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 |
|---|---|---|---|---|---|---|
MeasureTheory.Supermartingale.setIntegral_le | Mathlib/Probability/Martingale/Basic.lean | theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Supermartingale f ℱ μ)
{i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ | Ω : Type u_1
ι : Type u_3
inst✝¹ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
inst✝ : SigmaFiniteFiltration μ ℱ
f : ι → Ω → ℝ
hf : Supermartingale f ℱ μ
i j : ι
hij : i ≤ j
s : Set Ω
hs : MeasurableSet s
⊢ μ[f j|↑ℱ i] ≤ᶠ[ae μ] f i | filter_upwards [hf.2.1 i j hij] with _ heq using heq | no goals | d9882fe29a1eccc9 |
lowerCentralSeries_antitone | Mathlib/GroupTheory/Nilpotent.lean | theorem lowerCentralSeries_antitone : Antitone (lowerCentralSeries G) | case intro.intro.intro
G : Type u_1
inst✝ : Group G
n : ℕ
x : G
hx : x ∈ closure {x | ∃ p ∈ lowerCentralSeries G n, ∃ q, p * q * p⁻¹ * q⁻¹ = x}
y z : G
hz : z ∈ lowerCentralSeries G n
a : G
ha : z * a * z⁻¹ * a⁻¹ = y
⊢ z * (a * z⁻¹ * a⁻¹) ∈ lowerCentralSeries G n | exact mul_mem hz (Normal.conj_mem (lowerCentralSeries_normal n) z⁻¹ (inv_mem hz) a) | no goals | db060d010fc2a818 |
SeminormFamily.basisSets_smul_left | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U | case pos
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ... | use (s.sup p).ball 0 (r / ‖x‖) | case h
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ (s.sup p).ball 0 (r / ‖x‖) ∈ AddGroupFilterBasis.t... | 746096ac124c9ebb |
CategoryTheory.Functor.IsCoverDense.Types.naturality_apply | Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean | theorem naturality_apply [G.IsLocallyFull K] {X Y : C} (i : G.obj X ⟶ G.obj Y) (x) :
ℱ'.1.map i.op (α.app _ x) = α.app _ (ℱ.map i.op x) | C : Type u_1
inst✝² : Category.{u_5, u_1} C
D : Type u_2
inst✝¹ : Category.{u_6, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val
inst✝ : G.IsLocallyFull K
X Y : C
i : G.obj X ⟶ G.obj Y
x : (G.op ⋙ ℱ).obj (op Y)
⊢ ℱ'.val.map i.op (α.app (op Y) x) = α.app (o... | have {X Y} (i : X ⟶ Y) (x) :
ℱ'.1.map (G.map i).op (α.app _ x) = α.app _ (ℱ.map (G.map i).op x) := by
exact congr_fun (α.naturality i.op).symm x | C : Type u_1
inst✝² : Category.{u_5, u_1} C
D : Type u_2
inst✝¹ : Category.{u_6, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val
inst✝ : G.IsLocallyFull K
X Y : C
i : G.obj X ⟶ G.obj Y
x : (G.op ⋙ ℱ).obj (op Y)
this :
∀ {X Y : C} (i : X ⟶ Y) (x : (G.op ⋙... | 0e18cd3a80d5b906 |
MeasureTheory.integral_Ioi_deriv_mul_eq_sub | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integral_Ioi_deriv_mul_eq_sub
(hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
(huv : IntegrableOn (u' * v + u * v') (Ioi a))
(h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = ... | A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tends... | filter_upwards [eventually_ne_atTop a] with x hx | case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty ... | c1989a63bc42f58a |
minpoly.mem_range_of_degree_eq_one | Mathlib/FieldTheory/Minpoly/Basic.lean | theorem mem_range_of_degree_eq_one (hx : (minpoly A x).degree = 1) :
x ∈ (algebraMap A B).range | A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : (minpoly A x).degree = 1
h : IsIntegral A x
key : x = (algebraMap A B) (-(minpoly A x).coeff 0)
⊢ x ∈ (algebraMap A B).range | exact ⟨-(minpoly A x).coeff 0, key.symm⟩ | no goals | f3de3c81a71d387c |
IsSelfAdjoint.hasEigenvector_of_isMaxOn | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | theorem hasEigenvector_of_isMaxOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
(hextr : IsMaxOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
HasEigenvector (T : E →ₗ[𝕜] E) (↑(⨆ x : { x : E // x ≠ 0 }, T.rayleighQuotient x)) x₀ | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
hx : x ∈ sphere 0 ‖x₀‖
⊢ ‖x‖ ... | simpa using hx | no goals | ab8f63176b60c4a5 |
MixedCharZero.reduce_to_p_prime | Mathlib/Algebra/CharP/MixedCharZero.lean | theorem reduce_to_p_prime {P : Prop} :
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ p : ℕ, p.Prime → MixedCharZero R p → P | R : Type u_1
inst✝ : CommRing R
P : Prop
h : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
q : ℕ
q_pos : q > 0
q_mixedChar : MixedCharZero R q
I : Ideal R
hI_ne_top : I ≠ ⊤
right✝ : CharP (R ⧸ I) q
M : Ideal R
hM_max : M.IsMaximal
h_IM : I ≤ M
r : ℕ := ringChar (R ⧸ M)
q_zero : ↑q = 0
⊢ r ≠ 0 | apply ne_zero_of_dvd_ne_zero (ne_of_gt q_pos) | R : Type u_1
inst✝ : CommRing R
P : Prop
h : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
q : ℕ
q_pos : q > 0
q_mixedChar : MixedCharZero R q
I : Ideal R
hI_ne_top : I ≠ ⊤
right✝ : CharP (R ⧸ I) q
M : Ideal R
hM_max : M.IsMaximal
h_IM : I ≤ M
r : ℕ := ringChar (R ⧸ M)
q_zero : ↑q = 0
⊢ r ∣ q | c9cd4718e7a6656f |
ModelWithCorners.interior_disjointUnion | Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean | lemma interior_disjointUnion :
ModelWithCorners.interior (I := I) (M ⊕ M') =
Sum.inl '' (ModelWithCorners.interior (I := I) M)
∪ Sum.inr '' (ModelWithCorners.interior (I := I) M') | case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : Chart... | set x := Sum.getLeft p h with x_eq | case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : Chart... | a37cb5a73686bbc7 |
Lean.Grind.eqNDRec_heq | Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Lemmas.lean | theorem eqNDRec_heq.{u_1, u_2} {α : Sort u_2} {a : α}
{motive : α → Sort u_1} (v : motive a) {b : α} (h : a = b)
: HEq (@Eq.ndrec α a motive v b h) v | α : Sort u_2
a : α
motive : α → Sort u_1
v : motive a
⊢ HEq (⋯ ▸ v) v | rfl | no goals | 1a1ced6def61e073 |
Subgroup.isOpen_of_mem_nhds | Mathlib/Topology/Algebra/OpenSubgroup.lean | theorem isOpen_of_mem_nhds [ContinuousMul G] (H : Subgroup G) {g : G} (hg : (H : Set G) ∈ 𝓝 g) :
IsOpen (H : Set G) | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
⊢ ↑H ∈ 𝓝 x | have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg) | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
⊢ ↑H ∈ 𝓝 x | b78c6ecd314e40e6 |
IsPrimitiveRoot.arg | Mathlib/RingTheory/RootsOfUnity/Complex.lean | theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
∃ i : ℤ, ζ.arg = i / n * (2 * Real.pi) ∧ IsCoprime i n ∧ i.natAbs < n | case neg.convert_2
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ -Real.pi < (↑i - ↑n) * (2 * Real.pi) / ↑n ∧ (↑i - ↑n) * (2 * Real.pi) / ↑n ≤ Real.pi | refine ⟨?_, le_trans ?_ Real.pi_pos.le⟩ | case neg.convert_2.refine_1
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ -Real.pi < (↑i - ↑n) * (2 * Real.pi) / ↑n
case neg.convert_2.refine_2
