha : a.Infinite
hs : InjOn s a
x✝² : Infinite ↑a
x✝¹ x✝ : ↑a
i : ι
hi : i ∈ a
j : ι
hj : j ∈ a
hij : s ↑⟨i, hi⟩ = s ↑⟨j, hj⟩
⊢ ⟨i, hi⟩ = ⟨j, hj⟩
|
simp [hs hi hj hij]
|
no goals
|
383aa77bfb6d978a
|
Array.attach_push
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean
|
theorem attach_push {a : α} {l : Array α} :
(l.push a).attach =
(l.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩
|
α : Type ?u.10502
a : α
l : Array α
⊢ a ∈ l.push a
|
simp
|
no goals
|
e0c7040d38326641
|
MeasureTheory.lintegral_iSup_directed
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
(h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
simp_rw [← iSup_apply]
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
be3dd78af53e2dbc
|
Finset.mem_sym_iff
|
Mathlib/Data/Finset/Sym.lean
|
theorem mem_sym_iff {m : Sym α n} : m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s
|
case succ.refine_1.intro.intro
α : Type u_1
s : Finset α
inst✝ : DecidableEq α
n✝ n : ℕ
ih : ∀ {m : Sym α n}, m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s
m : Sym α (n + 1)
a : α
ha : a ∈ s
he : ∃ a_1 ∈ s.sym n, a ::ₛ a_1 = m
b : α
hb : b ∈ m
⊢ b ∈ s
|
obtain ⟨m, he, rfl⟩ := he
|
case succ.refine_1.intro.intro.intro.intro
α : Type u_1
s : Finset α
inst✝ : DecidableEq α
n✝ n : ℕ
ih : ∀ {m : Sym α n}, m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s
a : α
ha : a ∈ s
b : α
m : Sym α n
he : m ∈ s.sym n
hb : b ∈ a ::ₛ m
⊢ b ∈ s
|
2df449e77a9ba950
|
Nat.Prime.emultiplicity_factorial_mul_succ
|
Mathlib/Data/Nat/Multiplicity.lean
|
theorem emultiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
emultiplicity p (p * (n + 1))! = emultiplicity p (p * n)! + emultiplicity p (n + 1) + 1
|
n p : ℕ
hp : Prime p
hp' : _root_.Prime p
h0 : 2 ≤ p
h1 : 1 ≤ p * n + 1
h2 : p * n + 1 ≤ p * (n + 1)
⊢ p * n + 1 ≤ p * (n + 1) + 1
|
omega
|
no goals
|
097a3d59a78795b9
|
SimpleGraph.Walk.toSubgraph_reverse
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
|
theorem toSubgraph_reverse (p : G.Walk u v) : p.reverse.toSubgraph = p.toSubgraph
|
case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : p✝.reverse.toSubgraph = p✝.toSubgraph
⊢ G.subgraphOfAdj h✝ ⊔ G.singletonSubgraph u✝ ⊔ p✝.toSubgraph = G.subgraphOfAdj h✝ ⊔ p✝.toSubgraph
|
congr
|
case cons.e_a
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : p✝.reverse.toSubgraph = p✝.toSubgraph
⊢ G.subgraphOfAdj h✝ ⊔ G.singletonSubgraph u✝ = G.subgraphOfAdj h✝
|
1cddc7d273048f17
|
taylor_mean_remainder_bound
|
Mathlib/Analysis/Calculus/Taylor.lean
|
theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b)
(hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b)
(hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) :
‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !
