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 |
|---|---|---|---|---|---|---|
Ordinal.range_omega | Mathlib/SetTheory/Cardinal/Aleph.lean | theorem range_omega : range omega = {x | ω ≤ x ∧ IsInitial x} | case h.mpr
x : Ordinal.{u_1}
⊢ x ∈ {x | ω ≤ x ∧ x.IsInitial} → x ∈ range ⇑ω_ | rintro ⟨ha', ha⟩ | case h.mpr.intro
x : Ordinal.{u_1}
ha' : ω ≤ x
ha : x.IsInitial
⊢ x ∈ range ⇑ω_ | 2b8a699686971ec4 |
WeierstrassCurve.Projective.Point.toAffine_add | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma toAffine_add {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q) :
toAffine W (W.add P Q) = toAffine W P + toAffine W Q | case neg
F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z)
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q | have := toAffine_add_of_Z_ne_zero hP hQ hPz hQz hxy | case neg
F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z)
this :
toAffine W
![(Projective.toAffine W).addX (P x / P z) (Q x / Q z)
((Projective.toAffine W).slo... | 7a72ea913c00d8ec |
Nat.digits_add | Mathlib/Data/Nat/Digits.lean | theorem digits_add (b : ℕ) (h : 1 < b) (x y : ℕ) (hxb : x < b) (hxy : x ≠ 0 ∨ y ≠ 0) :
digits b (x + b * y) = x :: digits b y | case intro.succ
x b : ℕ
h : 1 < b + 2
hxb : x < b + 2
n✝ : ℕ
hxy : x ≠ 0 ∨ n✝ + 1 ≠ 0
⊢ (b + 2).digits (x + (b + 2) * (n✝ + 1)) = x :: (b + 2).digits (n✝ + 1) | dsimp [digits] | case intro.succ
x b : ℕ
h : 1 < b + 2
hxb : x < b + 2
n✝ : ℕ
hxy : x ≠ 0 ∨ n✝ + 1 ≠ 0
⊢ (b + 2).digitsAux ⋯ (x + (b + 2) * (n✝ + 1)) = x :: (b + 2).digitsAux ⋯ (n✝ + 1) | 5707ca6647bd00ed |
Polynomial.isLocalHom_expand | Mathlib/Algebra/Polynomial/Expand.lean | theorem isLocalHom_expand {p : ℕ} (hp : 0 < p) : IsLocalHom (expand R p) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 0 < p
f : R[X]
hf1 : IsUnit ((expand R p) f)
⊢ IsUnit f | have hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit hf1) | R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : 0 < p
f : R[X]
hf1 : IsUnit ((expand R p) f)
hf2 : (expand R p) f = C (((expand R p) f).coeff 0)
⊢ IsUnit f | 39ad0872e0293086 |
CompactIccSpace.mk' | Mathlib/Topology/Order/Compact.lean | lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
(h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
isCompact_Icc {a b} | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Preorder α
h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)
a b : α
hab : ¬a ≤ b
⊢ IsCompact ∅ | exact isCompact_empty | no goals | a8529e4e18c5748e |
Finset.powersetCard_map | Mathlib/Data/Finset/Powerset.lean | theorem powersetCard_map {β : Type*} (f : α ↪ β) (n : ℕ) (s : Finset α) :
powersetCard n (s.map f) = (powersetCard n s).map (mapEmbedding f).toEmbedding :=
ext fun t => by
simp only [card_map, mem_powersetCard, le_eq_subset, gt_iff_lt, mem_map, mapEmbedding_apply]
constructor
· classical
intro h... | case mp
α : Type u_1
β : Type u_2
f : α ↪ β
n : ℕ
s : Finset α
t : Finset β
h : t ⊆ map f s ∧ #t = n
this : map f (filter (fun x => f x ∈ t) s) = t
⊢ filter (fun x => f x ∈ t) s ⊆ s ∧ n = n | simp | no goals | dfde83e9b4eb52a7 |
CompHausLike.LocallyConstant.presheaf_ext | Mathlib/Condensed/Discrete/LocallyConstant.lean | /--
To check equality of two elements of `X(S)`, it suffices to check equality after composing with
each `X(S) → X(Sᵢ)`.
-/
lemma presheaf_ext (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w)
[PreservesFiniteProducts X] (x y : X.obj ⟨S⟩)
[HasExplicitFiniteCoproducts.{u} P]
(h : ∀ (a : Fiber f), X.map (sigmaIncl ... | case a.a.h
P : TopCat → Prop
inst✝³ : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝² : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑S.toTop) (Y.obj (op (of P PUnit.{u + 1})))
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝¹ : Preserve... | rw [← sigmaComparison_comp_sigmaIso] at h | case a.a.h
P : TopCat → Prop
inst✝³ : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝² : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑S.toTop) (Y.obj (op (of P PUnit.{u + 1})))
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝¹ : Preserve... | 4aad38c3a12edfe7 |
GaloisConnection.u_eq | Mathlib/Order/GaloisConnection/Defs.lean | theorem u_eq {z : α} {y : β} : u y = z ↔ ∀ x, x ≤ z ↔ l x ≤ y | α : Type u
β : Type v
inst✝¹ : PartialOrder α
inst✝ : Preorder β
l : α → β
u : β → α
gc : GaloisConnection l u
z : α
y : β
⊢ u y = z ↔ ∀ (x : α), x ≤ z ↔ l x ≤ y | constructor | case mp
α : Type u
β : Type v
inst✝¹ : PartialOrder α
inst✝ : Preorder β
l : α → β
u : β → α
gc : GaloisConnection l u
z : α
y : β
⊢ u y = z → ∀ (x : α), x ≤ z ↔ l x ≤ y
case mpr
α : Type u
β : Type v
inst✝¹ : PartialOrder α
inst✝ : Preorder β
l : α → β
u : β → α
gc : GaloisConnection l u
z : α
y : β
⊢ (∀ (x : α), x ≤... | f90e2defec673558 |
aux1 | Mathlib/Algebra/Jordan/Basic.lean | theorem aux1 {a b c : A} :
⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +
2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆
=
⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +
⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +
(⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅... | A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =
⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
⁅L a, 2 • L (b * c)⁆ +
(⁅L b, L... | rw [add_lie, add_lie] | A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
⁅L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L... | b5c23462450b58db |
Equiv.Perm.support_subtype_perm | Mathlib/GroupTheory/Perm/Support.lean | theorem support_subtype_perm [DecidableEq α] {s : Finset α} (f : Perm α) (h) :
(f.subtypePerm h : Perm s).support = ({x | f x ≠ x} : Finset s) | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
f : Perm α
h : ∀ (x : α), x ∈ s ↔ f x ∈ s
⊢ (f.subtypePerm h).support = filter (fun x => f ↑x ≠ ↑x) univ | ext | case h
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
f : Perm α
h : ∀ (x : α), x ∈ s ↔ f x ∈ s
a✝ : { x // x ∈ s }
⊢ a✝ ∈ (f.subtypePerm h).support ↔ a✝ ∈ filter (fun x => f ↑x ≠ ↑x) univ | ba506bad3e009e97 |
MeasureTheory.measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | theorem measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure {Ω : Type*}
[MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
[OpensMeasurableSpace Ω] {μ ν : Measure Ω} [IsFiniteMeasure μ]
(h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ x, f x ∂μ = ∫⁻ x, f x ∂ν) {F : Set Ω} (F_closed : IsClosed F) :
μ F... | case measure_univ_lt_top
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : HasOuterApproxClosed Ω
inst✝¹ : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝ : IsFiniteMeasure μ
h : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν
F : Set Ω
F_closed : IsClosed F
whole : μ univ = ν un... | simp [← whole] | no goals | 545c1e4b7e0daa43 |
Asymptotics.isLittleO_norm_left | Mathlib/Analysis/Asymptotics/Defs.lean | theorem isLittleO_norm_left : (fun x => ‖f' x‖) =o[l] g ↔ f' =o[l] g | α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
f' : α → E'
l : Filter α
