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 |
|---|---|---|---|---|---|---|
Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
{i₁ i₂ : Fin (n + 3)} :
⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
s.points i₁ -ᵥ s.points i₂⟫ =
0 | 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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcen... | repeat rw [← sum_subset fs.subset_univ _] | 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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcen... | 4dd62aff1c506de0 |
lTensor_injective_of_exact_of_exact_of_rTensor_injective | Mathlib/RingTheory/LocalRing/Module.lean | theorem lTensor_injective_of_exact_of_exact_of_rTensor_injective
{M₁ M₂ M₃ N₁ N₂ N₃}
[AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup M₃] [Module R M₃]
[AddCommGroup N₁] [Module R N₁] [AddCommGroup N₂] [Module R N₂] [AddCommGroup N₃] [Module R N₃]
{f₁ : M₁ →ₗ[R] M₂} {f₂ : M₂... | R : Type u_7
inst✝¹² : CommRing R
M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_4
N₂ : Type u_5
N₃ : Type u_6
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R M₁
inst✝⁹ : AddCommGroup M₂
inst✝⁸ : Module R M₂
inst✝⁷ : AddCommGroup M₃
inst✝⁶ : Module R M₃
inst✝⁵ : AddCommGroup N₁
inst✝⁴ : Module R N₁
inst✝³ : AddComm... | intro x hx | R : Type u_7
inst✝¹² : CommRing R
M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_4
N₂ : Type u_5
N₃ : Type u_6
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R M₁
inst✝⁹ : AddCommGroup M₂
inst✝⁸ : Module R M₂
inst✝⁷ : AddCommGroup M₃
inst✝⁶ : Module R M₃
inst✝⁵ : AddCommGroup N₁
inst✝⁴ : Module R N₁
inst✝³ : AddComm... | 806122d1b633f0e5 |
Ordinal.exists_fundamental_sequence | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem exists_fundamental_sequence (a : Ordinal.{u}) :
∃ f, IsFundamentalSequence a a.cof.ord f | case intro.intro.intro.intro.intro.refine_2
a : Ordinal.{u}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = a
hι : #ι = a.cof
r : ι → ι → Prop
wo : IsWellOrder ι r
hr : (#ι).ord = type r
this✝ : IsWellOrder ι r
r' : { i // ∀ (j : ι), r j i → f j < f i } → { i // ∀ (j : ι), r j i → f j < f i } → Prop :=
Subrel r fun i =>... | rwa [hrr'.2, @enum_lt_enum _ r'] | no goals | 116700a50c4a0ec3 |
zpow_right_strictMono | Mathlib/Algebra/Order/Group/Basic.lean | @[to_additive zsmul_left_strictMono]
lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n | α : Type u_1
inst✝ : OrderedCommGroup α
a : α
ha : 1 < a
⊢ StrictMono fun n => a ^ n | refine strictMono_int_of_lt_succ fun n ↦ ?_ | α : Type u_1
inst✝ : OrderedCommGroup α
a : α
ha : 1 < a
n : ℤ
⊢ a ^ n < a ^ (n + 1) | 5a37c30f3a6aaa4b |
IntermediateField.Lifts.union_isExtendible | Mathlib/FieldTheory/Extension.lean | theorem union_isExtendible [alg : Algebra.IsAlgebraic F E]
[Nonempty c] (hext : ∀ σ ∈ c, σ.IsExtendible) :
(union c hc).IsExtendible := fun S ↦ by
let Ω := adjoin F (S : Set E) →ₐ[F] K
have ⟨ω, hω⟩ : ∃ ω : Ω, ∀ π : c, ∃ θ ≥ π.1, ⟨_, ω⟩ ≤ θ ∧ θ.carrier = π.1.1 ⊔ adjoin F S | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin... | have := finiteDimensional_adjoin (S := (S : Set E)) fun _ _ ↦ (alg.isIntegral).1 _ | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin... | f447a1810182010c |
ComplexShape.Embedding.not_boundaryGE_next' | Mathlib/Algebra/Homology/Embedding/Boundary.lean | lemma not_boundaryGE_next' [e.IsRelIff] {j k : ι} (hj : ¬ e.BoundaryGE j) (hk : c.next j = k) :
¬ e.BoundaryGE k | case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j k : ι
hj : ¬e.BoundaryGE j
hk : c.next j = k
hjk : ¬c.Rel j k
⊢ ¬e.BoundaryGE k | subst hk | case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j : ι
hj : ¬e.BoundaryGE j
hjk : ¬c.Rel j (c.next j)
⊢ ¬e.BoundaryGE (c.next j) | 34a5e2fcf06d10d2 |
LinearMap.mul_eq_one_of_mul_eq_one | Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean | theorem mul_eq_one_of_mul_eq_one [FiniteDimensional K V] {f g : V →ₗ[K] V} (hfg : f * g = 1) :
g * f = 1 | K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
i : V →ₗ[K] V
hi : g ∘ₗ i = id
⊢ g * f = 1 | have : f * (g * i) = f * 1 := congr_arg _ hi | K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f g : V →ₗ[K] V
hfg : f * g = 1
ginj : Injective ⇑g
i : V →ₗ[K] V
hi : g ∘ₗ i = id
this : f * (g * i) = f * 1
⊢ g * f = 1 | eaf011920ab463bb |
CategoryTheory.HasProjectiveDimensionLT.subsingleton | Mathlib/CategoryTheory/Abelian/Projective/Dimension.lean | lemma subsingleton [hX : HasProjectiveDimensionLT X n] (i : ℕ) (hi : n ≤ i) (Y : C) :
Subsingleton (Ext.{w} X Y i) | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
n : ℕ
hX : HasProjectiveDimensionLT X n
i : ℕ
hi : n ≤ i
Y : C
this : HasExt C := HasExt.standard C
⊢ Subsingleton (Ext X Y i) | have := hX.subsingleton' i hi | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
n : ℕ
hX : HasProjectiveDimensionLT X n
i : ℕ
hi : n ≤ i
Y : C
this✝ : HasExt C := HasExt.standard C
this : ∀ ⦃Y : C⦄, Subsingleton (Ext X Y i)
⊢ Subsingleton (Ext X Y i) | 1e1a208acf7aaf73 |
dotProduct_self_eq_zero | Mathlib/LinearAlgebra/Matrix/DotProduct.lean | theorem dotProduct_self_eq_zero [LinearOrderedRing R] {v : n → R} : dotProduct v v = 0 ↔ v = 0 :=
(Finset.sum_eq_zero_iff_of_nonneg fun i _ => mul_self_nonneg (v i)).trans <| by
simp [funext_iff]
| n : Type u_2
R : Type u_4
inst✝¹ : Fintype n
inst✝ : LinearOrderedRing R
v : n → R
⊢ (∀ i ∈ Finset.univ, v i * v i = 0) ↔ v = 0 | simp [funext_iff] | no goals | 4400ac4f9efc75e0 |
Nat.psp_from_prime_gt_p | Mathlib/NumberTheory/FermatPsp.lean | theorem psp_from_prime_gt_p {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
(p_gt_two : 2 < p) : p < psp_from_prime b p | case h₁
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
AB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
h : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2
⊢ (b ^ 2) ^ (p - 1) * b ^ 2 - 1 ≥ ?b | exact tsub_le_tsub_left (one_le_of_lt p_gt_two) ((b ^ 2) ^ (p - 1) * b ^ 2) | no goals | 260daf6c75f986dc |
PrimeSpectrum.existsUnique_idempotent_basicOpen_eq_of_isClopen | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | @[stacks 00EE]
lemma existsUnique_idempotent_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃! e : R, IsIdempotentElem e ∧ s = basicOpen e | case right.intro.intro
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ ↑J
⊢ (fun x => x ^ n) x ∈ ↑(... | exact Ideal.mem_sup_right (Ideal.pow_mem_pow hx n) | no goals | f72a236e53847d1c |
CategoryTheory.extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types | Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean | lemma extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types [FinitaryPreExtensive C]
{F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
(h : Presheaf.IsLocallySurjective (extensiveTopology C) f) {X : C} :
Function.Surjective (f.app (op X)) | case intro.intro.intro.intro.intro
