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
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|---|---|---|---|---|---|---|
Set.SMulAntidiagonal.finite_of_isPWO
|
Mathlib/Data/Set/SMulAntidiagonal.lean
|
theorem finite_of_isPWO (hs : s.IsPWO) (ht : t.IsPWO) (a) : (smulAntidiagonal s t a).Finite
|
G : Type u_1
P : Type u_2
s : Set G
t : Set P
inst✝³ : PartialOrder G
inst✝² : PartialOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
hs : s.IsPWO
ht : t.IsPWO
a : P
h : (s.smulAntidiagonal t a).Infinite
h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x1 x2 => x1 ≤ x2)
h2 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x1 x2 => x1 ≤ x2)
⊢ False
|
have isrfl : IsRefl (G × P) (Prod.fst ⁻¹'o fun x x_1 ↦ x ≤ x_1) := by
refine { refl := ?refl }
simp_all only [Order.Preimage, le_refl, Prod.forall, implies_true]
|
G : Type u_1
P : Type u_2
s : Set G
t : Set P
inst✝³ : PartialOrder G
inst✝² : PartialOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
hs : s.IsPWO
ht : t.IsPWO
a : P
h : (s.smulAntidiagonal t a).Infinite
h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x1 x2 => x1 ≤ x2)
h2 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x1 x2 => x1 ≤ x2)
isrfl : IsRefl (G × P) (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)
⊢ False
|
7f1573a75cfc8a75
|
ConvexOn.continuousOn_tfae
|
Mathlib/Analysis/Convex/Continuous.lean
|
lemma ConvexOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConvexOn ℝ C f) : TFAE [
LocallyLipschitzOn C f,
ContinuousOn f C,
∃ x₀ ∈ C, ContinuousAt f x₀,
∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f|]
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 => x1 ≤ x2) x r
x : E
hx : x ∈ C
⊢ ContinuousAt (fun δ => (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0
|
fun_prop (disch := norm_num)
|
no goals
|
d7a82928a7cc2bf5
|
SigmaCompactSpace_iff_exists_compact_covering
|
Mathlib/Topology/Compactness/SigmaCompact.lean
|
/-- A topological space is σ-compact iff there exists a countable collection of compact
subspaces that cover the entire space. -/
lemma SigmaCompactSpace_iff_exists_compact_covering :
SigmaCompactSpace X ↔ ∃ K : ℕ → Set X, (∀ n, IsCompact (K n)) ∧ ⋃ n, K n = univ
|
X : Type u_1
inst✝ : TopologicalSpace X
⊢ SigmaCompactSpace X ↔ ∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = univ
|
rw [← isSigmaCompact_univ_iff, IsSigmaCompact]
|
no goals
|
a990befcd7cbd3b8
|
Rat.AbsoluteValue.one_lt_of_not_bounded
|
Mathlib/NumberTheory/Ostrowski.lean
|
/-- If `f n > 1` for some `n` then `f n > 1` for all `n ≥ 2` -/
lemma one_lt_of_not_bounded (notbdd : ¬ ∀ n : ℕ, f n ≤ 1) {n₀ : ℕ} (hn₀ : 1 < n₀) : 1 < f n₀
|
f : AbsoluteValue ℚ ℝ
n₀ : ℕ
hn₀ : 1 < n₀
h : f ↑n₀ ≤ 1
n m : ℕ
hm : 1 ≤ m
L : List ℕ := n₀.digits m
⊢ ↑n₀ * (↑(Nat.log n₀ m) + 1) ≤ ↑n₀ * (logb ↑n₀ ↑m + 1)
|
gcongr
|
case h.bc
f : AbsoluteValue ℚ ℝ
n₀ : ℕ
hn₀ : 1 < n₀
h : f ↑n₀ ≤ 1
n m : ℕ
hm : 1 ≤ m
L : List ℕ := n₀.digits m
⊢ ↑(Nat.log n₀ m) ≤ logb ↑n₀ ↑m
|
2244dd101a1df19f
|
exists_isIntegralCurveAt_of_contMDiffAt
|
Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean
|
theorem exists_isIntegralCurveAt_of_contMDiffAt [CompleteSpace E]
(hv : ContMDiffAt I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)) x₀)
(hx : I.IsInteriorPoint x₀) :
∃ γ : ℝ → M, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀
|
case intro.intro.intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 1 M
v : (x : M) → TangentSpace I x
t₀ : ℝ
x₀ : M
inst✝ : CompleteSpace E
hx : I.IsInteriorPoint x₀
left✝ : ContinuousAt (fun x => { proj := x, snd := v x }) x₀
hv :
ContDiffWithinAt ℝ 1
(↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(range ↑I) (↑(extChartAt I x₀) x₀)
f : ℝ → E
hf1 : f t₀ = ↑(extChartAt I x₀) x₀
hf2 :
∀ᶠ (y : ℝ) in 𝓝 t₀,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y
a : ℝ
ha : a > 0
hf2' :
∀ y ∈ Metric.ball t₀ a,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y
hcont : ∀ A ∈ 𝓝 (↑(extChartAt I x₀) x₀), f ⁻¹' A ∈ 𝓝 t₀
hnhds : ∀ᶠ (x' : ℝ) in 𝓝 t₀, f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 x'
s : Set ℝ
hs : s ∈ 𝓝 t₀
haux :
∀ y ∈ s,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y ∧
f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 y
t : ℝ
ht : t ∈ s
xₜ : M := ↑(extChartAt I x₀).symm (f t)
⊢ HasMFDerivAt (modelWithCornersSelf ℝ ℝ) I (↑(extChartAt I x₀).symm ∘ f) t
(ContinuousLinearMap.smulRight 1 (v ((↑(extChartAt I x₀).symm ∘ f) t)))
|
have h : HasDerivAt f (x := t) <| fderivWithin ℝ (extChartAt I x₀ ∘ (extChartAt I xₜ).symm)
(range I) (extChartAt I xₜ xₜ) (v xₜ) := (haux t ht).1
|
case intro.intro.intro.intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 1 M
v : (x : M) → TangentSpace I x
t₀ : ℝ
x₀ : M
inst✝ : CompleteSpace E
hx : I.IsInteriorPoint x₀
left✝ : ContinuousAt (fun x => { proj := x, snd := v x }) x₀
hv :
ContDiffWithinAt ℝ 1
(↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(range ↑I) (↑(extChartAt I x₀) x₀)
f : ℝ → E
hf1 : f t₀ = ↑(extChartAt I x₀) x₀
hf2 :
∀ᶠ (y : ℝ) in 𝓝 t₀,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y
a : ℝ
ha : a > 0
hf2' :
∀ y ∈ Metric.ball t₀ a,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y
hcont : ∀ A ∈ 𝓝 (↑(extChartAt I x₀) x₀), f ⁻¹' A ∈ 𝓝 t₀
hnhds : ∀ᶠ (x' : ℝ) in 𝓝 t₀, f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 x'
s : Set ℝ
hs : s ∈ 𝓝 t₀
haux :
∀ y ∈ s,
HasDerivAt f
((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘
(fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm)
(f y)).2
y ∧
f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 y
t : ℝ
ht : t ∈ s
xₜ : M := ↑(extChartAt I x₀).symm (f t)
h :
HasDerivAt f
((fderivWithin ℝ (↑(extChartAt I x₀) ∘ ↑(extChartAt I xₜ).symm) (range ↑I) (↑(extChartAt I xₜ) xₜ)) (v xₜ)) t
⊢ HasMFDerivAt (modelWithCornersSelf ℝ ℝ) I (↑(extChartAt I x₀).symm ∘ f) t
(ContinuousLinearMap.smulRight 1 (v ((↑(extChartAt I x₀).symm ∘ f) t)))
|
b2c5f907bce00300
|
AddChar.exists_divisor_of_not_isPrimitive
|
Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean
|
/-- If `e` is not primitive, then `e.mulShift d = 1` for some proper divisor `d` of `N`. -/
lemma exists_divisor_of_not_isPrimitive (he : ¬e.IsPrimitive) :
∃ d : ℕ, d ∣ N ∧ d < N ∧ e.mulShift d = 1
|
case intro.intro.intro.intro.intro.intro.refine_2
N : ℕ
inst✝¹ : NeZero N
R : Type u_1
inst✝ : CommRing R
e : AddChar (ZMod N) R
d : ℕ
hd : d ∣ N
u : ZMod N
hu : IsUnit u
hb_ne : u * ↑d ≠ 0
hb : e.mulShift (u * ↑d) = 1
⊢ (e.mulShift ↑d).mulShift u = e.mulShift (↑d * u)
|
ext1 y
|
case intro.intro.intro.intro.intro.intro.refine_2.h
N : ℕ
inst✝¹ : NeZero N
R : Type u_1
inst✝ : CommRing R
e : AddChar (ZMod N) R
d : ℕ
hd : d ∣ N
u : ZMod N
hu : IsUnit u
hb_ne : u * ↑d ≠ 0
hb : e.mulShift (u * ↑d) = 1
y : ZMod N
⊢ ((e.mulShift ↑d).mulShift u) y = (e.mulShift (↑d * u)) y
|
88a547542522f0dd
|
MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
|
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
|
theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite μ] (f : α → ℝ≥0)
(fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0∞,
(∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε
|
case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
|
have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
|
case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
|
2b831a0739b26da1
|
Ideal.span_singleton_mul_le_iff
|
Mathlib/RingTheory/Ideal/Operations.lean
|
theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J
|
case mp
R : Type u
inst✝ : CommSemiring R
x : R
I J : Ideal R
h : ∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J
zI : R
hzI : zI ∈ I
⊢ x * zI ∈ J
|
exact h x (dvd_refl x) zI hzI
|
no goals
|
4fbba6b14ba98442
|
CauchySeq.mem_entourage
|
Mathlib/Topology/UniformSpace/Cauchy.lean
|
theorem CauchySeq.mem_entourage {β : Type*} [SemilatticeSup β] {u : β → α} (h : CauchySeq u)
{V : Set (α × α)} (hV : V ∈ 𝓤 α) : ∃ k₀, ∀ i j, k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V
|
α : Type u
uniformSpace : UniformSpace α
β : Type u_1
inst✝ : SemilatticeSup β
u : β → α
h : CauchySeq u
V : Set (α × α)
hV : V ∈ 𝓤 α
this✝ : Nonempty β
this : Tendsto (Prod.map u u) atTop (𝓤 α)
⊢ ∃ k₀, ∀ (i j : β), k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V
|
rw [← prod_atTop_atTop_eq] at this
|
α : Type u
uniformSpace : UniformSpace α
β : Type u_1
inst✝ : SemilatticeSup β
u : β → α
h : CauchySeq u
V : Set (α × α)
hV : V ∈ 𝓤 α
this✝ : Nonempty β
this : Tendsto (Prod.map u u) (atTop ×ˢ atTop) (𝓤 α)
⊢ ∃ k₀, ∀ (i j : β), k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V
|
b15586cef43f8804