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x => ‖f' x‖) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g | exact forall₂_congr fun _ _ => isBigOWith_norm_left | no goals | 403b9aa052c68398 |
bernsteinPolynomial.sum_mul_smul | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem sum_mul_smul (n : ℕ) :
(∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν) =
(n * (n - 1)) • X ^ 2 | R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomi... | trans MvPolynomial.aeval e (pderiv true (pderiv true ((x + y) ^ n))) * X ^ 2 | R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomi... | 5a1ea5f98d0a949e |
isPreconnected_of_forall_constant | Mathlib/Topology/Connected/Clopen.lean | theorem isPreconnected_of_forall_constant {s : Set α}
(hs : ∀ f : α → Bool, ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y) : IsPreconnected s | α : Type u
inst✝ : TopologicalSpace α
s : Set α
hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y
u v : Set α
u_op : IsOpen u
v_op : IsOpen v
hsuv : s ⊆ u ∪ v
x : α
x_in_s : x ∈ s
x_in_u : x ∈ u
H : s ∩ (u ∩ v) = ∅
y : α
y_in_s : y ∈ s
y_in_v : y ∈ v
hy : y ∉ u
⊢ IsClosed (Subtype.val ⁻¹' v)ᶜ | exact (v_op.preimage continuous_subtype_val).isClosed_compl | no goals | 0b9dc97844bbc91c |
Function.Injective.of_comp_right | Mathlib/Logic/Function/Basic.lean | theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
| case intro
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : γ → α
I : Injective (f ∘ g)
hg : Surjective g
y : α
x : γ
h : f (g x) = f y
⊢ g x = y | obtain ⟨y, rfl⟩ := hg y | case intro.intro
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : γ → α
I : Injective (f ∘ g)
hg : Surjective g
x y : γ
h : f (g x) = f (g y)
⊢ g x = g y | 6d531765884ca3ef |
Lean.Order.admissible_pprod_snd | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Basic.lean | theorem admissible_pprod_snd {α : Sort u} {β : Sort v} [CCPO α] [CCPO β] (P : β → Prop)
(hadm : admissible P) : admissible (fun (x : α ×' β) => P x.2) | α : Sort u
β : Sort v
inst✝¹ : CCPO α
inst✝ : CCPO β
P : β → Prop
hadm : admissible P
c : α ×' β → Prop
hchain : chain c
h : ∀ (x : α ×' β), c x → (fun x => P x.snd) x
⊢ P (CCPO.csup c).snd | apply hadm _ (PProd.chain.chain_snd hchain) | α : Sort u
β : Sort v
inst✝¹ : CCPO α
inst✝ : CCPO β
P : β → Prop
hadm : admissible P
c : α ×' β → Prop
hchain : chain c
h : ∀ (x : α ×' β), c x → (fun x => P x.snd) x
⊢ ∀ (x : β), PProd.chain.snd c x → P x | ad0d9d2474ff8268 |
IsAssociatedPrime.annihilator_le | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | theorem IsAssociatedPrime.annihilator_le (h : IsAssociatedPrime I M) :
(⊤ : Submodule R M).annihilator ≤ I | case intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : M
hI : Ideal.IsPrime (ker (toSpanSingleton R M x))
⊢ ⊤.annihilator ≤ ker (toSpanSingleton R M x) | rw [← Submodule.annihilator_span_singleton] | case intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : M
hI : Ideal.IsPrime (ker (toSpanSingleton R M x))
⊢ ⊤.annihilator ≤ (Submodule.span R {x}).annihilator | 194886cc2bb570ce |
Order.krullDim_pos_iff | Mathlib/Order/KrullDimension.lean | lemma krullDim_pos_iff : 0 < krullDim α ↔ ∃ x y : α, x < y | α : Type u_1
inst✝ : Preorder α
⊢ krullDim α ≤ 0 ↔ ∀ (x y : α), ¬x < y | simp_rw [← isMax_iff_forall_not_lt, ← krullDim_nonpos_iff_forall_isMax] | no goals | dc766689907bf295 |
ENNReal.half_lt_self | Mathlib/Data/ENNReal/Inv.lean | theorem half_lt_self (hz : a ≠ 0) (ht : a ≠ ∞) : a / 2 < a | case intro
a : ℝ≥0
hz : a ≠ 0
⊢ a / 2 < a
case intro
a : ℝ≥0
hz : a ≠ 0
⊢ 2 ≠ 0 | exacts [NNReal.half_lt_self hz, two_ne_zero' _] | no goals | 4240ca8b153ca5da |
Submonoid.LocalizationMap.sec_spec' | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem sec_spec' {f : LocalizationMap S N} (z : N) :
f.toMap (f.sec z).1 = f.toMap (f.sec z).2 * z | M : Type u_1
inst✝¹ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝ : CommMonoid N
f : S.LocalizationMap N
z : N
⊢ f.toMap (f.sec z).1 = f.toMap ↑(f.sec z).2 * z | rw [mul_comm, sec_spec] | no goals | 9d8069dcf2618fd1 |
MeasureTheory.StronglyMeasurable.integral_kernel_prod_right | Mathlib/Probability/Kernel/MeasurableIntegral.lean | theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : To... | intro n | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : To... | 8e2e9c2ee13e9a95 |
Array.map_induction | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem map_induction (as : Array α) (f : α → β) (motive : Nat → Prop) (h0 : motive 0)
(p : Fin as.size → β → Prop) (hs : ∀ i, motive i.1 → p i (f as[i]) ∧ motive (i+1)) :
motive as.size ∧
∃ eq : (as.map f).size = as.size, ∀ i h, p ⟨i, h⟩ ((as.map f)[i]) | case neg
α : Type u_1
β : Type u_2
as : Array α
f : α → β
motive : Nat → Prop
h0 : motive 0
p : Fin as.size → β → Prop
hs : ∀ (i : Fin as.size), motive ↑i → p i (f as[i]) ∧ motive (↑i + 1)
i : Fin as.size
b : Array β
m : motive ↑i
eq : b.size = ↑i
w : ∀ (i : Fin as.size) (h2 : ↑i < b.size), p i b[↑i]
j : Fin as.size
h✝... | simp only [show j = i by omega] | case neg
α : Type u_1
β : Type u_2
as : Array α
f : α → β
motive : Nat → Prop
h0 : motive 0
p : Fin as.size → β → Prop
hs : ∀ (i : Fin as.size), motive ↑i → p i (f as[i]) ∧ motive (↑i + 1)
i : Fin as.size
b : Array β
m : motive ↑i
eq : b.size = ↑i
w : ∀ (i : Fin as.size) (h2 : ↑i < b.size), p i b[↑i]
j : Fin as.size
h✝... | 07b92593a639a2fd |
Orientation.oangle_eq_of_angle_eq_of_sign_eq | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
(h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
(hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z | case inr.inr
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : (o.oangle w x).sign = (o.oangle y 0).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle y 0).sign =... | simpa using hs | no goals | ec88cb80dbeb7f16 |
LinearMap.bot_lt_ker_of_det_eq_zero | Mathlib/LinearAlgebra/Determinant.lean | theorem bot_lt_ker_of_det_eq_zero {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] {f : M →ₗ[𝕜] M}
(hf : LinearMap.det f = 0) : ⊥ < LinearMap.ker f | M : Type u_2
inst✝² : AddCommGroup M
𝕜 : Type u_7
inst✝¹ : Field 𝕜
inst✝ : Module 𝕜 M
f : M →ₗ[𝕜] M
hf : LinearMap.det f = 0
this : FiniteDimensional 𝕜 M
⊢ ⊥ < ker f | contrapose hf | M : Type u_2
inst✝² : AddCommGroup M
𝕜 : Type u_7
inst✝¹ : Field 𝕜
inst✝ : Module 𝕜 M
f : M →ₗ[𝕜] M
this : FiniteDimensional 𝕜 M
hf : ¬⊥ < ker f
⊢ ¬LinearMap.det f = 0 | 92fd01011447bbe2 |
Ideal.ideal_prod_prime | Mathlib/RingTheory/Ideal/Prod.lean | theorem ideal_prod_prime (I : Ideal (R × S)) :
I.IsPrime ↔
(∃ p : Ideal R, p.IsPrime ∧ I = Ideal.prod p ⊤) ∨
∃ p : Ideal S, p.IsPrime ∧ I = Ideal.prod ⊤ p | case mp.inl
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal (R × S)
hI : ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime
h : map (RingHom.fst R S) I = ⊤
⊢ (∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = p.prod ⊤) ∨
∃ p, p.IsPrime ∧ (map (RingHom... | right | case mp.inl.h
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal (R × S)
hI : ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime
h : map (RingHom.fst R S) I = ⊤