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ (↑i - ↑n) * (2 * Real.pi) / ↑n ≤ 0 | 27e9c2564262c07b |
LocallyFinite.finite_nonempty_of_compact | Mathlib/Topology/Compactness/Compact.lean | theorem LocallyFinite.finite_nonempty_of_compact [CompactSpace X] {f : ι → Set X}
(hf : LocallyFinite f) : { i | (f i).Nonempty }.Finite | X : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
f : ι → Set X
hf : LocallyFinite f
⊢ {i | (f i).Nonempty}.Finite | simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ | no goals | 91a62b7c98a05f57 |
Int.isUnit_sq | Mathlib/Data/Int/Order/Units.lean | theorem isUnit_sq {a : ℤ} (ha : IsUnit a) : a ^ 2 = 1 | a : ℤ
ha : IsUnit a
⊢ a ^ 2 = 1 | rw [sq, isUnit_mul_self ha] | no goals | 5411e649b7b09c66 |
List.set_getElem_succ_eraseIdx_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Erase.lean | theorem set_getElem_succ_eraseIdx_succ
{l : List α} {i : Nat} (h : i + 1 < l.length) :
(l.eraseIdx (i + 1)).set i l[i + 1] = l.eraseIdx i | case h.isFalse.isTrue.isTrue
α : Type u_1
l : List α
i : Nat
h : i + 1 < l.length
n : Nat
h₁ : n < ((l.eraseIdx (i + 1)).set i l[i + 1]).length
h₂ : n < (l.eraseIdx i).length
h✝² : ¬i = n
h✝¹ : n < i + 1
h✝ : n < i
⊢ l[n] = l[n] | rfl | no goals | 31983ea669fb242b |
Subalgebra.finrank_sup_le_of_free | Mathlib/RingTheory/Adjoin/Dimension.lean | theorem finrank_sup_le_of_free : finrank R ↥(A ⊔ B) ≤ finrank R A * finrank R B | case pos
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : StrongRankCondition R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
inst✝¹ : Free R ↥A
inst✝ : Free R ↥B
h : Module.Finite R ↥A ∧ Module.Finite R ↥B
⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B | obtain ⟨_, _⟩ := h | case pos.intro
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : StrongRankCondition R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
inst✝¹ : Free R ↥A
inst✝ : Free R ↥B
left✝ : Module.Finite R ↥A
right✝ : Module.Finite R ↥B
⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B | 9b7b5a8cbdf91750 |
LinearMap.isClosed_or_dense_ker | Mathlib/Topology/Algebra/Module/Simple.lean | theorem LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M) | case inr
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑⊤ ∨ Dense ↑⊤ | left | case inr.h
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑⊤ | 8a89eedc2ab854fa |
Associates.eq_factors_of_eq_counts | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
a.factors = b.factors | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ∀... | intro p hp | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ∀... | 92d336a2c52c647d |
Associates.eq_pow_count_factors_of_dvd_pow | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α}
(hp : Irreducible p) {n : ℕ} (h : a ∣ p ^ n) : a = p ^ p.count a.factors | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
a✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ ∀ (p_1 : Associates α), Irreducible p_... | have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
Nat.eq_zero_of_le_zero <| by
convert count_le_count_of_le hph hq h
symm
rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero] | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
a✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Ir... | db8650aa84ceeb2f |
Batteries.RBNode.Balanced.append | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n | α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝² x✝¹ a✝ : RBNode α
x✝ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝ x✝ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
w✝ : RBColor
IH : (b.append c).Bal... | split | case h_1
α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝³ x✝² a✝¹ : RBNode α
x✝¹ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝¹ x✝¹ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝¹.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
w✝ : RBColor
IH : (b... | b3ecc8a07b6c9ec6 |
LinearMap.mapsTo_biSup_of_mapsTo | Mathlib/Algebra/DirectSum/LinearMap.lean | lemma mapsTo_biSup_of_mapsTo {ι : Type*} {N : ι → Submodule R M}
(s : Set ι) {f : Module.End R M} (hf : ∀ i, MapsTo f (N i) (N i)) :
MapsTo f ↑(⨆ i ∈ s, N i) ↑(⨆ i ∈ s, N i) | R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type u_4
N : ι → Submodule R M
s : Set ι
f : Module.End R M
hf : ∀ (i : ι), Submodule.map f (N i) ≤ N i
⊢ Submodule.map f (⨆ i ∈ s, N i) ≤ ⨆ i ∈ s, N i | simpa only [Submodule.map_iSup] using iSup₂_mono <| fun i _ ↦ hf i | no goals | b3d6df199b89a4ff |
CategoryTheory.Adjunction.localization_counit_app | Mathlib/CategoryTheory/Localization/Adjunction.lean | @[simp]
lemma localization_counit_app (X₂ : C₂) :
(adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =
G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫
(CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫
L₂.map (adj.counit.app X₂) | C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁷ : Category.{u_8, u_1} C₁
inst✝⁶ : Category.{u_7, u_2} C₂
inst✝⁵ : Category.{u_6, u_3} D₁
inst✝⁴ : Category.{u_5, u_4} D₂
G : C₁ ⥤ C₂
F : C₂ ⥤ C₁
adj : G ⊣ F
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProper... | apply Localization.η_app | no goals | c65ffa4546fd9d0a |
CategoryTheory.Limits.colim.exact_mapShortComplex | Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Colim.lean | /-- Assuming `HasExactColimitsOfShape J C`, this lemma rephrases the exactness
of the functor `colim : (J ⥤ C) ⥤ C` by saying that if `S : ShortComplex (J ⥤ C)`
is exact, then the short complex obtained by taking the colimits is exact,
where we allow the replacement of the chosen colimit cocones of the
colimit API by a... | C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃... | refine (ShortComplex.exact_iff_of_iso ?_).2 (hS.map colim) | C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃... | dc1c71373d793112 |
gramSchmidt_mem_span | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | theorem gramSchmidt_mem_span (f : ι → E) :
∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) | rw [gramSchmidt_def 𝕜 f i] | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ f i - ∑ i_1 ∈ Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i_1})) (... | 2db3f8db7f4c82a7 |
ContinuousLinearMap.exists_preimage_norm_le | Mathlib/Analysis/Normed/Operator/Banach.lean | theorem exists_preimage_norm_le (surj : Surjective f) :
∃ C > 0, ∀ y, ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖ | case succ
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPai... | rw [iterate_succ'] | case succ
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPai... | fbb58cd4816e6d13 |
