name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
one_lt_p : -1 < p - 1
⊢ ∀ (x : ℝ), ∫ (t : ℝ) in 0 ..x, t ^ (p - 1) = x ^ p / p | intro x | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
one_lt_p : -1 < p - 1
x : ℝ
⊢ ∫ (t : ℝ) in 0 ..x, t ^ (p - 1) = x ^ p / p | 720d77f4c064a30f |
Array.snd_unzip | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem snd_unzip (as : Array (α × β)) : (Array.unzip as).snd = as.map Prod.snd | α : Type u_1
β : Type u_2
as : Array (α × β)
⊢ as.unzip.snd = map Prod.snd as | simp only [unzip] | α : Type u_1
β : Type u_2
as : Array (α × β)
⊢ (foldl (fun x x_1 => (x.fst.push x_1.fst, x.snd.push x_1.snd)) (#[], #[]) as).snd = map Prod.snd as | c928f05c42aa012c |
Polynomial.iterate_derivative_natCast_mul | Mathlib/Algebra/Polynomial/Derivative.lean | theorem iterate_derivative_natCast_mul {n k : ℕ} {f : R[X]} :
derivative^[k] ((n : R[X]) * f) = n * derivative^[k] f | R : Type u
inst✝ : Semiring R
n k : ℕ
f : R[X]
⊢ (⇑derivative)^[k] (↑n * f) = ↑n * (⇑derivative)^[k] f | induction' k with k ih generalizing f <;> simp [*] | no goals | fdb090098f2bcf15 |
CategoryTheory.Square.IsPushout.epi_f₂₄ | Mathlib/CategoryTheory/Limits/Shapes/Pullback/Square.lean | lemma epi_f₂₄ [Epi sq.f₁₃] : Epi sq.f₂₄ :=
(MorphismProperty.epimorphisms C).of_isPushout h (by assumption)
| C : Type u
inst✝¹ : Category.{v, u} C
sq : Square C
h : sq.IsPushout
inst✝ : Epi sq.f₁₃
⊢ MorphismProperty.epimorphisms C sq.f₁₃ | assumption | no goals | b93ae7b7ac2b6aff |
OrderedFinpartition.index_extendMiddle_zero | Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean | lemma index_extendMiddle_zero (c : OrderedFinpartition n) (i : Fin c.length) :
(c.extendMiddle i).index 0 = i | n : ℕ
c : OrderedFinpartition n
i : Fin c.length
this :
(c.extendMiddle i).emb i 0 = (c.extendMiddle i).emb ((c.extendMiddle i).index 0) ((c.extendMiddle i).invEmbedding 0)
⊢ (c.extendMiddle i).index 0 = i | contrapose! this | n : ℕ
c : OrderedFinpartition n
i : Fin c.length
this : ¬(c.extendMiddle i).index 0 = i
⊢ ¬(c.extendMiddle i).emb i 0 = (c.extendMiddle i).emb ((c.extendMiddle i).index 0) ((c.extendMiddle i).invEmbedding 0) | 491affafcfcca4d4 |
PadicInt.zmod_congr_of_sub_mem_span_aux | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem zmod_congr_of_sub_mem_span_aux (n : ℕ) (x : ℤ_[p]) (a b : ℤ)
(ha : x - a ∈ (Ideal.span {(p : ℤ_[p]) ^ n}))
(hb : x - b ∈ (Ideal.span {(p : ℤ_[p]) ^ n})) : (a : ZMod (p ^ n)) = b | p : ℕ
hp_prime : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
a b : ℤ
ha : ↑p ^ n ∣ ↑a - x
hb : ↑p ^ n ∣ x - ↑b
this : ↑p ^ n ∣ ↑a - x + (x - ↑b)
⊢ ↑p ^ n ∣ a - b | rwa [sub_eq_add_neg, sub_eq_add_neg, add_assoc, neg_add_cancel_left, ← sub_eq_add_neg, ←
Int.cast_sub, pow_p_dvd_int_iff] at this | no goals | 541043e9b577f28a |
R_mul | Mathlib/Geometry/Manifold/Algebra/Monoid.lean | theorem R_mul {G : Type*} [Semigroup G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I ∞ G]