|
MeromorphicOn.isClopen_setOf_order_eq_top
|
Mathlib/Analysis/Meromorphic/Order.lean
|
theorem isClopen_setOf_order_eq_top : IsClopen { u : U | (hf u.1 u.2).order = ⊤ }
|
case right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
⊢ IsOpen {u | ⋯.order = ⊤}
|
apply isOpen_iff_forall_mem_open.mpr
|
case right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
⊢ ∀ x ∈ {u | ⋯.order = ⊤}, ∃ t ⊆ {u | ⋯.order = ⊤}, IsOpen t ∧ x ∈ t
|
7349d8844a909f0d
|
CategoryTheory.Presheaf.isSheaf_iff_multiequalizer
|
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
theorem isSheaf_iff_multiequalizer [∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)] :
IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P)
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
⊢ (∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P))) ↔
∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P)
|
refine forall₂_congr fun X S => ⟨?_, ?_⟩
|
case refine_1
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
⊢ Nonempty (IsLimit (S.multifork P)) → IsIso (S.toMultiequalizer P)
case refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
⊢ IsIso (S.toMultiequalizer P) → Nonempty (IsLimit (S.multifork P))
|
32f315e61f5eaf0d
|
Ordinal.cof_eq_cof_toType
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem cof_eq_cof_toType (o : Ordinal) : o.cof = @Order.cof o.toType (· ≤ ·)
|
o : Ordinal.{u_1}
⊢ o.cof = Order.cof fun x1 x2 => x1 ≤ x2
|
conv_lhs => rw [← type_toType o, cof_type_lt]
|
no goals
|
8163d29eb5b73544
|
Polynomial.splits_of_degree_le_one
|
Mathlib/Algebra/Polynomial/Splits.lean
|
theorem splits_of_degree_le_one {f : K[X]} (hf : degree f ≤ 1) : Splits i f :=
if hif : degree (f.map i) ≤ 0 then splits_of_map_eq_C i (degree_le_zero_iff.mp hif)
else by
push_neg at hif
rw [← Order.succ_le_iff, ← WithBot.coe_zero, WithBot.orderSucc_coe, Nat.succ_eq_succ] at hif
exact splits_of_map_degree_eq_one i ((degree_map_le.trans hf).antisymm hif)
|
K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
f : K[X]
hf : f.degree ≤ 1
hif : ↑(Nat.succ 0) ≤ (map i f).degree
⊢ Splits i f
|
exact splits_of_map_degree_eq_one i ((degree_map_le.trans hf).antisymm hif)
|
no goals
|
f333d26a19f7b9ea
|
associated_norm_prod_smith
|
Mathlib/LinearAlgebra/FreeModule/Norm.lean
|
theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i)
|
R : Type u_1
S : Type u_2
ι : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra R S
inst✝ : Fintype ι
b : Basis ι R S
f : S
hf : f ≠ 0
hI : ¬span {f} = ⊥
b' : Basis ι R S := ringBasis b (span {f}) hI
e : S ≃ₗ[R] S :=
b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl ι) ≪≫ₗ LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)
i : ι
⊢ (↑(Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) ⋯))) (b' i) =
(↑(Algebra.lmul R S).toRingHom f ∘ₗ
↑(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ∘ₗ
↑(b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl ι)))
(b' i)
|
simp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,
Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul,
zero_smul, Finset.sum_ite_eq', mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]
|
R : Type u_1
S : Type u_2
ι : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra R S
inst✝ : Fintype ι
b : Basis ι R S
f : S
hf : f ≠ 0
hI : ¬span {f} = ⊥
b' : Basis ι R S := ringBasis b (span {f}) hI
e : S ≃ₗ[R] S :=
b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl ι) ≪≫ₗ LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)
i : ι
⊢ smithCoeffs b (span {f}) ⋯ i • b' i =
(↑(Algebra.lmul R S).toRingHom f)
((LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ((selfBasis b (span {f}) hI) ((Equiv.refl ι) i)))
|
e517c670dc65b094
|
PresheafOfModules.Sheafify.smul_add
|
Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
|
protected lemma smul_add : smul α φ r (m + m') = smul α φ r m + smul α φ r m'
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
⊢ S ∈ J (Opposite.unop X)
|
refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_
|
case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
⊢ Presheaf.imageSieve α r ∈ J (Opposite.unop X)
case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
⊢ Presheaf.imageSieve φ m ∈ J (Opposite.unop X)
case refine_3
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
⊢ Presheaf.imageSieve φ m' ∈ J (Opposite.unop X)
|
e45a9bac6c91f436
|
isCofinal_empty_iff
|
Mathlib/Order/Cofinal.lean
|
theorem isCofinal_empty_iff : IsCofinal (∅ : Set α) ↔ IsEmpty α
|
α : Type u_1
inst✝ : LE α
⊢ IsCofinal ∅ ↔ IsEmpty α
|
refine ⟨fun h ↦ ⟨fun a ↦ ?_⟩, fun h ↦ .of_isEmpty _⟩
|
α : Type u_1
inst✝ : LE α
h : IsCofinal ∅
a : α
⊢ False
|
a2c35bd64ff9843c
|
CategoryTheory.Presieve.ofArrows_pullback
|
Mathlib/CategoryTheory/Sites/Sieves.lean
|
theorem ofArrows_pullback [HasPullbacks C] {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X) :
(ofArrows (fun i => pullback (g i) f) fun _ => pullback.snd _ _) =
pullbackArrows f (ofArrows Z g)
|
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y : C
f : Y ⟶ X
inst✝ : HasPullbacks C
ι : Type u_1
Z : ι → C
g : (i : ι) → Z i ⟶ X
⊢ (ofArrows (fun i => pullback (g i) f) fun x => pullback.snd (g x) f) = pullbackArrows f (ofArrows Z g)
|
funext T
|
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y : C
f : Y ⟶ X
inst✝ : HasPullbacks C
ι : Type u_1
Z : ι → C
g : (i : ι) → Z i ⟶ X
T : C
⊢ (ofArrows (fun i => pullback (g i) f) fun x => pullback.snd (g x) f) = pullbackArrows f (ofArrows Z g)
|
c1ab51a7f74dc585
|
MeasureTheory.eLpNorm_const_lt_top_iff
|
Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
|
theorem eLpNorm_const_lt_top_iff {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
eLpNorm (fun _ : α => c) p μ < ∞ ↔ c = 0 ∨ μ Set.univ < ∞
|
case neg.inl
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
c : F
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hp : 0 < p.toReal
hμ : ¬μ = 0
hc : ¬c = 0
hμ_top : μ Set.univ = ⊤
⊢ ‖c‖ₑ * μ Set.univ ^ (1 / p.toReal) < ⊤ ↔ c = 0 ∨ μ Set.univ < ⊤
|
simp [hc, hμ_top, hp]
|
no goals
|
3659c685f6a51069
|
Algebra.TensorProduct.map_range
|
Mathlib/RingTheory/TensorProduct/Basic.lean
|
theorem map_range (f : A →ₐ[R] C) (g : B →ₐ[R] D) :
(map f g).range = (includeLeft.comp f).range ⊔ (includeRight.comp g).range
|
case a.intro.intro.intro.intro
R : Type uR
A : Type uA
B : Type uB
C : Type uC
D : Type uD
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : Semiring D
inst✝ : Algebra R D
f : A →ₐ[R] C
g : B →ₐ[R] D
a : A
b : B
⊢ f a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] g b ∈ ↑((includeLeft.comp f).range ⊔ (includeRight.comp g).range)
|
exact mul_mem_sup (AlgHom.mem_range_self _ a) (AlgHom.mem_range_self _ b)
|
no goals
|
778b8efd2725c05c
|
EuclideanGeometry.dist_mul_of_eq_angle_of_dist_mul
|
Mathlib/Geometry/Euclidean/Triangle.lean
|
theorem dist_mul_of_eq_angle_of_dist_mul (a b c a' b' c' : P) (r : ℝ) (h : ∠ a' b' c' = ∠ a b c)
(hab : dist a' b' = r * dist a b) (hcb : dist c' b' = r * dist c b) :
dist a' c' = r * dist a c
|
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
h' : dist a' c' ^ 2 = (r * dist a c) ^ 2
⊢ dist a' c' = r * dist a c
|
by_cases hab₁ : a = b
|
case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
h' : dist a' c' ^ 2 = (r * dist a c) ^ 2
hab₁ : a = b
⊢ dist a' c' = r * dist a c
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
h' : dist a' c' ^ 2 = (r * dist a c) ^ 2
hab₁ : ¬a = b
⊢ dist a' c' = r * dist a c
|
21e37fd5b54d5f31
|
LinearMap.polyCharpolyAux_map_eq_charpoly
|
Mathlib/Algebra/Module/LinearMap/Polynomial.lean
|
@[simp]
lemma polyCharpolyAux_map_eq_charpoly [Module.Finite R M] [Module.Free R M]
(x : L) :
(polyCharpolyAux φ b bₘ).map (MvPolynomial.eval (b.repr x)) = (φ x).charpoly
|
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
inst✝¹ : Module.Finite R M
inst✝ : Module.Free R M
x : L
⊢ Polynomial.map (MvPolynomial.eval ⇑(b.repr x)) (φ.polyCharpolyAux b bₘ) = charpoly (φ x)
|
nontriviality R
|
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Basis ιM R M
inst✝¹ : Module.Finite R M
inst✝ : Module.Free R M
x : L
a✝ : Nontrivial R
⊢ Polynomial.map (MvPolynomial.eval ⇑(b.repr x)) (φ.polyCharpolyAux b bₘ) = charpoly (φ x)
|
d79cc8664c1c3ad2
|
measurableSet_bddAbove_range
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
|
lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
MeasurableSet {b | BddAbove (range (fun i ↦ f i b))}
|
case inr.intro
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
this : {b | BddAbove (range fun i => f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
