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 |
|---|---|---|---|---|---|---|
BoxIntegral.IntegrationParams.MemBaseSet.exists_common_compl | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | theorem MemBaseSet.exists_common_compl
(h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂)
(hU : π₁.iUnion = π₂.iUnion) :
∃ π : Prepartition I, π.iUnion = ↑I \ π₁.iUnion ∧
(l.bDistortion → π.distortion ≤ c₁) ∧ (l.bDistortion → π.distortion ≤ c₂) | ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c₁ c₂ : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ r₁ π₁
h₂ : l.MemBaseSet I c₂ r₂ π₂
hU : π₁.iUnion = π₂.iUnion
⊢ ∃ π,
π.iUnion = ↑I \ π₁.iUnion ∧ (l.bDistortion = true → π.distortion ≤ c₁) ∧ (l.bDistortion =... | wlog hc : c₁ ≤ c₂ with H | case inr
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c₁ c₂ : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ r₁ π₁
h₂ : l.MemBaseSet I c₂ r₂ π₂
hU : π₁.iUnion = π₂.iUnion
H :
∀ {ι : Type u_1} [inst : Fintype ι] {I : Box ι} {c₁ c₂ : ℝ≥0} {l : IntegrationParams}... | 2df11db01408e2ca |
tendsto_tsum_of_dominated_convergence | Mathlib/Analysis/Normed/Group/Tannery.lean | /-- **Tannery's theorem**: topological sums commute with termwise limits, when the norms of the
summands are eventually uniformly bounded by a summable function.
(This is the special case of the Lebesgue dominated convergence theorem for the counting measure
on a discrete set. However, we prove it under somewhat weake... | case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.N... | obtain ⟨T, hT⟩ : ∃ (T : Finset β), dist (∑ b ∈ T, bound b) S < ε / 3 := by
rw [HasSum, Metric.tendsto_nhds] at hS
classical exact Eventually.exists <| hS _ (by positivity) | case inr.inr.intro
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ ... | d4198172206cf97e |
toSubalgebra_toIntermediateField | Mathlib/FieldTheory/IntermediateField/Basic.lean | theorem toSubalgebra_toIntermediateField (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) :
(S.toIntermediateField inv_mem).toSubalgebra = S | K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
S : Subalgebra K L
inv_mem : ∀ x ∈ S, x⁻¹ ∈ S
⊢ (S.toIntermediateField inv_mem).toSubalgebra = S | ext | case h
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
S : Subalgebra K L
inv_mem : ∀ x ∈ S, x⁻¹ ∈ S
x✝ : L
⊢ x✝ ∈ (S.toIntermediateField inv_mem).toSubalgebra ↔ x✝ ∈ S | 2225d3d557e3190c |
lp.sum_rpow_le_of_tendsto | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem sum_rpow_le_of_tendsto (hp : p ≠ ∞) {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C)
{f : ∀ a, E a} (hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) (s : Finset α) :
∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C ^ p.toReal | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)
s : Finset α
⊢ ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C ^ p... | have hp' : p ≠ 0 := (zero_lt_one.trans_le _i.elim).ne' | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i => ↑(F i)) l (𝓝 f)
s : Finset α
hp' : p ≠ 0
⊢ ∑ i ∈ s, ‖f i‖ ^ p.to... | 3c76cdccdea461fc |
Orthonormal.sum_inner_products_le | Mathlib/Analysis/InnerProductSpace/Orthonormal.lean | theorem Orthonormal.sum_inner_products_le {s : Finset ι} (hv : Orthonormal 𝕜 v) :
∑ i ∈ s, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2 | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k ∈ s, inner (v k) x * inner x (v k)
⊢ ∑ i ∈ s, ‖inner (v i) x‖ ^ 2 ... | have h₃ : ∀ z : 𝕜, re (z * conj z) = ‖z‖ ^ 2 := by
intro z
simp only [mul_conj, normSq_eq_def']
norm_cast | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, inner (v i) x * inner x (v j) * inner (v j) (v i) = ∑ k ∈ s, inner (v k) x * inner x (v k)
h₃ : ∀ (z : 𝕜), re (z * (starR... | 991788cdd3b9c14c |
List.pairwise_replicate | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Pairwise.lean | theorem pairwise_replicate {n : Nat} {a : α} :
(replicate n a).Pairwise R ↔ n ≤ 1 ∨ R a a | case pos.h
α : Type u_1
R : α → α → Prop
a : α
n : Nat
ih : Pairwise R (replicate n a) ↔ n ≤ 1 ∨ R a a
h : ¬n = 0 → R a a
h' : n ≤ 1
w : n = 0
⊢ n + 1 ≤ 1 | subst w | case pos.h
α : Type u_1
R : α → α → Prop
a : α
ih : Pairwise R (replicate 0 a) ↔ 0 ≤ 1 ∨ R a a
h : ¬0 = 0 → R a a
h' : 0 ≤ 1
⊢ 0 + 1 ≤ 1 | 2d4c357c8a7b913e |
Filter.compl_mem_kernMap | Mathlib/Order/Filter/Map.lean | theorem compl_mem_kernMap {s : Set β} : sᶜ ∈ kernMap m f ↔ ∃ t, tᶜ ∈ f ∧ m '' t = s | α : Type u_1
β : Type u_2
m : α → β
f : Filter α
s : Set β
⊢ sᶜ ∈ kernMap m f ↔ ∃ t, tᶜ ∈ f ∧ m '' t = s | simp_rw [mem_kernMap_iff_compl, compl_compl] | no goals | 620ad1b8fac9b277 |
Convex.helly_theorem' | Mathlib/Analysis/Convex/Radon.lean | theorem helly_theorem' {F : ι → Set E} {s : Finset ι}
(h_convex : ∀ i ∈ s, Convex 𝕜 (F i))
(h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
(⋂ i ∈ s, F i).Nonempty | case h.h.intro.intro.a
𝕜 : Type u_2
E : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
(∀ i ∈ s, Convex 𝕜 (F i)) →
(∀ I ⊆ s, #I ≤ finrank 𝕜 E + ... | apply Nonempty.some_mem | no goals | d589b7c5c4686ede |
FirstOrder.Language.Theory.exists_large_model_of_infinite_model | Mathlib/ModelTheory/Satisfiability.lean | theorem exists_large_model_of_infinite_model (T : L.Theory) (κ : Cardinal.{w}) (M : Type w')
[L.Structure M] [M ⊨ T] [Infinite M] :
∃ N : ModelType.{_, _, max u v w} T, Cardinal.lift.{max u v w} κ ≤ #N | case intro
L : Language
T : L.Theory
κ : Cardinal.{w}
M : Type w'
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Infinite M
N : ((L.lhomWithConstants (Quotient.out κ)).onTheory T ∪ L.distinctConstantsTheory Set.univ).ModelType
this : ↑N ⊨ L.distinctConstantsTheory Set.univ
⊢ lift.{max u v w, w} κ ≤ #↑N | refine _root_.trans (lift_le.2 (le_of_eq (Cardinal.mk_out κ).symm)) ?_ | case intro
L : Language
T : L.Theory
κ : Cardinal.{w}
M : Type w'
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Infinite M
N : ((L.lhomWithConstants (Quotient.out κ)).onTheory T ∪ L.distinctConstantsTheory Set.univ).ModelType
this : ↑N ⊨ L.distinctConstantsTheory Set.univ
⊢ lift.{max (max u v) w, w} #(Quotient.out κ) ≤... | cb2210ff1dd443c9 |
AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus | Mathlib/AlgebraicGeometry/AffineScheme.lean | lemma IsAffineOpen.fromSpec_image_zeroLocus {X : Scheme.{u}} {U : X.Opens}
(hU : IsAffineOpen U) (s : Set Γ(X, U)) :
hU.fromSpec.base '' PrimeSpectrum.zeroLocus s = X.zeroLocus s ∩ U | X : Scheme
U : X.Opens
hU : IsAffineOpen U
s : Set ↑Γ(X, U)
⊢ ⇑(ConcreteCategory.hom hU.fromSpec.base) '' PrimeSpectrum.zeroLocus s = X.zeroLocus s ∩ ↑U | rw [← hU.fromSpec_preimage_zeroLocus, Set.image_preimage_eq_inter_range, range_fromSpec] | no goals | 4ba67f81fdf1eb9e |