LinearIndependent.map_pow_expChar_pow_of_isSeparable | Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean | theorem LinearIndependent.map_pow_expChar_pow_of_isSeparable [Algebra.IsSeparable F E]
(h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n) | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : ∀ (s : Finset ι), LinearIndependent F (v ∘ Subtype.val)
halg : Algebra.IsAlgebraic F E
s : Finset ι
⊢ LinearIndependent F ((fun x => v x ^ q ^ n) ∘ Subtype.val... | let E' := adjoin F (s.image v : Set E) | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : ∀ (s : Finset ι), LinearIndependent F (v ∘ Subtype.val)
halg : Algebra.IsAlgebraic F E
s : Finset ι
E' : IntermediateField F E := adjoin F ↑(Finset.image v s)
... | 8836e43fdea16b48 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem go_denote_eq (aig : AIG α) (distance : AIG.RefVec aig n) (curr : Nat)
(hcurr : curr ≤ n - 1) (acc : AIG.RefVec aig w)
(lhs : BitVec w) (rhs : BitVec n) (assign : α → Bool)
(hacc : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, acc.get idx hidx, assign⟧ = (BitVec.sshiftRightRec lhs rhs curr).getLsbD idx)
... | case isTrue.hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
n w : Nat
aig : AIG α
distance : aig.RefVec n
curr : Nat
hcurr : curr ≤ n - 1
acc : aig.RefVec w
lhs : BitVec w
rhs : BitVec n
assign : α → Bool
hacc :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := acc.get idx hidx }⟧ = (lhs.ssh... | simp [Ref.hgate] | no goals | a62c971c970b9dfe |
RCLike.nonUnitalContinuousFunctionalCalculus | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean | theorem RCLike.nonUnitalContinuousFunctionalCalculus :
NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) where
predicate_zero | 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunctionalCalculus 𝕜 ... | have : inr ∘ ψ = cfcₙAux hp₁ a ha := by ext1; rw [Function.comp_apply, coe_ψ] | 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunctionalCalculus 𝕜 ... | 95b319408dc7d905 |
integral_bernoulliFun_eq_zero | Mathlib/NumberTheory/ZetaValues.lean | theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in (0)..1, bernoulliFun k x = 0 | k : ℕ
hk : k ≠ 0
⊢ ∫ (x : ℝ) in 0 ..1, bernoulliFun k x = 0 | rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)] | k : ℕ
hk : k ≠ 0
⊢ bernoulliFun (k + 1) 1 / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0 | 6e6ae5d2aa4d445c |
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean | theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1 | case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls... | rw [AIG.LawfulVecOperator.decl_eq (f := blastArithShiftRight)] | case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls... | f92988d5a1595b39 |
Set.exists_subset_encard_eq | Mathlib/Data/Set/Card.lean | theorem exists_subset_encard_eq {k : ℕ∞} (hk : k ≤ s.encard) : ∃ t, t ⊆ s ∧ t.encard = k | α : Type u_1
s : Set α
k : ℕ∞
x✝ : 0 ≤ s.encard
⊢ ∅.encard = 0 | simp | no goals | 05b71bd015f46ce9 |
MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq | Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean | theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
(hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
f =ᵐ[μ] g | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyM... | simp only [Set.compl_union] | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyM... | 928eae5949838171 |
HomologicalComplex₂.D₂_D₁ | Mathlib/Algebra/Homology/TotalComplex.lean | @[reassoc (attr := simp)]
lemma D₂_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = - K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' | case pos
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁... | have h₆ : ComplexShape.π c₁ c₂ c₁₂ (c₁.next i₁, c₂.next i₂) = i₁₂'' := by
rw [← c₁₂.next_eq' h₂, ← ComplexShape.next_π₁ c₂ c₁₂ h₃, h₅] | case pos
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁... | 2d728f63029305d6 |
intervalIntegral.intervalIntegrable_cpow' | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem intervalIntegrable_cpow' {r : ℂ} (h : -1 < r.re) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) volume a b | case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x => ↑x ^ r) volume 0 c | rw [IntervalIntegrable.iff_comp_neg, neg_zero] | case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x => ↑(-x) ^ r) volume 0 (-c) | e33db8d106f7d67d |
NormedAddGroupHom.mkNormedAddGroupHom_norm_le' | Mathlib/Analysis/Normed/Group/Hom.lean | theorem mkNormedAddGroupHom_norm_le' (f : V₁ →+ V₂) {C : ℝ} (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mkNormedAddGroupHom C h‖ ≤ max C 0 :=
opNorm_le_bound _ (le_max_right _ _) fun x =>
(h x).trans <| by gcongr; apply le_max_left
| case h
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : V₁ →+ V₂
C : ℝ
h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖
x : V₁
⊢ C ≤ C ⊔ 0 | apply le_max_left | no goals | 9103a32bfa4bec78 |
PartENat.le_of_lt_add_one | Mathlib/Data/Nat/PartENat.lean | theorem le_of_lt_add_one {x y : PartENat} (h : x < y + 1) : x ≤ y | case a.intro
n m : ℕ
h : ↑m < ↑n + 1
⊢ m ≤ n | apply Nat.le_of_lt_succ | case a.intro.a
n m : ℕ
h : ↑m < ↑n + 1
⊢ m < n.succ | 941bfcc29fd6ae31 |
Subfield.mem_iSup_of_directed | Mathlib/Algebra/Field/Subfield/Basic.lean | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (· ≤ ·) S)
{x : K} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i | K : Type u
inst✝ : DivisionRing K
ι : Sort u_1
hι : Nonempty ι
S : ι → Subfield K
hS : Directed (fun x1 x2 => x1 ≤ x2) S
x : K
s : Subfield K :=
let __spread.0 := (⨆ i, (S i).toSubring).copy (⋃ i, ↑(S i).toSubring) ⋯;
{ toSubring := __spread.0, inv_mem' := ⋯ }
this : iSup S = s
⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i | exact this ▸ Set.mem_iUnion | no goals | e62260f3c816ead1 |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖... | rw [tsum_geometric_of_lt_one (by positivity) zn, ← div_eq_mul_inv] | no goals | e1f8d437480849f9 |
PowerBasis.repr_pow_isIntegral | Mathlib/RingTheory/Adjoin/PowerBasis.lean | theorem repr_pow_isIntegral [IsDomain S] (hB : IsIntegral R B.gen) {x : A}
(hx : ∀ i, IsIntegral R (B.basis.repr x i))
(hmin : minpoly S B.gen = (minpoly R B.gen).map (algebraMap R S)) (n : ℕ) :
∀ i, IsIntegral R (B.basis.repr (x ^ n) i) | S : Type u_2
inst✝⁷ : CommRing S
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
B : PowerBasis S A
inst✝ : IsDomain S
hB : IsIntegral R B.gen
x : A
hx : ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)
hmi... | revert hx | S : Type u_2
inst✝⁷ : CommRing S
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
B : PowerBasis S A