⊢ MeasurableSet (⋃ i, {x | ∀ (i_1 : ι), f i_1 x ≤ u i})
|
exact MeasurableSet.iUnion (fun n ↦ B (u n))
|
no goals
|
2ea9408e1624118d
|
Vector.attachWith_map_coe
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean
|
theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : Vector α n) (H : ∀ a ∈ l, p a) :
((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f
|
case mk
α : Type u_1
β : Type u_2
n : Nat
p : α → Prop
f : α → β
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
H : ∀ (a : α), a ∈ { toArray := toArray✝, size_toArray := size_toArray✝ } → p a
⊢ map (fun i => f i.val) ({ toArray := toArray✝, size_toArray := size_toArray✝ }.attachWith p H) =
map f { toArray := toArray✝, size_toArray := size_toArray✝ }
|
simp
|
no goals
|
bbc9845a1e972924
|
maximal_orthonormal_iff_orthogonalComplement_eq_bot
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 𝕜 ((↑) : v → E)) :
(∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ (span 𝕜 v)ᗮ = ⊥
|
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
⊢ u ⊆ v
|
intro x hxu
|
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
⊢ x ∈ v
|
ce0a185108c33220
|
exists_pos_lt_subset_ball
|
Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean
|
theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r'
|
case inr.intro.intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
hne : s.Nonempty
this : IsCompact s
y : α
hys : y ∈ s
hy : s ⊆ closedBall x (dist y x)
⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'
|
have hyr : dist y x < r := h hys
|
case inr.intro.intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
hne : s.Nonempty
this : IsCompact s
y : α
hys : y ∈ s
hy : s ⊆ closedBall x (dist y x)
hyr : dist y x < r
⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'
|
b60dc94a9e85c422
|
IsExtreme.inter
|
Mathlib/Analysis/Convex/Extreme.lean
|
theorem IsExtreme.inter (hAB : IsExtreme 𝕜 A B) (hAC : IsExtreme 𝕜 A C) :
IsExtreme 𝕜 A (B ∩ C)
|
case right.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A B C : Set E
hAB : IsExtreme 𝕜 A B
hAC : IsExtreme 𝕜 A C
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxB : x ∈ B
hxC : x ∈ C
hx : x ∈ openSegment 𝕜 x₁ x₂
hx₁B : x₁ ∈ B
hx₂B : x₂ ∈ B
⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C
|
obtain ⟨hx₁C, hx₂C⟩ := hAC.2 hx₁A hx₂A hxC hx
|
case right.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A B C : Set E
hAB : IsExtreme 𝕜 A B
hAC : IsExtreme 𝕜 A C
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxB : x ∈ B
hxC : x ∈ C
hx : x ∈ openSegment 𝕜 x₁ x₂
hx₁B : x₁ ∈ B
hx₂B : x₂ ∈ B
hx₁C : x₁ ∈ C
hx₂C : x₂ ∈ C
⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C
|
c3a399dad622271d
|
CategoryTheory.Sheaf.isSeparating
|
Mathlib/CategoryTheory/Generator/Sheaf.lean
|
lemma isSeparating {ι : Type w} {S : ι → A} (hS : IsSeparating (Set.range S)) :
IsSeparating (Set.range (fun (⟨X, i⟩ : C × ι) ↦ freeYoneda J X (S i)))
|
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasCoproducts A
inst✝ : HasWeakSheafify J A
ι : Type w
S : ι → A
hS : IsSeparating (Set.range S)
F G : Sheaf J A
f g : F ⟶ G
hfg :
∀
G_1 ∈
Set.range fun x =>
match x with
| (X, i) => freeYoneda J X (S i),
∀ (h : G_1 ⟶ F), h ≫ f = h ≫ g
⊢ ∀
G_1 ∈
Set.range fun x =>
match x with
| (X, i) => Presheaf.freeYoneda X (S i),
∀ (h : G_1 ⟶ (sheafToPresheaf J A).obj F), h ≫ (sheafToPresheaf J A).map f = h ≫ (sheafToPresheaf J A).map g
|
rintro _ ⟨⟨X, i⟩, rfl⟩ a
|
case intro.mk
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasCoproducts A
inst✝ : HasWeakSheafify J A
ι : Type w
S : ι → A
hS : IsSeparating (Set.range S)
F G : Sheaf J A
f g : F ⟶ G
hfg :
∀
G_1 ∈
Set.range fun x =>
match x with
| (X, i) => freeYoneda J X (S i),
∀ (h : G_1 ⟶ F), h ≫ f = h ≫ g
X : C
i : ι
a :
(fun x =>
match x with
| (X, i) => Presheaf.freeYoneda X (S i))
(X, i) ⟶
(sheafToPresheaf J A).obj F
⊢ a ≫ (sheafToPresheaf J A).map f = a ≫ (sheafToPresheaf J A).map g
|
57771cac286cd7c3
|
Nat.succ_dvd_centralBinom
|
Mathlib/Data/Nat/Choose/Central.lean
|
theorem succ_dvd_centralBinom (n : ℕ) : n + 1 ∣ n.centralBinom
|
n : ℕ
h_s : (n + 1).Coprime (2 * n + 1)
⊢ n + 1 ∣ n.centralBinom
|
apply h_s.dvd_of_dvd_mul_left
|
n : ℕ
h_s : (n + 1).Coprime (2 * n + 1)
⊢ n + 1 ∣ (2 * n + 1) * n.centralBinom
|
fb1b90bbcc5f1bf9
|
Finset.prod_dite_eq
|
Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
theorem prod_dite_eq [DecidableEq α] (s : Finset α) (a : α) (b : ∀ x : α, a = x → β) :
∏ x ∈ s, (if h : a = x then b x h else 1) = ite (a ∈ s) (b a rfl) 1
|
case neg
α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
a : α
b : (x : α) → a = x → β
h : a ∉ s
⊢ (∏ x ∈ s, if h : a = x then b x h else 1) = 1
|
rw [Finset.prod_eq_one]
|
case neg
α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
a : α
b : (x : α) → a = x → β
h : a ∉ s
⊢ ∀ x ∈ s, (if h : a = x then b x h else 1) = 1
|
7166f35cfa045e98
|
Fin.predAbove_last_of_ne_last
|
Mathlib/Data/Fin/Basic.lean
|
@[simp] lemma predAbove_last_of_ne_last {i : Fin (n + 2)} (hi : i ≠ last (n + 1)) :
predAbove (last n) i = castPred i hi
|
n : ℕ
i : Fin (n + 2)
hi✝ : i ≠ last (n + 1)
hi : ∃ j, j.castSucc = i
⊢ (last n).predAbove i = i.castPred hi✝
|
rcases hi with ⟨y, rfl⟩
|
case intro
n : ℕ
y : Fin (n + 1)
hi : y.castSucc ≠ last (n + 1)
⊢ (last n).predAbove y.castSucc = y.castSucc.castPred hi
|
ff637eb3a44a0cf5
|
Acc.cutExpand
|
Mathlib/Logic/Hydra.lean
|
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a}
|
case intro.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ : α
t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
a' : α
⊢ a' ∈ {a}.erase a + t → Acc (CutExpand r) {a'}
|
rw [erase_singleton, zero_add]
|
case intro.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ : α
t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
a' : α
⊢ a' ∈ t → Acc (CutExpand r) {a'}
|
cfe6e29617386e0f
|
Polynomial.X_sub_C_mul_removeFactor
|
Mathlib/FieldTheory/SplittingField/Construction.lean
|
theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f
|
K : Type v
inst✝ : Field K
f : K[X]
hf : f.natDegree ≠ 0
g : K[X]
hg : f = f.factor * g
⊢ (map (AdjoinRoot.of f.factor) f).IsRoot (AdjoinRoot.root f.factor)
|
rw [IsRoot.def, eval_map, hg, eval₂_mul, ← hg, AdjoinRoot.eval₂_root, zero_mul]
|
no goals
|
63fc58033fe1fb68
|
List.dropSlice_eq_dropSliceTR
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Basic.lean
|
theorem dropSlice_eq_dropSliceTR : @dropSlice = @dropSliceTR
|
α : Type u_1
n✝ : Nat
l : List α
m✝ m : Nat
acc : Array α
x : α
xs : List α
n : Nat
h : l = acc.toList ++ x :: xs
⊢ (acc.push x).toList ++ dropSlice n (m + 1) xs = acc.toList ++ x :: dropSlice n (m + 1) xs
case a
α : Type u_1
n✝ : Nat
l : List α
m✝ m : Nat
acc : Array α
x : α
xs : List α
n : Nat
h : l = acc.toList ++ x :: xs
⊢ l = (acc.push x).toList ++ xs
|
{simp}
|
case a
α : Type u_1
n✝ : Nat
l : List α
m✝ m : Nat
acc : Array α
x : α
xs : List α
n : Nat
h : l = acc.toList ++ x :: xs
⊢ l = (acc.push x).toList ++ xs
|
f858ea09a79c5819
|
Nat.descFactorial_mul_descFactorial
|
Mathlib/Data/Nat/Factorial/Basic.lean
|
theorem descFactorial_mul_descFactorial {k m n : ℕ} (hkm : k ≤ m) :
(n - k).descFactorial (m - k) * n.descFactorial k = n.descFactorial m
|
k m n : ℕ
hkm : k ≤ m
hmn : ¬m ≤ n
hkn : k ≤ n
⊢ n - k < m - k
|
omega
|
no goals
|
e320856096df3a95
|
List.count_filter
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Count.lean
|
theorem count_filter {l : List α} (h : p a) : count a (filter p l) = count a l
|
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
p : α → Bool
a : α
l : List α
h : p a = true
⊢ countP (fun a_1 => a_1 == a && p a_1) l = countP (fun x => x == a) l
|
congr
|
case e_p
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
p : α → Bool
a : α
l : List α
h : p a = true
⊢ (fun a_1 => a_1 == a && p a_1) = fun x => x == a
|
a19453790c8f5ea1
|
HallMarriageTheorem.hall_cond_of_compl
|
Mathlib/Combinatorics/Hall/Finite.lean
|
theorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(hus : #s = #(s.biUnion t)) (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(s' : Finset (sᶜ : Set ι)) : #s' ≤ #(s'.biUnion fun x' => t x' \ s.biUnion t)
|
case inr.intro.intro
α : Type v
inst✝ : DecidableEq α
ι : Type u
t✝ : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t✝)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t✝)
s' : Finset ↑(↑s)ᶜ
this✝ : DecidableEq ι
disj : Disjoint s (image (fun z => ↑z) s')
this : #s' = #(s ∪ image (fun z => ↑z) s') - #s
t : α
x' : ↑(↑s)ᶜ
hx' : x' ∈ s'
rat : t ∈ t✝ ↑x'
hs : ∀ (x : ι), ¬(x ∈ s ∧ t ∈ t✝ x)