MeasureTheory.Measure.haar.chaar_sup_eq | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem chaar_sup_eq {K₀ : PositiveCompacts G}
{K₁ K₂ : Compacts G} (h : Disjoint K₁.1 K₂.1) (h₂ : IsClosed K₂.1) :
chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ | case intro.intro.intro.intro.h.refine_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
h₂ : IsClosed K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆... | exact Subset.trans (inv_subset.mpr h1U) inter_subset_right | no goals | f819296e30b05cf2 |
mellin_comp_mul_left | Mathlib/Analysis/MellinTransform.lean | theorem mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (fun t => f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ¬a = 0 ∨ ¬s - 1 ≠ 0 | exact Or.inl ha.ne' | no goals | eba6520b29ea7f1f |
Ordinal.card_iSup_Iio_le_card_mul_iSup | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem card_iSup_Iio_le_card_mul_iSup {o : Ordinal.{u}} (f : Iio o → Ordinal.{max u v}) :
(⨆ a : Iio o, f a).card ≤ Cardinal.lift.{v} o.card * ⨆ a : Iio o, (f a).card | o : Ordinal.{u}
f : ↑(Iio o) → Ordinal.{max u v}
⊢ (sum fun i => (f (o.enumIsoToType.symm i)).card) ≤ Cardinal.lift.{v, u} o.card * ⨆ a, (f a).card | convert ← sum_le_iSup_lift _ | case h.e'_4.h.e'_5.h.e'_1
o : Ordinal.{u}
f : ↑(Iio o) → Ordinal.{max u v}
⊢ #o.toType = o.card
case h.e'_4.h.e'_6
o : Ordinal.{u}
f : ↑(Iio o) → Ordinal.{max u v}
⊢ ⨆ i, (f (o.enumIsoToType.symm i)).card = ⨆ a, (f a).card | c3d702eab5714f39 |
IsSimpleRing.isField_center | Mathlib/RingTheory/SimpleRing/Field.lean | lemma isField_center (A : Type*) [Ring A] [IsSimpleRing A] : IsField (Subring.center A) where
exists_pair_ne := ⟨0, 1, zero_ne_one⟩
mul_comm := mul_comm
mul_inv_cancel | A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x✝ : A
hx1✝ : x✝ ∈ Subring.center A
hx1 : ∀ (g : A), g * x✝ = x✝ * g
hx2 : x✝ ≠ 0
x : A
⊢ (fun x => x✝ * x) (-x) = -(fun x => x✝ * x) x | simp | no goals | 08d555f2a5a9e7b4 |
CategoryTheory.Equivalence.changeFunctor_refl | Mathlib/CategoryTheory/Equivalence.lean | theorem changeFunctor_refl (e : C ≌ D) : e.changeFunctor (Iso.refl _) = e | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
⊢ e.changeFunctor (Iso.refl e.functor) = e | aesop_cat | no goals | d0b11db58d671a14 |
WeierstrassCurve.Projective.toAffine_negAddY_of_eq | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | private lemma toAffine_negAddY_of_eq {P : Fin 3 → F} (hPz : P z ≠ 0) {n d : F} (hd : d ≠ 0) :
W.toAffine.negAddY (P x / P z) (P x / P z) (P y / P z) (-n / P z / d) =
(-n * (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2
- P x * P z * d ^ 2) + P y * P z ^ 2 * d ^ 3) / P z... | F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
n d : F
hd : d ≠ 0
⊢ (-(n *
((n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2) * d * P z -
P z * (P z * d ^ 3) * P x) *
P z) +
P y * (P z * d * (P z * (P z * d ^ 3) * P z... | ring1 | no goals | 236121ded01b30f2 |
DirectSum.coe_of_mul_apply_aux | Mathlib/Algebra/DirectSum/Internal.lean | theorem coe_of_mul_apply_aux [AddMonoid ι] [SetLike.GradedMonoid A] {i : ι} (r : A i)
(r' : ⨁ i, A i) {j n : ι} (H : ∀ x : ι, i + x = n ↔ x = j) :
((of (fun i => A i) i r * r') n : R) = r * r' j | ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : AddMonoid ι
inst✝ : SetLike.GradedMonoid A
i : ι
r : ↥(A i)
r' : ⨁ (i : ι), ↥(A i)
j n : ι
H : ∀ (x : ι), i + x = n ↔ x = j
⊢ ↑(((of (fun i => ↥(A i)) i) r * r') n) = ... | rw [coe_mul_apply_eq_dfinsupp_sum] | ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : Semiring R
inst✝³ : SetLike σ R
inst✝² : AddSubmonoidClass σ R
A : ι → σ
inst✝¹ : AddMonoid ι
inst✝ : SetLike.GradedMonoid A
i : ι
r : ↥(A i)
r' : ⨁ (i : ι), ↥(A i)
j n : ι
H : ∀ (x : ι), i + x = n ↔ x = j
⊢ (DFinsupp.sum ((of (fun i => ↥(A i)) i) r... | 9f8e036478d3d2eb |
BitVec.getLsbD_twoPow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_twoPow (i j : Nat) : (twoPow w i).getLsbD j = ((i < w) && (i = j)) | case pos
i j w : Nat
hj : j < i
⊢ i < w + 1 → ¬i = j | omega | no goals | ee3f3c24ee66326c |
ZFSet.rank_eq_wfRank | Mathlib/SetTheory/ZFC/Rank.lean | theorem rank_eq_wfRank : lift.{u + 1, u} (rank x) = IsWellFounded.rank (α := ZFSet) (· ∈ ·) x | x : ZFSet.{u}
⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 => x1 ∈ x2) x | induction' x using inductionOn with x ih | case h
x✝ x : ZFSet.{u}
ih : ∀ y ∈ x, lift.{u + 1, u} y.rank = IsWellFounded.rank (fun x1 x2 => x1 ∈ x2) y
⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 => x1 ∈ x2) x | 677b5f91f98d45d8 |
Ring.DirectLimit.lift_unique | Mathlib/Algebra/Colimit/Ring.lean | theorem lift_unique (F : DirectLimit G f →+* P) (x) :
F x = lift G f P (fun i ↦ F.comp <| of G f i) (fun i j hij x ↦ by simp) x | ι : Type u_1
inst✝² : Preorder ι
G : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝ : CommRing P
F : DirectLimit G f →+* P
x : DirectLimit G f
⊢ F x = (lift G f P (fun i => F.comp (of G f i)) ⋯) x | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | case intro
ι : Type u_1
inst✝² : Preorder ι
G : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝ : CommRing P
F : DirectLimit G f →+* P
x : FreeCommRing ((i : ι) × G i)
⊢ F
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ... | 051b292ba4e666cf |
LinearMap.ker_le_iff | Mathlib/Algebra/Module/Submodule/Range.lean | theorem ker_le_iff [RingHomSurjective τ₁₂] {p : Submodule R M} :
ker f ≤ p ↔ ∃ y ∈ range f, f ⁻¹' {y} ⊆ p | case mpr.intro.intro
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Ring R
inst✝⁷ : Ring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
f : F
inst✝ : RingHomSurjecti... | rw [← SetLike.mem_coe, range_coe, Set.mem_range] at h₁ | case mpr.intro.intro
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Ring R
inst✝⁷ : Ring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
f : F
inst✝ : RingHomSurjecti... | 4607f9408a9aa2c2 |
Matrix.adjugate_fin_succ_eq_det_submatrix | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | theorem adjugate_fin_succ_eq_det_submatrix {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) α) (i j) :
adjugate A i j = (-1) ^ (j + i : ℕ) * det (A.submatrix j.succAbove i.succAbove) | α : Type w
inst✝ : CommRing α
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) α
i j h : Fin n.succ
hjk : h ≠ i
⊢ (-1) ^ (↑j + ↑h) * Pi.single i 1 h * (A.submatrix j.succAbove h.succAbove).det = 0 | rw [Pi.single_eq_of_ne hjk, mul_zero, zero_mul] | no goals | fb700ffdfc785acd |