inst✝ : IsDomain S
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (min... | d2285bbb358047c1 |
LinearMap.nilRank_le_natTrailingDegree_charpoly | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | lemma nilRank_le_natTrailingDegree_charpoly (x : L) :
nilRank φ ≤ (φ x).charpoly.natTrailingDegree | case h
R : Type u_1
L : Type u_2
M : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Free R M
inst✝³ : Module.Finite R M
inst✝² : Module.Finite R L
inst✝¹ : Free R L
inst✝ : Nontrivial R
x : L
h : (φ.polyCharpoly (chooseBa... | apply_fun (MvPolynomial.eval ((chooseBasis R L).repr x)) at h | case h
R : Type u_1
L : Type u_2
M : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Free R M
inst✝³ : Module.Finite R M
inst✝² : Module.Finite R L
inst✝¹ : Free R L
inst✝ : Nontrivial R
x : L
h :
(MvPolynomial.eval ⇑((c... | 7d44b5dff95aaa8c |
AList.insertRec_insert | Mathlib/Data/List/AList.lean | theorem insertRec_insert {C : AList β → Sort*} (H0 : C ∅)
(IH : ∀ (a : α) (b : β a) (l : AList β), a ∉ l → C l → C (l.insert a b)) {c : Sigma β}
{l : AList β} (h : c.1 ∉ l) :
@insertRec α β _ C H0 IH (l.insert c.1 c.2) = IH c.1 c.2 l h (@insertRec α β _ C H0 IH l) | α : Type u
β : α → Type v
inst✝ : DecidableEq α
C : AList β → Sort u_1
H0 : C ∅
IH : (a : α) → (b : β a) → (l : AList β) → a ∉ l → C l → C (insert a b l)
c : Sigma β
l : AList β
h : c.fst ∉ l
⊢ insertRec H0 IH (insert c.fst c.snd l) = IH c.fst c.snd l h (insertRec H0 IH l) | obtain ⟨l, hl⟩ := l | case mk
α : Type u
β : α → Type v
inst✝ : DecidableEq α
C : AList β → Sort u_1
H0 : C ∅
IH : (a : α) → (b : β a) → (l : AList β) → a ∉ l → C l → C (insert a b l)
c : Sigma β
l : List (Sigma β)
hl : l.NodupKeys
h : c.fst ∉ { entries := l, nodupKeys := hl }
⊢ insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys ... | ac5fbba4f95f506f |
Real.Angle.sign_two_nsmul_eq_sign_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem sign_two_nsmul_eq_sign_iff {θ : Angle} :
((2 : ℕ) • θ).sign = θ.sign ↔ θ = π ∨ |θ.toReal| < π / 2 | case neg.refine_1
θ : Angle
hpi : ¬θ = ↑π
h : (2 • θ).sign = θ.sign
hle : π / 2 ≤ θ.toReal ∨ θ.toReal ≤ -(π / 2)
⊢ False | have hpi' : θ.toReal ≠ π := by simpa using hpi | case neg.refine_1
θ : Angle
hpi : ¬θ = ↑π
h : (2 • θ).sign = θ.sign
hle : π / 2 ≤ θ.toReal ∨ θ.toReal ≤ -(π / 2)
hpi' : θ.toReal ≠ π
⊢ False | 796c2e838b8abfa4 |
List.findIdx?_go_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
findIdx?.go p xs (i+1) = (findIdx?.go p xs 0).map fun k => k + (i + 1) | case cons
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ {i : Nat}, findIdx?.go p tail✝ (i + 1) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
i : Nat
⊢ Option.map (fun i => i + 1) (if p head✝ = true then some i else Option.map (fun i => i + 1) (findIdx?.go p tail✝ i)) =
Option.map (fu... | split | case cons.isTrue
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ {i : Nat}, findIdx?.go p tail✝ (i + 1) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
i : Nat
h✝ : p head✝ = true
⊢ Option.map (fun i => i + 1) (some i) = Option.map (fun k => k + (i + 1)) (some 0)
case cons.isFalse
α : Type ... | 00b11065e9d67637 |
VitaliFamily.exists_measurable_supersets_limRatio | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0 | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a)... | gcongr | case hinf
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ... | f2acb3dc8c3a0a55 |
Ideal.matricesOver_strictMono_of_nonempty | Mathlib/LinearAlgebra/Matrix/Ideal.lean | theorem matricesOver_strictMono_of_nonempty [Nonempty n] :
StrictMono (matricesOver (R := R) n) :=
matricesOver_monotone n |>.strictMono_of_injective <| fun I J eq => by
ext x
have : (∀ _ _, x ∈ I) ↔ (∀ _ _, x ∈ J) := congr((Matrix.of fun _ _ => x) ∈ $eq)
simpa only [forall_const] using this
| R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matricesOver n I = matricesOver n J
⊢ I = J | ext x | case h
R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matricesOver n I = matricesOver n J
x : R
⊢ x ∈ I ↔ x ∈ J | 0208c0bde6369ee2 |
List.forIn_eq_foldlM | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (init : β) (l : List α) :
forIn l init f = ForInStep.value <$>
l.foldlM (fun b a => match b with
| .yield b => f a b
| .done b => pure (.done b)) (ForInStep.yield init) | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
a : α
as : List α
ih :
∀ (init : β),
forIn as init f =
ForInStep.value <$>
List.foldlM
(fun b a =>
match b with
| ForInStep.yield b => f a b
... | simp [ih] | no goals | 5803c92344e4da54 |
List.map_attachWith | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Attach.lean | theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
(f : { x // P x } → β) :
(l.attachWith P H).map f =
l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) | case cons.h
α : Type u_1
β : Type u_2
P : α → Prop
f : { x // P x } → β
x : α
xs : List α
ih : ∀ {H : ∀ (a : α), a ∈ xs → P a}, map f (xs.attachWith P H) = pmap (fun a h => f ⟨a, ⋯⟩) xs ⋯
H : ∀ (a : α), a ∈ x :: xs → P a
⊢ ∀ (a : α), a ∈ xs → ∀ (h₁ : a ∈ xs ∧ P a) (h₂ : a ∈ x :: xs ∧ P a), f ⟨a, ⋯⟩ = f ⟨a, ⋯⟩ | simp | no goals | 5c127c9a27e15920 |
Ideal.sup_pow_add_le_pow_sup_pow | Mathlib/RingTheory/Ideal/Operations.lean | lemma sup_pow_add_le_pow_sup_pow {n m : ℕ} : (I ⊔ J) ^ (n + m) ≤ I ^ n ⊔ J ^ m | case neg
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
n m i : ℕ
hi : i ∈ Finset.range (n + m + 1)
hn : ¬n ≤ i
⊢ m ≤ n + m - i | omega | no goals | 690ef13be270f2c6 |
NonUnitalRing.ext | Mathlib/Algebra/Ring/Ext.lean | theorem ext ⦃inst₁ inst₂ : NonUnitalRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ | case mk.mk
R : Type u
toNonUnitalNonAssocRing✝¹ : NonUnitalNonAssocRing R
mul_assoc✝¹ : ∀ (a b c : R), a * b * c = a * (b * c)
toNonUnitalNonAssocRing✝ : NonUnitalNonAssocRing R
mul_assoc✝ : ∀ (a b c : R), a * b * c = a * (b * c)
h_add : HAdd.hAdd = HAdd.hAdd
h_mul : HMul.hMul = HMul.hMul
this : toNonUnitalNonAssocRing... | congr | no goals | 3a34c023b5fdf73d |
Finset.image₂_left_comm | Mathlib/Data/Finset/NAry.lean | theorem image₂_left_comm {γ : Type*} {u : Finset γ} {f : α → δ → ε} {g : β → γ → δ}
{f' : α → γ → δ'} {g' : β → δ' → ε} (h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
image₂ f s (image₂ g t u) = image₂ g' t (image₂ f' s u) :=
coe_injective <| by
push_cast
exact image2_left_comm h_left_comm
| α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ ↑... | push_cast | α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ i... | e951e1d0d2442dd3 |
map_wittPolynomial | Mathlib/RingTheory/WittVector/WittPolynomial.lean | theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n | p : ℕ