⊢ ∃ a ∈ s', t ∈ t✝ ↑a ∧ ∀ (x : ι), ¬(x ∈ s ∧ t ∈ t✝ x)
|
use x', hx', rat, hs
|
no goals
|
633ed149d2eeb7d3
|
isIntegral_quotientMap_iff
|
Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
|
theorem isIntegral_quotientMap_iff {I : Ideal S} :
(Ideal.quotientMap I f le_rfl).IsIntegral ↔
((Ideal.Quotient.mk I).comp f : R →+* S ⧸ I).IsIntegral
|
R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
I : Ideal S
g : R →+* R ⧸ Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)
this : (Ideal.quotientMap I f ⋯).comp g = (Ideal.Quotient.mk I).comp f
⊢ (Ideal.quotientMap I f ⋯).IsIntegral ↔ ((Ideal.Quotient.mk I).comp f).IsIntegral
|
refine ⟨fun h => ?_, fun h => RingHom.IsIntegral.tower_top g _ (this ▸ h)⟩
|
R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
I : Ideal S
g : R →+* R ⧸ Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)
this : (Ideal.quotientMap I f ⋯).comp g = (Ideal.Quotient.mk I).comp f
h : (Ideal.quotientMap I f ⋯).IsIntegral
⊢ ((Ideal.Quotient.mk I).comp f).IsIntegral
|
29162be1ec047379
|
ContinuousLinearMap.integral_comp_comm
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem integral_comp_comm [CompleteSpace E] (L : E →L[𝕜] F) {φ : X → E} (φ_int : Integrable φ μ) :
∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ)
|
case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] F
φ : X → E
φ_int : Integrable φ μ
⊢ ∀ (c : E) ⦃s : Set X⦄,
MeasurableSet s → μ s < ⊤ → (fun φ => ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)) (s.indicator fun x => c)
|
intro e s s_meas _
|
case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] F
φ : X → E
φ_int : Integrable φ μ
e : E
s : Set X
s_meas : MeasurableSet s
a✝ : μ s < ⊤
⊢ ∫ (x : X), L (s.indicator (fun x => e) x) ∂μ = L (∫ (x : X), s.indicator (fun x => e) x ∂μ)
|
a849e0b0325a5e6f
|
Affine.Simplex.Regular.equilateral
|
Mathlib/Analysis/Normed/Affine/Simplex.lean
|
lemma Regular.equilateral {s : Simplex R P n} (hr : s.Regular) : s.Equilateral
|
case pos
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : i = 1
⊢ dist (s.points i) (s.points j) = dist (s.points 0) (s.points 1)
|
rw [hi, dist_comm]
|
case pos
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : i = 1
⊢ dist (s.points j) (s.points 1) = dist (s.points 0) (s.points 1)
|
1804910b0c1433ab
|
Polynomial.IsSeparableContraction.degree_eq
|
Mathlib/RingTheory/Polynomial/SeparableDegree.lean
|
theorem IsSeparableContraction.degree_eq [hF : ExpChar F q] (g : F[X])
(hg : IsSeparableContraction q f g) : g.natDegree = hf.degree
|
case prime
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hg : IsSeparableContraction q f g
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
⊢ g.natDegree = hf.degree
|
rcases hg with ⟨hg, m, hm⟩
|
case prime.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
⊢ g.natDegree = hf.degree
|
e8a084ffc5e2adb4
|
MeasureTheory.FiniteMeasure.prod_zero
|
Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
|
@[simp] lemma prod_zero : μ.prod (0 : FiniteMeasure β) = 0
|
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
⊢ μ.prod 0 = 0
|
rw [← mass_zero_iff, mass_prod, zero_mass, mul_zero]
|
no goals
|
dfab91dc2fd62368
|
intervalIntegral.intervalIntegrable_rpow'
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b
|
case inl
a b r : ℝ
h : -1 < r
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun x => x ^ r) volume 0 c
|
exact this c hc
|
no goals
|
03dcc4f722838526
|
ConformalAt.comp
|
Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean
|
theorem comp {f : X → Y} {g : Y → Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
ConformalAt (g ∘ f) x
|
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : NormedAddCommGroup Z
inst✝² : NormedSpace ℝ X
inst✝¹ : NormedSpace ℝ Y
inst✝ : NormedSpace ℝ Z
f : X → Y
g : Y → Z
x : X
hg : ConformalAt g (f x)
hf : ConformalAt f x
⊢ ConformalAt (g ∘ f) x
|
rcases hf with ⟨f', hf₁, cf⟩
|
case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : NormedAddCommGroup Z
inst✝² : NormedSpace ℝ X
inst✝¹ : NormedSpace ℝ Y
inst✝ : NormedSpace ℝ Z
f : X → Y
g : Y → Z
x : X
hg : ConformalAt g (f x)
f' : X →L[ℝ] Y
hf₁ : HasFDerivAt f f' x
cf : IsConformalMap f'
⊢ ConformalAt (g ∘ f) x
|
92f1e4627d87ad06
|
summable_jacobiTheta₂_term_fderiv_iff
|
Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
|
lemma summable_jacobiTheta₂_term_fderiv_iff (z τ : ℂ) :
Summable (jacobiTheta₂_term_fderiv · z τ) ↔ 0 < im τ
|
case h
z τ : ℂ
h : Summable fun x => jacobiTheta₂_term_fderiv x z τ
this✝ : Summable fun x => ‖jacobiTheta₂_term_fderiv x z τ‖
this : ∀ᶠ (n : ℤ) in cofinite, n ≠ 0
n : ℤ
hn : n ≠ 0
⊢ 1 ≤ π * ↑|n| ^ 2
|
refine one_le_pi_div_two.trans (mul_le_mul_of_nonneg_left ?_ pi_pos.le)
|
case h
z τ : ℂ
h : Summable fun x => jacobiTheta₂_term_fderiv x z τ
this✝ : Summable fun x => ‖jacobiTheta₂_term_fderiv x z τ‖
this : ∀ᶠ (n : ℤ) in cofinite, n ≠ 0
n : ℤ
hn : n ≠ 0
⊢ 2⁻¹ ≤ ↑|n| ^ 2
|
d5a065cb2c1034b1
|
List.perm_ext_iff_of_nodup
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Perm.lean
|
theorem perm_ext_iff_of_nodup {l₁ l₂ : List α} (d₁ : Nodup l₁) (d₂ : Nodup l₂) :
l₁ ~ l₂ ↔ ∀ a, a ∈ l₁ ↔ a ∈ l₂
|
α : Type u_1
l₁ l₂ : List α
d₁ : l₁.Nodup
d₂ : l₂.Nodup
H : ∀ (a : α), a ∈ l₁ ↔ a ∈ l₂
⊢ l₁ ~ l₂
|
exact (subperm_of_subset d₁ fun a => (H a).1).antisymm <| subperm_of_subset d₂ fun a => (H a).2
|
no goals
|
60da269297319e2e
|
Nat.zero_ascFactorial
|
Mathlib/Data/Nat/Factorial/Basic.lean
|
theorem zero_ascFactorial : ∀ (k : ℕ), (0 : ℕ).ascFactorial k.succ = 0
| 0 => by
rw [ascFactorial_succ, ascFactorial_zero, Nat.zero_add, Nat.zero_mul]
| (k+1) => by
rw [ascFactorial_succ, zero_ascFactorial k, Nat.mul_zero]
|
⊢ ascFactorial 0 (succ 0) = 0
|
rw [ascFactorial_succ, ascFactorial_zero, Nat.zero_add, Nat.zero_mul]
|
no goals
|
7b66734ced605115
|
StructureGroupoid.LocalInvariantProp.liftPropAt_chart_symm
|
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
theorem liftPropAt_chart_symm [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
(hQ : ∀ y, Q id univ y) : LiftPropAt Q (chartAt (H := H) x).symm ((chartAt H x) x) :=
hG.liftPropAt_symm_of_mem_maximalAtlas hQ (chart_mem_maximalAtlas G x) (by simp)
|
H : Type u_1
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
G : StructureGroupoid H
x : M
Q : (H → H) → Set H → H → Prop
inst✝ : HasGroupoid M G
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
⊢ ↑(chartAt H x) x ∈ (chartAt H x).target
|
simp
|
no goals
|
ee361fe68864f72c
|
Matrix.conjTranspose_eq_one
|
Mathlib/Data/Matrix/ConjTranspose.lean
|
theorem conjTranspose_eq_one [DecidableEq n] [Semiring α] [StarRing α] {M : Matrix n n α} :
Mᴴ = 1 ↔ M = 1 :=
(Function.Involutive.eq_iff conjTranspose_conjTranspose).trans <|
by rw [conjTranspose_one]
|
n : Type u_3
α : Type v
inst✝² : DecidableEq n
inst✝¹ : Semiring α
inst✝ : StarRing α
M : Matrix n n α
⊢ M = 1ᴴ ↔ M = 1
|
rw [conjTranspose_one]
|
no goals
|
8d2c796611605995
|
DFinsupp.lex_lt_of_lt_of_preorder
|
Mathlib/Data/DFinsupp/Lex.lean
|
theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
|
case intro.intro
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt✝ : x < y
hle : ⇑x ≤ ⇑y
j : ι
hlt : x j < y j
⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
|
classical
have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn
obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩
refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩
exact of_not_not fun h ↦ hl ⟨k, mem_neLocus.2 (ne_of_not_le h).symm⟩ ((hle k).lt_of_not_le h) hk
|
no goals
|
b2be9adf1bae9d88
|
CategoryTheory.isIso_of_yoneda_map_bijective
|
Mathlib/CategoryTheory/Yoneda.lean
|
lemma isIso_of_yoneda_map_bijective {X Y : C} (f : X ⟶ Y)
(hf : ∀ (T : C), Function.Bijective (fun (x : T ⟶ X) => x ≫ f)) :
IsIso f
|
case intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
f : X ⟶ Y
hf : ∀ (T : C), Function.Bijective fun x => x ≫ f
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
⊢ IsIso f
|
exact ⟨g, (hf _).1 (by aesop_cat), hg⟩
|
no goals
|
3517b85a3fbd1c2e
|
HasFTaylorSeriesUpToOn.compContinuousLinearMap
|
Mathlib/Analysis/Calculus/ContDiff/Basic.lean
|
theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap
(hf : HasFTaylorSeriesUpToOn n f p s) (g : G →L[𝕜] E) :
HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g)
(g ⁻¹' s)
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →L[𝕜] E
⊢ HasFTaylorSeriesUpToOn n (f ∘ ⇑g) (fun x k => (p (g x) k).compContinuousLinearMap fun x => g) (⇑g ⁻¹' s)
|
let A : ∀ m : ℕ, (E[×m]→L[𝕜] F) → G[×m]→L[𝕜] F := fun m h => h.compContinuousLinearMap fun _ => g