(g h : G) : 𝑹 I (g * h) = (𝑹 I h).comp (𝑹 I g) | case h
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_8
inst✝³ : Semigroup G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I ∞ G
g h x✝ : G... | simp only [ContMDiffMap.comp_apply, R_apply, mul_assoc] | no goals | 781b321e679e31ba |
Ordinal.zero_sub | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem zero_sub (a : Ordinal) : 0 - a = 0 | a : Ordinal.{u_4}
⊢ 0 - a = 0 | rw [← Ordinal.le_zero] | a : Ordinal.{u_4}
⊢ 0 - a ≤ 0 | aab61f03cd3cce43 |
WittVector.verschiebung_mul_frobenius | Mathlib/RingTheory/WittVector/Identities.lean | theorem verschiebung_mul_frobenius (x y : 𝕎 R) :
verschiebung (x * frobenius y) = verschiebung x * y | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x y : 𝕎 R
⊢ verschiebung (x * frobenius y) = verschiebung x * y | have : IsPoly₂ p fun {R} [Rcr : CommRing R] x y ↦ verschiebung (x * frobenius y) :=
IsPoly.comp₂ (hg := verschiebung_isPoly)
(hf := IsPoly₂.comp (hh := mulIsPoly₂) (hf := idIsPolyI' p) (hg := frobenius_isPoly p)) | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x y : 𝕎 R
this : IsPoly₂ p fun {R} [CommRing R] x y => verschiebung (x * frobenius y)
⊢ verschiebung (x * frobenius y) = verschiebung x * y | ae1010327f1077fa |
Submodule.topologicalClosure_eq_top_iff | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem Submodule.topologicalClosure_eq_top_iff [CompleteSpace E] :
K.topologicalClosure = ⊤ ↔ Kᗮ = ⊥ | 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✝ : CompleteSpace E
⊢ K.topologicalClosure = ⊤ ↔ Kᗮ = ⊥ | rw [← Submodule.orthogonal_orthogonal_eq_closure] | 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✝ : CompleteSpace E
⊢ Kᗮᗮ = ⊤ ↔ Kᗮ = ⊥ | 2b2708d5ffbe295d |
Polynomial.splits_prod | Mathlib/Algebra/Polynomial/Splits.lean | theorem splits_prod {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, (s j).Splits i) → (∏ x ∈ t, s x).Splits i | K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t✝ : Finset ι
a : ι
t : Finset ι
hat : a ∉ t
ih : (∀ j ∈ t, Splits i (s j)) → Splits i (∏ x ∈ t, s x)
ht : Splits i (s a) ∧ ∀ x ∈ t, Splits i (s x)
⊢ Splits i (∏ x ∈ insert a t, s x) | rw [Finset.prod_insert hat] | K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t✝ : Finset ι
a : ι
t : Finset ι
hat : a ∉ t
ih : (∀ j ∈ t, Splits i (s j)) → Splits i (∏ x ∈ t, s x)
ht : Splits i (s a) ∧ ∀ x ∈ t, Splits i (s x)
⊢ Splits i (s a * ∏ x ∈ t, s x) | c734241f09c10f8e |
FirstOrder.Language.Term.realize_constantsVarsEquivLeft | Mathlib/ModelTheory/Semantics.lean | theorem realize_constantsVarsEquivLeft [L[[α]].Structure M]
[(lhomWithConstants L α).IsExpansionOn M] {n} {t : L[[α]].Term (β ⊕ (Fin n))} {v : β → M}
{xs : Fin n → M} :
(constantsVarsEquivLeft t).realize (Sum.elim (Sum.elim (fun a => ↑(L.con a)) v) xs) =
t.realize (Sum.elim v xs) | L : Language
M : Type w
inst✝² : L.Structure M
α : Type u'