⊢ ∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = ⊤.prod p | 5d31973d7dbcd867 |
LinearMap.det_smul | Mathlib/LinearAlgebra/Determinant.lean | theorem det_smul [Module.Free A M] (c : A) (f : M →ₗ[A] M) :
LinearMap.det (c • f) = c ^ Module.finrank A M * LinearMap.det f | case pos
M : Type u_2
inst✝³ : AddCommGroup M
A : Type u_5
inst✝² : CommRing A
inst✝¹ : Module A M
inst✝ : Module.Free A M
c : A
f : M →ₗ[A] M
a✝ : Nontrivial A
H : ∃ s, Nonempty (Basis { x // x ∈ s } A M)
this : Module.Finite A M
⊢ LinearMap.det (c • f) = c ^ Module.finrank A M * LinearMap.det f | simp only [← det_toMatrix (Module.finBasis A M), LinearEquiv.map_smul,
Fintype.card_fin, Matrix.det_smul] | no goals | 8fdbe434c6a19498 |
isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis | Mathlib/Topology/Compactness/Lindelof.lean | theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsLindelof (b i)) (U : Set X) :
IsLindelof U ∧ IsOpen U ↔ ∃ s : Set ι, s.Countable ∧ U = ⋃ i ∈ s, b i | case mp.intro.intro.intro.intro.intro.refine_1
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
Y : Type u
f' : Y → ι
h₁ : IsLindelof (⋃ i, (b ∘ f') i)
h₂ : IsOpen (⋃ i, (b ∘ f') i)
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Set Y
ht : ... | exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, mem_image_of_mem _ hi⟩ | no goals | 5ea85c22866d42e2 |
Function.update_eq_updateFinset | Mathlib/Data/Finset/Update.lean | theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) | ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
i : ι
y : π i
⊢ update x i y = updateFinset x {i} (uniqueElim y) | congr with j | case h
ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
i : ι
y : π i
j : ι
⊢ update x i y j = updateFinset x {i} (uniqueElim y) j | f513edede4d792eb |
MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : μ t < ⊤
⊢ ∀ᵐ (x : α) ∂μ.res... | apply ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter (hf_int_finite t t_meas t_lt_top) | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : μ t < ⊤
⊢ ∀ (s : Set α), Me... | c952d9bd03e52640 |
Filter.tendsto_iff_ptendsto | Mathlib/Order/Filter/Partial.lean | theorem tendsto_iff_ptendsto (l₁ : Filter α) (l₂ : Filter β) (s : Set α) (f : α → β) :
Tendsto f (l₁ ⊓ 𝓟 s) l₂ ↔ PTendsto (PFun.res f s) l₁ l₂ | α : Type u
β : Type v
l₁ : Filter α
l₂ : Filter β
s : Set α
f : α → β
⊢ Tendsto f (l₁ ⊓ 𝓟 s) l₂ ↔ PTendsto (PFun.res f s) l₁ l₂ | simp only [Tendsto, PTendsto, pmap_res] | no goals | 1ad49e0b0e6acd7e |
Metric.cthickening_subset_iUnion_closedBall_of_lt | Mathlib/Topology/MetricSpace/Thickening.lean | theorem cthickening_subset_iUnion_closedBall_of_lt {α : Type*} [PseudoMetricSpace α] (E : Set α)
{δ δ' : ℝ} (hδ₀ : 0 < δ') (hδδ' : δ < δ') : cthickening δ E ⊆ ⋃ x ∈ E, closedBall x δ' | α : Type u_2
inst✝ : PseudoMetricSpace α
E : Set α
δ δ' : ℝ
hδ₀ : 0 < δ'
hδδ' : δ < δ'
x : α
hx : x ∈ thickening δ' E
⊢ x ∈ ⋃ x ∈ E, closedBall x δ' | obtain ⟨y, hy₁, hy₂⟩ := mem_thickening_iff.mp hx | case intro.intro
α : Type u_2
inst✝ : PseudoMetricSpace α
E : Set α
δ δ' : ℝ
hδ₀ : 0 < δ'
hδδ' : δ < δ'
x : α
hx : x ∈ thickening δ' E
y : α
hy₁ : y ∈ E
hy₂ : dist x y < δ'
⊢ x ∈ ⋃ x ∈ E, closedBall x δ' | b330ebd17268c278 |
BoundedContinuousFunction.dist_lt_iff_of_compact | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | theorem dist_lt_iff_of_compact [CompactSpace α] (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C | case mpr
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
⊢ (∀ (x : α), dist (f x) (g x) < C) → dist f g < C | by_cases h : Nonempty α | case pos
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : Nonempty α
⊢ (∀ (x : α), dist (f x) (g x) < C) → dist f g < C
case neg
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝... | fb7b148d1f803c70 |
Ideal.Filtration.submodule_closure_single | Mathlib/RingTheory/Filtration.lean | theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid | case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
i : ℕ
m : M
hm : m ∈ ↑(F.N i)
j : ℕ
h : ¬i = j
⊢ 0 ∈ F.N j | exact (F.N j).zero_mem | no goals | bd336799b08b1c57 |
Multiset.map_count_True_eq_filter_card | Mathlib/Data/Multiset/Filter.lean | theorem map_count_True_eq_filter_card (s : Multiset α) (p : α → Prop) [DecidablePred p] :
(s.map p).count True = card (s.filter p) | α : Type u_1
s : Multiset α
p : α → Prop
inst✝ : DecidablePred p
⊢ count True (map p s) = (filter p s).card | simp only [count_eq_card_filter_eq, filter_map, card_map, Function.id_comp,
eq_true_eq_id, Function.comp_apply] | no goals | be5c1f40ec148462 |
Set.wellFoundedOn_range | Mathlib/Order/WellFoundedSet.lean | theorem wellFoundedOn_range : (range f).WellFoundedOn r ↔ WellFounded (r on f) | case mk.intro.refine_1
α : Type u_2
β : Type u_3
r : α → α → Prop
f : β → α
f' : β → ↑(range f) := fun c => ⟨f c, ⋯⟩
h : WellFounded (r on f)
c : β
⊢ ∀ {a : β} {b : ↑(range f)}, r ↑b ↑(f' a) → ∃ c, f' c = b | rintro _ ⟨_, c', rfl⟩ - | case mk.intro.refine_1.mk.intro
α : Type u_2
β : Type u_3
r : α → α → Prop
f : β → α
f' : β → ↑(range f) := fun c => ⟨f c, ⋯⟩
h : WellFounded (r on f)
c a✝ c' : β
⊢ ∃ c, f' c = ⟨f c', ⋯⟩ | cfcccc1dfe69da6c |
PowerSeries.map_surjective | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem map_surjective (f : S →+* T) (hf : Function.Surjective f) :
Function.Surjective (PowerSeries.map f) | case h.h
S : Type u_2
T : Type u_3
inst✝¹ : Semiring S
inst✝ : Semiring T
f : S →+* T
hf : Function.Surjective ⇑f
g : T⟦X⟧
k : ℕ
⊢ f (Classical.choose ⋯) = (coeff T k) g | exact Classical.choose_spec (hf ((coeff T k) g)) | no goals | 22da55e7c554d77c |
MeasureTheory.tilted_eq_withDensity_nnreal | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma tilted_eq_withDensity_nnreal (μ : Measure α) (f : α → ℝ) :
μ.tilted f = μ.withDensity (fun x ↦ ((↑) : ℝ≥0 → ℝ≥0∞)
(⟨exp (f x) / ∫ x, exp (f x) ∂μ, by positivity⟩ : ℝ≥0)) | case e_f.h
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
x : α
⊢ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) = ↑⟨rexp (f x) / ∫ (x : α), rexp (f x) ∂μ, ⋯⟩ | rw [ENNReal.ofReal_eq_coe_nnreal] | no goals | bedb4f144f9ee107 |
Ideal.absNorm_span_singleton | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem absNorm_span_singleton (r : S) :
absNorm (span ({r} : Set S)) = (Algebra.norm ℤ r).natAbs | case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : Nontrivial S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
r : S
hr : ¬r = 0
this : Fintype (S ⧸ span {r}) := (span {r}).fintypeQuotientOfFreeOfNeBot ⋯
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
⊢ (L... | congr | case neg.e_m.h.e_6.h
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : Nontrivial S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
r : S
hr : ¬r = 0
this : Fintype (S ⧸ span {r}) := (span {r}).fintypeQuotientOfFreeOfNeBot ⋯
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBa... | d5532931e84e3a93 |
Topology.RelCWComplex.cellFrontier_subset_skeletonLT | Mathlib/Topology/CWComplex/Classical/Basic.lean | lemma RelCWComplex.cellFrontier_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) :