Coalgebra.sum_map_tmul_tmul_eq | Mathlib/RingTheory/Coalgebra/Basic.lean | theorem sum_map_tmul_tmul_eq {B : Type*} [AddCommMonoid B] [Module R B]
{F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f g h : F) (a : A) {repr : Repr R a}
{a₁ : (i : repr.ι) → Repr R (repr.left i)} {a₂ : (i : repr.ι) → Repr R (repr.right i)} :
∑ i ∈ repr.index, ∑ j ∈ (a₂ i).index,
f (repr.left... | R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : Module R A
inst✝⁴ : Coalgebra R A
B : Type u_1
inst✝³ : AddCommMonoid B
inst✝² : Module R B
F : Type u_2
inst✝¹ : FunLike F A B
inst✝ : LinearMapClass F R A B
f g h : F
a : A
repr : Repr R a
a₁ : (i : repr.ι) → Repr R (repr.left i)
a₂ : (i ... | simp_all only [map_sum, TensorProduct.map_tmul, LinearMap.coe_coe] | no goals | b7ef1cfd9f54811e |
ZMod.eq_one_or_isUnit_sub_one | Mathlib/FieldTheory/Finite/Basic.lean | theorem ZMod.eq_one_or_isUnit_sub_one {n p k : ℕ} [Fact p.Prime] (hn : n = p ^ k) (a : ZMod n)
(ha : (orderOf a).Coprime n) : a = 1 ∨ IsUnit (a - 1) | case inr.inr
n p k : ℕ
inst✝ : Fact (Nat.Prime p)
hn : n = p ^ k
a : ZMod n
ha : (orderOf a).Coprime n
hn0 : n ≠ 0
ha0 : a ≠ 0
this : NeZero n
⊢ a = 1 ∨ IsUnit (a - 1) | obtain ⟨a, rfl⟩ := ZMod.natCast_zmod_surjective a | case inr.inr.intro
n p k : ℕ
inst✝ : Fact (Nat.Prime p)
hn : n = p ^ k
hn0 : n ≠ 0
this : NeZero n
a : ℕ
ha : (orderOf ↑a).Coprime n
ha0 : ↑a ≠ 0
⊢ ↑a = 1 ∨ IsUnit (↑a - 1) | e140f007bb03a360 |
NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_dvd_norm_le | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean | theorem card_isPrincipal_dvd_norm_le (s : ℝ) :
Nat.card {I : (Ideal (𝓞 K))⁰ // (J : Ideal (𝓞 K)) ∣ I ∧ IsPrincipal (I : Ideal (𝓞 K)) ∧
absNorm (I : Ideal (𝓞 K)) ≤ s} * torsionOrder K =
Nat.card {a : idealSet K J // mixedEmbedding.norm (a : mixedSpace K) ≤ s} | case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hs : 0 ≤ s
x✝ : { x // x ∈ Finset.Iic ⌊s⌋₊ }
i : ℕ
hi : i ∈ Finset.Iic ⌊s⌋₊
⊢ { a // (fun I => ⟨absNorm ↑↑I.1, ⋯⟩) a = ⟨i, hi⟩ } ≃
{ b // (fun a => ⟨intNorm ↑((idealSetEquiv K J) ↑a), ⋯⟩) b = ⟨i, hi⟩ } | simp_rw [Subtype.mk.injEq] | case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hs : 0 ≤ s
x✝ : { x // x ∈ Finset.Iic ⌊s⌋₊ }
i : ℕ
hi : i ∈ Finset.Iic ⌊s⌋₊
⊢ { a // absNorm ↑↑a.1 = i } ≃ { b // intNorm ↑((idealSetEquiv K J) ↑b) = i } | 676cab8481af674a |
convex_stdSimplex | Mathlib/Analysis/Convex/Basic.lean | theorem convex_stdSimplex : Convex 𝕜 (stdSimplex 𝕜 ι) | case refine_2
𝕜 : Type u_1
ι : Type u_5
inst✝¹ : OrderedSemiring 𝕜
inst✝ : Fintype ι
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a + b = 1 | exact hab | no goals | ecf6e08a2aba6b78 |
ClassGroup.exists_mk0_eq_mk0 | Mathlib/NumberTheory/ClassNumber/Finite.lean | theorem exists_mk0_eq_mk0 [IsDedekindDomain S] [Algebra.IsAlgebraic R S] (I : (Ideal S)⁰) :
∃ J : (Ideal S)⁰,
ClassGroup.mk0 I = ClassGroup.mk0 J ∧
algebraMap _ _ (∏ m ∈ finsetApprox bS adm, m) ∈ (J : Ideal S) | R : Type u_1
S : Type u_2
inst✝⁹ : EuclideanDomain R
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra R S
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁵ : DecidableEq ι
inst✝⁴ : Fintype ι
bS : Basis ι R S
adm : abv.IsAdmissible
inst✝³ : Infinite R
inst✝² : DecidableEq R
inst✝¹ : IsDedekindDomain S
inst✝ : Algebra.... | exact b_mem | no goals | b8ed2f844c81c819 |
CStarModule.norm_triangle | Mathlib/Analysis/CStarAlgebra/Module/Defs.lean | protected lemma norm_triangle (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ | A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x + y‖ ≤ ‖x‖ + ‖y‖ | have h : ‖x + y‖ ^ 2 ≤ (‖x‖ + ‖y‖) ^ 2 := by
calc _ ≤ ‖⟪x, x⟫ + ⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ := by
simp only [norm_eq_sqrt_norm_inner_self, inner_add_right, inner_add_left, ← add_assoc,
norm_nonneg, Real.sq_sqrt]
exact norm_add₃_le
_ ≤ ‖⟪x, x⟫‖ + ‖⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ := by gco... | A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : ‖x + y‖ ^ 2 ≤ (‖x‖ + ‖y‖) ^ 2
⊢ ‖x + y‖ ≤ ‖x‖ + ‖y‖ | 6038718c406b66b0 |
LeftOrdContinuous.comp | Mathlib/Order/OrdContinuous.lean | theorem comp (hg : LeftOrdContinuous g) (hf : LeftOrdContinuous f) : LeftOrdContinuous (g ∘ f) :=
fun s x h => by simpa only [image_image] using hg (hf h)
| α : Type u
β : Type v
γ : Type w
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
g : β → γ
f : α → β
hg : LeftOrdContinuous g
hf : LeftOrdContinuous f
s : Set α
x : α
h : IsLUB s x
⊢ IsLUB (g ∘ f '' s) ((g ∘ f) x) | simpa only [image_image] using hg (hf h) | no goals | 14a0b7b8640f6f40 |
List.find?_flatten_eq_some_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a : α} :
xs.flatten.find? p = some a ↔
p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
(∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) | α : Type u_1
p : α → Bool
a✝ : α
h : p a✝ = true
zs : List α
as : List (List α)
bs : List α
c : α
cs : List α
ds : List (List α)
h₁ : a✝ :: zs = c :: cs ++ ds.flatten
h₂ : ∀ (a : α), a ∈ as.flatten ++ bs → (!p a) = true
l : List α
ml : l ∈ as
a : α
m : a ∈ l
⊢ a ∈ as.flatten ++ bs | simpa using .inl ⟨l, ml, m⟩ | no goals | b3081438100b7db8 |
innerDualCone_univ | Mathlib/Analysis/Convex/Cone/InnerDual.lean | theorem innerDualCone_univ : (univ : Set H).innerDualCone = 0 | H : Type u_1
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
⊢ univ.innerDualCone = 0 | suffices ∀ x : H, x ∈ (univ : Set H).innerDualCone → x = 0 by
apply SetLike.coe_injective
exact eq_singleton_iff_unique_mem.mpr ⟨fun x _ => (inner_zero_right _).ge, this⟩ | H : Type u_1
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
⊢ ∀ x ∈ univ.innerDualCone, x = 0 | f5ace980151c53a2 |
MeasureTheory.withDensity_absolutelyContinuous' | Mathlib/MeasureTheory/Measure/WithDensity.lean | /-- If `f` is almost everywhere positive, then `μ ≪ μ.withDensity f`. See also
`withDensity_absolutelyContinuous` for the reverse direction, which always holds. -/
lemma withDensity_absolutelyContinuous' {μ : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ x ∂μ, f x ≠ 0) :
μ ≪ μ.withDensity f... | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
s : Set α
hs : MeasurableSet s
hf_ne_zero : μ {a | f a = 0} = 0
hμs : μ {a | a ∈ s ∧ ¬f a = 0} = 0
⊢ μ s = 0 | have hle : s ⊆ {a | a ∈ s ∧ ¬f a = 0} ∪ {a | f a = 0} :=
fun x hx ↦ or_iff_not_imp_right.mpr <| fun hnx ↦ ⟨hx, hnx⟩ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
s : Set α
hs : MeasurableSet s
hf_ne_zero : μ {a | f a = 0} = 0
hμs : μ {a | a ∈ s ∧ ¬f a = 0} = 0
hle : s ⊆ {a | a ∈ s ∧ ¬f a = 0} ∪ {a | f a = 0}
⊢ μ s = 0 | 5729584ad57cfef7 |