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk... | let y : (a : α) → (F.obj ⟨Y a⟩) := fun a ↦ (h' a).choose | case intro.intro.intro.intro.intro
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk... | f246db99c628b9b2 |
IsAlgebraic.of_ringHom_of_comp_eq | Mathlib/RingTheory/Algebraic/Basic.lean | theorem IsAlgebraic.of_ringHom_of_comp_eq (halg : IsAlgebraic S (g a))
(hf : Function.Surjective f) (hg : Function.Injective g)
(h : RingHom.comp (algebraMap S B) f = RingHom.comp g (algebraMap R A)) :
IsAlgebraic R a | case intro.intro.intro
R : Type u
S : Type u_1
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Ring A
inst✝⁶ : Algebra R A
B : Type u_2
inst✝⁵ : Ring B
inst✝⁴ : Algebra S B
FRS : Type u_3
FAB : Type u_4
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB ... | change (g : A →+* B) _ = _ | case intro.intro.intro
R : Type u
S : Type u_1
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Ring A
inst✝⁶ : Algebra R A
B : Type u_2
inst✝⁵ : Ring B
inst✝⁴ : Algebra S B
FRS : Type u_3
FAB : Type u_4
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB ... | 7bf9a2fe3767fdd6 |
AlgebraicGeometry.Scheme.PartialMap.exists_restrict_isOver | Mathlib/AlgebraicGeometry/RationalMap.lean | lemma PartialMap.exists_restrict_isOver [X.Over S] [Y.Over S] (f : X.PartialMap Y)
[f.toRationalMap.IsOver S] : ∃ U hU hU', (f.restrict U hU hU').IsOver S | X Y S : Scheme
inst✝² : X.Over S
inst✝¹ : Y.Over S
f : X.PartialMap Y
inst✝ : RationalMap.IsOver S f.toRationalMap
f' : X.PartialMap Y
hf₁ : IsOver S f'
hf₂ : f'.toRationalMap = f.toRationalMap
U : X.Opens
hU : Dense ↑U
hUl : U ≤ f'.domain
hUr : U ≤ f.domain
e : (f'.restrict U hU hUl).hom = (f.restrict U hU hUr).hom
⊢ ... | rw [IsOver, ← e] | X Y S : Scheme
inst✝² : X.Over S
inst✝¹ : Y.Over S
f : X.PartialMap Y
inst✝ : RationalMap.IsOver S f.toRationalMap
f' : X.PartialMap Y
hf₁ : IsOver S f'
hf₂ : f'.toRationalMap = f.toRationalMap
U : X.Opens
hU : Dense ↑U
hUl : U ≤ f'.domain
hUr : U ≤ f.domain
e : (f'.restrict U hU hUl).hom = (f.restrict U hU hUr).hom
⊢ ... | 0d55c4114c24f68e |
Finset.small_neg_pos_pos_mul | Mathlib/Combinatorics/Additive/SmallTripling.lean | @[to_additive]
private lemma small_neg_pos_pos_mul (hA : #(A ^ 3) ≤ K * #A) : #(A⁻¹ * A * A) ≤ K ^ 2 * #A | G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ 2 ≤ 3 | norm_num | no goals | 9ecf6558cb352719 |
Filter.HasBasis.hasBasis_self_subset | Mathlib/Order/Filter/Bases.lean | theorem HasBasis.hasBasis_self_subset {p : Set α → Prop} (h : l.HasBasis (fun s => s ∈ l ∧ p s) id)
{V : Set α} (hV : V ∈ l) : l.HasBasis (fun s => s ∈ l ∧ p s ∧ s ⊆ V) id | α : Type u_1
l : Filter α
p : Set α → Prop
h : l.HasBasis (fun s => s ∈ l ∧ p s) id
V : Set α
hV : V ∈ l
⊢ l.HasBasis (fun s => s ∈ l ∧ p s ∧ s ⊆ V) id | simpa only [and_assoc] using h.restrict_subset hV | no goals | dcd1fc7782ad5cc9 |
LinearIndependent.map_of_surjective_injectiveₛ | Mathlib/LinearAlgebra/LinearIndependent/Basic.lean | theorem LinearIndependent.map_of_surjective_injectiveₛ {R' M' : Type*}
[Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)
(i : R → R') (j : M →+ M') (hi : Surjective i) (hj : Injective j)
(hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R' (j ∘ v) | case intro.refine_1
ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
R' : Type u_6
M' : Type u_7
inst✝² : Semiring R'
inst✝¹ : AddCommMonoid M'
inst✝ : Module R' M'
hv : LinearIndependent R v
i : R → R'
j : M →+ M'
hi : Surjective i
hj : Injective ⇑j
hc : ... | rwa [hi', hi'] at h | no goals | 845e67a3f398d819 |
Ideal.coeff_zero_mem_comap_of_root_mem_of_eval_mem | Mathlib/RingTheory/Ideal/GoingUp.lean | theorem coeff_zero_mem_comap_of_root_mem_of_eval_mem {r : S} (hr : r ∈ I) {p : R[X]}
(hp : p.eval₂ f r ∈ I) : p.coeff 0 ∈ I.comap f | R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
I : Ideal S
r : S
hr : r ∈ I
p : R[X]
hp : eval₂ f r p.divX * r + f (p.coeff 0) ∈ I
⊢ p.coeff 0 ∈ comap f I | refine mem_comap.mpr ((I.add_mem_iff_right ?_).mp hp) | R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
I : Ideal S
r : S
hr : r ∈ I
p : R[X]
hp : eval₂ f r p.divX * r + f (p.coeff 0) ∈ I
⊢ eval₂ f r p.divX * r ∈ I | 71086f59edd4aee7 |
Cardinal.add_eq_right | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem add_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) : a + b = b | a b : Cardinal.{u_1}
hb : ℵ₀ ≤ b
ha : a ≤ b
⊢ a + b = b | rw [add_comm, add_eq_left hb ha] | no goals | 84d7ee4166484886 |
Sylow.characteristic_of_normal | Mathlib/GroupTheory/Sylow.lean | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : P.Normal) : P.Characteristic | G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : (↑P).Normal
this : Unique (Sylow p G)
⊢ ∀ (ϕ : G ≃* G), Subgroup.map ϕ.toMonoidHom ↑P = ↑P | intro Φ | G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : (↑P).Normal
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ Subgroup.map Φ.toMonoidHom ↑P = ↑P | 8c910d250da28df4 |
HomologicalComplex₂.d_comm | Mathlib/Algebra/Homology/HomologicalBicomplex.lean | @[reassoc]
lemma d_comm (K : HomologicalComplex₂ C c₁ c₂) (i₁ i₁' : I₁) (i₂ i₂' : I₂) :
(K.d i₁ i₁').f i₂ ≫ (K.X i₁').d i₂ i₂' = (K.X i₁).d i₂ i₂' ≫ (K.d i₁ i₁').f i₂' | C : Type u_1
inst✝¹ : Category.{u_4, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i₁ i₁' : I₁
i₂ i₂' : I₂
⊢ (K.d i₁ i₁').f i₂ ≫ (K.X i₁').d i₂ i₂' = (K.X i₁).d i₂ i₂' ≫ (K.d i₁ i₁').f i₂' | simp | no goals | 90d4ec86f38317c7 |
BitVec.getLsbD_rotateRightAux_of_geq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_rotateRightAux_of_geq {x : BitVec w} {r : Nat} {i : Nat} (hi : i ≥ w - r) :
(x.rotateRightAux r).getLsbD i = (decide (i < w) && x.getLsbD (i - (w - r))) | w : Nat
x : BitVec w
r i : Nat
hi : i ≥ w - r
⊢ ((x >>> r).getLsbD i || (x <<< (w - r)).getLsbD i) = (decide (i < w) && x.getLsbD (i - (w - r))) | suffices (x >>> r).getLsbD i = false by
simp only [this, getLsbD_shiftLeft, Bool.false_or]
by_cases hiw : i < w
<;> simp [hiw, hi] | w : Nat
x : BitVec w
r i : Nat
hi : i ≥ w - r
⊢ (x >>> r).getLsbD i = false | 85912949c4385521 |
PiTensorProduct.mul_assoc | Mathlib/RingTheory/PiTensorProduct.lean | protected lemma mul_assoc (x y z : ⨂[R] i, A i) : mul (mul x y) z = mul x (mul y z) | case H.H.H.H.H.H
ι : Type u_1
R : Type u_3
A : ι → Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : (i : ι) → NonUnitalSemiring (A i)
inst✝² : (i : ι) → Module R (A i)
inst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)
x✝ y✝ z✝ : ⨂[R] (i : ι), A i
x y z : (i : ι) → A i
⊢ (tprod R) x... | simpa only [tprod_mul_tprod] using congr_arg (tprod R) (mul_assoc x y z) | no goals | 4f6f0f2f471d605a |
Dioph.diophFn_compn | Mathlib/NumberTheory/Dioph.lean | theorem diophFn_compn :
∀ {n} {S : Set (α ⊕ (Fin2 n) → ℕ)} (_ : Dioph S) {f : Vector3 ((α → ℕ) → ℕ) n}
(_ : VectorAllP DiophFn f), Dioph {v : α → ℕ | (v ⊗ fun i => f i v) ∈ S}
| 0, S, d, f => fun _ =>