cellFrontier n j ⊆ skeletonLT C n | case intro.h
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
n : ℕ
j : cell C n
I : (m : ℕ) → Finset (cell C m)
hI : cellFrontier n j ⊆ D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j
⊢ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j ⊆ ⋃ m, ⋃ (_ : ↑m < ↑n), ⋃ j, closedCell m j | intro x xmem | case intro.h
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
n : ℕ
j : cell C n
I : (m : ℕ) → Finset (cell C m)
hI : cellFrontier n j ⊆ D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j
x : X
xmem : x ∈ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j
⊢ x ∈ ⋃ m, ⋃ (_ : ↑m < ↑n), ⋃ j, closedCell m ... | b66d81e5a34117fe |
Equiv.Perm.eq_sign_of_surjective_hom | Mathlib/GroupTheory/Perm/Sign.lean | theorem eq_sign_of_surjective_hom {s : Perm α →* ℤˣ} (hs : Surjective s) : s = sign :=
have : ∀ {f}, IsSwap f → s f = -1 := fun {f} ⟨x, y, hxy, hxy'⟩ =>
hxy'.symm ▸
by_contradiction fun h => by
have : ∀ f, IsSwap f → s f = 1 := fun f ⟨a, b, hab, hab'⟩ => by
rw [← isConj_iff_eq, ← Or.resolv... | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Perm α →* ℤˣ
hs : Surjective ⇑s
f✝ : Perm α
x✝¹ : f✝.IsSwap
x y : α
hxy : x ≠ y
hxy' : f✝ = swap x y
h : ¬s (swap x y) = -1
f : Perm α
x✝ : f.IsSwap
a b : α
hab : a ≠ b
hab' : f = swap a b
⊢ IsConj (s (swap a b)) (s (swap x y)) | exact s.map_isConj (isConj_swap hab hxy) | no goals | 8e0d79dcb0016e04 |
BitVec.getLsbD_sshiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_sshiftRight (x : BitVec w) (s i : Nat) :
getLsbD (x.sshiftRight s) i =
(!decide (w ≤ i) && if s + i < w then x.getLsbD (s + i) else x.msb) | case pos
w : Nat
x : BitVec w
s i : Nat
hmsb : x.msb = false
hi : i ≥ w
⊢ x.getLsbD (s + i) = (!decide (w ≤ i) && (decide (s + i < w) && x.getLsbD (s + i))) | simp only [hi, decide_true, Bool.not_true, Bool.false_and] | case pos
w : Nat
x : BitVec w
s i : Nat
hmsb : x.msb = false
hi : i ≥ w
⊢ x.getLsbD (s + i) = false | 3860ec5babbfcdfd |
EuclideanGeometry.dist_smul_vadd_eq_dist | Mathlib/Geometry/Euclidean/Basic.lean | theorem dist_smul_vadd_eq_dist {v : V} (p₁ p₂ : P) (hv : v ≠ 0) (r : ℝ) :
dist (r • v +ᵥ p₁) p₂ = dist p₁ p₂ ↔ r = 0 ∨ r = -2 * ⟪v, p₁ -ᵥ p₂⟫ / ⟪v, v⟫ | 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
v : V
p₁ p₂ : P
hv : v ≠ 0
r : ℝ
hvi : inner v v ≠ 0
hd : discrim (inner v v) (2 * inner v (p₁ -ᵥ p₂)) 0 = 2 * inner v (p₁ -ᵥ p₂) * (2 * inner v (p₁ -ᵥ p₂))
⊢ inner v v * (r * r) + 2... | rw [quadratic_eq_zero_iff hvi hd, neg_add_cancel, zero_div, neg_mul_eq_neg_mul, ←
mul_sub_right_distrib, sub_eq_add_neg, ← mul_two, mul_assoc, mul_div_assoc, mul_div_mul_left,
mul_div_assoc] | case hc
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
v : V
p₁ p₂ : P
hv : v ≠ 0
r : ℝ
hvi : inner v v ≠ 0
hd : discrim (inner v v) (2 * inner v (p₁ -ᵥ p₂)) 0 = 2 * inner v (p₁ -ᵥ p₂) * (2 * inner v (p₁ -ᵥ p₂))
⊢ 2 ≠ 0 | 44b00dbdc3b3f9e2 |
CategoryTheory.ShortComplex.HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | lemma HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso
(h : S.HomologyData) [S.HasHomology] :
h.right.homologyIso = h.left.homologyIso ≪≫ h.iso | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
inst✝ : S.HasHomology
⊢ h.iso = h.left.homologyIso.symm ≪≫ h.right.homologyIso | ext | case w
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
inst✝ : S.HasHomology
⊢ h.iso.hom = (h.left.homologyIso.symm ≪≫ h.right.homologyIso).hom | 17b29b5f5a626f43 |
ae_eq_zero_of_integral_smooth_smul_eq_zero | Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean | theorem ae_eq_zero_of_integral_smooth_smul_eq_zero [SigmaCompactSpace M]
(hf : LocallyIntegrable f μ)
(h : ∀ g : M → ℝ, ContMDiff I 𝓘(ℝ) ∞ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
∀ᵐ x ∂μ, f x = 0 | case h.refine_2
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M... | simp [this] | no goals | 28fe4c32a4a6619f |
WeierstrassCurve.Jacobian.smul_equiv_smul | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma smul_equiv_smul (P Q : Fin 3 → R) {u v : R} (hu : IsUnit u) (hv : IsUnit v) :
u • P ≈ v • Q ↔ P ≈ Q | R : Type r
inst✝ : CommRing R
P Q : Fin 3 → R
u v : R
hu : IsUnit u
hv : IsUnit v
⊢ u • P ≈ v • Q ↔ P ≈ Q | rw [← Quotient.eq_iff_equiv, ← Quotient.eq_iff_equiv, smul_eq P hu, smul_eq Q hv] | no goals | f39d76a9b9e30427 |
InformationTheory.not_differentiableWithinAt_klFun_Iio_zero | Mathlib/InformationTheory/KullbackLeibler/KLFun.lean | lemma not_differentiableWithinAt_klFun_Iio_zero : ¬ DifferentiableWithinAt ℝ klFun (Iio 0) 0 | ⊢ Tendsto (deriv klFun) (nhdsWithin 0 (Iio 0)) atBot | rw [deriv_klFun] | ⊢ Tendsto log (nhdsWithin 0 (Iio 0)) atBot | ca4c2cf48091d252 |
Equiv.Perm.pow_mod_card_support_cycleOf_self_apply | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem pow_mod_card_support_cycleOf_self_apply [DecidableEq α] [Fintype α]
(f : Perm α) (n : ℕ) (x : α) : (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
n : ℕ
x : α
⊢ (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x | by_cases hx : f x = x | case pos
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
n : ℕ
x : α
hx : f x = x
⊢ (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x
case neg
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
n : ℕ
x : α
hx : ¬f x = x
⊢ (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x | a2e5f09535723482 |
Profinite.NobelingProof.CC_exact | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem CC_exact {f : LocallyConstant C ℤ} (hf : Linear_CC' C hsC ho f = 0) :
∃ y, πs C o y = f | case refine_3.h.mk.inr.h.e_6.h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
f : LocallyConstant ↑C ℤ
hf :
⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun :... | exact C1_projOrd C hsC ho hx₁ | no goals | c7ebbbfd480d7eed |
Std.DHashMap.Internal.Raw₀.contains_of_contains_insertIfNew' | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem contains_of_contains_insertIfNew' [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {k a : α}
{v : β k} :
(m.insertIfNew k v).contains a → ¬((k == a) ∧ m.contains k = false) → m.contains a | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
k a : α
v : β k
⊢ (m.insertIfNew k v).contains a = true → ¬((k == a) = true ∧ m.contains k = false) → m.contains a = true | simp_to_model [insertIfNew] using List.containsKey_of_containsKey_insertEntryIfNew' | no goals | 129fb185f1b6dc87 |
Vitali.exists_disjoint_subfamily_covering_enlargement | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_subfamily_covering_enlargement (B : ι → Set α) (t : Set ι) (δ : ι → ℝ)
(τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R)
(hne : ∀ a ∈ t, (B a).Nonempty) :
∃ u ⊆ t,
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b | case intro.intro.intro
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
{u |
u ⊆ t ∧
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonem... | clear hat hcon a_disj a | case intro.intro.intro
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
{u |
u ⊆ t ∧