β : Type v'
inst✝¹ : L[[α]].Structure M
inst✝ : (L.lhomWithConstants α).IsExpansionOn M
n : ℕ
t : L[[α]].Term (β ⊕ Fin n)
v : β → M
xs : Fin n → M
⊢ realize (Sum.elim (Sum.elim (fun a => ↑(L.con a)) v) xs ∘ ⇑(Equiv.sumAssoc α β (Fin n)).symm) t.constantsToVars ... | rcongr x | case e_v.h
L : Language
M : Type w
inst✝² : L.Structure M
α : Type u'
β : Type v'
inst✝¹ : L[[α]].Structure M
inst✝ : (L.lhomWithConstants α).IsExpansionOn M
n : ℕ
t : L[[α]].Term (β ⊕ Fin n)
v : β → M
xs : Fin n → M
x : α ⊕ β ⊕ Fin n
⊢ (Sum.elim (Sum.elim (fun a => ↑(L.con a)) v) xs ∘ ⇑(Equiv.sumAssoc α β (Fin n)).sym... | cf988a541d87b2fd |
PiNat.exists_lipschitz_retraction_of_isClosed | Mathlib/Topology/MetricSpace/PiNat.lean | theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
(hne : s.Nonempty) :
∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f | case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : ... | by_cases H : longestPrefix x s < firstDiff x y ∨ longestPrefix y s < firstDiff x y | case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : ... | bd92edd65dec8c2a |
Homeomorph.isConnected_preimage | Mathlib/Topology/Homeomorph.lean | theorem isConnected_preimage {s : Set Y} (h : X ≃ₜ Y) :
IsConnected (h ⁻¹' s) ↔ IsConnected s | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
h : X ≃ₜ Y
⊢ IsConnected (⇑h ⁻¹' s) ↔ IsConnected s | rw [← image_symm, isConnected_image] | no goals | afbd4ab70deb5aa5 |
Batteries.TransCmp.cmp_congr_right | Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean | theorem cmp_congr_right (yz : cmp y z = .eq) : cmp x y = cmp x z | cmp✝ : ?m.4441 → ?m.4441 → Ordering
inst✝¹ : TransCmp cmp✝
x✝ : Sort ?u.4439
cmp : x✝ → x✝ → Ordering
inst✝ : TransCmp cmp
y z x : x✝
yz : cmp y z = Ordering.eq
⊢ cmp x y = cmp x z | rw [← Ordering.swap_inj, symm, symm, cmp_congr_left yz] | no goals | bebb35c008c6e2f4 |
Int.fract_eq_iff | Mathlib/Algebra/Order/Floor.lean | theorem fract_eq_iff {a b : α} : fract a = b ↔ 0 ≤ b ∧ b < 1 ∧ ∃ z : ℤ, a - b = z :=
⟨fun h => by
rw [← h]
exact ⟨fract_nonneg _, fract_lt_one _, ⟨⌊a⌋, sub_sub_cancel _ _⟩⟩,
by
rintro ⟨h₀, h₁, z, hz⟩
rw [← self_sub_floor, eq_comm, eq_sub_iff_add_eq, add_comm, ← eq_sub_iff_add_eq, hz,
Int.cast... | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
a b : α
h : fract a = b
⊢ 0 ≤ fract a ∧ fract a < 1 ∧ ∃ z, a - fract a = ↑z | exact ⟨fract_nonneg _, fract_lt_one _, ⟨⌊a⌋, sub_sub_cancel _ _⟩⟩ | no goals | 5ab27f9ba5053d60 |
HahnSeries.unit_aux | Mathlib/RingTheory/HahnSeries/Summable.lean | theorem unit_aux (x : HahnSeries Γ R) {r : R} (hr : r * x.leadingCoeff = 1)
(hxo : IsAddUnit x.order) : 0 < (1 - single (IsAddUnit.addUnit hxo).neg r * x).orderTop | case pos
Γ : Type u_1
R : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : CommRing R
x : HahnSeries Γ R
r : R
hr : r * x.leadingCoeff = 1
hxo : IsAddUnit x.order
y : HahnSeries Γ R := x - (single x.order) x.leadingCoeff