ext (reindex_dioph _ (id ⊗ Fin2.elim0) d) fun v => by
dsimp
-- Porting note: `congr` use to... | α : Type
n : ℕ
S : Set (α ⊕ Fin2 n.succ → ℕ)
d : Dioph S
f✝ : Vector3 ((α → ℕ) → ℕ) n.succ
f : (α → ℕ) → ℕ
fl : Vector3 ((α → ℕ) → ℕ) n
⊢ DiophFn f → VectorAllP DiophFn fl → Dioph {v | (v ⊗ fun i => (f :: fl) i v) ∈ S} | exact fun df dfl =>
have : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr) ∈ S} :=
ext (diophFn_comp1 (reindex_dioph _ (some ∘ inl ⊗ none::some ∘ inr) d) <|
reindex_diophFn inl df)
fun v => by
dsimp
suffices (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) =
v ∘ inl ⊗ ... | no goals | 7665905a9f4dbace |
Set.iUnion_unpair_prod | Mathlib/Data/Nat/Pairing.lean | theorem iUnion_unpair_prod {α β} {s : ℕ → Set α} {t : ℕ → Set β} :
⋃ n : ℕ, s n.unpair.fst ×ˢ t n.unpair.snd = (⋃ n, s n) ×ˢ ⋃ n, t n | α : Type u_1
β : Type u_2
s : ℕ → Set α
t : ℕ → Set β
⊢ ⋃ n, s (unpair n).1 ×ˢ t (unpair n).2 = ⋃ x, s x.1 ×ˢ t x.2 | exact surjective_unpair.iUnion_comp (fun x => s x.fst ×ˢ t x.snd) | no goals | e0b87f8249c7ea81 |
coprime_sq_sub_mul_of_even_odd | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem coprime_sq_sub_mul_of_even_odd {m n : ℤ} (h : Int.gcd m n = 1) (hm : m % 2 = 0)
(hn : n % 2 = 1) : Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1 | m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 0
hn : n % 2 = 1
⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1 | by_contra H | m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 0
hn : n % 2 = 1
H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1
⊢ False | 151cb5dfbae33a2a |
Vector.eq_push_append_of_mem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem eq_push_append_of_mem {xs : Vector α n} {x : α} (h : x ∈ xs) :
∃ (n₁ n₂ : Nat) (as : Vector α n₁) (bs : Vector α n₂) (h : n₁ + 1 + n₂ = n),
xs = (as.push x ++ bs).cast h ∧ x ∉ as | case mk.intro.intro.intro
α : Type u_1
x : α
as bs : Array α
w : ¬x ∈ as
h : x ∈ { toArray := as.push x ++ bs, size_toArray := ⋯ }
⊢ ∃ n₁ n₂ as_1 bs_1 h,
{ toArray := as.push x ++ bs, size_toArray := ⋯ } = Vector.cast h (as_1.push x ++ bs_1) ∧ ¬x ∈ as_1 | exact ⟨_, _, as.toVector, bs.toVector, by simp, by simp, by simpa using w⟩ | no goals | 10d47003372b5a57 |
Lean.Data.AC.Context.mergeIdem_head2 | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.mergeIdem_head2 (h : x ≠ y) : mergeIdem (x :: y :: ys) = x :: mergeIdem (y :: ys) | x y : Nat
ys : List Nat
h : x ≠ y
⊢ mergeIdem (x :: y :: ys) = x :: mergeIdem (y :: ys) | simp [mergeIdem, mergeIdem.loop, h] | no goals | 07a99fd825e8807d |
HomologicalComplex.from_single_hom_ext | Mathlib/Algebra/Homology/Single.lean | @[ext]
lemma from_single_hom_ext {K : HomologicalComplex V c} {j : ι} {A : V}
{f g : (single V c j).obj A ⟶ K} (hfg : f.f j = g.f j) : f = g | case h
V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
K : HomologicalComplex V c
j : ι
A : V
f g : (single V c j).obj A ⟶ K
hfg : f.f j = g.f j
i : ι
⊢ f.f i = g.f i | by_cases h : i = j | case pos
V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
K : HomologicalComplex V c
j : ι
A : V
f g : (single V c j).obj A ⟶ K
hfg : f.f j = g.f j
i : ι
h : i = j
⊢ f.f i = g.f i
case neg
V : Type u
inst✝³ : Category.{v, u... | 6caea7f306a5b768 |
ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | theorem norm_compContinuous_linearIsometryEquiv (g : ContinuousMultilinearMap 𝕜 E₁ G)
(f : ∀ i, E i ≃ₗᵢ[𝕜] E₁ i) :
‖g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i)‖ = ‖g‖ | 𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
ins... | ext1 m | case H
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGrou... | 45114b2521c18aa7 |
SetTheory.PGame.Relabelling.birthday_congr | Mathlib/SetTheory/Game/Birthday.lean | theorem Relabelling.birthday_congr : ∀ {x y : PGame.{u}}, x ≡r y → birthday x = birthday y
| ⟨xl, xr, xL, xR⟩, ⟨yl, yr, yL, yR⟩, r => by
unfold birthday
congr 1
all_goals
apply lsub_eq_of_range_eq.{u, u, u}
ext i; constructor
all_goals rintro ⟨j, rfl⟩
· exact ⟨_, (r.moveLeft j).birthda... | case e_a.h
xl xr : Type u
xL : xl → PGame
xR : xr → PGame
yl yr : Type u
yL : yl → PGame
yR : yr → PGame
r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR
i : Ordinal.{u}
⊢ (i ∈ Set.range fun i => (xR i).birthday) ↔ i ∈ Set.range fun i => (yR i).birthday | constructor | case e_a.h.mp
xl xr : Type u
xL : xl → PGame
xR : xr → PGame
yl yr : Type u
yL : yl → PGame
yR : yr → PGame
r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR
i : Ordinal.{u}
⊢ (i ∈ Set.range fun i => (xR i).birthday) → i ∈ Set.range fun i => (yR i).birthday
case e_a.h.mpr
xl xr : Type u
xL : xl → PGame
xR : xr → PGame
... | 11800ebcdb8e9617 |
Polynomial.exists_prod_multiset_X_sub_C_mul | Mathlib/Algebra/Polynomial/Roots.lean | theorem exists_prod_multiset_X_sub_C_mul (p : R[X]) :
∃ q,
(p.roots.map fun a => X - C a).prod * q = p ∧
Multiset.card p.roots + q.natDegree = p.natDegree ∧ q.roots = 0 | case right.inr.right
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
he : p = (Multiset.map (fun a => X - C a) p.roots).prod * q
hq : q ≠ 0
⊢ q.roots = 0 | replace he := congr_arg roots he.symm | case right.inr.right
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hq : q ≠ 0
he : ((Multiset.map (fun a => X - C a) p.roots).prod * q).roots = p.roots
⊢ q.roots = 0 | 121409544deb5742 |
Ideal.Quotient.maximal_of_isField | Mathlib/RingTheory/Ideal/Quotient/Basic.lean | theorem maximal_of_isField {R} [CommRing R] (I : Ideal R) (hqf : IsField (R ⧸ I)) :
I.IsMaximal | case left.intro.mk.intro.mk
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hqf : IsField (R ⧸ I)
h : 1 ∈ I
w✝¹ : R ⧸ I
x : R
w✝ : R ⧸ I
y : R
hxy : Quot.mk (⇑(Submodule.quotientRel I)) x ≠ Quot.mk (⇑(Submodule.quotientRel I)) y
⊢ False | exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h)) | no goals | 5e093035780dd060 |
Std.Sat.AIG.denote_mkGate | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Lemmas.lean | theorem denote_mkGate {aig : AIG α} {input : GateInput aig} :
⟦aig.mkGate input, assign⟧
=
((⟦aig, input.lhs.ref, assign⟧ ^^ input.lhs.inv) && (⟦aig, input.rhs.ref, assign⟧ ^^ input.rhs.inv)) | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.GateInput
⊢ ⟦assign, aig.mkGate input⟧ =
((⟦assign, { aig := aig, ref := input.lhs.ref }⟧ ^^ input.lhs.inv) &&
(⟦assign, { aig := aig, ref := input.rhs.ref }⟧ ^^ input.rhs.inv)) | conv =>
lhs
unfold denote denote.go | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.GateInput
⊢ (match elem : (aig.mkGate input).aig.decls[(aig.mkGate input).ref.gate] with
| Decl.const b => b
| Decl.atom v => assign v
| Decl.gate lhs rhs linv rinv =>
let_fun this := ⋯;
let lval := deno... | 61ad764200642dc9 |
LieSubmodule.ucs_le_of_normalizer_eq_self | Mathlib/Algebra/Lie/Nilpotent.lean | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ | R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N₁ : LieSubmodule R L M