|
case intro
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : ℝ
hay : a ≤ y
hyx : y < x
⊢ ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C
|
rw [norm_smul, Real.norm_eq_abs]
|
case intro
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : ℝ
hay : a ≤ y
hyx : y < x
⊢ |(↑n !)⁻¹ * (x - y) ^ n| * ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C
|
b246fe1248089a18
|
Polynomial.Chebyshev.algebraMap_eval_U
|
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
theorem algebraMap_eval_U [Algebra R R'] (x : R) (n : ℤ) :
algebraMap R R' ((U R n).eval x) = (U R' n).eval (algebraMap R R' x)
|
R : Type u_1
R' : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing R'
inst✝ : Algebra R R'
x : R
n : ℤ
⊢ (algebraMap R R') (eval x (U R n)) = eval ((algebraMap R R') x) (U R' n)
|
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_U]
|
no goals
|
a95f0580a7da75cf
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.safe_insert_of_performRupCheck_insertRat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean
|
theorem safe_insert_of_performRupCheck_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (rupHints : Array Nat) :
(performRupCheck (insertRatUnits f (negate c)).1 rupHints).2.2.1 = true
→
Limplies (PosFin n) f (f.insert c)
|
case inl
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
rupHints : Array Nat
performRupCheck_success :
(Array.foldl (confirmRupHint (f.insertRatUnits c.negate).1.clauses)
((f.insertRatUnits c.negate).1.assignments, [], false, false) rupHints).2.2.fst =
true
p : PosFin n → Bool
pf : p ⊨ f
c' : DefaultClause n
c'_eq_c : c' = c
⊢ p ⊨ c
|
exact sat_of_confirmRupHint_of_insertRat_fold f hf c rupHints p pf performRupCheck_success
|
no goals
|
7b923cb96f92e5d9
|
CStarModule.norm_inner_le
|
Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
|
/-- The Cauchy-Schwarz inequality for Hilbert C⋆-modules. -/
lemma norm_inner_le {x y : E} : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖
|
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
this : ‖⟪x, y⟫_A‖ ^ 2 ≤ (‖x‖ * ‖y‖) ^ 2
⊢ ‖⟪x, y⟫_A‖ ≤ ‖x‖ * ‖y‖
|
refine (pow_le_pow_iff_left₀ (norm_nonneg ⟪x, y⟫_A) ?_ (by norm_num)).mp this
|
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
this : ‖⟪x, y⟫_A‖ ^ 2 ≤ (‖x‖ * ‖y‖) ^ 2
⊢ 0 ≤ ‖x‖ * ‖y‖
|
c2dabe140e3c60d0
|
ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib_aux
|
Mathlib/Probability/IdentDistrib.lean
|
theorem MemLp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : MemLp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
(hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ
|
case pos.intro.intro
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
this : {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} = {x | C ≤ ‖f i x‖}
⊢ eLpNorm (fun x => ‖{x | C ≤ ‖f i x‖}.indicator (f i) x‖) p μ =
eLpNorm (fun x => ‖{x | C ≤ ↑‖f j x‖₊}.indicator (f j) x‖) p μ
|
simp_rw [norm_indicator_eq_indicator_norm, coe_nnnorm]
|
case pos.intro.intro
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
this : {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} = {x | C ≤ ‖f i x‖}
⊢ eLpNorm (fun x => {x | C ≤ ‖f i x‖}.indicator (fun a => ‖f i a‖) x) p μ =
eLpNorm (fun x => {x | C ≤ ‖f j x‖}.indicator (fun a => ‖f j a‖) x) p μ
|
e0093e34b057b36a
|
PiTensorProduct.norm_eval_le_injectiveSeminorm
|
Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
|
theorem norm_eval_le_injectiveSeminorm (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