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →L[𝕜] E
A : (m : ℕ) → ContinuousMultilinearMap 𝕜 (fun i => E) F → ContinuousMultilinearMap 𝕜 (fun i => G) F :=
fun m h => h.compContinuousLinearMap fun x => g
⊢ HasFTaylorSeriesUpToOn n (f ∘ ⇑g) (fun x k => (p (g x) k).compContinuousLinearMap fun x => g) (⇑g ⁻¹' s)
|
827bfe05b18c9500
|
ManyOneDegree.le_trans
|
Mathlib/Computability/Reduce.lean
|
theorem le_trans {d₁ d₂ d₃ : ManyOneDegree} : d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃
|
d₁ d₂ d₃ : ManyOneDegree
⊢ d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃
|
induction d₁ using ManyOneDegree.ind_on
|
case h
d₂ d₃ : ManyOneDegree
p✝ : Set ℕ
⊢ of p✝ ≤ d₂ → d₂ ≤ d₃ → of p✝ ≤ d₃
|
68814163b8da16d6
|
Polynomial.dickson_one_one_zmod_p
|
Mathlib/RingTheory/Polynomial/Dickson.lean
|
theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p) p = X ^ p
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
K : Type
w✝¹ : Field K
w✝ : CharP K p
H : Set.univ.Infinite
h : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite
x : K
x✝ : x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}
φ : K[X] := X ^ 2 - C x * X + 1
⊢ {y | x = y + y⁻¹ ∨ y = 0}.Finite
|
have hφ : φ ≠ 0 := by
intro H
have : φ.eval 0 = 0 := by rw [H, eval_zero]
simpa [φ, eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul,
mul_zero, sq, zero_add, one_ne_zero]
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
K : Type
w✝¹ : Field K
w✝ : CharP K p
H : Set.univ.Infinite
h : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite
x : K
x✝ : x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}
φ : K[X] := X ^ 2 - C x * X + 1
hφ : φ ≠ 0
⊢ {y | x = y + y⁻¹ ∨ y = 0}.Finite
|
84ee1673e653698a
|
BoxIntegral.Box.mk'_eq_coe
|
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper
|
case mk
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper
|
split_ifs with h
|
case pos
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ∀ (i : ι), l i < u i
⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper
case neg
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ¬∀ (i : ι), l i < u i
⊢ ⊥ = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper
|
5a138799670a8b97
|
LieModule.Weight.ext
|
Mathlib/Algebra/Lie/Weights/Basic.lean
|
@[ext] lemma ext {χ₁ χ₂ : Weight R L M} (h : ∀ x, χ₁ x = χ₂ x) : χ₁ = χ₂
|
case mk.mk
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
f₁ : L → R
genWeightSpace_ne_bot'✝¹ : genWeightSpace M f₁ ≠ ⊥
f₂ : L → R
genWeightSpace_ne_bot'✝ : genWeightSpace M f₂ ≠ ⊥
h :
∀ (x : L),
{ toFun := f₁, genWeightSpace_ne_bot' := genWeightSpace_ne_bot'✝¹ } x =
{ toFun := f₂, genWeightSpace_ne_bot' := genWeightSpace_ne_bot'✝ } x
⊢ { toFun := f₁, genWeightSpace_ne_bot' := genWeightSpace_ne_bot'✝¹ } =
{ toFun := f₂, genWeightSpace_ne_bot' := genWeightSpace_ne_bot'✝ }
|
aesop
|
no goals
|
b878d4f6634b8d02
|
EReal.toENNReal_of_nonpos
|
Mathlib/Data/Real/EReal.lean
|
@[simp]
lemma toENNReal_of_nonpos {x : EReal} (hx : x ≤ 0) : x.toENNReal = 0
|
x : EReal
hx : x ≤ 0
⊢ x.toENNReal = 0
|
rw [toENNReal, if_neg (fun h ↦ ?_)]
|
x : EReal
hx : x ≤ 0
⊢ ENNReal.ofReal x.toReal = 0
x : EReal
hx : x ≤ 0
h : x = ⊤
⊢ False
|
766efdc9ce84b5ce
|
Behrend.le_sqrt_log
|
Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
|
theorem le_sqrt_log (hN : 4096 ≤ N) : log (2 / (1 - 2 / exp 1)) * (69 / 50) ≤ √(log ↑N)
|
N : ℕ
hN : 4096 ≤ N
this : ↑12 * log 2 ≤ log ↑N
⊢ 2 ≤ 2.7182818283
|
norm_num1
|
no goals
|
5c4f54751ae43d03
|
IsAntichain.finite_of_wellQuasiOrdered
|
Mathlib/Order/WellQuasiOrder.lean
|
theorem IsAntichain.finite_of_wellQuasiOrdered {s : Set α} (hs : IsAntichain r s)
(hr : WellQuasiOrdered r) : s.Finite
|
case intro.intro.intro
α : Type u_1
r : α → α → Prop
s : Set α
hs : IsAntichain r s
hr : WellQuasiOrdered r
hi : s.Infinite
m n : ℕ
hmn : m < n
h : r ↑((Set.Infinite.natEmbedding s hi) m) ↑((Set.Infinite.natEmbedding s hi) n)
⊢ False
|
exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|
hs.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)
|
no goals
|
0a1ebf254e4abc46
|
IsPrimitiveRoot.neZero'
|
Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean
|
theorem neZero' {n : ℕ} [NeZero n] (hζ : IsPrimitiveRoot ζ n) : NeZero ((n : ℕ) : R)
|
case pos.intro
R : Type u_4
ζ : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
p : ℕ := ringChar R
hfin : FiniteMultiplicity p n
m : ℕ
hm : n = p ^ multiplicity p n * m ∧ ¬p ∣ m
hp : p ∣ n
k : ℕ
hk : multiplicity p n = k.succ
this✝ : NeZero p
hpri : Fact (Nat.Prime p)
this : (frobenius R p) (ζ ^ (p ^ k * m)) = (frobenius R p) 1
⊢ NeZero ↑n
|
exfalso
|
case pos.intro
R : Type u_4
ζ : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
p : ℕ := ringChar R
hfin : FiniteMultiplicity p n
m : ℕ
hm : n = p ^ multiplicity p n * m ∧ ¬p ∣ m
hp : p ∣ n
k : ℕ
hk : multiplicity p n = k.succ
this✝ : NeZero p
hpri : Fact (Nat.Prime p)
this : (frobenius R p) (ζ ^ (p ^ k * m)) = (frobenius R p) 1
⊢ False
|
9b1d4db2d0e199c4
|
Filter.map_comap
|
Mathlib/Order/Filter/Map.lean
|
theorem map_comap (f : Filter β) (m : α → β) : (f.comap m).map m = f ⊓ 𝓟 (range m)
|
α : Type u_1
β : Type u_2
f : Filter β
m : α → β
⊢ map m (comap m f) = f ⊓ 𝓟 (range m)
|
refine le_antisymm (le_inf map_comap_le <| le_principal_iff.2 range_mem_map) ?_
|
α : Type u_1
β : Type u_2
f : Filter β
m : α → β
⊢ f ⊓ 𝓟 (range m) ≤ map m (comap m f)
|
6d44131d149f58b8
|
Algebra.FormallyEtale.iff_exists_algEquiv_prod
|
Mathlib/RingTheory/Etale/Field.lean
|
theorem iff_exists_algEquiv_prod [EssFiniteType K A] :
FormallyEtale K A ↔
∃ (I : Type u) (_ : Finite I) (Ai : I → Type u) (_ : ∀ i, Field (Ai i))
(_ : ∀ i, Algebra K (Ai i)) (_ : A ≃ₐ[K] Π i, Ai i),
∀ i, Algebra.IsSeparable K (Ai i)
|
case mpr
K A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)) → FormallyEtale K A
|
intro ⟨I, _, Ai, _, _, e, _⟩
|
case mpr
K A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
I : Type u
w✝² : Finite I
Ai : I → Type u
w✝¹ : (i : I) → Field (Ai i)
w✝ : (i : I) → Algebra K (Ai i)
e : A ≃ₐ[K] (i : I) → Ai i
h✝ : ∀ (i : I), Algebra.IsSeparable K (Ai i)
⊢ FormallyEtale K A
|
bed71e21534c428f
|
Std.DHashMap.Raw.Const.getKey_insertMany_list_of_contains_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem getKey_insertMany_list_of_contains_eq_false [EquivBEq α] [LawfulHashable α] (h : m.WF)
{l : List (α × β)} {k : α}
(contains_eq_false : (l.map Prod.fst).contains k = false)
{h'} :
(insertMany m l).getKey k h' =
m.getKey k (mem_of_mem_insertMany_list h h' contains_eq_false)
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List (α × β)
k : α
contains_eq_false : (List.map Prod.fst l).contains k = false
h' : k ∈ insertMany m l
⊢ (insertMany m l).getKey k h' = m.getKey k ⋯
|
simp_to_raw using Raw₀.Const.getKey_insertMany_list_of_contains_eq_false
|
no goals
|
a4903e874350d005
|
MeasureTheory.Integrable.bdd_mul
|
Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
|
theorem Integrable.bdd_mul {F : Type*} [NormedDivisionRing F] {f g : α → F} (hint : Integrable g μ)
(hm : AEStronglyMeasurable f μ) (hfbdd : ∃ C, ∀ x, ‖f x‖ ≤ C) :
Integrable (fun x => f x * g x) μ
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
F : Type u_6
inst✝ : NormedDivisionRing F
f g : α → F
hint : Integrable g μ
hm : AEStronglyMeasurable f μ
hα : Nonempty α
C : ℝ
hC : ∀ (x : α), ‖f x‖ ≤ C
hCnonneg : 0 ≤ C
⊢ (fun x => ‖f x * g x‖₊) ≤ fun x => ⟨C, hCnonneg⟩ * ‖g x‖₊
|
intro x
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
F : Type u_6
inst✝ : NormedDivisionRing F
f g : α → F
hint : Integrable g μ
hm : AEStronglyMeasurable f μ
hα : Nonempty α
C : ℝ
hC : ∀ (x : α), ‖f x‖ ≤ C
hCnonneg : 0 ≤ C
x : α
⊢ (fun x => ‖f x * g x‖₊) x ≤ (fun x => ⟨C, hCnonneg⟩ * ‖g x‖₊) x
|
1c8516ef04187086
|
Subgroup.pi_le_iff
|
Mathlib/Algebra/Group/Subgroup/Finite.lean
|
theorem pi_le_iff [DecidableEq η] [Finite η] {H : ∀ i, Subgroup (f i)} {J : Subgroup (∀ i, f i)} :
pi univ H ≤ J ↔ ∀ i : η, map (MonoidHom.mulSingle f i) (H i) ≤ J
|
case mp.intro.intro
η : Type u_3
f : η → Type u_4
inst✝² : (i : η) → Group (f i)
inst✝¹ : DecidableEq η
inst✝ : Finite η
H : (i : η) → Subgroup (f i)
J : Subgroup ((i : η) → f i)
h : pi univ H ≤ J
i : η
x : f i
hx : x ∈ ↑(H i)
⊢ (MonoidHom.mulSingle f i) x ∈ J
|
apply h
|
case mp.intro.intro.a
η : Type u_3
f : η → Type u_4
inst✝² : (i : η) → Group (f i)
inst✝¹ : DecidableEq η
inst✝ : Finite η
H : (i : η) → Subgroup (f i)
J : Subgroup ((i : η) → f i)
h : pi univ H ≤ J
i : η
x : f i
hx : x ∈ ↑(H i)
⊢ (MonoidHom.mulSingle f i) x ∈ pi univ H
|
13d3a4bf3766cc30
|
Batteries.RBNode.lowerBound?_eq_find?