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonem... | e38a150de4970f9c |
SimplexCategory.iso_eq_iso_refl | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem iso_eq_iso_refl {x : SimplexCategory} (e : x ≅ x) : e = Iso.refl x | case w.a
x : SimplexCategory
e : x ≅ x
h : Finset.univ.card = x.len + 1
eq₁ : RelIso.toRelEmbedding (orderIsoOfIso e) = Finset.univ.orderEmbOfFin h
eq₂ : RelIso.toRelEmbedding (orderIsoOfIso (Iso.refl x)) = Finset.univ.orderEmbOfFin h
⊢ Hom.toOrderHom e.hom = Hom.toOrderHom (Iso.refl x).hom | convert congr_arg (fun φ => (OrderEmbedding.toOrderHom φ)) (eq₁.trans eq₂.symm) | case h.e'_2
x : SimplexCategory
e : x ≅ x
h : Finset.univ.card = x.len + 1
eq₁ : RelIso.toRelEmbedding (orderIsoOfIso e) = Finset.univ.orderEmbOfFin h
eq₂ : RelIso.toRelEmbedding (orderIsoOfIso (Iso.refl x)) = Finset.univ.orderEmbOfFin h
⊢ Hom.toOrderHom e.hom = OrderEmbedding.toOrderHom (RelIso.toRelEmbedding (orderIs... | c8a68730c47018a8 |
MeasureTheory.withDensity_eq_zero_iff | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem withDensity_eq_zero_iff {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
μ.withDensity f = 0 ↔ f =ᵐ[μ] 0 | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ μ.withDensity f = 0 ↔ f =ᶠ[ae μ] 0 | rw [← measure_univ_eq_zero, withDensity_apply _ .univ, restrict_univ, lintegral_eq_zero_iff' hf] | no goals | c9442168a7b3da0c |
Finset.filter_and | Mathlib/Data/Finset/Basic.lean | theorem filter_and (s : Finset α) : (s.filter fun a => p a ∧ q a) = s.filter p ∩ s.filter q :=
ext fun _ => by simp [mem_filter, mem_inter, and_comm, and_left_comm, and_self_iff, and_assoc]
| α : Type u_1
p q : α → Prop
inst✝² : DecidablePred p
inst✝¹ : DecidablePred q
inst✝ : DecidableEq α
s : Finset α
x✝ : α
⊢ x✝ ∈ filter (fun a => p a ∧ q a) s ↔ x✝ ∈ filter p s ∩ filter q s | simp [mem_filter, mem_inter, and_comm, and_left_comm, and_self_iff, and_assoc] | no goals | 9c62be78010bf282 |
Equiv.Perm.OnCycleFactors.kerParam_range_eq_centralizer_of_count_le_one | Mathlib/GroupTheory/SpecificGroups/Alternating/Centralizer.lean | theorem OnCycleFactors.kerParam_range_eq_centralizer_of_count_le_one
(h_count : ∀ i, g.cycleType.count i ≤ 1) :
(kerParam g).range = Subgroup.centralizer {g} | case h.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
x : Perm α
hx : x ∈ Subgroup.centralizer {g}
c : { x // x ∈ g.cycleFactorsFinset }
⊢ ↑(((toPermHom g) ⟨x, ⋯⟩) c) = ↑(1 c) | rw [← Multiset.nodup_iff_count_le_one, cycleType_def,
Multiset.nodup_map_iff_inj_on (cycleFactorsFinset g).nodup] at h_count | case h.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_count :
∀ x ∈ g.cycleFactorsFinset.val, ∀ y ∈ g.cycleFactorsFinset.val, (card ∘ support) x = (card ∘ support) y → x = y
x : Perm α
hx : x ∈ Subgroup.centralizer {g}
c : { x // x ∈ g.cycleFactorsFinset }
⊢ ↑(((toPermHom g) ⟨x, ⋯⟩) c) = ↑(1 c... | 50f9021ffb5b7a29 |
frattini_le_comap_frattini_of_surjective | Mathlib/GroupTheory/Frattini.lean | lemma frattini_le_comap_frattini_of_surjective (hφ : Function.Surjective φ) :
frattini G ≤ (frattini H).comap φ | G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
⊢ ∀ i ∈ {H_1 | IsCoatom H_1}, ⨅ a ∈ {H | IsCoatom H}, a ≤ comap φ i | intro M hM | G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM : M ∈ {H_1 | IsCoatom H_1}
⊢ ⨅ a ∈ {H | IsCoatom H}, a ≤ comap φ M | 07175ba0874095b4 |
int_prod_range_nonneg | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) | case intro.succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ ∏ k ∈ Finset.range (n + 1 + (n + 1)), (m - ↑k) | rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc] | case intro.succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ (∏ x ∈ Finset.range ((1 + 1) * n), (m - ↑x)) * ((m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1))) | 845865b3fc92c079 |
SimpleGraph.isClique_map_finset_iff_of_nontrivial | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem isClique_map_finset_iff_of_nontrivial (ht : t.Nontrivial) :
(G.map f).IsClique t ↔ ∃ (s : Finset α), G.IsClique s ∧ s.map f = t | case mpr
α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
t : Finset β
ht : t.Nontrivial
⊢ (∃ s, G.IsClique ↑s ∧ map f s = t) → (SimpleGraph.map f G).IsClique ↑t | rintro ⟨s, hs, rfl⟩ | case mpr.intro.intro
α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
s : Finset α
hs : G.IsClique ↑s
ht : (map f s).Nontrivial
⊢ (SimpleGraph.map f G).IsClique ↑(map f s) | 666a9cd7c64db5fe |
min_assoc | Mathlib/Order/Defs/LinearOrder.lean | lemma min_assoc (a b c : α) : min (min a b) c = min a (min b c) | case h₃.h₂
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : d ≤ a
h₂ : d ≤ min b c
⊢ d ≤ c | apply le_trans h₂ | case h₃.h₂
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : d ≤ a
h₂ : d ≤ min b c
⊢ min b c ≤ c | c74984dcee9d4eb0 |
MeasureTheory.unifIntegrable_subsingleton | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem unifIntegrable_subsingleton [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
{f : ι → α → β} (hf : ∀ i, MemLp (f i) p μ) : UnifIntegrable f p μ | case pos.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
i : ι
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s →... | convert hδ s hs hμs | no goals | b4be6a9aa99176d2 |
MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : MeasurableSet s)
(x : E) (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1))
(t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x... | case e_a
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t... | rw [inter_comm _ u, inter_comm _ u, eq_comm] | case e_a
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t... | a092845449b3addf |
Array.filterMap_flatMap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem filterMap_flatMap {β γ} (l : Array α) (g : α → Array β) (f : β → Option γ) :
(l.flatMap g).filterMap f = l.flatMap fun a => (g a).filterMap f | α : Type u_1
β : Type u_2
γ : Type u_3
l : Array α
g : α → Array β
f : β → Option γ
⊢ filterMap f (flatMap g l) = flatMap (fun a => filterMap f (g a)) l | rcases l with ⟨l⟩ | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
g : α → Array β
f : β → Option γ
l : List α
⊢ filterMap f (flatMap g { toList := l }) = flatMap (fun a => filterMap f (g a)) { toList := l } | 2e88c183d4eabbce |
SimplexCategory.mkOfSucc_δ_gt | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | /-- If `i + 1 > j`, `mkOfSucc i ≫ δ j` is the morphism `⦋1⦌ ⟶ ⦋n⦌` that
sends `0` and `1` to `i + 1` and `i + 2`, respectively. -/
lemma mkOfSucc_δ_gt {n : ℕ} {i : Fin n} {j : Fin (n + 2)}
(h : j < i.succ.castSucc) :
mkOfSucc i ≫ δ j = mkOfSucc i.succ | case a.h.h.h.«_@».Mathlib.AlgebraicTopology.SimplexCategory.Defs._hyg.913.«1».h
n : ℕ
i : Fin n
j : Fin (n + 2)
h : j < i.succ.castSucc
⊢ j ≤ ((Hom.toOrderHom (mkOfSucc i)) ((fun i => i) ⟨1, ⋯⟩)).castSucc | exact Nat.le_of_lt h | no goals | d2722ed2284db630 |
List.mk_mem_sym2 | Mathlib/Data/List/Sym.lean | theorem mk_mem_sym2 {xs : List α} {a b : α} (ha : a ∈ xs) (hb : b ∈ xs) :
s(a, b) ∈ xs.sym2 | case cons.inl.inl.h
α : Type u_1
b : α
xs : List α