‖lift f.toMultilinearMap x‖ ≤ ‖f‖ * injectiveSeminorm x
|
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜]
↥(LinearMap.range (lift f.toMultilinearMap)) :=
(lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) :=
lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) :=
f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
⊢ ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * injectiveSeminorm x
|
have hle : Seminorm.comp (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G))
(toDualContinuousMultilinearMap G (𝕜 := 𝕜) (E := E)) ≤ injectiveSeminorm := by
simp only [injectiveSeminorm]
refine le_csSup dualSeminorms_bounded ?_
rw [Set.mem_setOf]
existsi G, inferInstance, inferInstance
rfl
|
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜]
↥(LinearMap.range (lift f.toMultilinearMap)) :=
(lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) :=
lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) :=
f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
hle :
(normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm
⊢ ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * injectiveSeminorm x
|
83d83ebebe9757c9
|
Matrix.blockDiagonal'_add
|
Mathlib/Data/Matrix/Block.lean
|
theorem blockDiagonal'_add [AddZeroClass α] (M N : ∀ i, Matrix (m' i) (n' i) α) :
blockDiagonal' (M + N) = blockDiagonal' M + blockDiagonal' N
|
case a
o : Type u_4
m' : o → Type u_7
n' : o → Type u_8
α : Type u_12
inst✝¹ : DecidableEq o
inst✝ : AddZeroClass α
M N : (i : o) → Matrix (m' i) (n' i) α
i✝ : (i : o) × m' i
j✝ : (i : o) × n' i
⊢ (if h : i✝.fst = j✝.fst then M i✝.fst i✝.snd (cast ⋯ j✝.snd) + N i✝.fst i✝.snd (cast ⋯ j✝.snd) else 0) =
(if h : i✝.fst = j✝.fst then M i✝.fst i✝.snd (cast ⋯ j✝.snd) else 0) +
if h : i✝.fst = j✝.fst then N i✝.fst i✝.snd (cast ⋯ j✝.snd) else 0
|
split_ifs <;> simp
|
no goals
|
d0a4d0c8e1c2828c
|
InnerProductSpace.Core.inner_mul_inner_self_le
|
Mathlib/Analysis/InnerProductSpace/Defs.lean
|
theorem inner_mul_inner_self_le (x y : F) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫
|
case h.e'_4.h.e'_5.h.e'_6
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
hzero : ¬⟪x, y⟫_𝕜 = 0
hzero' : ‖⟪x, y⟫_𝕜‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫_𝕜‖ * t * ‖⟪x, y⟫_𝕜‖ = 2 * ‖⟪x, y⟫_𝕜‖ ^ 2 * t
|
ring
|
no goals
|
3490ee6e9e583d18
|
Metric.diam_union'
|
Mathlib/Topology/MetricSpace/Bounded.lean
|
theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t
|
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
h : (s ∩ t).Nonempty
⊢ diam (s ∪ t) ≤ diam s + diam t
|
rcases h with ⟨x, ⟨xs, xt⟩⟩
|
case intro.intro
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
x : α
xs : x ∈ s
xt : x ∈ t
⊢ diam (s ∪ t) ≤ diam s + diam t
|
70dfeb046e69a232
|
SimpleGraph.Walk.exists_length_eq_zero_iff
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
theorem exists_length_eq_zero_iff {u v : V} : (∃ p : G.Walk u v, p.length = 0) ↔ u = v
|
case mp
V : Type u
G : SimpleGraph V
u v : V
⊢ (∃ p, p.length = 0) → u = v
|
rintro ⟨p, hp⟩
|
case mp.intro
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.length = 0
⊢ u = v
|
620215c56eb5f21e
|
IsCompact.elim_finite_subcover_image
|
Mathlib/Topology/Compactness/Compact.lean
|
theorem IsCompact.elim_finite_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsCompact s)
(hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i
|
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set X