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ ∑ x ∈ range (n + 1), (map f) ((monomial (single x (p ^ (n - x)))) (↑p ^ x)) =
∑ i ∈ range (n + 1), (monomial (single i (p ^ (n - i)))) (↑p ^ i) | refine sum_congr rfl fun i _ => ?_ | p : ℕ
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
n i : ℕ
x✝ : i ∈ range (n + 1)
⊢ (map f) ((monomial (single i (p ^ (n - i)))) (↑p ^ i)) = (monomial (single i (p ^ (n - i)))) (↑p ^ i) | 812afc8e6e227a9a |
CategoryTheory.isCardinalPresentable_of_equivalence | Mathlib/CategoryTheory/Presentable/Basic.lean | lemma isCardinalPresentable_of_equivalence
{C' : Type u₃} [Category.{v₃} C'] [IsCardinalPresentable X κ] (e : C ≌ C') :
IsCardinalPresentable (e.functor.obj X) κ | case h.up
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
Y : J ⥤ C'
this : PreservesColimitsOfShape J (coyoneda.obj (op X))
X✝ Y... | apply Equiv.ulift.injective | case h.up.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
Y : J ⥤ C'
this : PreservesColimitsOfShape J (coyoneda.obj (op X))
X✝... | fd601d9188daee0d |
List.append_eq_appendTR | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean | theorem append_eq_appendTR : @List.append = @appendTR | case h.h.h.cons
α : Type u_1
bs : List α
a : α
as : List α
ih : as.append bs = (as.reverseAux nil).reverseAux bs
⊢ (a :: as).append bs = (a :: nil).reverseAux ((as.reverseAux nil).reverseAux bs) | simp [List.append, ih, reverseAux] | no goals | aae20de0c8888126 |
Finset.kruskal_katona_lovasz_form | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | theorem kruskal_katona_lovasz_form (hir : i ≤ r) (hrk : r ≤ k) (hkn : k ≤ n)
(h₁ : (𝒜 : Set (Finset (Fin n))).Sized r) (h₂ : k.choose r ≤ #𝒜) :
k.choose (r - i) ≤ #(∂^[i] 𝒜) | n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ #(powersetCard (r - i) range'k) = #(∂ ^[i] 𝒞) | congr! | case h.e'_2
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ powersetCard (r - i) range'k = ∂ ^[i] 𝒞 | edb5f2f186d17abf |
WeierstrassCurve.b₆_of_isCharTwoJNeZeroNF | Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean | theorem b₆_of_isCharTwoJNeZeroNF : W.b₆ = 4 * W.a₆ | R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJNeZeroNF
⊢ W.b₆ = 4 * W.a₆ | rw [b₆, a₃_of_isCharTwoJNeZeroNF] | R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJNeZeroNF
⊢ 0 ^ 2 + 4 * W.a₆ = 4 * W.a₆ | 62cfcc7187a72866 |
MeasureTheory.setToFun_congr_measure_of_integrable | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞)
(hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) :
setToFun μ T hT f = setToFun μ' T hT' f | case h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : Domin... | have hμ's : μ' s ≠ ∞ := by
refine ((hμ'_le s).trans_lt ?_).ne
rw [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_lt_top hc'.lt_top hμs | case h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : Domin... | 65e35cd00eda3e5b |
Algebra.Presentation.aux_surjective | Mathlib/RingTheory/Presentation.lean | private lemma aux_surjective : Function.Surjective (Q.aux P) := fun p ↦ by
induction' p using MvPolynomial.induction_on with a p q hp hq p i h
· use rename Sum.inr <| P.σ a
simp only [aux, aeval_rename, Sum.elim_comp_inr]
have (p : MvPolynomial P.vars R) :
aeval (C ∘ P.val) p = (C (aeval P.val p) : ... | case h_add.intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra S T
Q : Presentation S T
P : Presentation R S
q : MvPolynomial Q.vars S
hq : ∃ a, (Algebra.Presentation.aux Q P) a = q
a : MvPolynomial (Q.vars ⊕ P.vars) R
⊢ ∃ a_1, (Algeb... | obtain ⟨b, rfl⟩ := hq | case h_add.intro.intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra S T
Q : Presentation S T
P : Presentation R S
a b : MvPolynomial (Q.vars ⊕ P.vars) R
⊢ ∃ a_1, (Algebra.Presentation.aux Q P) a_1 = (Algebra.Presentation.aux Q P) a +... | 6a819377477f7ce9 |
smul_eq_self_of_preimage_zpow_eq_self | Mathlib/Data/Set/Pointwise/Iterate.lean | theorem smul_eq_self_of_preimage_zpow_eq_self {G : Type*} [CommGroup G] {n : ℤ} {s : Set G}
(hs : (fun x => x ^ n) ⁻¹' s = s) {g : G} {j : ℕ} (hg : g ^ n ^ j = 1) : g • s = s | case intro.intro
G : Type u_1
inst✝ : CommGroup G
n : ℤ
s : Set G
hs : (fun x => x ^ n) ⁻¹' s = s
g : G
j : ℕ
hg : g ^ n ^ j = 1
g' : G
hg' : g' ^ n ^ j = 1
y : G
hy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s
⊢ (⇑(zpowGroupHom n))^[j] (g' * y) ∈ s | replace hg' : (zpowGroupHom n)^[j] g' = 1 := by simpa [zpowGroupHom] | case intro.intro
G : Type u_1
inst✝ : CommGroup G
n : ℤ
s : Set G
hs : (fun x => x ^ n) ⁻¹' s = s
g : G
j : ℕ
hg : g ^ n ^ j = 1
g' y : G
hy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s
hg' : (⇑(zpowGroupHom n))^[j] g' = 1
⊢ (⇑(zpowGroupHom n))^[j] (g' * y) ∈ s | c18d60c68b83e353 |
isAlgebraic_of_isLocalization | Mathlib/RingTheory/Localization/Integral.lean | lemma isAlgebraic_of_isLocalization {R} [CommRing R] (M : Submonoid R) (S) [CommRing S]
[Nontrivial R] [Algebra R S] [IsLocalization M S] : Algebra.IsAlgebraic R S | case isAlgebraic.intro.intro
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontrivial R
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
s : ↥M
⊢ IsAlgebraic R (mk' S x s) | by_cases hs : (s : R) = 0 | case pos
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontrivial R
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
s : ↥M
hs : ↑s = 0
⊢ IsAlgebraic R (mk' S x s)
case neg
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontr... | 5abd58f488e94823 |
Convex.convex_remove_iff_not_mem_convexHull_remove | Mathlib/Analysis/Convex/Hull.lean | theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x}) | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
y : E
hy : y ∈ (convexHull 𝕜) (s \ {x})
⊢ y ∉ {x} | rintro (rfl : y = x) | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
y : E
hx : y ∉ (convexHull 𝕜) (s \ {y})
hy : y ∈ (convexHull 𝕜) (s \ {y})
⊢ False | 2f0adfe15b540bae |
AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma ideal_le_ker_glueDataObjι (U V : X.affineOpens) :
I.ideal V ≤ RingHom.ker (U.1.ι.app V.1 ≫ (I.glueDataObjι U).app _).hom | X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
⊢ I.ideal V ≤ RingHom.ker (CommRingCat.Hom.hom (Hom.app (↑U).ι ↑V ≫ Hom.app (I.glueDataObjι U) ((↑U).ι ⁻¹ᵁ ↑V))) | intro x hx | X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑Γ(X, ↑V)
hx : x ∈ I.ideal V
⊢ x ∈ RingHom.ker (CommRingCat.Hom.hom (Hom.app (↑U).ι ↑V ≫ Hom.app (I.glueDataObjι U) ((↑U).ι ⁻¹ᵁ ↑V))) | 9da7a9bdc95919a0 |
le_hasProd | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | theorem le_hasProd (hf : HasProd f a) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ a :=