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem lowerBound?_eq_find? {t : RBNode α} {cut} (lb) (H : t.find? cut = some x) :
t.lowerBound? cut lb = some x
|
α : Type u_1
x : α
t : RBNode α
cut : α → Ordering
lb : Option α
H : find? cut t = some x
⊢ lowerBound? cut t lb = some x
|
rw [← reverse_reverse t] at H ⊢
|
α : Type u_1
x : α
t : RBNode α
cut : α → Ordering
lb : Option α
H : find? cut t.reverse.reverse = some x
⊢ lowerBound? cut t.reverse.reverse lb = some x
|
6d75f266b35cca6f
|
Complex.norm_sub_mem_Icc_angle
|
Mathlib/Analysis/Complex/Angle.lean
|
/-- Chord-length is a multiple of arc-length up to constants. -/
lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y)
|
case intro.intro.refine_2
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2 ≤ θ ^ 2
|
calc
_ = 2 * (1 - θ.cos) := by linear_combination θ.cos_sq_add_sin_sq
_ ≤ 2 * (1 - (1 - θ ^ 2 / 2)) := by gcongr; exact Real.one_sub_sq_div_two_le_cos
_ = _ := by ring
|
no goals
|
5f1a2d709d27832f
|
LinearMap.det_restrictScalars
|
Mathlib/RingTheory/Norm/Transitivity.lean
|
theorem LinearMap.det_restrictScalars [AddCommGroup A] [Module R A] [Module S A]
[IsScalarTower R S A] [Module.Free S A] {f : A →ₗ[S] A} :
(f.restrictScalars R).det = Algebra.norm R f.det
|
case inr.inl.inl
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Fintype ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
case inr.inl.inr
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Infinite ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
|
swap
|
case inr.inl.inr
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Infinite ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
case inr.inl.inl
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Fintype ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
|
e61a35e4b25ff2ca
|
Order.height_coe_withBot
|
Mathlib/Order/KrullDimension.lean
|
@[simp] lemma height_coe_withBot (x : α) : height (x : WithBot α) = height x + 1
|
α✝ : Type u_1
inst✝¹ : Preorder α✝
α : Type u_1
inst✝ : Preorder α
x : α
p : LTSeries (WithBot α)
hlast : RelSeries.last p = ↑x
hlenpos : p.length ≠ 0
p' : LTSeries α :=
{ length := p.length - 1,
toFun := fun x =>
match x with
| ⟨i, hi⟩ => (p.toFun ⟨i + 1, ⋯⟩).unbot ⋯,
step := ⋯ }
hlast' : RelSeries.last p' = x
⊢ ↑p.length ≤ height x + 1
|
suffices p'.length ≤ height p'.last by
simpa [p', hlast'] using this
|
α✝ : Type u_1
inst✝¹ : Preorder α✝
α : Type u_1
inst✝ : Preorder α
x : α
p : LTSeries (WithBot α)
hlast : RelSeries.last p = ↑x
hlenpos : p.length ≠ 0
p' : LTSeries α :=
{ length := p.length - 1,
toFun := fun x =>
match x with
| ⟨i, hi⟩ => (p.toFun ⟨i + 1, ⋯⟩).unbot ⋯,
step := ⋯ }
hlast' : RelSeries.last p' = x
⊢ ↑p'.length ≤ height (RelSeries.last p')
|
16287c0e56f7b584
|
Int.bmod_zero
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem bmod_zero : Int.bmod 0 m = 0
|
m✝ m : Nat
h : ¬0 ≤ ↑(m / 2)
⊢ False
m✝ m : Nat
h : ↑m / 2 + 2 / 2 ≤ 0
⊢ 2 ≠ 0
m✝ m : Nat
h : (↑m + (1 + 1)) / 2 ≤ 0
⊢ 2 ∣ 1 + 1
|
exact h (ofNat_nonneg _)
|
m✝ m : Nat
h : ↑m / 2 + 2 / 2 ≤ 0
⊢ 2 ≠ 0
m✝ m : Nat
h : (↑m + (1 + 1)) / 2 ≤ 0
⊢ 2 ∣ 1 + 1
|
aafdd0ab2162fca6
|
Nat.zero_pow
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean
|
theorem zero_pow {n : Nat} (H : 0 < n) : 0 ^ n = 0
|
n✝ n : Nat
H : 0 < n + 1
⊢ 0 ^ (n + 1) = 0
|
rw [Nat.pow_succ, Nat.mul_zero]
|
no goals
|
7b218e9a4a4d43d8
|
Behrend.sphere_zero_right
|
Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
|
theorem sphere_zero_right (n k : ℕ) : sphere (n + 1) 0 k = ∅
|
n k : ℕ
⊢ sphere (n + 1) 0 k = ∅
|
simp [sphere]
|
no goals
|
f70f7de7732573a0
|
CategoryTheory.Subobject.finset_inf_arrow_factors
|
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
theorem finset_inf_arrow_factors {I : Type*} {B : C} (s : Finset I) (P : I → Subobject B) (i : I)
(m : i ∈ s) : (P i).Factors (s.inf P).arrow
|
case insert
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
I : Type u_1
B : C
P : I → Subobject B
a✝¹ : I
s✝ : Finset I
a✝ : a✝¹ ∉ s✝
ih : ∀ i ∈ s✝, (P i).Factors (s✝.inf P).arrow
i✝ : I
m : i✝ = a✝¹ ∨ i✝ ∈ s✝
⊢ (P i✝).Factors (P a✝¹ ⊓ s✝.inf P).arrow
|
rcases m with (rfl | m)
|
case insert.inl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
I : Type u_1
B : C
P : I → Subobject B
s✝ : Finset I
ih : ∀ i ∈ s✝, (P i).Factors (s✝.inf P).arrow
i✝ : I
a✝ : i✝ ∉ s✝
⊢ (P i✝).Factors (P i✝ ⊓ s✝.inf P).arrow
case insert.inr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
I : Type u_1
B : C
P : I → Subobject B
a✝¹ : I
s✝ : Finset I
a✝ : a✝¹ ∉ s✝
ih : ∀ i ∈ s✝, (P i).Factors (s✝.inf P).arrow
i✝ : I
m : i✝ ∈ s✝
⊢ (P i✝).Factors (P a✝¹ ⊓ s✝.inf P).arrow
|
2e69cd1837536d26
|
Complex.uniformContinuous_ringHom_eq_id_or_conj
|
Mathlib/Topology/Instances/Complex.lean
|
theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype
|
case refine_2.inl.h.h.intro
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ofRealHom.fieldRange
j : ↥K.topologicalClosure ≃+* ↥ofRealHom.fieldRange := RingEquiv.subfieldCongr h
ψ₁ : ℝ →+* ℂ := extψ.comp (j.symm.toRingHom.comp ofRealHom.rangeRestrict)
hψ₁ : Continuous ⇑ψ₁
x : ↥K
r : ℝ
hr : ofRealHom.rangeRestrict r = j (ι x)
⊢ (↑↑ψ).toFun x = K.subtype x
|
have :=
RingHom.congr_fun (ringHom_eq_ofReal_of_continuous hψ₁) r
|
case refine_2.inl.h.h.intro
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝¹ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this✝ : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ofRealHom.fieldRange
j : ↥K.topologicalClosure ≃+* ↥ofRealHom.fieldRange := RingEquiv.subfieldCongr h
ψ₁ : ℝ →+* ℂ := extψ.comp (j.symm.toRingHom.comp ofRealHom.rangeRestrict)
hψ₁ : Continuous ⇑ψ₁
x : ↥K
r : ℝ
hr : ofRealHom.rangeRestrict r = j (ι x)
this : ψ₁ r = ofRealHom r
⊢ (↑↑ψ).toFun x = K.subtype x
|
f78bcdf9d891573b
|
exists_lt_mul_right_of_nonneg
|
Mathlib/Algebra/Order/Field/Basic.lean
|
private lemma exists_lt_mul_right_of_nonneg {a b c : α} (ha : 0 ≤ a) (hc : 0 ≤ c) (h : c < a * b) :
∃ b' ∈ Set.Ico 0 b, c < a * b'
|
α : Type u_2
inst✝ : LinearOrderedField α
a b c : α
ha : 0 ≤ a
hc : 0 ≤ c
hb : 0 < b
h : c < b * a
⊢ ∃ b' ∈ Set.Ico 0 b, c < b' * a
|
exact exists_lt_mul_left_of_nonneg hb.le hc h
|
no goals
|
1c69017d4ad8f569
|
CommGroup.exists_apply_ne_one_aux
|
Mathlib/GroupTheory/FiniteAbelian/Duality.lean
|
private
lemma exists_apply_ne_one_aux
(H : ∀ n : ℕ, n ∣ Monoid.exponent G → ∀ a : ZMod n, a ≠ 0 →
∃ φ : Multiplicative (ZMod n) →* M, φ (.ofAdd a) ≠ 1)
{a : G} (ha : a ≠ 1) :
∃ φ : G →* M, φ a ≠ 1
|
G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
ha : ∀ (i : ι), e a i = 1
⊢ a = 1
|
exact (MulEquiv.map_eq_one_iff e).mp <| funext ha
|
no goals
|
ca48e086b93104fd
|
PythagoreanTriple.isPrimitiveClassified_of_coprime_of_pos
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
theorem isPrimitiveClassified_of_coprime_of_pos (hc : Int.gcd x y = 1) (hzpos : 0 < z) :
h.IsPrimitiveClassified
|
case inr.intro.intro
x y z : ℤ
h : PythagoreanTriple x y z
hc : y.gcd x = 1
hzpos : 0 < z
h2 : x % 2 = 1 ∧ y % 2 = 0
m n : ℤ
H :
(y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧
m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)
⊢ h.IsPrimitiveClassified
|
use m, n
|
case h
x y z : ℤ
h : PythagoreanTriple x y z
hc : y.gcd x = 1
hzpos : 0 < z
h2 : x % 2 = 1 ∧ y % 2 = 0
m n : ℤ
H :
(y = m ^ 2 - n ^ 2 ∧ x = 2 * m * n ∨ y = 2 * m * n ∧ x = m ^ 2 - n ^ 2) ∧
m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)
⊢ (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧
m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)
|
d8eb1778fac552d9
|
List.lex_eq_false_iff_exists
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean
|
theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α → Bool)
(lt_irrefl : ∀ x y, x == y → lt x y = false)
(lt_asymm : ∀ x y, lt x y = true → lt y x = false)
(lt_antisymm : ∀ x y, lt x y = false → lt y x = false → x == y) :
lex l₁ l₂ lt = false ↔
(l₂.isEqv (l₁.take l₂.length) (· == ·)) ∨
(∃ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length),
(∀ j, (hj : j < i) →
l₁[j]'(Nat.lt_trans hj h₁) == l₂[j]'(Nat.lt_trans hj h₂)) ∧ lt l₂[i] l₁[i])
|
case cons.cons.mpr.inr.intro.intro.intro.intro.succ.refine_1
α : Type u_1
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
lt : α → α → Bool
lt_irrefl : ∀ (x y : α), (x == y) = true → lt x y = false
lt_asymm : ∀ (x y : α), lt x y = true → lt y x = false
lt_antisymm : ∀ (x y : α), lt x y = false → lt y x = false → (x == y) = true
a : α
l₁ : List α
ih :
∀ {l₂ : List α},
l₁.lex l₂ lt = false ↔
(l₂.isEqv (take l₂.length l₁) fun x1 x2 => x1 == x2) = true ∨
∃ i h₁ h₂, (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₂[i] l₁[i] = true
b : α
l₂ : List α
i : Nat
h₁ : i + 1 < l₁.length + 1
h₂ : i + 1 < l₂.length + 1
w₁ : ∀ (j : Nat) (hj : j < i + 1), ((a :: l₁)[j] == (b :: l₂)[j]) = true
w₂ : lt (b :: l₂)[i + 1] (a :: l₁)[i + 1] = true
x✝ : (a == b) = true
⊢ ∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true
|
intro j hj
|
case cons.cons.mpr.inr.intro.intro.intro.intro.succ.refine_1
α : Type u_1
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
lt : α → α → Bool
lt_irrefl : ∀ (x y : α), (x == y) = true → lt x y = false
lt_asymm : ∀ (x y : α), lt x y = true → lt y x = false
lt_antisymm : ∀ (x y : α), lt x y = false → lt y x = false → (x == y) = true
a : α
l₁ : List α
ih :
∀ {l₂ : List α},
l₁.lex l₂ lt = false ↔
(l₂.isEqv (take l₂.length l₁) fun x1 x2 => x1 == x2) = true ∨
∃ i h₁ h₂, (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₂[i] l₁[i] = true
b : α
l₂ : List α
i : Nat
h₁ : i + 1 < l₁.length + 1
h₂ : i + 1 < l₂.length + 1