b : Set ι
c : ι → Set X
hs : IsCompact s
hc₁ : ∀ i ∈ b, IsOpen (c i)
hc₂ : s ⊆ ⋃ i ∈ b, c i
⊢ ∃ b' ⊆ b, b'.Finite ∧ s ⊆ ⋃ i ∈ b', c i
|
simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂
|
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set X
b : Set ι
c : ι → Set X
hs : IsCompact s
hc₁ : ∀ (x : { a // a ∈ b }), IsOpen (c ↑x)
hc₂ : s ⊆ ⋃ x, c ↑x
⊢ ∃ b' ⊆ b, b'.Finite ∧ s ⊆ ⋃ i ∈ b', c i
|
6b726d862ddbfba3
|
List.IsRotated.trans
|
Mathlib/Data/List/Rotate.lean
|
theorem IsRotated.trans : ∀ {l l' l'' : List α}, l ~r l' → l' ~r l'' → l ~r l''
| _, _, _, ⟨n, rfl⟩, ⟨m, rfl⟩ => ⟨n + m, by rw [rotate_rotate]⟩
|
α : Type u
l✝ : List α
n m : ℕ
⊢ l✝.rotate (n + m) = (l✝.rotate n).rotate m
|
rw [rotate_rotate]
|
no goals
|
a8f0f74c1c3f1e73
|
Ordinal.lt_limit
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem lt_limit {o} (h : IsLimit o) {a} : a < o ↔ ∃ x < o, a < x
|
o : Ordinal.{u_4}
h : o.IsLimit
a : Ordinal.{u_4}
⊢ a < o ↔ ∃ x < o, a < x
|
simpa only [not_forall₂, not_le, bex_def] using not_congr (@limit_le _ h a)
|
no goals
|
6a5b23f82c8b384b
|
Rat.AbsoluteValue.exists_minimal_nat_zero_lt_and_lt_one
|
Mathlib/NumberTheory/Ostrowski.lean
|
/-- There exists a minimal positive integer with absolute value smaller than 1. -/
lemma exists_minimal_nat_zero_lt_and_lt_one :
∃ p : ℕ, (0 < f p ∧ f p < 1) ∧ ∀ m : ℕ, 0 < f m ∧ f m < 1 → p ≤ m
|
case intro.intro
f : AbsoluteValue ℚ ℝ
hf_nontriv : f.IsNontrivial
bdd : ∀ (n : ℕ), f ↑n ≤ 1
n : ℕ
hn1 : n ≠ 0
hn2 : f ↑n ≠ 1
P : Set ℕ := {m | 0 < f ↑m ∧ f ↑m < 1}
⊢ ∃ p, (0 < f ↑p ∧ f ↑p < 1) ∧ ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
|
have hP : P.Nonempty :=
⟨n, map_pos_of_ne_zero f (Nat.cast_ne_zero.mpr hn1), lt_of_le_of_ne (bdd n) hn2⟩
|
case intro.intro
f : AbsoluteValue ℚ ℝ
hf_nontriv : f.IsNontrivial
bdd : ∀ (n : ℕ), f ↑n ≤ 1
n : ℕ
hn1 : n ≠ 0
hn2 : f ↑n ≠ 1
P : Set ℕ := {m | 0 < f ↑m ∧ f ↑m < 1}
hP : P.Nonempty
⊢ ∃ p, (0 < f ↑p ∧ f ↑p < 1) ∧ ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
|
856d5d85add25372
|
AlgebraicTopology.DoldKan.Γ₀_obj_termwise_mapMono_comp_PInfty
|
Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean
|
theorem Γ₀_obj_termwise_mapMono_comp_PInfty (X : SimplicialObject C) {Δ Δ' : SimplexCategory}
(i : Δ ⟶ Δ') [Mono i] :
Γ₀.Obj.Termwise.mapMono (AlternatingFaceMapComplex.obj X) i ≫ PInfty.f Δ.len =
PInfty.f Δ'.len ≫ X.map i.op
|
case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ 0 = PInfty.f n' ≫ X.map i.op
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ⦋n'⦌ ≠ ⦋n⦌
|
swap
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ⦋n'⦌ ≠ ⦋n⦌
case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ 0 = PInfty.f n' ≫ X.map i.op
|
8c91ae5c330900ef
|
MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval
|
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
theorem continuous_testAgainstNN_eval (f : Ω →ᵇ ℝ≥0) :
Continuous fun μ : FiniteMeasure Ω ↦ μ.testAgainstNN f
|
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ Continuous fun φ => φ f
|
exact WeakBilin.eval_continuous _ _
|
no goals
|
82b00458343c302b
|
TsirelsonInequality.sqrt_two_inv_mul_self
|
Mathlib/Algebra/Star/CHSH.lean
|
theorem sqrt_two_inv_mul_self : (√2)⁻¹ * (√2)⁻¹ = (2⁻¹ : ℝ)
|
⊢ (√2)⁻¹ * (√2)⁻¹ = 2⁻¹
|
rw [← mul_inv]
|
⊢ (√2 * √2)⁻¹ = 2⁻¹
|
af12444dc2596475
|
Pell.Solution₁.eq_zero_of_d_neg
|
Mathlib/NumberTheory/Pell.lean
|
theorem eq_zero_of_d_neg (h₀ : d < 0) (a : Solution₁ d) : a.x = 0 ∨ a.y = 0