hy : y = 0
hrx : (single hxo.addUnit.neg) r * x = 1
⊢ 0 < (1 - (single hxo.addUnit.neg)... | simp only [hrx, sub_self, orderTop_zero, WithTop.top_pos] | no goals | b85024fcc7f13365 |
RatFunc.liftMonoidWithZeroHom_apply_div | Mathlib/FieldTheory/RatFunc/Basic.lean | theorem liftMonoidWithZeroHom_apply_div {L : Type*} [CommGroupWithZero L]
(φ : MonoidWithZeroHom K[X] L) (hφ : K[X]⁰ ≤ L⁰.comap φ) (p q : K[X]) :
liftMonoidWithZeroHom φ hφ (algebraMap _ _ p / algebraMap _ _ q) = φ p / φ q | K : Type u
inst✝² : CommRing K
inst✝¹ : IsDomain K
L : Type u_1
inst✝ : CommGroupWithZero L
φ : K[X] →*₀ L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
p q : K[X]
⊢ (liftMonoidWithZeroHom φ hφ) ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q) = φ p / φ q | rcases eq_or_ne q 0 with (rfl | hq) | case inl
K : Type u
inst✝² : CommRing K
inst✝¹ : IsDomain K
L : Type u_1
inst✝ : CommGroupWithZero L
φ : K[X] →*₀ L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
p : K[X]
⊢ (liftMonoidWithZeroHom φ hφ) ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) 0) = φ p / φ 0
case inr
K : Type u
inst✝² : CommRing K
inst✝¹ : ... | 8948efb0ead77193 |
List.reverseRecOn_concat | Mathlib/Data/List/Induction.lean | theorem reverseRecOn_concat {motive : List α → Sort*} (x : α) (xs : List α) (nil : motive [])
(append_singleton : ∀ (l : List α) (a : α), motive l → motive (l ++ [a])) :
reverseRecOn (motive := motive) (xs ++ [x]) nil append_singleton =
append_singleton _ _ (reverseRecOn (motive := motive) xs nil append_s... | α : Type u_1
motive : List α → Sort u_2
x : α
xs : List α
nil : motive []
append_singleton : (l : List α) → (a : α) → motive l → motive (l ++ [a])
ys : List α
h : xs.reverse.reverse = ys
⊢ motive (ys ++ [x]) = motive (xs ++ [x]) | simp [(reverse_reverse _).symm.trans h] | no goals | d14aaefefb052f03 |
Real.quadratic_root_cos_pi_div_five | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem quadratic_root_cos_pi_div_five :
letI c := cos (π / 5)
4 * c ^ 2 - 2 * c - 1 = 0 | θ : ℝ := π / 5
hθ : θ = π / 5
⊢ 4 * cos θ ^ 2 - 2 * cos θ - 1 = 0 | set c := cos θ | θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
⊢ 4 * c ^ 2 - 2 * c - 1 = 0 | 9a015197b0eb09bb |
Matrix.BlockTriangular.toBlock_inverse_mul_toBlock_eq_one | Mathlib/LinearAlgebra/Matrix/Block.lean | theorem BlockTriangular.toBlock_inverse_mul_toBlock_eq_one [LinearOrder α] [Invertible M]
(hM : BlockTriangular M b) (k : α) :
((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) *
M.toBlock (fun i => b i < k) fun i => b i < k) =
1 | α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | let p i := b i < k | α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fu... | f05aada380a2baba |
Batteries.UnionFind.root_link | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | theorem root_link {self : UnionFind} {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) :