inst✝ : LieModule R L M
h : N₁.normalizer = N₁
k : ℕ
⊢ ucs k ⊥ ≤ ucs k N₁ | gcongr | case h
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N₁ : LieSubmodule R L M
inst✝ : LieModule R L M
h : N₁.normalizer = N₁
k : ℕ
⊢ ⊥ ≤ N₁ | 14ab1f8caa3b3baf |
ProbabilityTheory.IndepFun.integral_mul | Mathlib/Probability/Integration.lean | theorem IndepFun.integral_mul (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
⊢ integral μ (X * Y) = integral μ X * integral μ Y | by_cases h'X : X =ᵐ[μ] 0 | case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'X : X =ᶠ[ae μ] 0
⊢ integral μ (X * Y) = integral μ X * integral μ Y
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : ... | d5b07299b613cdd4 |
Array.eraseP_eq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem eraseP_eq_iff {p} {l : Array α} :
l.eraseP p = l' ↔
((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁.push a ++ l₂ ∧ l' = l₁ ++ l₂ | α : Type u_1
p : α → Bool
a : α
l₁ : List α
h₁ : ∀ (b : α), b ∈ { toList := l₁ } → p b = false
h₂ : p a = true
x : List α
⊢ { toList := l₁ }.toList ++ a :: { toList := x }.toList = l₁ ++ a :: x ∧
{ toList := l₁ }.toList ++ { toList := x }.toList = l₁ ++ x | simp | no goals | 348942f3ee07de42 |
PerfectClosure.natCast_eq_iff | Mathlib/FieldTheory/PerfectClosure.lean | theorem natCast_eq_iff (x y : ℕ) : (x : PerfectClosure K p) = y ↔ (x : K) = y | K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y : ℕ
⊢ ↑x = ↑y ↔ ↑x = ↑y | constructor <;> intro H | case mp
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y : ℕ
H : ↑x = ↑y
⊢ ↑x = ↑y
case mpr
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y : ℕ
H : ↑x = ↑y
⊢ ↑x = ↑y | f69df36b38bd9cb7 |
Int.negOnePow_odd | Mathlib/Algebra/Ring/NegOnePow.lean | lemma negOnePow_odd (n : ℤ) (hn : Odd n) : n.negOnePow = -1 | case intro
k : ℤ
⊢ (2 * k + 1).negOnePow = -1 | simp only [negOnePow_add, negOnePow_two_mul, negOnePow_one, mul_neg, mul_one] | no goals | d3df0ebb2872ddae |
Computation.eq_empty_of_not_terminates | Mathlib/Data/Seq/Computation.lean | theorem eq_empty_of_not_terminates {s} (H : ¬Terminates s) : s = empty α | case a.h.some
α : Type u
s : Computation α
H : ¬s.Terminates
n : ℕ
val✝ : α
h : ↑s n = some val✝
⊢ some val✝ = ↑(empty α) n | refine absurd ?_ H | case a.h.some
α : Type u
s : Computation α
H : ¬s.Terminates
n : ℕ
val✝ : α
h : ↑s n = some val✝
⊢ s.Terminates | 2288ee80b5ba77cf |
FixedPoints.linearIndependent_smul_of_linearIndependent | Mathlib/FieldTheory/Fixed.lean | theorem linearIndependent_smul_of_linearIndependent {s : Finset F} :
(LinearIndepOn (FixedPoints.subfield G F) id (s : Set F)) →
LinearIndepOn F (MulAction.toFun G F) s | G : Type u
inst✝² : Group G
F : Type v
inst✝¹ : Field F
inst✝ : MulSemiringAction G F
s✝ : Finset F
this✝ : IsEmpty ↑↑∅
a : F
s : Finset F
has : a ∉ s
ih : LinearIndepOn (↥(subfield G F)) id ↑s → LinearIndepOn F ⇑(toFun G F) ↑s
hs : LinearIndepOn (↥(subfield G F)) id ↑s ∧ id a ∉ Submodule.span (↥(subfield G F)) (id '' ... | exact Submodule.smul_mem _ _ <| Submodule.subset_span <| by simpa | no goals | 04111f46c8f406c8 |
Trivialization.preimageSingletonHomeomorph_symm_apply | Mathlib/Topology/FiberBundle/Trivialization.lean | theorem preimageSingletonHomeomorph_symm_apply {b : B} (hb : b ∈ e.baseSet) (p : F) :
(e.preimageSingletonHomeomorph hb).symm p =
⟨e.symm (b, p), by rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]⟩ :=
rfl
| B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e : Trivialization F proj
x : Z
b : B
hb : b ∈ e.baseSet
p : F
⊢ ↑e.symm (b, p) ∈ proj ⁻¹' {b} | rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff] | no goals | 2eebc993cf605e9e |
ProbabilityTheory.Kernel.measurableSet_absolutelyContinuous | Mathlib/Probability/Kernel/RadonNikodym.lean | /-- The set of points `a : α` such that `κ a ≪ η a` is measurable. -/
@[measurability]
lemma measurableSet_absolutelyContinuous (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
MeasurableSet {a | κ a ≪ η a} | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
⊢ MeasurableSet {a | κ a ≪ η a} | simp_rw [← singularPart_eq_zero_iff_absolutelyContinuous,
singularPart_eq_zero_iff_measure_eq_zero] | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
⊢ MeasurableSet {a | (κ a) (κ.mutuallySingularSetSlice η a) = 0} | 3b23a80313df013f |
exists_smul_not_mem_of_subset_orbit_closure | Mathlib/GroupTheory/Perm/ClosureSwap.lean | theorem exists_smul_not_mem_of_subset_orbit_closure (S : Set G) (T : Set α) {a : α}
(hS : ∀ g ∈ S, g⁻¹ ∈ S) (subset : T ⊆ orbit (closure S) a) (not_mem : a ∉ T)
(nonempty : T.Nonempty) : ∃ σ ∈ S, ∃ a ∈ T, σ • a ∉ T | G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
S : Set G
T : Set α
a : α
hS : ∀ g ∈ S, g⁻¹ ∈ S
subset : T ⊆ orbit (↥(closure S)) a
not_mem : a ∉ T
nonempty : T.Nonempty
key0 : ∀ σ ∈ S, ∀ a ∈ T, σ • a ∈ T
⊢ closure S ≤ stabilizer G T | refine (closure_le _).mpr fun σ hσ ↦ ?_ | G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
S : Set G
T : Set α
a : α
hS : ∀ g ∈ S, g⁻¹ ∈ S
subset : T ⊆ orbit (↥(closure S)) a
not_mem : a ∉ T
nonempty : T.Nonempty
key0 : ∀ σ ∈ S, ∀ a ∈ T, σ • a ∈ T
σ : G
hσ : σ ∈ S
⊢ σ ∈ ↑(stabilizer G T) | 3fc3ff0b0e667e8f |
Complex.stolzSet_empty | Mathlib/Analysis/Complex/AbelLimit.lean | theorem stolzSet_empty {M : ℝ} (hM : M ≤ 1) : stolzSet M = ∅ | M : ℝ
hM : M ≤ 1
z : ℂ
zn : 0 < 1 - ‖z‖
⊢ 1 * (1 - ‖z‖) = ‖1‖ - ‖z‖ | rw [one_mul, norm_one] | no goals | e2c2fd4395a83b42 |
contDiffWithinAt_succ_iff_hasFDerivWithinAt | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt (hn : n ≠ ∞) :
ContDiffWithinAt 𝕜 (n + 1) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, (n = ω → AnalyticOn 𝕜 f u) ∧
∃ f' : E → E →L[𝕜] F,
(∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u x | case mp.intro.intro.intro.intro.refine_1
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜... | exact Hp.analyticOn (H'p rfl 0) | no goals | 954f60033717bd9b |
Ordnode.Sized.rotateL | Mathlib/Data/Ordmap/Ordset.lean | theorem Sized.rotateL {l x r} (hl : @Sized α l) (hr : Sized r) : Sized (rotateL l x r) | case node
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
hr : (node size✝ l✝ x✝ r✝).Sized
⊢ (l.rotateL x (node size✝ l✝ x✝ r✝)).Sized | rw [Ordnode.rotateL_node] | case node
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
hr : (node size✝ l✝ x✝ r✝).Sized
⊢ (if l✝.size < ratio * r✝.size then l.node3L x l✝ x✝ r✝ else l.node4L x l✝ x✝ r✝).Sized | 485e683ec0a48852 |
CategoryTheory.Adjunction.CommShift.compatibilityUnit_unique_right | Mathlib/CategoryTheory/Shift/Adjunction.lean | /-- Given an adjunction `adj : F ⊣ G`, `a` in `A` and commutation isomorphisms
`e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a` and
`e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a`, if `e₁` and `e₂` are compatible with the
unit of the adjunction `adj`, then `e₁` uniquely determines `e₂`.