|
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
⊢ a.x ^ 2 - d * a.y ^ 2 ≠ 1
|
have h1 := sq_pos_of_ne_zero h.1
|
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
h1 : 0 < a.x ^ 2
⊢ a.x ^ 2 - d * a.y ^ 2 ≠ 1
|
50a8ed2852157cdb
|
MeasureTheory.condExp_ae_eq_restrict_zero
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
|
theorem condExp_ae_eq_restrict_zero (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict s] 0) :
μ[f|m] =ᵐ[μ.restrict s] 0
|
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᶠ[ae (μ.restrict s)] 0
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
⊢ SigmaFinite ((μ.restrict s).trim hm)
|
rw [← restrict_trim hm _ hs]
|
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᶠ[ae (μ.restrict s)] 0
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
⊢ SigmaFinite ((μ.trim hm).restrict s)
|
1f7750ee72b97131
|
Finsupp.toMultiset_map
|
Mathlib/Data/Finsupp/Multiset.lean
|
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g)
|
case refine_2
α : Type u_1
β : Type u_2
f✝ : α →₀ ℕ
g : α → β
a : α
n : ℕ
f : α →₀ ℕ
a✝¹ : a ∉ f.support
a✝ : n ≠ 0
ih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)
⊢ Multiset.map g (toMultiset (single a n + f)) = toMultiset (mapDomain g (single a n + f))
|
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
|
case refine_2
α : Type u_1
β : Type u_2
f✝ : α →₀ ℕ
g : α → β
a : α
n : ℕ
f : α →₀ ℕ
a✝¹ : a ∉ f.support
a✝ : n ≠ 0
ih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)
⊢ n • (Multiset.mapAddMonoidHom g) {a} + toMultiset (mapDomain g f) = n • {g a} + toMultiset (mapDomain g f)
|
b734ca1aed5316da
|
IsAlgebraic.algebraMap
|
Mathlib/RingTheory/Algebraic/Basic.lean
|
theorem IsAlgebraic.algebraMap {a : S} :
IsAlgebraic R a → IsAlgebraic R (algebraMap S A a) := fun ⟨f, hf₁, hf₂⟩ =>
⟨f, hf₁, by rw [aeval_algebraMap_apply, hf₂, map_zero]⟩
|
R : Type u
S : Type u_1
A : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
a : S
x✝ : IsAlgebraic R a
f : R[X]
hf₁ : f ≠ 0
hf₂ : (aeval a) f = 0
⊢ (aeval ((algebraMap S A) a)) f = 0
|
rw [aeval_algebraMap_apply, hf₂, map_zero]
|
no goals
|
5749bbfed23b4e12
|
Algebra.discr_powerBasis_eq_prod'
|
Mathlib/RingTheory/Discriminant.lean
|
theorem discr_powerBasis_eq_prod' [Algebra.IsSeparable K L] (e : Fin pb.dim ≃ (L →ₐ[K] E)) :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, -((e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen))
|
case e_f.h.e_f.h
K : Type u
L : Type v
E : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
e : Fin pb.dim ≃ (L →ₐ[K] E)
i j : Fin pb.dim
⊢ ((e j) pb.gen - (e i) pb.gen) ^ 2 = -(((e j) pb.gen - (e i) pb.gen) * ((e i) pb.gen - (e j) pb.gen))
|
ring
|
no goals
|
ebd88c8ab922c15a
|
Multiset.union_add_inter
|
Mathlib/Data/Multiset/UnionInter.lean
|
lemma union_add_inter (s t : Multiset α) : s ∪ t + s ∩ t = s + t
|
case a
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
⊢ t + s ≤ (s ∪ t + s) ∩ (s ∪ t + t)
|
refine le_inter (Multiset.add_le_add_right le_union_right) ?_
|
case a
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
⊢ t + s ≤ s ∪ t + t
|
0b8ef669b6694030
|
MeasureTheory.Measure.measure_sInf_le
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
theorem measure_sInf_le (h : μ ∈ m) : sInf m ≤ μ :=
have : sInf (toOuterMeasure '' m) ≤ μ.toOuterMeasure := sInf_le (mem_image_of_mem _ h)
le_iff.2 fun s hs => by rw [sInf_apply hs]; exact this s
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m : Set (Measure α)