CommGrp.isZero_of_subsingleton | Mathlib/Algebra/Category/Grp/Zero.lean | theorem isZero_of_subsingleton (G : CommGrp) [Subsingleton G] : IsZero G | case refine_1
G : CommGrp
inst✝ : Subsingleton ↑G
X : CommGrp
f : G ⟶ X
⊢ f = default | ext x | case refine_1.hf.h
G : CommGrp
inst✝ : Subsingleton ↑G
X : CommGrp
f : G ⟶ X
x : ↑G
⊢ (Hom.hom f) x = (Hom.hom default) x | 0109e12937cbcbd9 |
Prod.mk_lt_mk_of_le_of_lt | Mathlib/Order/Basic.lean | lemma mk_lt_mk_of_le_of_lt (h₁ : a₁ ≤ a₂) (h₂ : b₁ < b₂) : (a₁, b₁) < (a₂, b₂) | α : Type u_2
β : Type u_3
inst✝¹ : Preorder α
inst✝ : Preorder β
a₁ a₂ : α
b₁ b₂ : β
h₁ : a₁ ≤ a₂
h₂ : b₁ < b₂
⊢ (a₁, b₁) < (a₂, b₂) | simp [lt_iff, *] | no goals | 5cc46f1809ad3b04 |
continuous_inf_dom_left₂ | Mathlib/Topology/Constructions.lean | theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
{tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
(h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by
haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f ... | X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 t... | exact Continuous fun p : X × Y => f p.1 p.2 | no goals | 997f870b6f34fc9d |
StrictMonoOn.eq_iff_eq | Mathlib/Order/Monotone/Basic.lean | theorem StrictMonoOn.eq_iff_eq (hf : StrictMonoOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a = f b ↔ a = b :=
⟨fun h ↦ le_antisymm ((hf.le_iff_le ha hb).mp h.le) ((hf.le_iff_le hb ha).mp h.ge), by
rintro rfl
rfl⟩
| α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s : Set α
hf : StrictMonoOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ a = b → f a = f b | rintro rfl | α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s : Set α
hf : StrictMonoOn f s
a : α
ha hb : a ∈ s
⊢ f a = f a | 1781a1b6aa54cbbe |
Cardinal.ord_le | Mathlib/SetTheory/Ordinal/Basic.lean | theorem ord_le {c o} : ord c ≤ o ↔ c ≤ o.card :=
inductionOn c fun α =>
Ordinal.inductionOn o fun β s _ => by
let ⟨r, _, e⟩ := ord_eq α
simp only [card_type]; constructor <;> intro h
· rw [e] at h
exact
let ⟨f⟩ := h
⟨f.toEmbedding⟩
· obtain ⟨f⟩ := h
have... | c : Cardinal.{u_1}
o : Ordinal.{u_1}
α β : Type u_1
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ (type s).card | simp only [card_type] | c : Cardinal.{u_1}
o : Ordinal.{u_1}
α β : Type u_1
s : β → β → Prop
x✝ : IsWellOrder β s
r : α → α → Prop
w✝ : IsWellOrder α r
e : (#α).ord = type r
⊢ (#α).ord ≤ type s ↔ #α ≤ #β | 76676751ae2ef84e |
ContinuousLinearEquiv.comp_right_differentiableOn_iff | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | theorem comp_right_differentiableOn_iff {f : F → G} {s : Set F} :
DifferentiableOn 𝕜 (f ∘ iso) (iso ⁻¹' s) ↔ DifferentiableOn 𝕜 f s | case a
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
iso : E ≃L[𝕜] F
f : F → G
s : Set F
H : DifferentiableOn... | simpa only [mem_preimage, apply_symm_apply] using hy | no goals | dcf460cd45203f0a |
AffineSubspace.coe_direction_eq_vsub_set_right | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem coe_direction_eq_vsub_set_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) :
(s.direction : Set V) = (· -ᵥ p) '' s | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hp : p ∈ s
⊢ ↑s -ᵥ ↑s = (fun x => x -ᵥ p) '' ↑s | refine le_antisymm ?_ ?_ | case refine_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hp : p ∈ s
⊢ ↑s -ᵥ ↑s ≤ (fun x => x -ᵥ p) '' ↑s
case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Mod... | d2d0dab6124cb47b |
Std.Sat.AIG.LawfulVecOperator.isPrefix_aig | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/LawfulVecOperator.lean | theorem isPrefix_aig (aig : AIG α) (input : β aig len) :
IsPrefix aig.decls (f aig input).aig.decls | α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
β : AIG α → Nat → Type
f : {len : Nat} → (aig : AIG α) → β aig len → RefVecEntry α len
inst✝ : LawfulVecOperator α β fun {len} => f
len : Nat
aig : AIG α
input : β aig len
⊢ IsPrefix aig.decls (f aig input).aig.decls | apply IsPrefix.of | case idx_eq
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
β : AIG α → Nat → Type
f : {len : Nat} → (aig : AIG α) → β aig len → RefVecEntry α len
inst✝ : LawfulVecOperator α β fun {len} => f
len : Nat
aig : AIG α
input : β aig len
⊢ ∀ (idx : Nat) (h : idx < aig.decls.size), (f aig input).aig.decls[idx] = aig.decls... | 5c23bef39116f766 |
cfcₙ_mono | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean | lemma cfcₙ_mono {f g : R → R} {a : A} (h : ∀ x ∈ σₙ R a, f x ≤ g x)
(hf : ContinuousOn f (σₙ R a) | R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁵ : OrderedCommSemiring R
inst✝¹⁴ : Nontrivial R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : IsTopologicalSemiring R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : ∀ (α : Type ?u.940962) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝⁸ : Topolog... | by_cases ha : p a | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁵ : OrderedCommSemiring R
inst✝¹⁴ : Nontrivial R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : IsTopologicalSemiring R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : ∀ (α : Type ?u.940962) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝⁸ ... | 7bbcd08f70e05623 |
MeasureTheory.condExp_bot' | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_bot' [hμ : NeZero μ] (f : α → E) :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ | case pos.intro
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hμ : NeZero μ
f : α → E
hμ_finite : IsFiniteMeasure μ
h_meas : StronglyMeasurable (μ[f|⊥])
c : E
h_eq : μ[f|⊥] = fun x => c
h_integral : (μ Set.univ).toReal • c = ... | exact ⟨NeZero.ne _, measure_ne_top μ Set.univ⟩ | no goals | 46233d8f5b2c46f3 |
MeasureTheory.withDensity_tsum | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem withDensity_tsum {ι : Type*} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
μ.withDensity (∑' n, f n) = sum fun n => μ.withDensity (f n) | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ (μ.withDensity (∑' (n : ι), f n)) s = (sum fun n => μ.withDensity (f n)) s | simp_rw [sum_apply _ hs, withDensity_apply _ hs] | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (∑' (n : ι), f n) a ∂μ = ∑' (i : ι), ∫⁻ (a : α) in s, f i a ∂μ | 64c96120ee87d086 |
IsDedekindDomainInv.dimensionLEOne | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem dimensionLEOne : DimensionLEOne A := ⟨by
-- We're going to show that `P` is maximal because any (maximal) ideal `M`
-- that is strictly larger would be `⊤`.
rintro P P_ne hP
refine Ideal.isMaximal_def.mpr ⟨hP.ne_top, fun M hM => ?_⟩
-- We may assume `P` and `M` (as fractional ideals) are nonzero.