ih : b ∈ xs → b ∈ xs → s(b, b) ∈ xs.sym2
⊢ s(b, b) = s(b, b) | rfl | no goals | 2e917f6e418f8710 |
MeasureTheory.integral_exp_pos | Mathlib/MeasureTheory/Integral/Bochner.lean | lemma integral_exp_pos {μ : Measure α} {f : α → ℝ} [hμ : NeZero μ]
(hf : Integrable (fun x ↦ Real.exp (f x)) μ) :
0 < ∫ x, Real.exp (f x) ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hμ : NeZero μ
hf : Integrable (fun x => Real.exp (f x)) μ
⊢ 0 < μ (Function.support fun x => Real.exp (f x)) | suffices (Function.support fun x ↦ Real.exp (f x)) = Set.univ by simp [this, hμ.out] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hμ : NeZero μ
hf : Integrable (fun x => Real.exp (f x)) μ
⊢ (Function.support fun x => Real.exp (f x)) = univ | 2514b71f6019debc |
MeasureTheory.SignedMeasure.findExistsOneDivLT_min | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | theorem findExistsOneDivLT_min (hi : ¬s ≤[i] 0) {m : ℕ}
(hm : m < findExistsOneDivLT s i) : ¬ExistsOneDivLT s i m | α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ¬s ≤[i] 0
m : ℕ
hm : m < Nat.find ⋯
⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m | exact Nat.find_min _ hm | no goals | c1ddab5c74d4c5c1 |
BitVec.getMsbD_concat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getMsbD_concat {i w : Nat} {b : Bool} {x : BitVec w} :
(x.concat b).getMsbD i = if i < w then x.getMsbD i else decide (i = w) && b | case neg
i w : Nat
b : Bool
x : BitVec w
h₀ : ¬i = w
h₁ : ¬i < w
a✝ : i < w + 1
⊢ b = false | omega | no goals | d05db0a79a970455 |
IsNilpotent.exp_add_of_commute | Mathlib/RingTheory/Nilpotent/Exp.lean | theorem exp_add_of_commute {a b : A} (h₁ : Commute a b) (h₂ : IsNilpotent a) (h₃ : IsNilpotent b) :
exp (a + b) = exp a * exp b | case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)... | rw [z₁, add_zero] at split₁ | case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)... | 925becc4d11e37f5 |
Matrix.vecMul_surjective_iff_isUnit | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | theorem vecMul_surjective_iff_isUnit {A : Matrix m m R} :
Function.Surjective A.vecMul ↔ IsUnit A | m : Type u
inst✝² : DecidableEq m
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Fintype m
A : Matrix m m R
⊢ (Function.Surjective fun v => v ᵥ* A) ↔ IsUnit A | rw [vecMul_surjective_iff_exists_left_inverse, exists_left_inverse_iff_isUnit] | no goals | 68d65ce87950505f |
ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left' | Mathlib/Probability/Kernel/MeasurableLIntegral.lean | theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
(a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set (β × γ)
hs : MeasurableSet s
a : α
this : ∀ (b : β), Prod.mk b ⁻¹' s = {c | ((a, b), c) ∈ {p | (p.1.2, p.2) ∈ s}}
⊢ Measurable fun b => (η (a, b)) (Prod.mk b... | simp_rw [this] | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set (β × γ)
hs : MeasurableSet s
a : α
this : ∀ (b : β), Prod.mk b ⁻¹' s = {c | ((a, b), c) ∈ {p | (p.1.2, p.2) ∈ s}}
⊢ Measurable fun b => (η (a, b)) {c | ((a, ... | 294fa3b0e5a4ba28 |
Complex.sin_pi_div_two | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem sin_pi_div_two : sin (π / 2) = 1 :=
calc
sin (π / 2) = Real.sin (π / 2) | ⊢ sin (↑π / 2) = sin ↑(π / 2) | simp | no goals | 13ffcbe8ac348cd9 |
RingQuot.mkAlgHom_rel | Mathlib/Algebra/RingQuot.lean | theorem mkAlgHom_rel {s : A → A → Prop} {x y : A} (w : s x y) :
mkAlgHom S s x = mkAlgHom S s y | S : Type uS
inst✝² : CommSemiring S
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
s : A → A → Prop
x y : A
w : s x y
⊢ (mkAlgHom S s) x = (mkAlgHom S s) y | simp [mkAlgHom_def, mkRingHom_def, Quot.sound (Rel.of w)] | no goals | 19211ff2cf6e7d06 |
MeasureTheory.condExp_congr_ae | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] | case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
h : f =ᶠ[ae μ] g
hm : m ≤ m₀
⊢ μ[f|m] =ᶠ[ae μ] μ[g|m] | by_cases hμm : SigmaFinite (μ.trim hm) | case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
h : f =ᶠ[ae μ] g
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ μ[f|m] =ᶠ[ae μ] μ[g|m]
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measur... | e0e9f7f4839f4674 |
AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | theorem comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ)
(hnaq : n = a + q) :
φ ≫ (Hσ q).f (n + 1) =
-φ ≫ X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩ | case a
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
⊢ Od... | exact ⟨a, by omega⟩ | no goals | 964779d74b03b839 |
ZMod.isSquare_neg_one_iff' | Mathlib/NumberTheory/SumTwoSquares.lean | theorem ZMod.isSquare_neg_one_iff' {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 | n : ℕ
hn : Squarefree n
help : ∀ (a b : ZMod 4), a ≠ 3 → b ≠ 3 → a * b ≠ 3
⊢ IsSquare (-1) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 | rw [ZMod.isSquare_neg_one_iff hn] | n : ℕ
hn : Squarefree n
help : ∀ (a b : ZMod 4), a ≠ 3 → b ≠ 3 → a * b ≠ 3
⊢ (∀ {q : ℕ}, Nat.Prime q → q ∣ n → q % 4 ≠ 3) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 | 2086e5554fd0b596 |
Set.iUnion_prod_of_monotone | Mathlib/Data/Set/Lattice.lean | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | case h.mk.left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
x : α
hz : z ∈ s x
hw : w ∈ t x
⊢ (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i | exact ⟨⟨x, hz⟩, x, hw⟩ | no goals | 6c8a94814fe2d781 |
Option.all_guard | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean | theorem all_guard (p : α → Prop) [DecidablePred p] (a : α) :
Option.all q (guard p a) = (!p a || q a) | α : Type u_1
q : α → Bool
p : α → Prop
inst✝ : DecidablePred p
a : α
⊢ Option.all q (guard p a) = (!decide (p a) || q a) | simp only [guard] | α : Type u_1
q : α → Bool
p : α → Prop
inst✝ : DecidablePred p
a : α
⊢ Option.all q (if p a then some a else none) = (!decide (p a) || q a) | 9b1dd0ac49b2d23d |
CategoryTheory.Localization.Monoidal.rightUnitor_hom_app | Mathlib/CategoryTheory/Localization/Monoidal.lean | lemma rightUnitor_hom_app (X : C) :
(ρ_ ((L').obj X)).hom =
(L').obj X ◁ (ε' L W ε).inv ≫ (μ _ _ _ _ _).hom ≫
(L').map (ρ_ X).hom | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
X : C
⊢ (ρ_ (L'.obj X)).hom = L'.obj X ◁ (ε' L W ε).inv ≫ (μ L W ε X (𝟙_ C)).hom ≫ L'.m... | dsimp [monoidalCategoryStruct, rightUnitor] | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
X : C
⊢ ((tensorBifunctor L W ε).obj (L'.obj X)).map ε.inv ≫
(liftNatTrans L' W (t... | 57a5c56b0a9e267c |
Topology.IsScott.scott_eq_upper_of_completeLinearOrder | Mathlib/Topology/Order/ScottTopology.lean | lemma scott_eq_upper_of_completeLinearOrder : scott α univ = upper α | case a.h.a
α : Type u_1
inst✝ : CompleteLinearOrder α
this✝ : TopologicalSpace α := upper α
U : Set α
this : TopologicalSpace α := scott α univ
⊢ IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U | rw [@isOpen_iff_Iic_compl_or_univ _ _ (scott α univ) ({ topology_eq_scott := rfl }) U] | no goals | feb5d9ba88e61dd3 |
Submodule.baseChange_span | Mathlib/LinearAlgebra/TensorProduct/Tower.lean | @[simp]