calc
f i = ∏ i ∈ {i}, f i | ι : Type u_1
α : Type u_3
inst✝² : OrderedCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
f : ι → α
a : α
hf : HasProd f a
i : ι
hb : ∀ (j : ι), j ≠ i → 1 ≤ f j
⊢ f i = ∏ i ∈ {i}, f i | rw [prod_singleton] | no goals | 19ef94c7e446040f |
Complex.norm_one_add_mul_inv_le | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | /-- Give a bound on `‖(1 + t * z)⁻¹‖` for `0 ≤ t ≤ 1` and `‖z‖ < 1`. -/
lemma norm_one_add_mul_inv_le {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz : ‖z‖ < 1) :
‖(1 + t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹ | t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ ‖1 + ↑t * z‖⁻¹ ≤ (1 - ‖z‖)⁻¹ | refine inv_anti₀ (by linarith) ?_ | t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - ‖z‖ ≤ ‖1 + ↑t * z‖ | 3995f690a9b2d2b3 |
Module.End.isNilpotent_restrict_of_le | Mathlib/RingTheory/Nilpotent/Lemmas.lean | lemma isNilpotent_restrict_of_le {f : End R M} {p q : Submodule R M}
{hp : MapsTo f p p} {hq : MapsTo f q q} (h : p ≤ q) (hf : IsNilpotent (f.restrict hq)) :
IsNilpotent (f.restrict hp) | case h.h.mk.a
R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
p q : Submodule R M
hp : MapsTo ⇑f ↑p ↑p
hq : MapsTo ⇑f ↑q ↑q
h : p ≤ q
n : ℕ
x : M
hx : x ∈ p
hn : (LinearMap.restrict f hq ^ n) ⟨x, ⋯⟩ = 0 ⟨x, ⋯⟩
⊢ ↑((LinearMap.restrict f hp ^ n) ⟨x, hx⟩) = ↑(0 ⟨x, hx⟩... | simp_rw [LinearMap.zero_apply, ZeroMemClass.coe_zero, ZeroMemClass.coe_eq_zero] at hn ⊢ | case h.h.mk.a
R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
p q : Submodule R M
hp : MapsTo ⇑f ↑p ↑p
hq : MapsTo ⇑f ↑q ↑q
h : p ≤ q
n : ℕ
x : M
hx : x ∈ p
hn : (LinearMap.restrict f hq ^ n) ⟨x, ⋯⟩ = 0
⊢ (LinearMap.restrict f hp ^ n) ⟨x, hx⟩ = 0 | 0c6b0966b16b1e56 |
sSup_eq_bot' | Mathlib/Order/CompleteLattice.lean | lemma sSup_eq_bot' {s : Set α} : sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥} | α : Type u_1
inst✝ : CompleteLattice α
s : Set α
⊢ sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥} | rw [sSup_eq_bot, ← subset_singleton_iff_eq, subset_singleton_iff] | no goals | 1cefef35bb8747b5 |
FiberBundle.totalSpaceMk_isClosedEmbedding | Mathlib/Topology/FiberBundle/Basic.lean | theorem totalSpaceMk_isClosedEmbedding [T1Space B] (x : B) :
IsClosedEmbedding (@TotalSpace.mk B F E x) :=
⟨totalSpaceMk_isEmbedding F E x, by
rw [TotalSpace.range_mk]
exact isClosed_singleton.preimage <| continuous_proj F E⟩
| B : Type u_2
F : Type u_3
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace F
E : B → Type u_5
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (b : B) → TopologicalSpace (E b)
inst✝¹ : FiberBundle F E
inst✝ : T1Space B
x : B
⊢ IsClosed (range (TotalSpace.mk x)) | rw [TotalSpace.range_mk] | B : Type u_2
F : Type u_3
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace F
E : B → Type u_5
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (b : B) → TopologicalSpace (E b)
inst✝¹ : FiberBundle F E
inst✝ : T1Space B
x : B
⊢ IsClosed (TotalSpace.proj ⁻¹' {x}) | afde15389e7a6a5b |
MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0∞,
(∀ x, (f x : ℝ≥0∞) < g x) ∧
LowerSemicontinuous g ∧
(∀ᵐ x ∂μ, g x < ⊤) ∧
Integrable (fun x => (g x).toReal) μ ∧ (∫ x,... | case intro.intro.intro.intro.intro.intro.refine_2.hfm
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
fint : Integrable (fun x => ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 <... | apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable | no goals | 3f39b00d374e3aed |
Finset.eq_one_of_prod_eq_one | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem eq_one_of_prod_eq_one {s : Finset α} {f : α → β} {a : α} (hp : ∏ x ∈ s, f x = 1)
(h1 : ∀ x ∈ s, x ≠ a → f x = 1) : ∀ x ∈ s, f x = 1 | case pos
α : Type u_3
β : Type u_4
inst✝ : CommMonoid β
s : Finset α
f : α → β
a : α
hp : ∏ x ∈ s, f x = 1
h1 : ∀ x ∈ s, x ≠ a → f x = 1
x : α
hx : x ∈ s
h : x = a
⊢ f x = 1 | rw [h] | case pos
α : Type u_3
β : Type u_4
inst✝ : CommMonoid β
s : Finset α
f : α → β
a : α
hp : ∏ x ∈ s, f x = 1
h1 : ∀ x ∈ s, x ≠ a → f x = 1
x : α
hx : x ∈ s
h : x = a
⊢ f a = 1 | 15f6948a1ae4c501 |
Real.hasSum_log_sub_log_of_abs_lt_one | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | theorem hasSum_log_sub_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
HasSum (fun k : ℕ => (2 : ℝ) * (1 / (2 * k + 1)) * x ^ (2 * k + 1))
(log (1 + x) - log (1 - x)) | x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n => -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x => 2 * x) = fun k => 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
⊢ HasSum term (log (1 + x) - log (1 - x)) | have h₁ := (hasSum_pow_div_log_of_abs_lt_one (Eq.trans_lt (abs_neg x) h)).mul_left (-1) | x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n => -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x => 2 * x) = fun k => 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
h₁ : HasSum (fun i => -1 * ((-x) ^ (i + 1) / (↑i + 1))) (-1 * -log (1 - -x))
⊢ HasSum term (log (1 + x) - log (1 - x)) | ad64e8a77fc6c38e |
AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | theorem coequalizer_π_app_isLocalHom
(U : TopologicalSpace.Opens (coequalizer f.toShHom g.toShHom).carrier) :
IsLocalHom ((coequalizer.π f.toShHom g.toShHom :).c.app (op U)).hom | X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace
this✝¹ :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom f))
(SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom g)) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToP... | infer_instance | no goals | f9571eebfdddf194 |
CoxeterSystem.getElem_leftInvSeq_alternatingWord | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem getElem_leftInvSeq_alternatingWord
(i j : B) (p k : ℕ) (h : k < 2 * p) :
(lis (alternatingWord i j (2 * p)))[k]'(by simp; omega) =
π alternatingWord j i (2 * k + 1) | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
p : ℕ
i j : B
h : 0 < 2 * p
⊢ 0 < (alternatingWord i j (2 * p)).length | simp [h] | no goals | d39ee0dd3fb7776c |
Std.Sat.CNF.any_not_isEmpty_iff_exists_mem | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean | theorem any_not_isEmpty_iff_exists_mem {f : CNF α} :
(List.any f fun c => !List.isEmpty c) = true ↔ ∃ v, Mem v f | case mpr
α : Type u_1
f : CNF α
⊢ (∃ v c, c ∈ f ∧ ((v, false) ∈ c ∨ (v, true) ∈ c)) → ∃ x, x ∈ f ∧ ∃ x_1, x_1 ∈ x | intro h | case mpr
α : Type u_1
f : CNF α
h : ∃ v c, c ∈ f ∧ ((v, false) ∈ c ∨ (v, true) ∈ c)
⊢ ∃ x, x ∈ f ∧ ∃ x_1, x_1 ∈ x | cfda4c463f240c75 |