ha... | A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
⊢ M = ⊤ | have P'_ne : (P : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr P_ne | A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
⊢ M = ⊤ | ec848de7999ea57e |
DirichletCharacter.BadChar.F_eq_LSeries | Mathlib/NumberTheory/LSeries/Nonvanishing.lean | /-- `B.F` agrees with the L-series of `zetaMul χ` on `1 < s.re`. -/
private lemma F_eq_LSeries (B : BadChar N) {s : ℂ} (hs : 1 < s.re) :
B.F s = LSeries B.χ.zetaMul s | case hf
N : ℕ
inst✝ : NeZero N
B : DirichletCharacter.BadChar N
s : ℂ
hs : 1 < s.re
⊢ LSeriesSummable (⇑↑ζ) s | exact LSeriesSummable_zeta_iff.mpr hs | no goals | 1a2b47fd23bb26b9 |
CategoryTheory.Limits.preservesBinaryBiproduct_of_preservesBiproduct | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | /-- A functor that preserves biproducts of a pair preserves binary biproducts. -/
lemma preservesBinaryBiproduct_of_preservesBiproduct (F : C ⥤ D)
[PreservesZeroMorphisms F] (X Y : C) [PreservesBiproduct (pairFunction X Y) F] :
PreservesBinaryBiproduct X Y F where
preserves {b} hb := ⟨{
isLimit :=
... | C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
X Y : C
inst✝ : PreservesBiproduct (pairFunction X Y) F
b : BinaryBicone X Y
hb : b.IsBilimit
j : Discrete WalkingPair
⊢ ((Cocones.precomp... | rcases j with ⟨⟨⟩⟩ <;> simp | no goals | 2ed16a28f912a87b |
MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
(ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 | case intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ... | refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt | case intro.intro.refine_1
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) i... | e9059c5fe2b96d38 |
GenContFract.IntFractPair.coe_of_rat_eq | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | theorem coe_of_rat_eq (v_eq_q : v = (↑q : K)) :
((IntFractPair.of q).mapFr (↑) : IntFractPair K) = IntFractPair.of v | K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
⊢ mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v | simp [IntFractPair.of, v_eq_q] | no goals | 4cb0dce41cf6fec4 |
MeasureTheory.tendsto_lintegral_nn_filter_of_le_const | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | theorem tendsto_lintegral_nn_filter_of_le_const {ι : Type*} {L : Filter ι} [L.IsCountablyGenerated]
(μ : Measure Ω) [IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
(fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c) {f : Ω → ℝ≥0}
(fs_lim : ∀ᵐ ω : Ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω))) :
Tendsto (... | case refine_2
Ω : Type u_1
inst✝⁴ : TopologicalSpace Ω
inst✝³ : MeasurableSpace Ω
inst✝² : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
f : Ω → ℝ≥0
fs_lim : ∀ᵐ... | simpa only [Function.comp_apply, ENNReal.tendsto_coe] using fs_lim | no goals | e702473d2b1c276c |
reflection_orthogonal | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem reflection_orthogonal : reflection Kᗮ = .trans (reflection K) (.neg _) | case h
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
x✝ : E
⊢ (reflection Kᗮ) x✝ = ((reflection K).trans (LinearIsometryEquiv.neg 𝕜)) x✝ | apply reflection_orthogonal_apply | no goals | f2144d29979d50da |
ProbabilityTheory.strong_law_aux5 | Mathlib/Probability/StrongLaw.lean | theorem strong_law_aux5 :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range n, truncation (X i) i ω - ∑ i ∈ range n, X i ω) =o[atTop]
fun n : ℕ => (n : ℝ) | case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ... | have : -(n : ℝ) < X n ω := by
apply lt_of_lt_of_le _ (hnonneg n ω)
simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos | case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ... | 89dd352ac6ec4b05 |
MeasureTheory.lintegral_prod_of_measurable | Mathlib/MeasureTheory/Measure/Prod.lean | theorem lintegral_prod_of_measurable :
∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ z, f z ∂μ.prod ν = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ | α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
⊢ ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ | have m := @measurable_prod_mk_left | α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
m :
∀ {α : Type ?u.165217} {β : Type ?u.165216} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},
Measurable (Prod.mk x)
⊢ ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ (z : α × β), f z ... | 4d25298bbc46afab |
CategoryTheory.Limits.colimitLimitToLimitColimit_surjective | Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean | theorem colimitLimitToLimitColimit_surjective :
Function.Surjective (colimitLimitToLimitColimit F) | case intro.intro.h.w
J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e : ∀ (j : J), colimit.ι ((curry.obj F).obj j) (k j) (y j) = l... | intro j | case intro.intro.h.w
J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e : ∀ (j : J), colimit.ι ((curry.obj F).obj j) (k j) (y j) = l... | 1be585fb623fc3d9 |
Language.mem_pow | Mathlib/Computability/Language.lean | theorem mem_pow {l : Language α} {x : List α} {n : ℕ} :
x ∈ l ^ n ↔ ∃ S : List (List α), x = S.flatten ∧ S.length = n ∧ ∀ y ∈ S, y ∈ l | case succ.mp
α : Type u_1
l : Language α
n : ℕ
ihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = S.flatten ∧ S.length = n ∧ ∀ y ∈ S, y ∈ l
x : List α
⊢ (∃ a ∈ l, ∃ b, (∃ S, b = S.flatten ∧ S.length = n ∧ ∀ y ∈ S, y ∈ l) ∧ a ++ b = x) →
∃ S, x = S.flatten ∧ S.length = n + 1 ∧ ∀ y ∈ S, y ∈ l | rintro ⟨a, ha, b, ⟨S, rfl, rfl, hS⟩, rfl⟩ | case succ.mp.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
l : Language α
a : List α
ha : a ∈ l
S : List (List α)
hS : ∀ y ∈ S, y ∈ l
ihn : ∀ {x : List α}, x ∈ l ^ S.length ↔ ∃ S_1, x = S_1.flatten ∧ S_1.length = S.length ∧ ∀ y ∈ S_1, y ∈ l
⊢ ∃ S_1, a ++ S.flatten = S_1.flatten ∧ S_1.length = S.length + 1 ∧ ∀ ... | e5967997d999f9f2 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftRight.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.ushiftRightRec lhs rhs curr).getLsbD idx)
... | case isTrue.hacc.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 }⟧ = (lh... | simp [hright] | no goals | fc604db047a9b54b |
Seminorm.convex_ball | Mathlib/Analysis/Seminorm.lean | theorem convex_ball : Convex ℝ (ball p x r) | case h.e'_6.h
𝕜 : Type u_3
E : Type u_7
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : NormedSpace ℝ 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : Module ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
p : Seminorm 𝕜 E
x : E
r : ℝ
y : E
⊢ p (y + -x) < r ↔ y ∈ {x_1 | (⇑p ∘ fun z => z + -x) x_1 < r} | rfl | no goals | 79b4594360c61d7c |
IsProperMap.comp | Mathlib/Topology/Maps/Proper/Basic.lean | /-- The composition of two proper maps is proper. -/
lemma IsProperMap.comp (hf : IsProperMap f) (hg : IsProperMap g) :
IsProperMap (g ∘ f) | X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : IsProperMap f
hg : IsProperMap g
ℱ : Filter X
z : Z
h : MapClusterPt z (map f ℱ) g
⊢ ∃ x, (g ∘ f) x = z ∧ ClusterPt x ℱ | rcases hg.clusterPt_of_mapClusterPt h with ⟨y, rfl, hy⟩ | case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : IsProperMap f
hg : IsProperMap g
ℱ : Filter X
y : Y
hy : ClusterPt y (map f ℱ)
h : MapClusterPt (g y) (map f ℱ) g
⊢ ∃ x, (g ∘ f) x = g y ∧ ClusterPt x ℱ | 634c3b9b89655115 |
affineIndependent_iff_of_fintype | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔
∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 | case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Fintype ι
p : ι → P
⊢ AffineIndependent k p → ∀ (w : ι → k), ∑ i : ι, w i = 0 → (univ.weightedVSub p) w = 0 → ∀ (i : ι), w i = 0 | exact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _) | no goals | 3193bfa1279ee234 |