-/
lemma compatibilityUnit_... | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ e₂' : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : Compatibili... | rw [← Iso.symm_eq_iff] | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ e₂' : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : Compatibili... | 34f605425cffdef1 |
edist_le_Ico_sum_edist | Mathlib/Topology/EMetricSpace/Basic.lean | theorem edist_le_Ico_sum_edist (f : ℕ → α) {m n} (h : m ≤ n) :
edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1)) | α : Type u
inst✝ : PseudoEMetricSpace α
f : ℕ → α
m n : ℕ
h : m ≤ n
⊢ edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1)) | induction n, h using Nat.le_induction with
| base => rw [Finset.Ico_self, Finset.sum_empty, edist_self]
| succ n hle ihn =>
calc
edist (f m) (f (n + 1)) ≤ edist (f m) (f n) + edist (f n) (f (n + 1)) := edist_triangle _ _ _
_ ≤ (∑ i ∈ Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
_ = ∑ i ∈ Finset.Ico m (... | no goals | b091463af02f4290 |
Set.finite_iUnion_iff | Mathlib/Data/Set/Finite/Lattice.lean | /-- An indexed union of pairwise disjoint sets is finite iff all sets are finite, and all but
finitely many are empty. -/
lemma finite_iUnion_iff {ι : Type*} {s : ι → Set α} (hs : Pairwise fun i j ↦ Disjoint (s i) (s j)) :
(⋃ i, s i).Finite ↔ (∀ i, (s i).Finite) ∧ {i | (s i).Nonempty}.Finite where
mp h | α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j => Disjoint (s i) (s j)
h : (∀ (i : ι), (s i).Finite) ∧ {i | (s i).Nonempty}.Finite
⊢ ∀ i ∉ {i | (s i).Nonempty}, s i = ∅ | simp [not_nonempty_iff_eq_empty] | no goals | 11fa585e2b7f3d1f |
LieModule.iSup_genWeightSpaceOf_eq_top | Mathlib/Algebra/Lie/Weights/Basic.lean | @[simp]
lemma iSup_genWeightSpaceOf_eq_top [IsTriangularizable R L M] (x : L) :
⨆ (φ : R), genWeightSpaceOf M φ x = ⊤ | R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : LieRing.IsNilpotent L
inst✝ : IsTriangularizable R L M
x : L
⊢ ⨆ i, ((toEnd R L M) x).maxGenEigenspace i = ⊤ | exact IsTriangularizable.maxGenEigenspace_eq_top x | no goals | 9c7402b30bdd7d3e |
NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem mem_span_fractionalIdealLatticeBasis {x : (mixedSpace K)} :
x ∈ Submodule.span ℤ (Set.range (fractionalIdealLatticeBasis K I)) ↔
x ∈ mixedEmbedding K '' I | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(basisOfFractionalIdeal K I) | rw [← Set.range_comp] | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
Set.range (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(basisOfFractionalIdeal K I)) | cacf208e6e2e0cb3 |
Batteries.HashMap.Imp.insert_WF | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem insert_WF [BEq α] [Hashable α] {m : Imp α β} {k v}
(h : m.buckets.WF) : (insert m k v).buckets.WF | α : Type u_1
β : Type u_2
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
k : α
v : β
h : m.buckets.WF
c✝ : Bool
h₁ : ∀ (x : α × β), x ∈ m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat].toList → ¬(x.fst == k) = true
this : ?m.58846
⊢ (if numBucketsForCapacity (m.size + 1) ≤ m.buckets.val.size then
{ size := m.... | split <;> [exact this; refine expand_WF this] | no goals | ea3fade3a424c296 |
hasFDerivAt_jacobiTheta₂ | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | lemma hasFDerivAt_jacobiTheta₂ (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
HasFDerivAt (fun p : ℂ × ℂ ↦ jacobiTheta₂ p.1 p.2) (jacobiTheta₂_fderiv z τ) (z, τ) | z τ : ℂ
hτ : 0 < τ.im
T : ℝ
hT : 0 < T
hτ' : T < τ.im
S : ℝ
hz : |z.im| < S
V : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}
hVo : IsOpen V
hVmem : (z, τ) ∈ V
hVp : IsPreconnected V
f : ℤ → ℂ × ℂ → ℂ := fun n p => jacobiTheta₂_term n p.1 p.2
f' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p => jacobiTheta₂_term_fderiv n p... | refine mul_le_mul_of_nonneg_left ?_ (by positivity) | z τ : ℂ
hτ : 0 < τ.im
T : ℝ
hT : 0 < T
hτ' : T < τ.im
S : ℝ
hz : |z.im| < S
V : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}
hVo : IsOpen V
hVmem : (z, τ) ∈ V
hVp : IsPreconnected V
f : ℤ → ℂ × ℂ → ℂ := fun n p => jacobiTheta₂_term n p.1 p.2
f' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p => jacobiTheta₂_term_fderiv n p... | a8fb035d7bff2b99 |
IsPrimitiveRoot.norm_sub_one_two | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem norm_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ k)) (hk : 2 ≤ k)
[H : IsCyclotomicExtension {(2 : ℕ+) ^ k} K L] (hirr : Irreducible (cyclotomic (2 ^ k) K)) :
norm K (ζ - 1) = 2 | K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ k)
hk : 2 ≤ k
H : IsCyclotomicExtension {2 ^ k} K L
hirr : Irreducible (cyclotomic (2 ^ k) K)
this : 2 < 2 ^ k
⊢ (Algebra.norm K) (ζ - 1) = 2 | replace hirr : Irreducible (cyclotomic ((2 : ℕ+) ^ k : ℕ+) K) := by simp [hirr] | K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ k)
hk : 2 ≤ k
H : IsCyclotomicExtension {2 ^ k} K L
this : 2 < 2 ^ k
hirr : Irreducible (cyclotomic (↑(2 ^ k)) K)
⊢ (Algebra.norm K) (ζ - 1) = 2 | 06a533966d791d57 |
ZNum.cast_inj | Mathlib/Data/Num/Lemmas.lean | theorem cast_inj [LinearOrderedRing α] {m n : ZNum} : (m : α) = n ↔ m = n | α : Type u_1
inst✝ : LinearOrderedRing α
m n : ZNum
⊢ ↑m = ↑n ↔ m = n | rw [← cast_to_int m, ← cast_to_int n, Int.cast_inj (α := α), to_int_inj] | no goals | 12f01162eef397bc |
MeasureTheory.SimpleFunc.integral_piecewise_zero | Mathlib/MeasureTheory/Integral/BochnerL1.lean | theorem integral_piecewise_zero {m : MeasurableSpace α} (f : α →ₛ F) (μ : Measure α) {s : Set α}
(hs : MeasurableSet s) : (piecewise s hs f 0).integral μ = f.integral (μ.restrict s) | case refine_1
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ filter (fun x => x ≠ 0) (piecewise s hs f 0).range ⊆ filter (fun x => x ≠ 0) f.range | intro y hy | case refine_1
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ filter (fun x => x ≠ 0) (piecewise s hs f 0).range
⊢ y ∈ filter (fun x => x ≠ 0) f.range | 81410ec8ec0b4f37 |
lp.norm_eq_zero_iff | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem norm_eq_zero_iff {f : lp E p} : ‖f‖ = 0 ↔ f = 0 | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
h : ‖f‖ = 0
⊢ f = 0 | rcases p.trichotomy with (rfl | rfl | hp) | case inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E 0)
h : ‖f‖ = 0
⊢ f = 0
case inr.inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E ⊤)
h : ‖f‖ = 0
⊢ f = 0
case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCom... | b5b6304c5b4b6762 |
FirstOrder.Language.Substructure.fg_def | Mathlib/ModelTheory/FinitelyGenerated.lean | theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
exact ⟨t, rfl⟩⟩
| L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
⊢ (∃ S, S.Finite ∧ (closure L).toFun S = N) → N.FG | rintro ⟨t', h, rfl⟩ | case intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
t' : Set M
h : t'.Finite
⊢ ((closure L).toFun t').FG | ece7697297eb418a |
Matrix.etaExpand_eq | Mathlib/Data/Matrix/Reflection.lean | theorem etaExpand_eq {m n} (A : Matrix (Fin m) (Fin n) α) : etaExpand A = A | α : Type u_1
m n : ℕ
A : Matrix (Fin m) (Fin n) α
⊢ ((Equiv.refl (Fin m → Fin n → α)) fun i j => A i j) = A | erw [Equiv.refl_apply] | no goals | eaddf2ea7b6e46ad |
PadicInt.valuation_coe_nonneg | Mathlib/NumberTheory/Padics/PadicIntegers.lean | lemma valuation_coe_nonneg : 0 ≤ (x : ℚ_[p]).valuation | case inl
p : ℕ
hp : Fact (Nat.Prime p)
⊢ 0 ≤ (↑0).valuation | simp | no goals | 0e57645605d81c5d |
MaximalSpectrum.iInf_localization_eq_bot | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | theorem iInf_localization_eq_bot : (⨅ v : MaximalSpectrum R,
Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = ⊥ | case h.mp
R : Type u_4
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K
hrange : x ∉ range ⇑(algebraMap R K)
hlocal : ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯
⊢ False | let denom : Ideal R := (1 : Submodule R K).comap (LinearMap.toSpanSingleton R K x) | case h.mp
R : Type u_4
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K