w₁ : ∀ (j : Nat) (hj : j < i + 1), ((a :: l₁)[j] == (b :: l₂)[j]) = true
w₂ : lt (b :: l₂)[i + 1] (a :: l₁)[i + 1] = true
x✝ : (a == b) = true
j : Nat
hj : j < i
⊢ (l₁[j] == l₂[j]) = true
|
0c9240a9b47ee0a9
|
Polynomial.jacobson_bot_of_integral_localization
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
theorem jacobson_bot_of_integral_localization
{R : Type*} [CommRing R] [IsDomain R] [IsJacobsonRing R]
(Rₘ Sₘ : Type*) [CommRing Rₘ] [CommRing Sₘ] (φ : R →+* S) (hφ : Function.Injective ↑φ) (x : R)
(hx : x ≠ 0) [Algebra R Rₘ] [IsLocalization.Away x Rₘ] [Algebra S Sₘ]
[IsLocalization ((Submonoid.powers x).map φ : Submonoid S) Sₘ]
(hφ' :
RingHom.IsIntegral (IsLocalization.map Sₘ φ (Submonoid.powers x).le_comap_map : Rₘ →+* Sₘ)) :
(⊥ : Ideal S).jacobson = (⊥ : Ideal S)
|
S : Type u_2
inst✝¹⁰ : CommRing S
inst✝⁹ : IsDomain S
R : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : IsJacobsonRing R
Rₘ : Type u_6
Sₘ : Type u_7
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
φ : R →+* S
hφ : Function.Injective ⇑φ
x : R
hx : x ≠ 0
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization.Away x Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ
hφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral
hM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S
this✝⁴ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
I : Ideal Sₘ
hI : I.IsMaximal
this✝³ : (comap (algebraMap S Sₘ) I).IsPrime
this✝² : (comap φ' I).IsPrime
this✝¹ : ⊥.IsPrime
hcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ
f : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=
quotientMap (comap (algebraMap S Sₘ) I) φ ⋯
g : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯
this✝ : (comap φ' I).IsMaximal
this : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal
⊢ (comap (algebraMap S Sₘ) I).IsMaximal
|
have : ((I.comap (algebraMap S Sₘ)).comap φ).IsMaximal := by
rwa [comap_comap, hcomm, ← comap_comap] at this
|
S : Type u_2
inst✝¹⁰ : CommRing S
inst✝⁹ : IsDomain S
R : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : IsJacobsonRing R
Rₘ : Type u_6
Sₘ : Type u_7
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
φ : R →+* S
hφ : Function.Injective ⇑φ
x : R
hx : x ≠ 0
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization.Away x Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ
hφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral
hM : Submonoid.map φ (Submonoid.powers x) ≤ nonZeroDivisors S
this✝⁵ : IsDomain Sₘ := IsLocalization.isDomain_of_le_nonZeroDivisors S hM
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
I : Ideal Sₘ
hI : I.IsMaximal
this✝⁴ : (comap (algebraMap S Sₘ) I).IsPrime
this✝³ : (comap φ' I).IsPrime
this✝² : ⊥.IsPrime
hcomm : φ'.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp φ
f : R ⧸ comap φ (comap (algebraMap S Sₘ) I) →+* S ⧸ comap (algebraMap S Sₘ) I :=
quotientMap (comap (algebraMap S Sₘ) I) φ ⋯
g : S ⧸ comap (algebraMap S Sₘ) I →+* Sₘ ⧸ I := quotientMap I (algebraMap S Sₘ) ⋯
this✝¹ : (comap φ' I).IsMaximal
this✝ : (comap (algebraMap R Rₘ) (comap φ' I)).IsMaximal
this : (comap φ (comap (algebraMap S Sₘ) I)).IsMaximal
⊢ (comap (algebraMap S Sₘ) I).IsMaximal
|
b75924b89ba7dfc0
|
ContMDiffWithinAt.mdifferentiableWithinAt
|
Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
|
theorem ContMDiffWithinAt.mdifferentiableWithinAt (hf : ContMDiffWithinAt I I' n f s x)
(hn : 1 ≤ n) : MDifferentiableWithinAt I I' f s x
|
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
n : WithTop ℕ∞
hf : ContMDiffWithinAt I I' n f s x
hn : 1 ≤ n
h : MDifferentiableWithinAt I I' f (s ∩ f ⁻¹' (extChartAt I' (f x)).source) x
⊢ (extChartAt I' (f x)).source ∈ 𝓝 (f x)
|
exact extChartAt_source_mem_nhds (f x)
|
no goals
|
d6329dd89b798ab7
|
Ordinal.principal_mul_iff_mul_left_eq
|
Mathlib/SetTheory/Ordinal/Principal.lean
|
theorem principal_mul_iff_mul_left_eq : Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o
|
case refine_1.inl
o : Ordinal.{u}
h : Principal (fun x1 x2 => x1 * x2) o
a : Ordinal.{u}
ha₀ : 0 < a
hao : a < o
ho : o ≤ 2
⊢ a * o = o
|
convert one_mul o
|
case h.e'_2.h.e'_5
o : Ordinal.{u}
h : Principal (fun x1 x2 => x1 * x2) o
a : Ordinal.{u}
ha₀ : 0 < a
hao : a < o
ho : o ≤ 2
⊢ a = 1
|
5ae376ccba37fa39
|
DenseRange.comp
|
Mathlib/Topology/Basic.lean
|
theorem DenseRange.comp {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f)
(cg : Continuous g) : DenseRange (g ∘ f)
|
Y : Type u_2
Z : Type u_3
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
α : Type u_4
g : Y → Z
f : α → Y
hg : DenseRange g
hf : DenseRange f
cg : Continuous g
⊢ DenseRange (g ∘ f)
|
rw [DenseRange, range_comp]
|
Y : Type u_2
Z : Type u_3
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
α : Type u_4
g : Y → Z
f : α → Y
hg : DenseRange g
hf : DenseRange f
cg : Continuous g
⊢ Dense (g '' range f)
|
c91accec9eddfc92
|
PowerSeries.order_monomial_of_ne_zero
|
Mathlib/RingTheory/PowerSeries/Order.lean
|
theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monomial R n a) = n
|
R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
h : a ≠ 0
⊢ ((monomial R n) a).order = ↑n
|
classical
rw [order_monomial, if_neg h]
|
no goals
|
6e33ccdbc1551010
|
Array.getElem_zipWith
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem getElem_zipWith (as : Array α) (bs : Array β) (f : α → β → γ) (i : Nat)
(hi : i < (zipWith f as bs).size) :
(zipWith f as bs)[i] = f (as[i]'(by simp at hi; omega)) (bs[i]'(by simp at hi; omega))
|
α : Type ?u.533946
β : Type ?u.533949
γ : Type ?u.533961
as : Array α
bs : Array β
f : α → β → γ
i : Nat
hi : i < min as.size bs.size
⊢ i < bs.size
|
omega
|
no goals
|
8ad9913a0188b980
|
MeasureTheory.lintegral_rpow_enorm_eq_rpow_eLpNorm'
|
Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
|
theorem lintegral_rpow_enorm_eq_rpow_eLpNorm' {f : α → ε} (hq0_lt : 0 < q) :
∫⁻ a, ‖f a‖ₑ ^ q ∂μ = eLpNorm' f q μ ^ q
|
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hq0_lt : 0 < q
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ = eLpNorm' f q μ ^ q
|
rw [eLpNorm'_eq_lintegral_enorm, ← ENNReal.rpow_mul, one_div, inv_mul_cancel₀, ENNReal.rpow_one]
|
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hq0_lt : 0 < q
⊢ q ≠ 0
|
ba3d8f7541fcdfdc
|
BitVec.msb_neg
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean
|
theorem msb_neg {w : Nat} {x : BitVec w} :
(-x).msb = ((x != 0#w && x != intMin w) ^^ x.msb)
|
w : Nat
x : BitVec w
hmin : x = intMin w
⊢ (x.getMsbD 0 ^^ decide (∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true)) = (x != 0#w && x != intMin w ^^ x.getMsbD 0)
|
have : (∃ j, j < w ∧ 0 < j ∧ 0 < w ∧ j = 0) ↔ False := by
simp; omega
|
w : Nat
x : BitVec w
hmin : x = intMin w
this : (∃ j, j < w ∧ 0 < j ∧ 0 < w ∧ j = 0) ↔ False
⊢ (x.getMsbD 0 ^^ decide (∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true)) = (x != 0#w && x != intMin w ^^ x.getMsbD 0)
|
a0abc44cd1acd615
|
CategoryTheory.bijection_natural
|
Mathlib/CategoryTheory/Closed/Ideal.lean
|
theorem bijection_natural (A B : C) (X X' : D) (f : (reflector i).obj (A ⊗ B) ⟶ X) (g : X ⟶ X') :
bijection i _ _ _ (f ≫ g) = bijection i _ _ _ f ≫ g
|
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : ChosenFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
X X' : D
f : (reflector i).obj (A ⊗ B) ⟶ X
g : X ⟶ X'
⊢ i.fullyFaithfulOfReflective.preimage
(prodComparison i ((reflector i).obj A) ((reflector i).obj B) ≫
uncurry
((unitCompPartialBijective B ⋯)
(curry
((β_ B (i.obj ((reflector i).obj A))).inv ≫
uncurry
((unitCompPartialBijective A ⋯)
(curry ((β_ A B).inv ≫ ((reflectorAdjunction i).homEquiv (A ⊗ B) X') (f ≫ g) ≫ 𝟙 (i.obj X')))) ≫
𝟙 (i.obj X')))) ≫
𝟙 (i.obj X')) =
i.fullyFaithfulOfReflective.preimage
(prodComparison i ((reflector i).obj A) ((reflector i).obj B) ≫
uncurry
((unitCompPartialBijective B ⋯)
(curry
((β_ B (i.obj ((reflector i).obj A))).inv ≫
uncurry
((unitCompPartialBijective A ⋯)
(curry ((β_ A B).inv ≫ ((reflectorAdjunction i).homEquiv (A ⊗ B) X) f ≫ 𝟙 (i.obj X)))) ≫
𝟙 (i.obj X)))) ≫
𝟙 (i.obj X)) ≫
g
|
apply i.map_injective
|
case a
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : ChosenFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
X X' : D
f : (reflector i).obj (A ⊗ B) ⟶ X
g : X ⟶ X'
⊢ i.map
(i.fullyFaithfulOfReflective.preimage
(prodComparison i ((reflector i).obj A) ((reflector i).obj B) ≫
uncurry
((unitCompPartialBijective B ⋯)
(curry
((β_ B (i.obj ((reflector i).obj A))).inv ≫
uncurry
((unitCompPartialBijective A ⋯)
(curry
((β_ A B).inv ≫ ((reflectorAdjunction i).homEquiv (A ⊗ B) X') (f ≫ g) ≫ 𝟙 (i.obj X')))) ≫
𝟙 (i.obj X')))) ≫
𝟙 (i.obj X'))) =
i.map
(i.fullyFaithfulOfReflective.preimage
(prodComparison i ((reflector i).obj A) ((reflector i).obj B) ≫
uncurry
((unitCompPartialBijective B ⋯)
(curry
((β_ B (i.obj ((reflector i).obj A))).inv ≫
uncurry
((unitCompPartialBijective A ⋯)
(curry ((β_ A B).inv ≫ ((reflectorAdjunction i).homEquiv (A ⊗ B) X) f ≫ 𝟙 (i.obj X)))) ≫
𝟙 (i.obj X)))) ≫
𝟙 (i.obj X)) ≫
g)
|
5120d0f0b0ffc665
|
CategoryTheory.Sieve.effectiveEpimorphic_singleton
|
Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean
|
theorem Sieve.effectiveEpimorphic_singleton {X Y : C} (f : Y ⟶ X) :
(Presieve.singleton f).EffectiveEpimorphic ↔ (EffectiveEpi f)
|
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : Y ⟶ X
h : Nonempty (IsColimit (generateSingleton f).arrows.cocone)
⊢ EffectiveEpi f
|
constructor
|
case mp.effectiveEpi
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : Y ⟶ X
h : Nonempty (IsColimit (generateSingleton f).arrows.cocone)
⊢ Nonempty (EffectiveEpiStruct f)
|
c9c9a1eadcf45899
|