PointedCone.toConvexCone_injective | Mathlib/Analysis/Convex/Cone/Pointed.lean | theorem toConvexCone_injective : Injective ((↑) : PointedCone 𝕜 E → ConvexCone 𝕜 E) :=
fun _ _ => by simp [toConvexCone]
| 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x✝¹ x✝ : PointedCone 𝕜 E
⊢ ↑x✝¹ = ↑x✝ → x✝¹ = x✝ | simp [toConvexCone] | no goals | 7918ed4c536c3d15 |
MeasureTheory.setIntegral_condExpL2_indicator | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | theorem setIntegral_condExpL2_indicator (hs : MeasurableSet[m] s) (ht : MeasurableSet t)
(hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) :
∫ x in s, (condExpL2 ℝ ℝ hm (indicatorConstLp 2 ht hμt 1) : α → ℝ) x ∂μ = (μ (t ∩ s)).toReal :=
calc
∫ x in s, (condExpL2 ℝ ℝ hm (indicatorConstLp 2 ht hμt 1) : α → ℝ) x ∂μ =
... | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ⊤
hμt : μ t ≠ ⊤
⊢ (μ (t ∩ s)).toReal • 1 = (μ (t ∩ s)).toReal | rw [smul_eq_mul, mul_one] | no goals | de7ef7724338c15b |
WeierstrassCurve.natDegree_coeff_preΨ' | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | private lemma natDegree_coeff_preΨ' (n : ℕ) :
(W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = expCoeff n | case odd.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of... | split_ifs <;>
simp only [apply_ite natDegree, natDegree_one.le, dp W.natDegree_Ψ₂Sq_le] | no goals | 9ededa4b854063c6 |
CoxeterSystem.exists_rightDescent_of_ne_one | Mathlib/GroupTheory/Coxeter/Length.lean | theorem exists_rightDescent_of_ne_one {w : W} (hw : w ≠ 1) : ∃ i : B, cs.IsRightDescent w i | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
hw : w ≠ 1
⊢ ∃ i, cs.IsLeftDescent w⁻¹ i | apply exists_leftDescent_of_ne_one | case hw
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
hw : w ≠ 1
⊢ w⁻¹ ≠ 1 | bbf0e2f284ab3b9a |
GaussianFourier.integrable_cexp_neg_mul_sq_norm_add | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem integrable_cexp_neg_mul_sq_norm_add (hb : 0 < b.re) (c : ℂ) (w : V) :
Integrable (fun (v : V) ↦ cexp (-b * ‖v‖^2 + c * ⟪w, v⟫)) | case h.e'_6.h
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
hb : 0 < b.re
c : ℂ
w : V
e : EuclideanSpace ℝ (Fin (Module.finrank ℝ V)) ≃ₗᵢ[ℝ] V := (stdOrthonormalBasis ℝ V).repr.symm
v : EuclideanSpace ℝ (Fin... | simp only [neg_mul, Function.comp_apply, LinearIsometryEquiv.norm_map,
LinearIsometryEquiv.symm_symm, conj_trivial, ofReal_sum,
ofReal_mul, LinearIsometryEquiv.inner_map_eq_flip] | no goals | 3e4da7d6638a8196 |
ContMDiffWithinAt.mfderivWithin | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | theorem ContMDiffWithinAt.mfderivWithin {x₀ : N} {f : N → M → M'} {g : N → M}
{t : Set N} {u : Set M}
(hf : ContMDiffWithinAt (J.prod I) I' n (Function.uncurry f) (t ×ˢ u) (x₀, g x₀))
(hg : ContMDiffWithinAt J I m g t x₀) (hx₀ : x₀ ∈ t)
(hu : MapsTo g t u) (hmn : m + 1 ≤ n) (h'u : UniqueMDiffOn I u) :
... | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup ... | convert hf'.2.mono this | case h.e'_11
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : Normed... | b011e1f26afdb375 |
ContMDiffAt.comp_of_eq | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | theorem ContMDiffAt.comp_of_eq {g : M' → M''} {x : M} {y : M'} (hg : ContMDiffAt I' I'' n g y)
(hf : ContMDiffAt I I' n f x) (hx : f x = y) : ContMDiffAt I I'' n (g ∘ f) x | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6... | exact hg.comp x hf | no goals | 57724353c0bc4d14 |
Ordnode.Sized.induction | Mathlib/Data/Ordmap/Ordset.lean | theorem Sized.induction {t} (hl : @Sized α t) {C : Ordnode α → Prop} (H0 : C nil)
(H1 : ∀ l x r, C l → C r → C (.node' l x r)) : C t | α : Type u_1
t : Ordnode α
hl : t.Sized
C : Ordnode α → Prop
H0 : C nil
H1 : ∀ (l : Ordnode α) (x : α) (r : Ordnode α), C l → C r → C (l.node' x r)
⊢ C t | induction t with
| nil => exact H0
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2) | no goals | 613d76c6108cb818 |
JacobsonNoether.exist_pow_eq_zero_of_le | Mathlib/FieldTheory/JacobsonNoether.lean | /-- If `D` is a purely inseparable extension of `k` of characteristic `p`,
then for every element `a` of `D \ k`, there exists a natural number `m`
greater than 0 such that `(a * x - x * a) ^ n = 0` (as linear maps) for
every `n` greater than `(p ^ m)`. -/
lemma exist_pow_eq_zero_of_le (p : ℕ) [hchar : ExpChar D ... | case intro
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
inter : (⇑((ad (↥k) D) a))^[p ^ m] = 0
⊢ (⇑((ad (↥k) D) a))^[n] = 0 | rw [(Nat.sub_eq_iff_eq_add hn).1 rfl, Function.iterate_add, inter, Pi.comp_zero,
iterate_map_zero, Function.const_zero] | no goals | 6921e21d57128a1f |
CategoryTheory.MorphismProperty.iSup_iff | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | @[simp]
lemma iSup_iff {ι : Sort*} (W : ι → MorphismProperty C) {X Y : C} (f : X ⟶ Y) :
iSup W f ↔ ∃ i, W i f | C : Type u
inst✝ : Category.{v, u} C
ι : Sort u_2
W : ι → MorphismProperty C
X Y : C
f : X ⟶ Y
⊢ iSup W f ↔ ∃ i, W i f | apply (sSup_iff (Set.range W) f).trans | C : Type u
inst✝ : Category.{v, u} C
ι : Sort u_2
W : ι → MorphismProperty C
X Y : C
f : X ⟶ Y
⊢ (∃ W_1, ↑W_1 f) ↔ ∃ i, W i f | 7d58f46f1390332b |
Real.BohrMollerup.tendsto_logGammaSeq_of_le_one | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem tendsto_logGammaSeq_of_le_one (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hx' : x ≤ 1) :
Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1) | f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1)) | refine tendsto_of_tendsto_of_tendsto_of_le_of_le' (f := logGammaSeq x)
(g := fun n ↦ f x - f 1 - x * (log (n + 1) - log n)) ?_ tendsto_const_nhds ?_ ?_ | case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
⊢ Tendsto (fun n => f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1))
case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) ... | 2a2b50b4ebb10e89 |
Std.DHashMap.Internal.List.containsKey_insertList | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem containsKey_insertList [BEq α] [PartialEquivBEq α] {l toInsert : List ((a : α) × β a)}
{k : α} : containsKey k (List.insertList l toInsert) =
(containsKey k l || (toInsert.map Sigma.fst).contains k) | case cons
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k : α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih :
∀ {l : List ((a : α) × β a)},
containsKey k (insertList l tl) = (containsKey k l || (List.map Sigma.fst tl).contains k)
l : List ((a : α) × β a)
⊢ (k == hd.fst || containsKey k l ||... | conv => left; left; rw [Bool.or_comm] | case cons
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k : α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih :
∀ {l : List ((a : α) × β a)},
containsKey k (insertList l tl) = (containsKey k l || (List.map Sigma.fst tl).contains k)
l : List ((a : α) × β a)
⊢ (containsKey k l || k == hd.fst ||... | 0e1771d36311393a |
Polynomial.natDegree_eq_zero_of_derivative_eq_zero | Mathlib/Algebra/Polynomial/Derivative.lean | theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
(h : derivative f = 0) : f.natDegree = 0 | case inr
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), (derivative f).coeff n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < f.natDegree
m : ℕ := f.natDegree - 1
hm : m + 1 = f.natDegree
h2 : (derivative f).coeff m = f.coeff (m + 1) * (↑m + 1)
⊢ False | rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2 | case inr
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), (derivative f).coeff n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < f.natDegree
m : ℕ := f.natDegree - 1
hm : m + 1 = f.natDegree
h2 : m + 1 = 0 ∨ f.coeff (m + 1) = 0
⊢ False | 1a3451df08199868 |
Submodule.finite_quotient_smul | Mathlib/RingTheory/Ideal/Quotient/Index.lean | /-- Let `N` be a finite index f.g. `R`-submodule, and `I` be a finite index ideal.