hrange : x ∉ range ⇑(algebraMap R K)
hlocal : ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯
denom : Ideal R := Submodule.comap (Lin... | de9a014edf6d9bd5 |
CategoryTheory.MorphismProperty.multiplicativeClosure_le_iff | Mathlib/CategoryTheory/MorphismProperty/Composition.lean | /-- The multiplicative closure of `W` is the smallest multiplicative property greater than or equal
to `W`. -/
@[simp]
lemma multiplicativeClosure_le_iff (W' : MorphismProperty C) [W'.IsMultiplicative] :
multiplicativeClosure W ≤ W' ↔ W ≤ W' where
mp h := le_multiplicativeClosure W |>.trans h
mpr h | case id
C : Type u
inst✝¹ : Category.{v, u} C
W W' : MorphismProperty C
inst✝ : W'.IsMultiplicative
h : W ≤ W'
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
x : C
⊢ W' (𝟙 x) | exact W'.id_mem _ | no goals | eaad7e68c7e5f5d0 |
Std.Tactic.BVDecide.LRAT.Internal.Literal.unsat_of_limplies_complement | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/CNF.lean | theorem unsat_of_limplies_complement [Entails α t] (x : t) (l : Literal α) :
Limplies α x l → Limplies α x (Literal.negate l) → Unsatisfiable α x | α : Type u_1
t : Type u_2
inst✝ : Entails α t
x : t
l : Literal α
p : α → Bool
px : p ⊨ x
h1 : p ⊨ l
h2 : ¬(p ⊨ l)
⊢ False | exact h2 h1 | no goals | c0ec2a0ae699a166 |
Submodule.closure_induction | Mathlib/LinearAlgebra/Span/Defs.lean | theorem closure_induction {p : (x : M) → x ∈ span R s → Prop}
(zero : p 0 (Submodule.zero_mem _))
(add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›))
(smul_mem : ∀ (r x) (h : x ∈ s), p (r • x) (Submodule.smul_mem _ _ <| subset_span h)) {x}
(hx : x ∈ span R s) : p x hx | R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
p : (x : M) → x ∈ span R s → Prop
zero : p 0 ⋯
add : ∀ (x y : M) (hx : x ∈ span R s) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯
smul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯
x : M
hx : x ∈ span R s
v : M
... | simp [← span_eq_closure] | no goals | 3007157e1dc12f6a |
Complex.continuousAt_ofReal_cpow | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem continuousAt_ofReal_cpow (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0) :
ContinuousAt (fun p => (p.1 : ℂ) ^ p.2 : ℝ × ℂ → ℂ) (x, y) | case inr.inr.hg
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : x < 0
A : ContinuousAt (fun p => (-↑p.1, p.2)) (x, y)
⊢ ContinuousAt (fun x => cexp (↑π * I * x.2)) (x, y) | exact (continuous_exp.comp (continuous_const.mul continuous_snd)).continuousAt | no goals | f2161d3cdb933e68 |
le_rpow_one_add_norm_iff_norm_le | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 | E : Type u_1
inst✝ : NormedAddCommGroup E
r t : ℝ
hr : 0 < r
ht : 0 < t
x : E
⊢ t ≤ (1 + ‖x‖) ^ (-r) ↔ 1 + ‖x‖ ≤ t ^ (-r)⁻¹ | exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm | no goals | 0c3cb296245ac15c |
MvQPF.liftR_map_last | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | theorem liftR_map_last [lawful : LawfulMvFunctor F]
{α : TypeVec n} {ι ι'} (R : ι' → ι' → Prop)
(x : F (α ::: ι)) (f g : ι → ι') (hh : ∀ x : ι, R (f x) (g x)) :
LiftR' (RelLast' _ R) ((id ::: f) <$$> x) ((id ::: g) <$$> x) :=
let h : ι → { x : ι' × ι' // uncurry R x } := fun x => ⟨(f x, g x), hh x⟩
let ... | n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
lawful : LawfulMvFunctor F
α : TypeVec.{u} n
ι ι' : Type u
R : ι' → ι' → Prop
x : F (α ::: ι)
f g : ι → ι'
hh : ∀ (x : ι), R (f x) (g x)
h : ι → { x // uncurry R x } := fun x => ⟨(f x, g x), ⋯⟩
b : α ::: ι ⟹ Subtype_ α.repeatEq ::: { x // uncurry R x } := diagSub ::: h... | apply eq_of_drop_last_eq | case h₀
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
lawful : LawfulMvFunctor F
α : TypeVec.{u} n
ι ι' : Type u
R : ι' → ι' → Prop
x : F (α ::: ι)
f g : ι → ι'
hh : ∀ (x : ι), R (f x) (g x)
h : ι → { x // uncurry R x } := fun x => ⟨(f x, g x), ⋯⟩
b : α ::: ι ⟹ Subtype_ α.repeatEq ::: { x // uncurry R x } := diagS... | fb5eb40d1609bc2c |
nilpotencyClass_eq_quotient_center_plus_one | Mathlib/GroupTheory/Nilpotent.lean | theorem nilpotencyClass_eq_quotient_center_plus_one [hH : IsNilpotent G] [Nontrivial G] :
Group.nilpotencyClass G = Group.nilpotencyClass (G ⧸ center G) + 1 | G : Type u_1
inst✝¹ : Group G
hH : IsNilpotent G
inst✝ : Nontrivial G
⊢ nilpotencyClass G = nilpotencyClass (G ⧸ center G) + 1 | rw [nilpotencyClass_quotient_center] | G : Type u_1
inst✝¹ : Group G
hH : IsNilpotent G
inst✝ : Nontrivial G
⊢ nilpotencyClass G = nilpotencyClass G - 1 + 1 | 00de20c2847c157d |
Std.Tactic.BVDecide.LRAT.Internal.lratCheckerSound | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/LRATCheckerSound.lean | theorem lratCheckerSound [DecidableEq α] [Clause α β] [Entails α σ] [Formula α β σ] (f : σ)
(f_readyForRupAdd : ReadyForRupAdd f) (f_readyForRatAdd : ReadyForRatAdd f)
(prf : List (Action β α)) (prfWellFormed : ∀ a : Action β α, a ∈ prf → WellFormedAction a) :
lratChecker f prf = success → Unsatisfiable α f... | case intro
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝³ : DecidableEq α
inst✝² : Clause α β
inst✝¹ : Entails α σ
inst✝ : Formula α β σ
action : Action β α
restPrf : List (Action β α)
ih :
∀ (f : σ),
ReadyForRupAdd f →
ReadyForRatAdd f →
(∀ (a : Action β α), a ∈ restPrf → WellFormedAction a) → lratC... | unfold lratChecker | case intro
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝³ : DecidableEq α
inst✝² : Clause α β
inst✝¹ : Entails α σ
inst✝ : Formula α β σ
action : Action β α
restPrf : List (Action β α)
ih :
∀ (f : σ),
ReadyForRupAdd f →
ReadyForRatAdd f →
(∀ (a : Action β α), a ∈ restPrf → WellFormedAction a) → lratC... | daa30b736e5664b3 |
MvPolynomial.coeff_homogeneousComponent | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | theorem coeff_homogeneousComponent (d : σ →₀ ℕ) :
coeff d (homogeneousComponent n φ) = if d.degree = n then coeff d φ else 0 | σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
n : ℕ
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ coeff d ((homogeneousComponent n) φ) = if d.degree = n then coeff d φ else 0 | rw [degree_eq_weight_one] | σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
n : ℕ
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ coeff d ((homogeneousComponent n) φ) = if (weight fun x => 1) d = n then coeff d φ else 0 | 78fdb2dfb6a93a06 |
EquicontinuousOn.tendsto_uniformOnFun_iff_pi' | Mathlib/Topology/UniformSpace/Ascoli.lean | theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 <| (⋃₀ 𝔖).r... | case a
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
this : CompactSpace ↑K
⊢ Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(... | rw [← (equicontinuous_restrict_iff _ |>.mpr <| F_eqcont K hK).tendsto_uniformFun_iff_pi] | case a
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
this : CompactSpace ↑K
⊢ Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(... | df8bc1e377ab037c |
Ring.isField_iff_isSimpleOrder_ideal | Mathlib/RingTheory/Ideal/Basic.lean | theorem isField_iff_isSimpleOrder_ideal : IsField R ↔ IsSimpleOrder (Ideal R) | case inr
R : Type u_5
inst✝ : CommSemiring R
h✝ : Nontrivial R
⊢ (∀ (I : Ideal R), ⊥ < I → ¬I < ⊤) ↔ IsSimpleOrder (Ideal R) | simp_rw [lt_top_iff_ne_top, bot_lt_iff_ne_bot, ← or_iff_not_imp_left, not_ne_iff] | case inr
R : Type u_5
inst✝ : CommSemiring R
h✝ : Nontrivial R
⊢ (∀ (I : Ideal R), I = ⊥ ∨ I = ⊤) ↔ IsSimpleOrder (Ideal R) | c6361a19122a16e4 |
Matrix.Pivot.mul_listTransvecRow_last_col_take | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem mul_listTransvecRow_last_col_take (i : Fin r ⊕ Unit) {k : ℕ} (hk : k ≤ r) :
(M * ((listTransvecRow M).take k).prod) i (inr unit) = M i (inr unit) | case succ
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
i : Fin r ⊕ Unit
k : ℕ
IH : k ≤ r → (M * (List.take k (listTransvecRow M)).prod) i (inr ()) = M i (inr ())
hk : k + 1 ≤ r
⊢ (M * (List.take (k + 1) (listTransvecRow M)).prod) i (inr ()) = M i (inr ()) | have hkr : k < r := hk | case succ
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
i : Fin r ⊕ Unit
k : ℕ