∃ r, (r = x ∨ r = y) ∧ ∀ i,
(link self x y yroot).rootD i =
if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i | case isFalse
self : UnionFind
x✝ y✝ : Fin self.size
xroot✝ : self.parent ↑x✝ = ↑x✝
yroot✝ : self.parent ↑y✝ = ↑y✝
h✝¹ : ¬↑x✝ = ↑y✝
x y : Fin self.size
xroot : self.parent ↑x = ↑x
yroot : self.parent ↑y = ↑y
m : UnionFind
hm : ∀ (i : Nat), m.parent i = if ↑y = i then ↑x else self.parent i
i : Nat
h✝ : ¬↑y = i
h : self.p... | split <;> [skip; rfl] | case isFalse.isTrue
self : UnionFind
x✝ y✝ : Fin self.size
xroot✝ : self.parent ↑x✝ = ↑x✝
yroot✝ : self.parent ↑y✝ = ↑y✝
h✝² : ¬↑x✝ = ↑y✝
x y : Fin self.size
xroot : self.parent ↑x = ↑x
yroot : self.parent ↑y = ↑y
m : UnionFind
hm : ∀ (i : Nat), m.parent i = if ↑y = i then ↑x else self.parent i
i : Nat
h✝¹ : ¬↑y = i
h ... | d17902f7f8b46530 |
MeasureTheory.SimpleFunc.measure_support_lt_top_of_lintegral_ne_top | Mathlib/MeasureTheory/Function/SimpleFunc.lean | lemma measure_support_lt_top_of_lintegral_ne_top {f : α →ₛ ℝ≥0∞} (hf : f.lintegral μ ≠ ∞) :
μ (support f) < ∞ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
hf : f.lintegral μ ≠ ⊤
⊢ f.FinMeasSupp μ | exact FinMeasSupp.of_lintegral_ne_top hf | no goals | 67e61b2ff326adaf |
cauchy_davenport_minOrder_mul | Mathlib/Combinatorics/Additive/CauchyDavenport.lean | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
∀ (a b : Finset α),
a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
⊢ op (b⁻¹ * a) • b = a | simp | no goals | 095ce4d72a361376 |
LightCondensed.isColimitLocallyConstantPresheafDiagram_desc_apply | Mathlib/Condensed/Discrete/Colimit.lean | @[simp]
lemma isColimitLocallyConstantPresheafDiagram_desc_apply (S : LightProfinite)
(s : Cocone (S.diagram.rightOp ⋙ locallyConstantPresheaf X))
(n : ℕ) (f : LocallyConstant (S.diagram.obj ⟨n⟩) X) :
(isColimitLocallyConstantPresheafDiagram X S).desc s (f.comap (S.asLimitCone.π.app ⟨n⟩).hom) =
s.ι.ap... | X : Type u
S : LightProfinite
s : Cocone (S.diagram.rightOp ⋙ locallyConstantPresheaf X)
n : ℕ
f : LocallyConstant (↑(S.diagram.obj (Opposite.op n)).toTop) X
⊢ (isColimitLocallyConstantPresheafDiagram X S).desc s
(LocallyConstant.comap (TopCat.Hom.hom (S.asLimitCone.π.app (Opposite.op n))) f) =
s.ι.app n f | change ((((locallyConstantPresheaf X).mapCocone (coconeRightOpOfCone S.asLimitCone)).ι.app n) ≫
(isColimitLocallyConstantPresheafDiagram X S).desc s) _ = _ | X : Type u
S : LightProfinite
s : Cocone (S.diagram.rightOp ⋙ locallyConstantPresheaf X)
n : ℕ
f : LocallyConstant (↑(S.diagram.obj (Opposite.op n)).toTop) X
⊢ (((locallyConstantPresheaf X).mapCocone (coconeRightOpOfCone S.asLimitCone)).ι.app n ≫
(isColimitLocallyConstantPresheafDiagram X S).desc s)
f =
... | 61a9706187c6c6ce |
Num.gcd_to_nat | Mathlib/Data/Num/Lemmas.lean | theorem gcd_to_nat : ∀ a b, (gcd a b : ℕ) = Nat.gcd a b | a✝ b✝ : Num
⊢ (a✝ * b✝).natSize ≤ a✝.natSize + b✝.natSize | simp only [natSize_to_nat, cast_mul] | a✝ b✝ : Num