Set.OrdConnected.image_hasDerivWithinAt
|
Mathlib/Analysis/Calculus/Darboux.lean
|
theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
(hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s)
|
case hs.intro.intro.intro.intro.intro.inr.intro.intro
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
hab : b ≤ a
this : Icc b a ⊆ s
c : ℝ
cmem : c ∈ Ioo b a
hc : f' c = m
⊢ m ∈ f' '' s
|
exact ⟨c, this <| Ioo_subset_Icc_self cmem, hc⟩
|
no goals
|
2ad4e8c6a69413a7
|
SimpleGraph.edgeDisjointTriangles_iff_mem_sym2_subsingleton
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
lemma edgeDisjointTriangles_iff_mem_sym2_subsingleton :
G.EdgeDisjointTriangles ↔
∀ ⦃e : Sym2 α⦄, ¬ e.IsDiag → {s ∈ G.cliqueSet 3 | e ∈ (s : Finset α).sym2}.Subsingleton
|
case h.mpr.intro.intro.intro.intro
α : Type u_1
G : SimpleGraph α
a b : α
hab✝ : a ≠ b
hab : G.Adj a b
c : α
hac : G.Adj a c
hbc : G.Adj b c
⊢ (∃ a_1 b_1 c_1, G.Adj a_1 b_1 ∧ G.Adj a_1 c_1 ∧ G.Adj b_1 c_1 ∧ {a, b, c} = {a_1, b_1, c_1}) ∧
a ∈ {a, b, c} ∧ b ∈ {a, b, c}
|
refine ⟨⟨a, b, c, ?_⟩, ?_⟩ <;> simp [*]
|
no goals
|
dd5ab32164f7deff
|
IsMulFreimanHom.mono
|
Mathlib/Combinatorics/Additive/FreimanHom.lean
|
@[to_additive]
lemma IsMulFreimanHom.mono (hmn : m ≤ n) (hf : IsMulFreimanHom n A B f) :
IsMulFreimanHom m A B f where
mapsTo := hf.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h
|
case inr.intro.refine_5
α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CancelCommMonoid β
A : Set α
B : Set β
f : α → β
m n : ℕ
hmn : m ≤ n
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = m
ht : t.card = m
h : s.prod = t.prod
a : α
ha : a ∈ A
⊢ (s + replicate (n - m) a).prod = (t + replicate (n - m) a).prod
|
rw [prod_add, prod_add, h]
|
no goals
|
b2311f9c87a69848
|
Profinite.NobelingProof.CC_exact
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem CC_exact {f : LocallyConstant C ℤ} (hf : Linear_CC' C hsC ho f = 0) :
∃ y, πs C o y = f
|
case refine_2
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
f : LocallyConstant ↑C ℤ
hf :
⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) =
⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
⊢ ∀ (x : I → Bool) (hx : x ∈ C0 C ho ∩ π (C1 C ho) fun x => ord I x < o),
(LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f) ⟨x, ⋯⟩ =
(LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⟨x, ⋯⟩
|
intro x hx
|
case refine_2
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
f : LocallyConstant ↑C ℤ
hf :
⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) =
⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
x : I → Bool
hx : x ∈ C0 C ho ∩ π (C1 C ho) fun x => ord I x < o
⊢ (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f) ⟨x, ⋯⟩ =
(LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⟨x, ⋯⟩
|
c7ebbbfd480d7eed
|
Ordinal.cof_eq_one_iff_is_succ
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem cof_eq_one_iff_is_succ {o} : cof.{u} o = 1 ↔ ∃ a, o = succ a :=
⟨inductionOn o fun α r _ z => by
rcases cof_eq r with ⟨S, hl, e⟩; rw [z] at e
obtain ⟨a⟩ := mk_ne_zero_iff.1 (by rw [e]; exact one_ne_zero)
refine
⟨typein r a,
Eq.symm <|
Quotient.sound
⟨RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ fun x y => ?_) fun x => ?_⟩⟩
· apply Sum.rec <;> [exact Subtype.val; exact fun _ => a]
· rcases x with (x | ⟨⟨⟨⟩⟩⟩) <;> rcases y with (y | ⟨⟨⟨⟩⟩⟩) <;>
simp [Subrel, Order.Preimage, EmptyRelation]
exact x.2
· suffices r x a ∨ ∃ _ : PUnit.{u}, ↑a = x by
convert this
dsimp [RelEmbedding.ofMonotone]; simp
rcases trichotomous_of r x a with (h | h | h)
· exact Or.inl h
· exact Or.inr ⟨PUnit.unit, h.symm⟩
· rcases hl x with ⟨a', aS, hn⟩
refine absurd h ?_
convert hn
change (a : α) = ↑(⟨a', aS⟩ : S)
have := le_one_iff_subsingleton.1 (le_of_eq e)
congr!,
fun ⟨a, e⟩ => by simp [e]⟩
|
case intro.intro.intro.refine_2.inl.inr.unit
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
x : { x // r x ↑a }
⊢ r ↑x ↑a
|
exact x.2
|
no goals
|
7afdd5212ddba93b
|
CategoryTheory.Limits.Multicofork.condition
|
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
|
theorem condition (a) : I.fst a ≫ K.π (J.fst a) = I.snd a ≫ K.π (J.snd a)
|
C : Type u
inst✝ : Category.{v, u} C
J : MultispanShape
I : MultispanIndex J C
K : Multicofork I
a : J.L
⊢ I.fst a ≫ K.π (J.fst a) = I.snd a ≫ K.π (J.snd a)
|
rw [← K.snd_app_right, ← K.fst_app_right]
|
no goals
|
b4fbf300c1fb170b
|
sum_range_pow
|
Mathlib/NumberTheory/Bernoulli.lean
|
theorem sum_range_pow (n p : ℕ) :
(∑ k ∈ range n, (k : ℚ) ^ p) =
∑ i ∈ range (p + 1), bernoulli i * ((p + 1).choose i) * (n : ℚ) ^ (p + 1 - i) / (p + 1)
|
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
h_cauchy :
((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => (coeff ℚ (q + 1)) (exp ℚ ^ n)) =
PowerSeries.mk fun p => ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!
hps :
∑ k ∈ range n, ↑k ^ p = (∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !
x : ℕ
x✝ : x ∈ range (p + 1)
⊢ bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ↑p ! * (↑p + 1) =
bernoulli x * ↑((p + 1).choose x) * ↑n ^ (p + 1 - x) * ((↑p + 1) * ↑p !)
|
ring
|
no goals
|
fc82cef36f6524aa
|
isPreconnected_closed_iff
|
Mathlib/Topology/Connected/Basic.lean
|
theorem isPreconnected_closed_iff {s : Set α} :
IsPreconnected s ↔ ∀ t t', IsClosed t → IsClosed t' →
s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty :=
⟨by
rintro h t t' ht ht' htt' ⟨x, xs, xt⟩ ⟨y, ys, yt'⟩
rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter]
intro h'
have xt' : x ∉ t' := (h' xs).resolve_left (absurd xt)
have yt : y ∉ t := (h' ys).resolve_right (absurd yt')
have := h _ _ ht.isOpen_compl ht'.isOpen_compl h' ⟨y, ys, yt⟩ ⟨x, xs, xt'⟩
rw [← compl_union] at this
exact this.ne_empty htt'.disjoint_compl_right.inter_eq,
by
rintro h u v hu hv huv ⟨x, xs, xu⟩ ⟨y, ys, yv⟩
rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter]
intro h'
have xv : x ∉ v := (h' xs).elim (absurd xu) id
have yu : y ∉ u := (h' ys).elim id (absurd yv)
have := h _ _ hu.isClosed_compl hv.isClosed_compl h' ⟨y, ys, yu⟩ ⟨x, xs, xv⟩
rw [← compl_union] at this
exact this.ne_empty huv.disjoint_compl_right.inter_eq⟩
|
case intro.intro.intro.intro
α : Type u
inst✝ : TopologicalSpace α
s : Set α
h :
∀ (t t' : Set α),
IsClosed t → IsClosed t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty
u v : Set α
hu : IsOpen u
hv : IsOpen v
huv : s ⊆ u ∪ v
x : α
xs : x ∈ s
xu : x ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
h' : s ⊆ uᶜ ∪ vᶜ
xv : x ∉ v
yu : y ∉ u
this : (s ∩ (u ∪ v)ᶜ).Nonempty
⊢ False
|
exact this.ne_empty huv.disjoint_compl_right.inter_eq
|
no goals
|
20f33475245c9747
|
FDerivMeasurableAux.isOpen_A_with_param
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
lemma isOpen_A_with_param {r s : ℝ} (hf : Continuous f.uncurry) (L : E →L[𝕜] F) :
IsOpen {p : α × E | p.2 ∈ A (f p.1) L r s}
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
∀ (y : E),
dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
ha : Continuous (f a)
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
⊢ Continuous fun p => ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖
|
fun_prop
|
no goals
|
e0d09b738019f2bb
|
mem_codiscreteWithin
|
Mathlib/Topology/DiscreteSubset.lean
|
lemma mem_codiscreteWithin {S T : Set X} :
S ∈ codiscreteWithin T ↔ ∀ x ∈ T, Disjoint (𝓝[≠] x) (𝓟 (T \ S))
|
X : Type u_1
inst✝ : TopologicalSpace X
S T : Set X
⊢ (∀ i ∈ T, ∃ u, IsOpen u ∧ i ∈ u ∧ ∀ (x : X), x ∈ u ∧ x ∈ T ∧ x ∉ {i} → x ∈ S) ↔
∀ x ∈ T, ∃ u, IsOpen u ∧ x ∈ u ∧ ∀ (x_1 : X), x_1 ∈ u ∧ x_1 ∉ {x} → ¬(x_1 ∈ T ∧ x_1 ∉ S)
|
congr! 7 with x - u y
|
case a.h.h'.h.e'_2.h.h.e'_2.h.e'_2.h.a
X : Type u_1
inst✝ : TopologicalSpace X
S T : Set X
x : X
u : Set X
y : X
⊢ y ∈ u ∧ y ∈ T ∧ y ∉ {x} → y ∈ S ↔ y ∈ u ∧ y ∉ {x} → ¬(y ∈ T ∧ y ∉ S)
|
4d99fdc431c6318a
|
MeasureTheory.extend_iUnion_le_tsum_nat
|
Mathlib/MeasureTheory/OuterMeasure/Induced.lean
|
theorem extend_iUnion_le_tsum_nat : ∀ s : ℕ → Set α,
extend m (⋃ i, s i) ≤ ∑' i, extend m (s i)
|
α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
f : ℕ → Set α
h : ∀ (i : ℕ), MeasurableSet (f i)
⊢ m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
|
simp (config := { singlePass := true }) only [iUnion_disjointed.symm]
|
α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
f : ℕ → Set α
h : ∀ (i : ℕ), MeasurableSet (f i)
⊢ m (⋃ i, disjointed f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
|
8e7b78a07d78bb51
|
Subsemiring.coe_pow
|
Mathlib/Algebra/Ring/Subsemiring/Defs.lean
|
theorem coe_pow {R} [Semiring R] (s : Subsemiring R) (x : s) (n : ℕ) :
((x ^ n : s) : R) = (x : R) ^ n
|
R : Type u_1
inst✝ : Semiring R
s : Subsemiring R
x : ↥s
n : ℕ
⊢ ↑(x ^ n) = ↑x ^ n
|
induction n with
| zero => simp
| succ n ih => simp [pow_succ, ih]
|
no goals
|
70e3ce2f427b52a7
|
MeasureTheory.Submartingale.zero_le_of_predictable
|
Mathlib/Probability/Martingale/Basic.lean
|
theorem Submartingale.zero_le_of_predictable [Preorder E] [SigmaFiniteFiltration μ 𝒢]
{f : ℕ → Ω → E} (hfmgle : Submartingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1)) (n : ℕ) :
f 0 ≤ᵐ[μ] f n
|
case succ
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
𝒢 : Filtration ℕ m0
inst✝¹ : Preorder E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hfmgle : Submartingale f 𝒢 μ
hfadp : Adapted 𝒢 fun n => f (n + 1)
k : ℕ
ih : f 0 ≤ᶠ[ae μ] f k
⊢ f 0 ≤ᶠ[ae μ] f (k + 1)
|
exact ih.trans ((hfmgle.2.1 k (k + 1) k.le_succ).trans_eq <| Germ.coe_eq.mp <|
congr_arg Germ.ofFun <| condExp_of_stronglyMeasurable (𝒢.le _) (hfadp _) <| hfmgle.integrable _)
|
no goals
|
cca69cb54a86659f
|
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