IH : k ≤ r → (M * (List.take k (listTransvecRow M)).prod) i (inr ()) = M i (inr ())
hk : k + 1 ≤ r
hkr : k < r
⊢ (M * (List.take (k + 1) (listTransvecRow M)).prod) i (inr ()) = M i (inr ()) | 2f9310f51d6173b3 |
Std.Tactic.BVDecide.LRAT.Internal.Assignment.has_both | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Assignment.lean | theorem has_both (b : Bool) : hasAssignment b both = true | b : Bool
⊢ (if b = true then hasPosAssignment else hasNegAssignment) both = true | split <;> decide | no goals | 30680deabda19239 |
ENNReal.lintegral_prod_norm_pow_le | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
(s : Finset ι) {f : ι → α → ℝ≥0∞} (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
{p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) :
∫⁻ a, ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i | α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
(∀ i ∈ s, AEMeasurable (f i) μ) →
∀ {p : ι → ℝ},
∑ i ∈ s, p i = 1 →
(∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ ins... | simpa [hi₀, h2i₀, sum_eq_zero_iff_of_nonneg (fun i hi ↦ h2p i <| mem_insert_of_mem hi)]
using hp | no goals | c6067aa6ebb0db88 |
Ideal.injective_algebraMap_quotient_residueField | Mathlib/RingTheory/LocalRing/ResidueField/Ideal.lean | lemma Ideal.injective_algebraMap_quotient_residueField :
Function.Injective (algebraMap (R ⧸ I) I.ResidueField) | R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ map (Quotient.mk I) (RingHom.ker (algebraMap R I.ResidueField)) = ⊥ | rw [Ideal.ker_algebraMap_residueField, map_quotient_self] | no goals | a3bc889e6f8e164d |
Submodule.map_smul' | Mathlib/Algebra/Module/Submodule/Map.lean | theorem map_smul' (f : V →ₗ[K] V₂) (p : Submodule K V) (a : K) :
p.map (a • f) = ⨆ _ : a ≠ 0, map f p | K : Type u_9
V : Type u_10
V₂ : Type u_11
inst✝⁴ : Semifield K
inst✝³ : AddCommMonoid V
inst✝² : Module K V
inst✝¹ : AddCommMonoid V₂
inst✝ : Module K V₂
f : V →ₗ[K] V₂
p : Submodule K V
a : K
⊢ map (a • f) p = ⨆ (_ : a ≠ 0), map f p | by_cases h : a = 0 <;> simp [h, Submodule.map_smul] | no goals | 6f25a92f67697705 |
summable_mul_of_summable_norm' | Mathlib/Analysis/Normed/Field/InfiniteSum.lean | theorem summable_mul_of_summable_norm' {f : ι → R} {g : ι' → R}
(hf : Summable fun x => ‖f x‖) (h'f : Summable f)
(hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
Summable fun x : ι × ι' => f x.1 * g x.2 | R : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : NormedRing R
f : ι → R
g : ι' → R
hf : Summable fun x => ‖f x‖
h'f : Summable f
hg : Summable fun x => ‖g x‖
h'g : Summable g
s : Finset ι × Finset ι' → Finset (ι × ι') := fun p => p.1 ×ˢ p.2
⊢ HasSum (fun x => f x.1 * g x.2) ((∑' (i : ι), f i) * ∑' (j : ι'), g j) | apply hasSum_of_subseq_of_summable (hf.mul_norm hg) tendsto_finset_prod_atTop | R : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : NormedRing R
f : ι → R
g : ι' → R
hf : Summable fun x => ‖f x‖
h'f : Summable f
hg : Summable fun x => ‖g x‖
h'g : Summable g
s : Finset ι × Finset ι' → Finset (ι × ι') := fun p => p.1 ×ˢ p.2
⊢ Tendsto (fun b => ∑ i ∈ b.1 ×ˢ b.2, f i.1 * g i.2) atTop (𝓝 ((∑' (i : ι), f i... | 50537692fdc3b17a |
ack_add_one_sq_lt_ack_add_three | Mathlib/Computability/Ackermann.lean | theorem ack_add_one_sq_lt_ack_add_three : ∀ m n, (ack m n + 1) ^ 2 ≤ ack (m + 3) n
| 0, n => by simpa using sq_le_two_pow_add_one_minus_three (n + 2)
| m + 1, 0 => by
rw [ack_succ_zero, ack_succ_zero]
apply ack_add_one_sq_lt_ack_add_three
| m + 1, n + 1 => by
rw [ack_succ_succ, ack_succ_succ]
appl... | m n : ℕ
⊢ (ack (m + 1) (n + 1) + 1) ^ 2 ≤ ack (m + 1 + 3) (n + 1) | rw [ack_succ_succ, ack_succ_succ] | m n : ℕ
⊢ (ack m (ack (m + 1) n) + 1) ^ 2 ≤ ack (m + 3) (ack (m + 3 + 1) n) | 194740b4445cd1a9 |
Ideal.IsMaximal.comap_piEvalRingHom | Mathlib/RingTheory/Ideal/Maps.lean | theorem IsMaximal.comap_piEvalRingHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
{i : ι} {I : Ideal (R i)} (h : I.IsMaximal) : (I.comap <| Pi.evalRingHom R i).IsMaximal | ι : Type u_4
R : ι → Type u_5
inst✝ : (i : ι) → Semiring (R i)
i : ι
I : Ideal (R i)
h : I.IsMaximal
⊢ (comap (Pi.evalRingHom R i) I).IsMaximal | refine isMaximal_iff.mpr ⟨I.ne_top_iff_one.mp h.ne_top, fun J x le hxI hxJ ↦ ?_⟩ | ι : Type u_4
R : ι → Type u_5
inst✝ : (i : ι) → Semiring (R i)
i : ι
I : Ideal (R i)
h : I.IsMaximal
J : Ideal ((i : ι) → R i)
x : (i : ι) → R i
le : comap (Pi.evalRingHom R i) I ≤ J
hxI : x ∉ comap (Pi.evalRingHom R i) I
hxJ : x ∈ J
⊢ 1 ∈ J | f7c735e2e46e1091 |
CategoryTheory.ShortComplex.RightHomologyData.exact_iff_mono_g' | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma RightHomologyData.exact_iff_mono_g' [S.HasHomology] (h : RightHomologyData S) :
S.Exact ↔ Mono h.g' | case mpr
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
a✝ : Mono h.g'
⊢ S.Exact | simp only [h.exact_iff, IsZero.iff_id_eq_zero, ← cancel_mono h.ι, ← cancel_mono h.g',
id_comp, h.ι_g', zero_comp] | no goals | 4bba4cb9b4ae6113 |
Profinite.exists_isClopen_of_cofiltered | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V | case refine_3.intro.intro.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat.mapCone ... | refine ⟨j0, ⋃ (s : S) (_ : s ∈ G), W s, ?_, ?_⟩ | case refine_3.intro.intro.intro.intro.refine_1
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat... | 0fb8cdb8a6309b67 |
continuousAt_cpow | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem continuousAt_cpow {p : ℂ × ℂ} (hp_fst : p.fst ∈ slitPlane) :
ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) p | p : ℂ × ℂ
hp_fst : p.1 ∈ slitPlane
⊢ ContinuousAt (fun x => Complex.log x.1 * x.2) p | exact
ContinuousAt.mul
(ContinuousAt.comp (continuousAt_clog hp_fst) continuous_fst.continuousAt)
continuous_snd.continuousAt | no goals | 3f2ee431d8cc86e4 |
IsPrimitiveRoot.sub_one_norm_isPrimePow | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem sub_one_norm_isPrimePow (hn : IsPrimePow (n : ℕ)) [IsCyclotomicExtension {n} K L]
(hirr : Irreducible (cyclotomic (n : ℕ) K)) (h : n ≠ 2) : norm K (ζ - 1) = (n : ℕ).minFac | n : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
hζ : IsPrimitiveRoot ζ ↑n
hn : IsPrimePow ↑n
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic (↑n) K)
h : n ≠ 2
this : 2 < n
hprime : Fact (Nat.Prime (↑n).minFac) := { out := minFac_prime (IsPrimePow.ne_one hn)... | obtain ⟨k, hk⟩ : ∃ k, (n : ℕ).factorization (n : ℕ).minFac = k + 1 :=
exists_eq_succ_of_ne_zero
(((n : ℕ).factorization.mem_support_toFun (n : ℕ).minFac).1 <|
mem_primeFactors_iff_mem_primeFactorsList.2 <|
(mem_primeFactorsList (IsPrimePow.ne_zero hn)).2 ⟨hprime.out, minFac_dvd _⟩) | case intro
n : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
hζ : IsPrimitiveRoot ζ ↑n
hn : IsPrimePow ↑n
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic (↑n) K)
h : n ≠ 2
this : 2 < n
hprime : Fact (Nat.Prime (↑n).minFac) := { out := minFac_prime (IsPrimePow... | c894a93f498c96ce |
MeasureTheory.Measure.bind_dirac_eq_map | Mathlib/MeasureTheory/Measure/GiryMonad.lean | @[simp]
lemma bind_dirac_eq_map (m : Measure α) {f : α → β} (hf : Measurable f) :
m.bind (fun x ↦ Measure.dirac (f x)) = m.map f | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (dirac (f a)) s ∂m = (map f m) s
case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f... | swap | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable fun x => dirac (f x)
case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → β
hf : Measurable f
s : Set β
hs... | bc190068b855212f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRupUnits_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem assignmentsInvariant_insertRupUnits_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRupUnits f units).1 | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.ins... | intro c cf | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.ins... | 7537b9cac0fe9952 |
HasFTaylorSeriesUpToOn.comp | Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean | theorem HasFTaylorSeriesUpToOn.comp {n : WithTop ℕ∞} {g : F → G} {f : E → F}