⊢ (↑a✝ * ↑b✝).size ≤ (↑a✝).size + (↑b✝).size | f3263983514ace5b |
CochainComplex.HomComplex.δ_δ | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | lemma δ_δ (n₀ n₁ n₂ : ℤ) (z : Cochain F G n₀) : δ n₁ n₂ (δ n₀ n₁ z) = 0 | case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0 | ext p q hpq | case pos.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = Cochain.v 0 p q hpq | 4c427934df1777bf |
Polynomial.natDegree_monomial_le | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem natDegree_monomial_le (a : R) {m : ℕ} : (monomial m a).natDegree ≤ m | R : Type u
inst✝ : Semiring R
a : R
m : ℕ
⊢ ((monomial m) a).natDegree ≤ m | classical
rw [Polynomial.natDegree_monomial]
split_ifs
exacts [Nat.zero_le _, le_rfl] | no goals | 937cac9b658e14b1 |
intrinsicInterior_nonempty | Mathlib/Analysis/Convex/Intrinsic.lean | theorem intrinsicInterior_nonempty (hs : Convex ℝ s) :
(intrinsicInterior ℝ s).Nonempty ↔ s.Nonempty :=
⟨by simp_rw [nonempty_iff_ne_empty]; rintro h rfl; exact h intrinsicInterior_empty,
Set.Nonempty.intrinsicInterior hs⟩
| V : Type u_2
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
hs : Convex ℝ s
⊢ (intrinsicInterior ℝ s).Nonempty → s.Nonempty | simp_rw [nonempty_iff_ne_empty] | V : Type u_2
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
hs : Convex ℝ s
⊢ intrinsicInterior ℝ s ≠ ∅ → s ≠ ∅ | 78ab5b8c3c019458 |
List.Perm.foldr_eq' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem Perm.foldr_eq' {f : α → β → β} {l₁ l₂ : List α} (p : l₁ ~ l₂)
(comm : ∀ x ∈ l₁, ∀ y ∈ l₁, ∀ (z), f y (f x z) = f x (f y z))
(init) : foldr f init l₁ = foldr f init l₂ | case cons
α : Type u_1
β : Type u_2
f : α → β → β
l₁ l₂ : List α
x : α
l₁✝ l₂✝ : List α
_p : l₁✝ ~ l₂✝
IH :
(∀ (x : α), x ∈ l₁✝ → ∀ (y : α), y ∈ l₁✝ → ∀ (z : β), f y (f x z) = f x (f y z)) →
∀ (init : β), foldr f init l₁✝ = foldr f init l₂✝
comm : ∀ (x_1 : α), x_1 ∈ x :: l₁✝ → ∀ (y : α), y ∈ x :: l₁✝ → ∀ (z : β),... | simp only [foldr] | case cons
α : Type u_1
β : Type u_2
f : α → β → β
l₁ l₂ : List α
x : α
l₁✝ l₂✝ : List α
_p : l₁✝ ~ l₂✝
IH :
(∀ (x : α), x ∈ l₁✝ → ∀ (y : α), y ∈ l₁✝ → ∀ (z : β), f y (f x z) = f x (f y z)) →
∀ (init : β), foldr f init l₁✝ = foldr f init l₂✝
comm : ∀ (x_1 : α), x_1 ∈ x :: l₁✝ → ∀ (y : α), y ∈ x :: l₁✝ → ∀ (z : β),... | 433831e1091382d7 |
SimpleGraph.EdgeDisjointTriangles.map | Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | lemma EdgeDisjointTriangles.map (f : α ↪ β) (hG : G.EdgeDisjointTriangles) :
(G.map f).EdgeDisjointTriangles | α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
hG : G.EdgeDisjointTriangles
⊢ (G.cliqueSet 3).Pairwise (Function.onFun (fun x y => (↑x ∩ ↑y).Subsingleton) (Finset.map f)) | rintro s hs t ht hst | α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
hG : G.EdgeDisjointTriangles
s : Finset α
hs : s ∈ G.cliqueSet 3
t : Finset α
ht : t ∈ G.cliqueSet 3
hst : s ≠ t