lemma baseChange_span (s : Set M) :
(span R s).baseChange A = span A (TensorProduct.mk R A M 1 '' s) | case intro.intro
R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
m : M
hm : m ∈ span R s
⊢ ((TensorProduct.mk R A M) 1) m ∈ ↑(span A (⇑((TensorProduct.mk R A M) 1) '' s)) | apply span_induction (p := fun m' _ ↦ (1 : A) ⊗ₜ[R] m' ∈ span A (TensorProduct.mk R A M 1 '' s))
(hx := hm) | case intro.intro.mem
R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
m : M
hm : m ∈ span R s
⊢ ∀ (x : M) (h : x ∈ s), (fun m' x => 1 ⊗ₜ[R] m' ∈ span A (⇑((TensorProduct.mk R A M) 1) '' s)) x ⋯
case intro.intro... | dcaedd5200da7dda |
LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete_of_ne_zero | Mathlib/GroupTheory/ArchimedeanDensely.lean | lemma LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete_of_ne_zero
{G₀ : Type*} [LinearOrderedCommGroupWithZero G₀] [Nontrivial G₀ˣ] {g : G₀} (hg : g ≠ 0) :
Set.WellFoundedOn {x : G₀ | g ≤ x} (· < ·) ↔ Nonempty (G₀ ≃*o ℤₘ₀) | case refine_1
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
this : ({x | g ≤ x}.WellFoundedOn fun x1 x2 => x1 < x2) ↔ {x | Units.mk0 g hg ≤ x}.WellFoundedOn fun x1 x2 => x1 < x2
f : G₀ˣ ≃*o Multiplicative ℤ
⊢ G₀ ≃* ℤₘ₀ | exact WithZero.withZeroUnitsEquiv.symm.trans f.withZero | no goals | dd175bef7c0a0720 |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case pos.intro
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible... | have hu0 : u ≠ 0 := by
rw [← pow_ne_zero_iff four_ne_zero, hu, div_ne_zero_iff]
exact ⟨ha₄, ha₄'⟩ | case pos.intro
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible... | 5599f8482dc6841e |
ClassGroup.normBound_pos | Mathlib/NumberTheory/ClassNumber/Finite.lean | theorem normBound_pos : 0 < normBound abv bS | case intro.a.a
R : Type u_1
S : Type u_2
inst✝⁵ : EuclideanDomain R
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : Algebra R S
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
bS : Basis ι R S
h : ∀ (i j k : ι), (Algebra.leftMulMatrix bS) (bS i) j k = 0
i j k : ι
⊢ (Algebra.leftMulMatrix ... | simp [h, DMatrix.zero_apply] | no goals | 963a57f6107a0e82 |
Set.chainHeight_union_eq | Mathlib/Order/Height.lean | theorem chainHeight_union_eq (s t : Set α) (H : ∀ a ∈ s, ∀ b ∈ t, a < b) :
(s ∪ t).chainHeight = s.chainHeight + t.chainHeight | α : Type u_1
inst✝ : Preorder α
s t : Set α
H : ∀ a ∈ s, ∀ b ∈ t, a < b
l : List α
hl : l ∈ s.subchain
l' : List α
hl' : l' ∈ t.subchain
h : t.chainHeight = ↑l'.length
⊢ l.length + l'.length ≤ (l ++ l').length + 0 | simp | no goals | 5b4afeceba92ad8b |
MvPolynomial.comp_aeval | Mathlib/Algebra/MvPolynomial/Eval.lean | theorem comp_aeval {B : Type*} [CommSemiring B] [Algebra R B] (φ : S₁ →ₐ[R] B) :
φ.comp (aeval f) = aeval fun i => φ (f i) | R : Type u
S₁ : Type v
σ : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S₁
inst✝² : Algebra R S₁
f : σ → S₁
B : Type u_2
inst✝¹ : CommSemiring B
inst✝ : Algebra R B
φ : S₁ →ₐ[R] B
⊢ φ.comp (aeval f) = aeval fun i => φ (f i) | ext i | case hf
R : Type u
S₁ : Type v
σ : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S₁
inst✝² : Algebra R S₁
f : σ → S₁
B : Type u_2
inst✝¹ : CommSemiring B
inst✝ : Algebra R B
φ : S₁ →ₐ[R] B
i : σ
⊢ (φ.comp (aeval f)) (X i) = (aeval fun i => φ (f i)) (X i) | f133e2b6967db436 |
IsCyclotomicExtension.Rat.Three.eq_one_or_neg_one_of_unit_of_congruent | Mathlib/NumberTheory/Cyclotomic/Three.lean | theorem eq_one_or_neg_one_of_unit_of_congruent
[NumberField K] [IsCyclotomicExtension {3} ℚ K] (hcong : ∃ n : ℤ, λ ^ 2 ∣ (u - n : 𝓞 K)) :
u = 1 ∨ u = -1 | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑u - ↑n
⊢ u = 1 ∨ u = -1 | have hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K | K : Type u_1
inst✝² : Field K
ζ : K
hζ✝ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, 3 ∣ ↑u - ↑n
hζ : IsPrimitiveRoot (zeta 3 ℚ K) ↑3
⊢ u = 1 ∨ u = -1 | 220139a45fa38b2d |
AddMonoidAlgebra.Monic.mul | Mathlib/Algebra/MonoidAlgebra/Degree.lean | lemma Monic.mul
(hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
(hp : p.Monic D) (hq : q.Monic D) : (p * q).Monic D | R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : LinearOrder B
inst✝⁴ : OrderBot B
p q : R[A]
D : A → B
inst✝³ : AddZeroClass A
inst✝² : Add B
inst✝¹ : AddLeftStrictMono B
inst✝ : AddRightStrictMono B
hD : Function.Injective D
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
hp : Monic D p
hq : Monic ... | exact hp | no goals | 17d05e24584ac857 |
Finset.small_alternating_pow_of_small_tripling | Mathlib/Combinatorics/Additive/SmallTripling.lean | /-- If `A` has small tripling, say with constant `K`, then `A` has small alternating powers, in the
sense that `|A^±1 * ... * A^±1|` is at most `|A|` times a constant exponential in the number of
terms in the product.
When `A` is symmetric (`A⁻¹ = A`), the base of the exponential can be lowered from `K ^ 3` to `K`,
wh... | case inr
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
m : ℕ
hm : 3 ≤ m
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin m), ε i ≠ 0
hA₀ : A.Nonempty
hK₁ : 1 ≤ K
δ : Fin 3 → ℤ
hδ : ∀ (i : Fin 3), |δ i| = 1
⊢ ↑(#(A ^ δ 0 * A ^ δ 1 * A ^ δ 2)) ... | simp only [zero_le_one, abs_eq, Int.reduceNeg, forall_iff_succ, isValue, succ_zero_eq_one,
succ_one_eq_two, IsEmpty.forall_iff, and_true] at hδ | case inr
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
m : ℕ
hm : 3 ≤ m
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin m), ε i ≠ 0
hA₀ : A.Nonempty
hK₁ : 1 ≤ K
δ : Fin 3 → ℤ
hδ : (δ 0 = 1 ∨ δ 0 = -1) ∧ (δ 1 = 1 ∨ δ 1 = -1) ∧ (δ 2 = 1 ∨ δ 2 ... | 647fdb3ae1591fd5 |
PFunctor.M.ext_aux | Mathlib/Data/PFunctor/Univariate/M.lean | theorem ext_aux [Inhabited (M F)] [DecidableEq F.A] {n : ℕ} (x y z : M F) (hx : Agree' n z x)
(hy : Agree' n z y) (hrec : ∀ ps : Path F, n = ps.length → iselect ps x = iselect ps y) :
x.approx (n + 1) = y.approx (n + 1) | case succ.step.step.f.f
F : PFunctor.{u}
inst✝¹ : Inhabited F.M
inst✝ : DecidableEq F.A
n : ℕ
n_ih :
∀ (x y z : F.M),
Agree' n z x →
Agree' n z y →
(∀ (ps : Path F), n = length ps → iselect ps x = iselect ps y) → x.approx (n + 1) = y.approx (n + 1)
z : F.M
a✝⁹ : F.A
x✝¹ y✝¹ : F.B a✝⁹ → F.M
a✝⁸ : ∀ (... | subst z | case succ.step.step.f.f
F : PFunctor.{u}
inst✝¹ : Inhabited F.M
inst✝ : DecidableEq F.A
n : ℕ
n_ih :
∀ (x y z : F.M),
Agree' n z x →
Agree' n z y →
(∀ (ps : Path F), n = length ps → iselect ps x = iselect ps y) → x.approx (n + 1) = y.approx (n + 1)
a✝⁸ : F.A
x✝¹ y✝¹ : F.B a✝⁸ → F.M
a✝⁷ : ∀ (i : F.B ... | 4a16a3c615b0519f |
CategoryTheory.unit_mateEquiv | Mathlib/CategoryTheory/Adjunction/Mates.lean | theorem unit_mateEquiv (α : TwoSquare G L₁ L₂ H) (c : C) :
G.map (adj₁.unit.app c) ≫ (mateEquiv adj₁ adj₂ α).app _ =
adj₂.unit.app _ ≫ R₂.map (α.app _) | C : Type u₁
D : Type u₂
E : Type u₃
F : Type u₄
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
inst✝ : Category.{v₄, u₄} F
G : C ⥤ E
H : D ⥤ F
L₁ : C ⥤ D
R₁ : D ⥤ C
L₂ : E ⥤ F
R₂ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
α : TwoSquare G L₁ L₂ H
c : C
⊢ adj₂.unit.app (G.obj c) ≫
... | rw [R₂.map_comp] | C : Type u₁