UniformConcaveOn.neg | Mathlib/Analysis/Convex/Strong.lean | lemma UniformConcaveOn.neg (hf : UniformConcaveOn s φ f) : UniformConvexOn s φ (-f) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ : ℝ → ℝ
s : Set E
f : E → ℝ
hf : UniformConcaveOn s φ f
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ -(a • (-f) x + b • (-f) y - a * b * φ ‖x - y‖) ≤ -(-f) (a • x + b • y) | simpa [add_comm, -neg_le_neg_iff, ← le_sub_iff_add_le', sub_eq_add_neg, neg_add]
using hf.2 hx hy ha hb hab | no goals | 4dfbf23a0715fa0c |
Geometry.SimplicialComplex.vertex_mem_convexHull_iff | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | theorem vertex_mem_convexHull_iff (hx : x ∈ K.vertices) (hs : s ∈ K.faces) :
x ∈ convexHull 𝕜 (s : Set E) ↔ x ∈ s | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
s : Finset E
x : E
hx : x ∈ K.vertices
hs : s ∈ K.faces
h : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ (convexHull 𝕜) ↑{x} | simp | no goals | cac78c48089373e1 |
Rat.fract_inv_num_lt_num_of_pos | Mathlib/Data/Rat/Floor.lean | theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
⊢ ↑q.den - q.num * ⌊q_inv⌋ < q.num | have q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * q⁻¹.den < q⁻¹.den := by
have : q⁻¹.num < (⌊q⁻¹⌋ + 1) * q⁻¹.den := Rat.num_lt_succ_floor_mul_den q⁻¹
have : q⁻¹.num < ⌊q⁻¹⌋ * q⁻¹.den + q⁻¹.den := by rwa [right_distrib, one_mul] at this
rwa [← sub_lt_iff_lt_add'] at this | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
⊢ ↑q.den - q.num * ⌊q_inv⌋ < q.num | d2870a8c7e295972 |
LinearEquiv.charpoly_conj | Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean | @[simp]
lemma LinearEquiv.charpoly_conj (e : M₁ ≃ₗ[R] M₂) (φ : Module.End R M₁) :
(e.conj φ).charpoly = φ.charpoly | R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : Nontrivial R
inst✝⁷ : AddCommGroup M₁
inst✝⁶ : Module R M₁
inst✝⁵ : Module.Finite R M₁
inst✝⁴ : Module.Free R M₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Module.Free R M₂
e : M₁ ≃ₗ[R] M₂
φ : Module.End R M₁
⊢ ... | let b := chooseBasis R M₁ | R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : Nontrivial R
inst✝⁷ : AddCommGroup M₁
inst✝⁶ : Module R M₁
inst✝⁵ : Module.Finite R M₁
inst✝⁴ : Module.Free R M₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Module.Free R M₂
e : M₁ ≃ₗ[R] M₂
φ : Module.End R M₁
b ... | b91ddefebec43a56 |
Finsupp.single_smul | Mathlib/Data/Finsupp/SMul.lean | theorem single_smul (a b : α) (f : α → M) (r : R) : single a r b • f a = single a (r • f b) b | α : Type u_1
M : Type u_3
R : Type u_6
inst✝² : Zero M
inst✝¹ : MonoidWithZero R
inst✝ : MulActionWithZero R M
a b : α
f : α → M
r : R
⊢ (single a r) b • f a = (single a (r • f b)) b | by_cases h : a = b <;> simp [h] | no goals | 01076ce47b0d8e31 |
CoalgebraCat.MonoidalCategoryAux.rightUnitor_hom_toLinearMap | Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean | theorem rightUnitor_hom_toLinearMap :
(ρ_ (CoalgebraCat.of R M)).hom.1.toLinearMap = (TensorProduct.rid R M).toLinearMap :=
TensorProduct.ext <| by ext; rfl
| R : Type u
inst✝³ : CommRing R
M : Type u
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Coalgebra R M
⊢ (TensorProduct.mk R
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.o... | ext | case h.h
R : Type u
inst✝³ : CommRing R
M : Type u
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Coalgebra R M
x✝ :
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
⊢ (((TensorProduct.mk ... | 2cd05fc798c201cd |
exists_norm_eq_iInf_of_complete_convex | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ... | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u ... | have eq₁ : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u ... | 1aed30586c16c1ed |
Array.any_toList | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem any_toList {p : α → Bool} (as : Array α) : as.toList.any p = as.any p | α : Type u_1
p : α → Bool
as : Array α
⊢ (∃ x, (∃ i h, as[i] = x) ∧ p x = true) ↔ ∃ i x, p as[i] = true | exact ⟨fun ⟨_, ⟨i, w, rfl⟩, h⟩ => ⟨i, w, h⟩, fun ⟨i, w, h⟩ => ⟨_, ⟨i, w, rfl⟩, h⟩⟩ | no goals | d0770c9f684b3b7f |
Con.comap_conGen_equiv | Mathlib/GroupTheory/Congruence/Basic.lean | theorem comap_conGen_equiv {M N : Type*} [Mul M] [Mul N] (f : MulEquiv M N) (rel : N → N → Prop) :
Con.comap f (map_mul f) (conGen rel) = conGen (fun x y ↦ rel (f x) (f y)) | M : Type u_4
N : Type u_5
inst✝¹ : Mul M
inst✝ : Mul N
f : M ≃* N
rel : N → N → Prop
a✝² b✝ : M
h : (conGen rel) (f a✝²) (f b✝)
n1 n2 w x y z : N
a✝¹ : ConGen.Rel rel w x
a✝ : ConGen.Rel rel y z
ih : ∀ (a b : M), f a = w → f b = x → (conGen fun x y => rel (f x) (f y)) a b
ih1 : ∀ (a b : M), f a = y → f b = z → (conGen ... | simp | no goals | d315948435eb07b2 |
Polynomial.bernoulli_three_eval_one_quarter | Mathlib/NumberTheory/ZetaValues.lean | theorem Polynomial.bernoulli_three_eval_one_quarter :
(Polynomial.bernoulli 3).eval (1 / 4) = 3 / 64 | ⊢ 2 ≠ 1 | decide | no goals | 729db16dc9d12f16 |
isJacobsonRing_iff_prime_eq | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isJacobsonRing_iff_prime_eq :
IsJacobsonRing R ↔ ∀ P : Ideal R, IsPrime P → P.jacobson = P | R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ x ∈ I | rw [← hI.radical, radical_eq_sInf I, mem_sInf] | R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsPrime} → x ∈ I_1 | 4be15f6834fa58b0 |
Finset.sum_Ico_by_parts | Mathlib/Algebra/BigOperators/Module.lean | theorem sum_Ico_by_parts (hmn : m < n) :
∑ i ∈ Ico m n, f i • g i =
f (n - 1) • G n - f m • G m - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i
h₂ :
∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g i =
∑ i ∈ Ico m (n - 1)... | abel | no goals | 35456e90ec4d4807 |
Cubic.c_eq_three_roots | Mathlib/Algebra/CubicDiscriminant.lean | theorem c_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.c = φ P.a * (x * y + x * z + y * z) | F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ φ P.c = φ P.a * (x * y + x * z + y * z) | injection eq_sum_three_roots ha h3 | no goals | bbbbb08af3a215b6 |
MonomialOrder.degree_lt_iff | Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean | theorem degree_lt_iff {f : MvPolynomial σ R} {d : σ →₀ ℕ} (hd : 0 ≺[m] d) :
m.degree f ≺[m] d ↔ ∀ c ∈ f.support, c ≺[m] d | σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
d : σ →₀ ℕ
hd : 0 < m.toSyn d
⊢ m.toSyn (m.degree f) < m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c < m.toSyn d | unfold degree | σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
d : σ →₀ ℕ
hd : 0 < m.toSyn d
⊢ m.toSyn (m.toSyn.symm (f.support.sup ⇑m.toSyn)) < m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c < m.toSyn d | 6c45fc57d2fc6548 |