⊢ Function.onFun (fun x y => (↑x ∩ ↑y).Subsingleton) (Finset.map f) s t | 655018fdc5f70372 |
Hopf_.mul_antipode₂ | Mathlib/CategoryTheory/Monoidal/Hopf_.lean | theorem mul_antipode₂ (A : Hopf_ C) :
(A.X.comul.hom ⊗ A.X.comul.hom) ≫
(α_ A.X.X.X A.X.X.X (A.X.X.X ⊗ A.X.X.X)).hom ≫
A.X.X.X ◁ (α_ A.X.X.X A.X.X.X A.X.X.X).inv ≫
A.X.X.X ◁ (β_ A.X.X.X A.X.X.X).hom ▷ A.X.X.X ≫
(α_ A.X.X.X (A.X.X.X ⊗ A.X.X.X) A.X.X.X).inv ≫
(α_ A.X.X.X A.X.X.X A.X.X.X)... | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
⊢ A.X.comul.hom ▷ A.X.X.X ≫
(α_ A.X.X.X A.X.X.X A.X.X.X).hom ≫
((((((A.X.X.X ◁ (β_ A.X.X.X A.X.X.X).hom ≫ A.X.X.X ◁ A.X.comul.hom ▷ A.X.X.X) ≫
A.X.X.X ◁ (α_ A.X.X.X A.X.X.X A.X.X.... | slice_lhs 5 6 =>
simp only [← MonoidalCategory.whiskerLeft_comp]
rw [← associator_naturality_right]
simp only [MonoidalCategory.whiskerLeft_comp] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
⊢ A.X.comul.hom ▷ A.X.X.X ≫
(α_ A.X.X.X A.X.X.X A.X.X.X).hom ≫
A.X.X.X ◁ (β_ A.X.X.X A.X.X.X).hom ≫
A.X.X.X ◁ A.X.comul.hom ▷ A.X.X.X ≫
(((((A.X.X.X ◁ (A.X.X.X ⊗ A.X.X.X) ◁ A.an... | d5483e989581d854 |
AlgebraicGeometry.Scheme.image_zeroLocus | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lemma image_zeroLocus {U : X.Opens} (s : Set Γ(X, U)) :
f.base '' X.zeroLocus s =
Y.zeroLocus (U := f ''ᵁ U) ((f.appIso U).inv.hom '' s) ∩ Set.range f.base | case neg
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
U : X.Opens
s : Set ↑Γ(X, U)
x : ↑↑Y.toPresheafedSpace
hx : x ∉ Set.range ⇑(ConcreteCategory.hom f.base)
⊢ x ∉ (fun a => (ConcreteCategory.hom f.base) a) '' X.zeroLocus s | exact fun H ↦ hx (Set.image_subset_range _ _ H) | no goals | 44eb1a12f6d6380b |
edist_triangle_right | Mathlib/Topology/EMetricSpace/Defs.lean | theorem edist_triangle_right (x y z : α) : edist x y ≤ edist x z + edist y z | α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
⊢ edist x y ≤ edist x z + edist z y | apply edist_triangle | no goals | ad3310d70aea9f85 |
MellinConvergent.comp_rpow | Mathlib/Analysis/MellinTransform.lean | theorem MellinConvergent.comp_rpow {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0) :
MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / a) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / ↑a) | refine Iff.trans ?_ (integrableOn_Ioi_comp_rpow_iff' _ ha) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ MellinConvergent (fun t => f (t ^ a)) s ↔
IntegrableOn (fun x => x ^ (a - 1) • ↑(x ^ a) ^ (s / ↑a - 1) • f (x ^ a)) (Ioi 0) volume | c0df53a68e81d44e |
LieIdeal.map_le_iff_le_comap | Mathlib/Algebra/Lie/Submodule.lean | theorem map_le_iff_le_comap : map f I ≤ J ↔ I ≤ comap f J | R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
⊢ map f I ≤ J ↔ I ≤ comap f J | rw [map_le] | R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
⊢ ⇑f '' ↑I ⊆ ↑J ↔ I ≤ comap f J | d2ab7ea0bbe9d309 |
Set.WellFoundedOn.induction | Mathlib/Order/WellFoundedSet.lean | theorem induction (hs : s.WellFoundedOn r) (hx : x ∈ s) {P : α → Prop}