Then `I • N` also has finite index. -/
lemma Submodule.finite_quotient_smul [Finite (R ⧸ I)] [Finite (M ⧸ N)] (hN : N.FG) :
Finite (M ⧸ I • N) | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
N : Submodule R M
inst✝¹ : Finite (R ⧸ I)
inst✝ : Finite (M ⧸ N)
hN : N.FG
e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=
(comap N.subtype (I • N)).quotEquivOfEq (I • ⊤) ⋯ ≪≫ₗ (quotTensorEquivQuotSMul (↥... | have : Module.Finite R N := Module.Finite.iff_fg.mpr hN | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
N : Submodule R M
inst✝¹ : Finite (R ⧸ I)
inst✝ : Finite (M ⧸ N)
hN : N.FG
e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=
(comap N.subtype (I • N)).quotEquivOfEq (I • ⊤) ⋯ ≪≫ₗ (quotTensorEquivQuotSMul (↥... | f225d1fd83c53ea4 |
Computation.LiftRel.equiv | Mathlib/Data/Seq/Computation.lean | theorem LiftRel.equiv (R : α → α → Prop) : Equivalence R → Equivalence (LiftRel R)
| ⟨refl, symm, trans⟩ => ⟨LiftRel.refl R refl, by apply LiftRel.symm; apply symm,
by apply LiftRel.trans; apply trans⟩
| α : Type u
R : α → α → Prop
refl : ∀ (x : α), R x x
symm : ∀ {x y : α}, R x y → R y x
trans : ∀ {x y z : α}, R x y → R y z → R x z
⊢ ∀ {x y : Computation α}, LiftRel R x y → LiftRel R y x | apply LiftRel.symm | case H
α : Type u
R : α → α → Prop
refl : ∀ (x : α), R x x
symm : ∀ {x y : α}, R x y → R y x
trans : ∀ {x y z : α}, R x y → R y z → R x z
⊢ Symmetric R | fa7d5dbe914e8b11 |
wbtw_iff_sameRay_vsub | Mathlib/Analysis/Convex/Between.lean | theorem wbtw_iff_sameRay_vsub {x y z : P} : Wbtw R x y z ↔ SameRay R (y -ᵥ x) (z -ᵥ y) | case inr.inr.intro.intro.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y z : P
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (y -ᵥ x) = r₂ • (z -ᵥ y)
⊢ Wbtw R x y z | refine
⟨r₂ / (r₁ + r₂),
⟨div_nonneg hr₂.le (add_nonneg hr₁.le hr₂.le),
div_le_one_of_le₀ (le_add_of_nonneg_left hr₁.le) (add_nonneg hr₁.le hr₂.le)⟩,
?_⟩ | case inr.inr.intro.intro.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y z : P
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (y -ᵥ x) = r₂ • (z -ᵥ y)
⊢ (lineMap x z) (r₂ / (r₁ + r₂)) = y | 3cdc41d133486434 |
Finset.Ico_add_one_left_eq_Ioo | Mathlib/Algebra/Order/Interval/Finset/SuccPred.lean | lemma Ico_add_one_left_eq_Ioo (a b : α) : Ico (a + 1) b = Ioo a b | α : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : One α
inst✝² : LocallyFiniteOrder α
inst✝¹ : Add α
inst✝ : SuccAddOrder α
a b : α
⊢ Ico (a + 1) b = Ioo a b | simpa [succ_eq_add_one] using Ico_succ_left_eq_Ioo a b | no goals | d78c1d7db28e005a |
SetTheory.PGame.ext | Mathlib/SetTheory/Game/PGame.lean | lemma ext {x y : PGame} (hl : x.LeftMoves = y.LeftMoves) (hr : x.RightMoves = y.RightMoves)
(hL : ∀ i j, HEq i j → x.moveLeft i = y.moveLeft j)
(hR : ∀ i j, HEq i j → x.moveRight i = y.moveRight j) :
x = y | x y : PGame
hl : x.LeftMoves = y.LeftMoves
hr : x.RightMoves = y.RightMoves
hL : ∀ (i : x.LeftMoves) (j : y.LeftMoves), HEq i j → x.moveLeft i = y.moveLeft j
hR : ∀ (i : x.RightMoves) (j : y.RightMoves), HEq i j → x.moveRight i = y.moveRight j
⊢ x = y | cases x | case mk
y : PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
hl : (mk α✝ β✝ a✝¹ a✝).LeftMoves = y.LeftMoves
hr : (mk α✝ β✝ a✝¹ a✝).RightMoves = y.RightMoves
hL : ∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : y.LeftMoves), HEq i j → (mk α✝ β✝ a✝¹ a✝).moveLeft i = y.moveLeft j
hR : ∀ (i : (mk α✝ β✝ a✝¹ a✝).RightMoves) ... | c76b2fd227da4270 |
NumberField.mixedEmbedding.exists_primitive_element_lt_of_isComplex | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀ : IsComplex w₀)
{B : ℝ≥0} (hB : minkowskiBound K ↑1 < convexBodyLT'Factor K * B) :
∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
∀ w : InfinitePlace K, w a < Real.sqrt (1 + B ^ 2) | case pos.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w => if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : ... | exact h_le₀.1 | no goals | b1817ae1b2a5d1e1 |
Polynomial.tendsto_div_exp_atTop | Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | case h_monomial
n : ℕ
c : ℝ
⊢ Tendsto (fun x => eval x ((monomial n) c) / rexp x) atTop (𝓝 0) | simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n) | no goals | 3682270aa5dd2711 |
FirstOrder.Language.DirectLimit.exists_fg_substructure_in_Sigma | Mathlib/ModelTheory/DirectLimit.lean | theorem exists_fg_substructure_in_Sigma (S : L.Substructure (DirectLimit G f)) (S_fg : S.FG) :
∃ i, ∃ T : L.Substructure (G i), T.map (of L ι G f i).toHom = S | case h
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectL... | simp only [Embedding.coe_toHom, of_apply] | case h
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectL... | 133e2af1566ef3d4 |
hasSum_mellin_pi_mul_sq | Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean | /-- Tailored version for even Jacobi theta functions. -/
lemma hasSum_mellin_pi_mul_sq {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * rexp (-π * r i ^ 2 * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re) :
HasSum (fun i ↦... | case h.e'_5.h.h.e'_6
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑|r i| ^ s = ↑|r i| ^ (↑2 * (s / 2)) | ring_nf | no goals | 96f69737f30abb4f |
Subalgebra.LinearDisjoint.of_linearDisjoint_finite_left | Mathlib/RingTheory/LinearDisjoint.lean | theorem of_linearDisjoint_finite_left [Algebra.IsIntegral R A]
(H : ∀ A' : Subalgebra R S, A' ≤ A → [Module.Finite R A'] → A'.LinearDisjoint B) :
A.LinearDisjoint B | case intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (t... | congr | case intro.intro.intro.intro.intro.intro.h.e_6.h
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).m... | 1d56e99b263dd6b5 |
PiNat.iUnion_cylinder_update | Mathlib/Topology/MetricSpace/PiNat.lean | theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
⋃ k, cylinder (update x n k) (n + 1) = cylinder x n | case h
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
⊢ y ∈ ⋃ k, cylinder (update x n k) (n + 1) ↔ y ∈ cylinder x n | simp only [mem_cylinder_iff, mem_iUnion] | case h
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
⊢ (∃ i, ∀ i_1 < n + 1, y i_1 = update x n i i_1) ↔ ∀ i < n, y i = x i | 822a9c224bc12ada |
Set.univ_pi_ite | Mathlib/Data/Set/Prod.lean | theorem univ_pi_ite (s : Set ι) [DecidablePred (· ∈ s)] (t : ∀ i, Set (α i)) :
(pi univ fun i => if i ∈ s then t i else univ) = s.pi t | case h
ι : Type u_1
α : ι → Type u_2
s : Set ι
inst✝ : DecidablePred fun x => x ∈ s
t : (i : ι) → Set (α i)
x✝ : (i : ι) → α i
⊢ (x✝ ∈ univ.pi fun i => if i ∈ s then t i else univ) ↔ x✝ ∈ s.pi t | simp_rw [mem_univ_pi] | case h
ι : Type u_1
α : ι → Type u_2
s : Set ι
inst✝ : DecidablePred fun x => x ∈ s
t : (i : ι) → Set (α i)
x✝ : (i : ι) → α i
⊢ (∀ (i : ι), x✝ i ∈ if i ∈ s then t i else univ) ↔ x✝ ∈ s.pi t | c7cef90aa3ee129f |
CFC.monotoneOn_one_sub_one_add_inv | Mathlib/Analysis/CStarAlgebra/ApproximateUnit.lean | lemma CFC.monotoneOn_one_sub_one_add_inv :
MonotoneOn (cfcₙ (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹)) (Set.Ici (0 : A)) | case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : autoParam (0 ≤ c) _auto✝
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 c
⊢ 1 + x✝ ≠ 0 | positivity | no goals | 66ae445c802b501c |
PythagoreanTriple.isPrimitiveClassified_of_coprime_of_odd_of_pos | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
(hzpos : 0 < z) : h.IsPrimitiveClassified | x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
⊢ ↑x * ↑x + ↑y * ↑y = ↑z * ↑z | norm_cast | no goals | e128c623fada3541 |
ENNReal.inv_mul_cancel_right' | Mathlib/Data/ENNReal/Inv.lean | /-- See `ENNReal.inv_mul_cancel_right` for a simpler version assuming `b ≠ 0`, `b ≠ ∞`. -/