D : Type u₂
E : Type u₃
F : Type u₄
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
inst✝ : Category.{v₄, u₄} F
G : C ⥤ E
H : D ⥤ F
L₁ : C ⥤ D
R₁ : D ⥤ C
L₂ : E ⥤ F
R₂ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
α : TwoSquare G L₁ L₂ H
c : C
⊢ adj₂.unit.app (G.obj c) ≫
... | 75158210b8e0ab27 |
PowerSeries.coeff_succ_X_mul | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ | R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
⊢ (coeff R (n + 1)) (X * φ) = (coeff R n) φ | simp only [coeff, Finsupp.single_add, add_comm n 1] | R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
⊢ (MvPowerSeries.coeff R (single () 1 + single () n)) (X * φ) = (MvPowerSeries.coeff R (single () n)) φ | 98d9de423b9dd68f |
Algebra.norm_eq_zero_iff | Mathlib/RingTheory/Norm/Basic.lean | theorem norm_eq_zero_iff [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} :
norm R x = 0 ↔ x = 0 | case mp.intro.intro.refine_2
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := C... | contrapose! v_ne with sum_eq | case mp.intro.intro.refine_2
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := C... | b22c4acf9f81550f |
Monotone.tendsto_le_alternating_series | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem Monotone.tendsto_le_alternating_series
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfm : Monotone f) (k : ℕ) : l ≤ ∑ i ∈ range (2 * k), (-1) ^ i * f i | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
ha : Antitone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
⊢ l ≤ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f... | exact ha.le_of_tendsto (hfl.comp (tendsto_atTop_mono (fun n ↦ by dsimp; omega) tendsto_id)) _ | no goals | 8204a47cf4df7c65 |
Ordinal.bsup_comp | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem bsup_comp {o o' : Ordinal.{max u v}} {f : ∀ a < o, Ordinal.{max u v w}}
(hf : ∀ {i j} (hi) (hj), i ≤ j → f i hi ≤ f j hj) {g : ∀ a < o', Ordinal.{max u v}}
(hg : blsub.{_, u} o' g = o) :
(bsup.{_, w} o' fun a ha => f (g a ha) (by rw [← hg]; apply lt_blsub)) = bsup.{_, w} o f | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o o' : Ordinal.{max u v}
f : (a : Ordinal.{max u v}) → a < o → Ordinal.{max u v w}
hf : ∀ {i j : Ordinal.{max u v}} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj
g : (a : Ordinal.{max u v}) → a < o' → Ordinal.{max u v}
hg : o... | rw [← hg] | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o o' : Ordinal.{max u v}
f : (a : Ordinal.{max u v}) → a < o → Ordinal.{max u v w}
hf : ∀ {i j : Ordinal.{max u v}} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj
g : (a : Ordinal.{max u v}) → a < o' → Ordinal.{max u v}
hg : o... | 2780919a3a9c7fa6 |
seminormFromBounded_of_mul_is_mul | Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean | theorem seminormFromBounded_of_mul_is_mul (f_nonneg : 0 ≤ f)
(f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) {x : R}
(hx : ∀ y : R, f (x * y) = f x * f y) (y : R) :
seminormFromBounded' f (x * y) = seminormFromBounded' f x * seminormFromBounded' f y | R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
y : R
⊢ seminormFromBounded' f (x * y) = f x * seminormFromBounded' f y | simp only [seminormFromBounded', mul_assoc, hx, mul_div_assoc,
Real.mul_iSup_of_nonneg (f_nonneg _)] | no goals | ef647cab5d35766f |
Algebra.FormallySmooth.of_isLocalization | Mathlib/RingTheory/Smooth/Basic.lean | theorem of_isLocalization : FormallySmooth R Rₘ | case h
R Rₘ : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing Rₘ
M : Submonoid R
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
this : Rₘ →ₐ[R] Q :=
let __src := IsLocaliza... | apply AlgHom.coe_ringHom_injective | case h.a
R Rₘ : Type u
inst✝⁵ : CommRing R
inst✝⁴ : CommRing Rₘ
M : Submonoid R
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
this : Rₘ →ₐ[R] Q :=
let __src := IsLocali... | c255f8ab1df303c7 |
orthogonalProjection_orthogonal_val | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem orthogonalProjection_orthogonal_val (u : E) :
(orthogonalProjection Kᗮ u : E) = u - orthogonalProjection K u :=
eq_orthogonalProjection_of_mem_orthogonal' (sub_orthogonalProjection_mem_orthogonal _)
(K.le_orthogonal_orthogonal (orthogonalProjection K u).2) <| by simp
| 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
u : E
⊢ u = u - ↑((orthogonalProjection K) u) + ↑((orthogonalProjection K) u) | simp | no goals | c46bfc1d61ea2db4 |
Finset.gcd_mul_left | Mathlib/Algebra/GCDMonoid/Finset.lean | theorem gcd_mul_left {a : α} : (s.gcd fun x ↦ a * f x) = normalize a * s.gcd f | case refine_1
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
a : α
⊢ (∅.gcd fun x => a * f x) = normalize a * ∅.gcd f | simp | no goals | 2fbe9fbe832756a5 |
Finset.shadow_singleton | Mathlib/Combinatorics/SetFamily/Shadow.lean | theorem shadow_singleton (a : α) : ∂ {{a}} = {∅} | α : Type u_1
inst✝ : DecidableEq α
a : α
⊢ ∂ {{a}} = {∅} | simp [shadow] | no goals | 4e99d1d02a66f08a |
Filter.Tendsto.if | Mathlib/Order/Filter/Tendsto.lean | theorem Tendsto.if {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop}
[∀ x, Decidable (p x)] (h₀ : Tendsto f (l₁ ⊓ 𝓟 { x | p x }) l₂)
(h₁ : Tendsto g (l₁ ⊓ 𝓟 { x | ¬p x }) l₂) :
Tendsto (fun x => if p x then f x else g x) l₁ l₂ | case h
α : Type u_1
β : Type u_2
l₁ : Filter α
l₂ : Filter β
f g : α → β
p : α → Prop
inst✝ : (x : α) → Decidable (p x)
h₀ : ∀ s ∈ l₂, {x | x ∈ {x | p x} → x ∈ f ⁻¹' s} ∈ l₁
h₁ : ∀ s ∈ l₂, {x | x ∈ {x | ¬p x} → x ∈ g ⁻¹' s} ∈ l₁
s : Set β
hs : s ∈ l₂
x : α
hp₀ : p x → x ∈ f ⁻¹' s
hp₁ : ¬p x → x ∈ g ⁻¹' s
⊢ x ∈ (fun x =... | rw [mem_preimage] | case h
α : Type u_1
β : Type u_2
l₁ : Filter α
l₂ : Filter β
f g : α → β
p : α → Prop
inst✝ : (x : α) → Decidable (p x)
h₀ : ∀ s ∈ l₂, {x | x ∈ {x | p x} → x ∈ f ⁻¹' s} ∈ l₁
h₁ : ∀ s ∈ l₂, {x | x ∈ {x | ¬p x} → x ∈ g ⁻¹' s} ∈ l₁
s : Set β
hs : s ∈ l₂
x : α
hp₀ : p x → x ∈ f ⁻¹' s
hp₁ : ¬p x → x ∈ g ⁻¹' s
⊢ (if p x then... | 75dfb26837786deb |
CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_hom_π_π | Mathlib/CategoryTheory/Limits/Fubini.lean | theorem limitCurrySwapCompLimIsoLimitCurryCompLim_hom_π_π {j} {k} :
(limitCurrySwapCompLimIsoLimitCurryCompLim G).hom ≫ limit.π _ j ≫ limit.π _ k =
(limit.π _ k ≫ limit.π _ j : limit (_ ⋙ lim) ⟶ _) | J : Type u_1
K : Type u_2
inst✝⁵ : Category.{u_6, u_1} J
inst✝⁴ : Category.{u_5, u_2} K
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
G : J × K ⥤ C
inst✝² : HasLimitsOfShape K C
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasLimit (curry.obj G ⋙ lim)
j : J
k : K
⊢ (limitIsoLimitCurryCompLim (Prod.swap K J ⋙ G)).inv ≫
lim... | dsimp [Equivalence.counit] | J : Type u_1
K : Type u_2
inst✝⁵ : Category.{u_6, u_1} J
inst✝⁴ : Category.{u_5, u_2} K
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
G : J × K ⥤ C
inst✝² : HasLimitsOfShape K C
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasLimit (curry.obj G ⋙ lim)
j : J
k : K
⊢ (limitIsoLimitCurryCompLim (Prod.swap K J ⋙ G)).inv ≫
lim... | 8a461539900f6f44 |
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