h : μ ∈ m
this : sInf (toOuterMeasure '' m) ≤ μ.toOuterMeasure
s : Set α
hs : MeasurableSet s
⊢ (sInf (toOuterMeasure '' m)) s ≤ μ s
|
exact this s
|
no goals
|
2793aa93cc3ffec1
|
div_div_div_cancel_left'
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
lemma div_div_div_cancel_left' (a b : G₀) (hc : c ≠ 0) : c / a / (c / b) = b / a
|
G₀ : Type u_3
inst✝ : CommGroupWithZero G₀
c a b : G₀
hc : c ≠ 0
⊢ c / a / (c / b) = b / a
|
rw [div_div_div_eq, mul_comm, mul_div_mul_right _ _ hc]
|
no goals
|
cbc620c447f3f824
|
IsLowerSet.mem_interior_of_forall_lt
|
Mathlib/Analysis/Normed/Order/UpperLower.lean
|
theorem IsLowerSet.mem_interior_of_forall_lt (hs : IsLowerSet s) (hx : x ∈ closure s)
(h : ∀ i, y i < x i) : y ∈ interior s
|
case intro.intro.intro.intro.intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : dist x z < ε
⊢ y ∈ interior s
|
rw [dist_pi_lt_iff hε] at hxz
|
case intro.intro.intro.intro.intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
⊢ y ∈ interior s
|
cddc3cf8c9427d15
|
MeasureTheory.IsStoppingTime.measurableSet_inter_le
|
Mathlib/Probability/Process/Stopping.lean
|
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
[MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
(s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω})
|
case h
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
⊢ (ω ∈ s ∧ τ ω ≤ π ω) ∧ (τ ω ≤ i ∨ π ω ≤ i) ↔
((ω ∈ s ∧ τ ω ≤ i) ∧ (τ ω ≤ i ∨ π ω ≤ i)) ∧ (τ ω ≤ π ω ∨ i ≤ π ω) ∧ (τ ω ≤ i ∨ True)
|
by_cases hτi : τ ω ≤ i
|
case pos
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
⊢ (ω ∈ s ∧ τ ω ≤ π ω) ∧ (τ ω ≤ i ∨ π ω ≤ i) ↔
((ω ∈ s ∧ τ ω ≤ i) ∧ (τ ω ≤ i ∨ π ω ≤ i)) ∧ (τ ω ≤ π ω ∨ i ≤ π ω) ∧ (τ ω ≤ i ∨ True)
case neg
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : ¬τ ω ≤ i
⊢ (ω ∈ s ∧ τ ω ≤ π ω) ∧ (τ ω ≤ i ∨ π ω ≤ i) ↔
((ω ∈ s ∧ τ ω ≤ i) ∧ (τ ω ≤ i ∨ π ω ≤ i)) ∧ (τ ω ≤ π ω ∨ i ≤ π ω) ∧ (τ ω ≤ i ∨ True)
|
93587e3bd5de43f7
|
integrable_mulExpNegMulSq_comp
|
Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.lean
|
theorem integrable_mulExpNegMulSq_comp (f : C(E, ℝ)) (hε : 0 < ε) :
Integrable (fun x => mulExpNegMulSq ε (f x)) P
|
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
ε : ℝ
f : C(E, ℝ)
hε : 0 < ε
⊢ Integrable (fun x => ε.mulExpNegMulSq (f x)) P
|
apply integrable P ⟨⟨fun x => mulExpNegMulSq ε (f x), by fun_prop⟩, ⟨2 * (sqrt ε)⁻¹, _⟩⟩
|
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
ε : ℝ
f : C(E, ℝ)
hε : 0 < ε
⊢ ∀ (x y : E),
dist ({ toFun := fun x => ε.mulExpNegMulSq (f x), continuous_toFun := ⋯ }.toFun x)
({ toFun := fun x => ε.mulExpNegMulSq (f x), continuous_toFun := ⋯ }.toFun y) ≤
2 * (√ε)⁻¹
|
79676fc5c6c76a89
|
FirstOrder.Language.equiv_between_cg
|
Mathlib/ModelTheory/PartialEquiv.lean
|
theorem equiv_between_cg (M_cg : Structure.CG L M) (N_cg : Structure.CG L N)
(g : L.FGEquiv M N)
(ext_dom : L.IsExtensionPair M N)
(ext_cod : L.IsExtensionPair N M) :
∃ f : M ≃[L] N, g ≤ f.toEmbedding.toPartialEquiv
|
case mk.intro.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝¹ : Countable ↑X
x✝ : Encodable ↑X
⊢ ∃ f, ↑g ≤ f.toEmbedding.toPartialEquiv
|
have _ : Countable (↑Y : Type _) := by simpa only [countable_coe_iff]
|
case mk.intro.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝² : Countable ↑X
x✝¹ : Encodable ↑X
x✝ : Countable ↑Y
⊢ ∃ f, ↑g ≤ f.toEmbedding.toPartialEquiv
|
d1b972cd720b6e06
|
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