AList.empty_lookupFinsupp | Mathlib/Data/Finsupp/AList.lean | theorem empty_lookupFinsupp : lookupFinsupp (∅ : AList fun _x : α => M) = 0 | α : Type u_1
M : Type u_2
inst✝ : Zero M
⊢ ∅.lookupFinsupp = 0 | classical
ext
simp | no goals | b378aa077b932512 |
FixedPoints.minpoly.eval₂ | Mathlib/FieldTheory/Fixed.lean | theorem eval₂ :
Polynomial.eval₂ (Subring.subtype <| (FixedPoints.subfield G F).toSubring) x (minpoly G F x) =
0 | G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Fintype G
x : F
⊢ Polynomial.eval₂ (subfield G F).subtype x (minpoly G F x) = 0 | rw [← prodXSubSMul.eval G F x, Polynomial.eval₂_eq_eval_map] | G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Fintype G
x : F
⊢ Polynomial.eval x (Polynomial.map (subfield G F).subtype (minpoly G F x)) = Polynomial.eval x (prodXSubSMul G F x) | a7d25a02e432b675 |
CategoryTheory.Functor.pi'_eval | Mathlib/CategoryTheory/Pi/Basic.lean | theorem pi'_eval (f : ∀ i, A ⥤ C i) (i : I) : pi' f ⋙ Pi.eval C i = f i | case h_map
I : Type w₀
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ autoParam (∀ (X Y : A) (f_1 : X ⟶ Y), (pi' f ⋙ Pi.eval C i).map f_1 = eqToHom ⋯ ≫ (f i).map f_1 ≫ eqToHom ⋯) _auto✝ | intro _ _ _ | case h_map
I : Type w₀
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ Y✝ : A
f✝ : X✝ ⟶ Y✝
⊢ (pi' f ⋙ Pi.eval C i).map f✝ = eqToHom ⋯ ≫ (f i).map f✝ ≫ eqToHom ⋯ | 44f316420514ba3a |
TendstoUniformly.comp | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem TendstoUniformly.comp (h : TendstoUniformly F f p) (g : γ → α) :
TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p | α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
h : TendstoUniformlyOnFilter F f p ⊤
g : γ → α
⊢ TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p ⊤ | simpa [principal_univ, comap_principal] using h.comp g | no goals | b08bd153e5f5b4a4 |
AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
(carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
(carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
show (∀ _, _ ∈ _) ∨ ∀ _, _ ∈ _ by
rw [← and_forall_ne nx, and_iff_left, ← and_forall... | case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm ... | rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_zero] | case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm ... | 5ba35f6f859f61e0 |
IsCompact.image_of_continuousOn | Mathlib/Topology/Compactness/Compact.lean | theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s) | X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
⊢ IsCompact (f '' s) | intro l lne ls | X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
ls : l ≤ 𝓟 (f '' s)
⊢ ∃ x ∈ f '' s, ClusterPt x l | 2cb55c7533cef93b |
fderivWithin_fderivWithin_eq_of_mem_nhdsWithin | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | lemma fderivWithin_fderivWithin_eq_of_mem_nhdsWithin (h : t ∈ 𝓝[s] x)
(hf : ContDiffWithinAt 𝕜 2 f t x) (hs : UniqueDiffOn 𝕜 s) (ht : UniqueDiffOn 𝕜 t) (hx : x ∈ s) :
fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x = fderivWithin 𝕜 (fderivWithin 𝕜 f t) t x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s... | exact fderivWithin_of_mem_nhdsWithin h (hs x hx) (hf.differentiableWithinAt one_le_two) | no goals | 338e0eff11ac573d |
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}) | 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), Measurab... | rw [this] | 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), Measurab... | 93587e3bd5de43f7 |
CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_fac | Mathlib/CategoryTheory/Limits/IsLimit.lean | theorem coconeOfHom_fac {Y : C} (f : X ⟶ Y) : coconeOfHom h f = (colimitCocone h).extend f | case e_ι.w.h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₃
inst✝ : Category.{v₃, u₃} C
F : J ⥤ C
X : C
h : coyoneda.obj (op X) ⋙ uliftFunctor.{u₁, v₃} ≅ F.cocones
Y : C
f : X ⟶ Y
j : J
t : h.hom.app Y { down := 𝟙 X ≫ f } = F.cocones.map f (h.hom.app X { down := 𝟙 X })
⊢ (h.hom.app Y { down := f }).app j = (h.h... | simp only [id_comp] at t | case e_ι.w.h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₃
inst✝ : Category.{v₃, u₃} C
F : J ⥤ C
X : C
h : coyoneda.obj (op X) ⋙ uliftFunctor.{u₁, v₃} ≅ F.cocones
Y : C
f : X ⟶ Y
j : J
t : h.hom.app Y { down := f } = F.cocones.map f (h.hom.app X { down := 𝟙 X })
⊢ (h.hom.app Y { down := f }).app j = (h.hom.app ... | 4193177dc0262cb9 |
le_inv_iff_mul_le_one_right | Mathlib/Algebra/Order/Group/Unbundled/Basic.lean | theorem le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel]
| α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a * b ≤ b⁻¹ * b ↔ a * b ≤ 1 | rw [inv_mul_cancel] | no goals | b5a254400e7116b9 |
ContinuousLinearMap.hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear | Mathlib/Analysis/Analytic/CPolynomial.lean | theorem hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear :
HasFiniteFPowerSeriesOnBall (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2)
f.toFormalMultilinearSeriesOfMultilinear 0 (Fintype.card (Option ι) + 1) ⊤ | 𝕜 : Type u_1
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype... | rw [toFormalMultilinearSeriesOfMultilinear, dif_pos rfl] | 𝕜 : Type u_1
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype... | 9e79f3aed59fcb73 |
HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn
(hf : HasFPowerSeriesWithinOnBall f p s x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
((x + ·)⁻¹' (insert x s) ∩ EMetric.ball (0 : E) r) | case intro.intro.refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p ... | simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu | no goals | 2a880b1d843248d7 |
MeasureTheory.eLpNorm_le_of_ae_nnnorm_bound | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | theorem eLpNorm_le_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) :
eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹ | case pos
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
C : ℝ≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C
hμ : NeZero μ
hp : p = 0
⊢ eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹ | simp [hp] | no goals | e13c88a1505d5b94 |
IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
IsDiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P | case mpr.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
Punique : ∃! P, P ≠ ⊥ ∧ P.IsPrime
this : IsLocalRing R
⊢ IsDiscreteValuationRing R | refine { not_a_field' := ?_ } | case mpr.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
Punique : ∃! P, P ≠ ⊥ ∧ P.IsPrime
this : IsLocalRing R
⊢ maximalIdeal R ≠ ⊥ | f7521cb0caad85a8 |
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