protected lemma inv_mul_cancel_right' (hb₀ : b = 0 → a = 0) (hb : b = ∞ → a = 0) :
a * b⁻¹ * b = a | case inr
a b : ℝ≥0∞
hb₀✝ : b = 0 → a = 0
hb : b = ⊤ → a = 0
hb₀ : b ≠ 0
⊢ a * b⁻¹ * b = a | obtain rfl | hb := eq_or_ne b ⊤ | case inr.inl
a : ℝ≥0∞
hb₀✝ : ⊤ = 0 → a = 0
hb : ⊤ = ⊤ → a = 0
hb₀ : ⊤ ≠ 0
⊢ a * ⊤⁻¹ * ⊤ = a
case inr.inr
a b : ℝ≥0∞
hb₀✝ : b = 0 → a = 0
hb✝ : b = ⊤ → a = 0
hb₀ : b ≠ 0
hb : b ≠ ⊤
⊢ a * b⁻¹ * b = a | f8034e58fad9e03c |
AlgebraicGeometry.RingedSpace.basicOpen_res | Mathlib/Geometry/RingedSpace/Basic.lean | theorem basicOpen_res {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) (f : X.presheaf.obj U) :
@basicOpen X (unop V) (X.presheaf.map i f) = unop V ⊓ @basicOpen X (unop U) f | case h.h.mpr.intro.intro
X : RingedSpace
U V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ
i : U ⟶ V
f : ↑(X.presheaf.obj U)
x : ↑↑X.toPresheafedSpace
hxV : x ∈ ↑(unop V)
w✝ : x ∈ unop U
hx : IsUnit ((ConcreteCategory.hom (X.presheaf.germ (unop U) x w✝)) f)
⊢ IsUnit ((ConcreteCategory.hom (X.presheaf.germ (unop U) x ⋯)) f) | exact hx | no goals | 95747598eb7ff13b |
AlgebraicGeometry.universallyClosed_eq_universallySpecializing | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | lemma universallyClosed_eq_universallySpecializing :
@UniversallyClosed = (topologically @SpecializingMap).universally ⊓ @QuasiCompact | ⊢ @UniversallyClosed = (topologically @SpecializingMap ⊓ @QuasiCompact).universally | apply le_antisymm | case a
⊢ @UniversallyClosed ≤ (topologically @SpecializingMap ⊓ @QuasiCompact).universally
case a
⊢ (topologically @SpecializingMap ⊓ @QuasiCompact).universally ≤ @UniversallyClosed | 862e4b8e16a6249b |
AlgebraicGeometry.StructureSheaf.germ_toOpen | Mathlib/AlgebraicGeometry/StructureSheaf.lean | theorem germ_toOpen
(U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R) (hx : x ∈ U) (f : R) :
(structureSheaf R).presheaf.germ U x hx (toOpen R U f) = toStalk R x f | R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
x : ↑(PrimeSpectrum.Top R)
hx : x ∈ U
f : R
⊢ (ConcreteCategory.hom ((structureSheaf R).presheaf.germ U x hx)) ((ConcreteCategory.hom (toOpen R U)) f) =
(ConcreteCategory.hom (toOpen R ?U ≫ (structureSheaf R).presheaf.germ ?U x ?hx)) f
case U
R : Type ... | rfl | no goals | 6df7d707c53e07ef |
LieAlgebra.zeroRootSubalgebra_normalizer_eq_self | Mathlib/Algebra/Lie/Weights/Cartan.lean | theorem zeroRootSubalgebra_normalizer_eq_self :
(zeroRootSubalgebra R L H).normalizer = zeroRootSubalgebra R L H | case h
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
inst✝ : LieRing.IsNilpotent ↥H
x y : L
hy : y ∈ H
hx : ∀ (y_1 : ↥H), ∃ k, ((toEnd R (↥H) L) y_1 ^ k) ⁅x, y⁆ = 0
k : ℕ
hk : ((toEnd R (↥H) L) ⟨y, hy⟩ ^ k) ⁅y, x⁆ = 0
⊢ ((toEnd R (↥H) L) ⟨y, hy⟩ ^ (k + 1)... | rw [LinearMap.iterate_succ, LinearMap.coe_comp, Function.comp_apply, toEnd_apply_apply,
LieSubalgebra.coe_bracket_of_module, Submodule.coe_mk, hk] | no goals | 02d44c0a4b4e91a4 |
CategoryTheory.eqToHom_comp_iff | Mathlib/CategoryTheory/EqToHom.lean | theorem eqToHom_comp_iff {X X' Y : C} (p : X = X') (f : X ⟶ Y) (g : X' ⟶ Y) :
eqToHom p ≫ g = f ↔ g = eqToHom p.symm ≫ f :=
{ mp := fun h => h ▸ by simp
mpr := fun h => h ▸ by simp [whisker_eq _ h] }
| C : Type u₁
inst✝ : Category.{v₁, u₁} C
X X' Y : C
p : X = X'
f : X ⟶ Y
g : X' ⟶ Y
h : g = eqToHom ⋯ ≫ f
⊢ eqToHom p ≫ eqToHom ⋯ ≫ f = f | simp [whisker_eq _ h] | no goals | ba8ae4b6aa96e40a |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id) | case neg
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_b... | simp only [Prod.exists, Bool.exists_bool, not_exists, not_or, unit] at hl | case neg
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_b... | 0ab20a88767ce69e |
nsmul_eq_mul | Mathlib/Data/Nat/Cast/Basic.lean | @[simp] lemma _root_.nsmul_eq_mul (n : ℕ) (a : α) : n • a = n * a | case succ
α : Type u_1
inst✝ : NonAssocSemiring α
a : α
n : ℕ
ih : n • a = ↑n * a
⊢ (n + 1) • a = ↑(n + 1) * a | rw [succ_nsmul, ih, Nat.cast_succ, add_mul, one_mul] | no goals | 8fc2fe51086d4864 |
Nat.all_eq_finRange_all | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Fold.lean | theorem all_eq_finRange_all {n : Nat} (f : (i : Nat) → i < n → Bool) :
all n f = (List.finRange n).all (fun ⟨i, h⟩ => f i h) | case succ
n : Nat
ih :
∀ (f : (i : Nat) → i < n → Bool),
n.all f =
(List.finRange n).all fun x =>
match x with
| ⟨i, h⟩ => f i h
f : (i : Nat) → i < n + 1 → Bool
⊢ (n + 1).all f =
(List.finRange (n + 1)).all fun x =>
match x with
| ⟨i, h⟩ => f i h | simp [ih, List.finRange_succ_last, List.all_map, Function.comp_def] | no goals | bb52d533e39be240 |
Finset.mem_finsuppAntidiag' | Mathlib/Algebra/Order/Antidiag/Finsupp.lean | lemma mem_finsuppAntidiag' :
f ∈ finsuppAntidiag s n ↔ f.sum (fun _ x ↦ x) = n ∧ f.support ⊆ s | ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
n : μ
f : ι →₀ μ
hf : f.support ⊆ s
⊢ s.sum ⇑f = n ↔ (f.sum fun x x => x) = n | rw [sum_of_support_subset (N := μ) f hf (fun _ x ↦ x) fun _ _ ↦ rfl] | no goals | e81896c5da58f374 |
LinearIndependent.finite_of_isNoetherian | Mathlib/RingTheory/Noetherian/Basic.lean | theorem LinearIndependent.finite_of_isNoetherian [Nontrivial R] {ι} {v : ι → M}
(hv : LinearIndependent R v) : Finite ι | R : Type u_1
M : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
ι : Type u_5
v : ι → M
hv : LinearIndependent R v
i : ι
contra : span R {v i} = ⊥
⊢ False | apply hv.ne_zero i | R : Type u_1
M : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
ι : Type u_5
v : ι → M
hv : LinearIndependent R v
i : ι
contra : span R {v i} = ⊥
⊢ v i = 0 | 1f9db6538c2d4f91 |
List.Forall.imp | Mathlib/Data/List/Basic.lean | theorem Forall.imp (h : ∀ x, p x → q x) : ∀ {l : List α}, Forall p l → Forall q l
| [] => id
| x :: l => by
simp only [forall_cons, and_imp]
rw [← and_imp]
exact And.imp (h x) (Forall.imp h)
| α : Type u
p q : α → Prop
h : ∀ (x : α), p x → q x
x : α
l : List α
⊢ Forall p (x :: l) → Forall q (x :: l) | simp only [forall_cons, and_imp] | α : Type u
p q : α → Prop
h : ∀ (x : α), p x → q x
x : α
l : List α
⊢ p x → Forall p l → q x ∧ Forall q l | d3e3250726717199 |
Ordinal.nmul_assoc | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem nmul_assoc (a b c : Ordinal) : a ⨳ b ⨳ c = a ⨳ (b ⨳ c) | case a
a b c a' : Ordinal.{u_1}
ha : a' < a
b' : Ordinal.{u_1}
hb : b' < b
c' : Ordinal.{u_1}
hc : c' < c
⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ (b' ⨳ c') | rw [← nmul_assoc] | case a
a b c a' : Ordinal.{u_1}
ha : a' < a
b' : Ordinal.{u_1}
hb : b' < b
c' : Ordinal.{u_1}
hc : c' < c
⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' | af6a958d5cf6546a |
Cardinal.aleph0_le | Mathlib/SetTheory/Cardinal/Basic.lean | theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
⟨fun h _ => (nat_lt_aleph0 _).le.trans h, fun h =>
le_of_not_lt fun hn => by
rcases lt_aleph0.1 hn with ⟨n, rfl⟩
exact (Nat.lt_succ_self _).not_le (Nat.cast_le.1 (h (n + 1)))⟩
| c : Cardinal.{u_1}
h : ∀ (n : ℕ), ↑n ≤ c
hn : c < ℵ₀
⊢ False | rcases lt_aleph0.1 hn with ⟨n, rfl⟩ | case intro
n : ℕ
h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n
hn : ↑n < ℵ₀
⊢ False | 8a8d523c3adcb040 |
AdjoinRoot.of.injective_of_degree_ne_zero | Mathlib/RingTheory/AdjoinRoot.lean | theorem of.injective_of_degree_ne_zero [IsDomain R] (hf : f.degree ≠ 0) :
Function.Injective (AdjoinRoot.of f) | R : Type u
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
p : R
hp : f ∣ C p
h : ¬f = 0
h_contra : p ≠ 0
⊢ (C p).degree ≤ f.degree | rwa [degree_C h_contra, zero_le_degree_iff] | no goals | f750c2be586c9c7b |
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