(hg : HasFTaylorSeriesUpToOn n g q t) (hf : HasFTaylorSeriesUpToOn n f p s) (h : MapsTo f s t) :
HasFTaylorSeriesUpToOn n (g ∘ f) (fun x ↦ (q (f x)).taylorComp (p x)) s | case fderivWithin
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearS... | have B : HasFDerivWithinAt (fun x ↦ (q (f x)).taylorComp (p x) m)
(∑ c : OrderedFinpartition m, ∑ i : Option (Fin c.length),
((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x :=
HasFDerivWithinAt.sum (fun c _ ↦ A c) | case fderivWithin
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearS... | 1f8ef3617bd6b6ce |
Multiset.mem_pi | Mathlib/Data/Multiset/Pi.lean | theorem mem_pi (m : Multiset α) (t : ∀ a, Multiset (β a)) :
∀ f : ∀ a ∈ m, β a, f ∈ pi m t ↔ ∀ (a) (h : a ∈ m), f a h ∈ t a | case neg
α : Type u_1
inst✝ : DecidableEq α
β : α → Type u_2
t : (a : α) → Multiset (β a)
a : α
m : Multiset α
ih : ∀ (f : (a : α) → a ∈ m → β a), f ∈ m.pi t ↔ ∀ (a : α) (h : a ∈ m), f a h ∈ t a
b : β a
hb : b ∈ t a
f' : (a : α) → a ∈ m → β a
hf' : ∀ (a : α) (h : a ∈ m), f' a h ∈ t a
a' : α
ha' : a' ∈ a ::ₘ m
h : ¬a' =... | apply hf' | no goals | 1643be7189404780 |
Polynomial.isUnit_or_eq_zero_of_separable_expand | Mathlib/FieldTheory/Separable.lean | theorem isUnit_or_eq_zero_of_separable_expand {f : F[X]} (n : ℕ) (hp : 0 < p)
(hf : (expand F (p ^ n) f).Separable) : IsUnit f ∨ n = 0 | case intro.intro
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
n : ℕ
hp : 0 < p
hn : n ≠ 0
hf2 : derivative ((expand F (p ^ n)) f) = 0
r : F
hr : IsUnit r
hrf : C r = (expand F (p ^ n)) f
⊢ IsUnit f | rw [eq_comm, expand_eq_C (pow_pos hp _)] at hrf | case intro.intro
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
n : ℕ
hp : 0 < p
hn : n ≠ 0
hf2 : derivative ((expand F (p ^ n)) f) = 0
r : F
hr : IsUnit r
hrf : f = C r
⊢ IsUnit f | 720e5d3b175c804d |
AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom | Mathlib/AlgebraicGeometry/Scheme.lean | theorem Scheme.Spec_map_presheaf_map_eqToHom {X : Scheme} {U V : X.Opens} (h : U = V) (W) :
(Spec.map (X.presheaf.map (eqToHom h).op)).app W = eqToHom (by cases h; dsimp; simp) | case refl
X : Scheme
U : X.Opens
this : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 (Scheme.Spec.obj (op Γ(X, U)))
W : (Spec Γ(X, U)).Opens
⊢ Hom.app (𝟙 (Scheme.Spec.obj (op Γ(X, U)))) W ≫ (Scheme.Spec.obj (op Γ(X, U))).presheaf.map (eqToHom ⋯).op = eqToHom ⋯ | simp [eqToHom_map] | no goals | 6819d9d83e818d30 |
HasFPowerSeriesWithinOnBall.unshift | Mathlib/Analysis/Analytic/Constructions.lean | theorem HasFPowerSeriesWithinOnBall.unshift (hf : HasFPowerSeriesWithinOnBall f pf s x r) :
HasFPowerSeriesWithinOnBall (fun y ↦ z + f y (y - x)) (pf.unshift z) s x r where
r_le | 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → E →L[𝕜] F
pf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
s : Set E
x : E
r : ℝ≥0∞
z : F
hf : HasFPowerSeriesWithinOnBall f p... | exact hf.r_le | no goals | e49bc65afdf730fa |
Continuous.matrix_det | Mathlib/Topology/Instances/Matrix.lean | theorem Continuous.matrix_det [Fintype n] [DecidableEq n] [CommRing R] [IsTopologicalRing R]
{A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).det | X : Type u_1
n : Type u_5
R : Type u_8
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace R
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : IsTopologicalRing R
A : X → Matrix n n R
hA : Continuous A
l : Equiv.Perm n
x✝ : l ∈ Finset.univ
⊢ Continuous fun x => ∏ i : n, A x (l i) i | exact continuous_finset_prod _ fun l _ => hA.matrix_elem _ _ | no goals | be69dc6c9d5a4b6b |
equicontinuousWithinAt_iff_pair | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem equicontinuousWithinAt_iff_pair {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S) :
EquicontinuousWithinAt F S x₀ ↔
∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U | case mpr.intro.intro
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
hx₀ : x₀ ∈ S
H : ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set X
hV : V ∈ 𝓝[S] x₀
hVU : ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι),... | filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhdsWithin hx₀ hV) x hx i | no goals | fef20fe686908e6e |
Finset.prod_add | Mathlib/Algebra/BigOperators/Ring/Finset.lean | theorem prod_add (f g : ι → α) (s : Finset ι) :
∏ i ∈ s, (f i + g i) = ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, g i | ι : Type u_1
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq ι
f g : ι → α
s : Finset ι
⊢ ∏ i ∈ s, (f i + g i) = ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, g i | classical
calc
∏ i ∈ s, (f i + g i) =
∏ i ∈ s, ∑ p ∈ ({True, False} : Finset Prop), if p then f i else g i := by simp
_ = ∑ p ∈ (s.pi fun _ => {True, False} : Finset (∀ a ∈ s, Prop)),
∏ a ∈ s.attach, if p a.1 a.2 then f a.1 else g a.1 := prod_sum _ _ _
_ = ∑ t ∈ s.powerset, (∏ a ∈ t, f a) * ∏ a ∈ s ... | no goals | bfeb78d0575b25b6 |
UniformSpace.metrizable_uniformity | Mathlib/Topology/Metrizable/Uniformity.lean | theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]
[IsCountablyGenerated (𝓤 X)] : ∃ I : PseudoMetricSpace X, I.toUniformSpace = ‹_› | X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → Set (X × X)
hU_symm : ∀ (n : ℕ), SymmetricRel (U n)
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
x y : X
⊢ (if... | split_ifs with h | case pos
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → Set (X × X)
hU_symm : ∀ (n : ℕ), SymmetricRel (U n)
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
x y ... | 6b4b13aa9c62ee27 |
RingQuot.mkRingHom_surjective | Mathlib/Algebra/RingQuot.lean | theorem mkRingHom_surjective (r : R → R → Prop) : Function.Surjective (mkRingHom r) | R : Type uR
inst✝ : Semiring R
r : R → R → Prop
⊢ Function.Surjective ⇑(mkRingHom r) | simp only [mkRingHom_def, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk] | R : Type uR
inst✝ : Semiring R
r : R → R → Prop
⊢ Function.Surjective fun x => { toQuot := Quot.mk (Rel r) x } | 9802ddc4f3e92c65 |
Finset.preimage_subset_of_subset_image | Mathlib/Data/Finset/Preimage.lean | lemma preimage_subset_of_subset_image [DecidableEq β] {f : α → β} {s : Finset β} {t : Finset α}
(hs : s ⊆ t.image f) {hf} : s.preimage f hf ⊆ t | α : Type u
β : Type v
inst✝ : DecidableEq β
f : α → β
s : Finset β
t : Finset α
hs : s ⊆ image f t
hf : InjOn f (f ⁻¹' ↑s)
⊢ f ⁻¹' ↑s ⊆ ↑t | exact Set.preimage_subset (mod_cast hs) hf | no goals | 85f1757466c99d85 |
DedekindDomain.ProdAdicCompletions.IsFiniteAdele.neg | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | theorem neg {x : K_hat R K} (hx : x.IsFiniteAdele) : (-x).IsFiniteAdele | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K_hat R K
hx : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, x v ∈ adicCompletionIntegers K v
h : ∀ (v : HeightOneSpectrum R), -x v ∈ adicCompletionIntegers K v ↔ x v ∈ adicC... | convert hx using 2 with v | case h.e'_2.h.a
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K_hat R K
hx : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, x v ∈ adicCompletionIntegers K v
h : ∀ (v : HeightOneSpectrum R), -x v ∈ adicCompletionIntegers K... | ec8f2cb785d1d049 |
IsCompact.exists_open_superset_measure_lt_top' | Mathlib/MeasureTheory/Measure/Typeclasses.lean | theorem exists_open_superset_measure_lt_top' (h : IsCompact s)
(hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)) : ∃ U ⊇ s, IsOpen U ∧ μ U < ∞ | case refine_4
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : IsCompact s
hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)
x : α
hx : x ∈ s
⊢ ∃ t ∈ 𝓝[s] x, ∃ U ⊇ t, IsOpen U ∧ μ U < ⊤ | rcases (hμ x hx).exists_mem_basis (nhds_basis_opens _) with ⟨U, ⟨hx, hUo⟩, hU⟩ | case refine_4.intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : IsCompact s
hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)
x : α
hx✝ : x ∈ s
U : Set α
hU : μ U < ⊤
hx : x ∈ U
hUo : IsOpen U
⊢ ∃ t ∈ 𝓝[s] x, ∃ U ⊇ t, IsOpen U ∧ μ U < ⊤ | 316e20d6f16448eb |
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