(hP : ∀ y ∈ s, (∀ z ∈ s, r z y → P z) → P y) : P x | α : Type u_2
r : α → α → Prop
s : Set α
x : α
hs : s.WellFoundedOn r
hx : x ∈ s
P : α → Prop
hP : ∀ y ∈ s, (∀ z ∈ s, r z y → P z) → P y
Q : ↑s → Prop := fun y => P ↑y
⊢ P x | change Q ⟨x, hx⟩ | α : Type u_2
r : α → α → Prop
s : Set α
x : α
hs : s.WellFoundedOn r
hx : x ∈ s
P : α → Prop
hP : ∀ y ∈ s, (∀ z ∈ s, r z y → P z) → P y
Q : ↑s → Prop := fun y => P ↑y
⊢ Q ⟨x, hx⟩ | 2068bdfdecd91454 |
List.pairwise_iff_get | Mathlib/.lake/packages/batteries/Batteries/Data/List/Pairwise.lean | theorem pairwise_iff_get : Pairwise R l ↔ ∀ (i j) (_hij : i < j), R (get l i) (get l j) | case mpr
α✝ : Type u_1
R : α✝ → α✝ → Prop
l : List α✝
h : ∀ (i j : Fin l.length), i < j → R (l.get i) (l.get j)
⊢ ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j] | intros i j hi hj h' | case mpr
α✝ : Type u_1
R : α✝ → α✝ → Prop
l : List α✝
h : ∀ (i j : Fin l.length), i < j → R (l.get i) (l.get j)
i j : Nat
hi : i < l.length
hj : j < l.length
h' : i < j
⊢ R l[i] l[j] | 826f6ec497e5fd5d |
MeasureTheory.SignedMeasure.exists_compl_positive_negative | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | theorem exists_compl_positive_negative (s : SignedMeasure α) :
∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 | case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n,... | refine le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (Eventually.of_forall fun n => ?_) | case a
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), s ≤[B n] 0
A : Set α := ⋃ n,... | 583feb8b06b6c249 |
Array.map_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lex/Lemmas.lean | theorem map_lt [LT α] [LT β]
{l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
map f l₁ < map f l₂ | case mk.mk
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
f : α → β
w : ∀ (x y : α), x < y → f x < f y
toList✝¹ toList✝ : List α
h : { toList := toList✝¹ } < { toList := toList✝ }
⊢ map f { toList := toList✝¹ } < map f { toList := toList✝ } | simpa using List.map_lt w h | no goals | d51cd655ac83969f |
FormalMultilinearSeries.comp_coeff_one | Mathlib/Analysis/Analytic/Composition.lean | theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 v | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologica... | simp | no goals | 2ac1709b5f4213f3 |
Int.add_le_add_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem add_le_add_left {a b : Int} (h : a ≤ b) (c : Int) : c + a ≤ c + b :=
let ⟨n, hn⟩ := le.dest h; le.intro n <| by rw [Int.add_assoc, hn]
| a b : Int
h : a ≤ b
c : Int
n : Nat
hn : a + ↑n = b
⊢ c + a + ↑n = c + b | rw [Int.add_assoc, hn] | no goals | 2721ebabb31203d0 |
MvPolynomial.C_mul_X_pow_eq_monomial | Mathlib/Algebra/MvPolynomial/Basic.lean | theorem C_mul_X_pow_eq_monomial {s : σ} {a : R} {n : ℕ} :
C a * X s ^ n = monomial (Finsupp.single s n) a | R : Type u
σ : Type u_1
inst✝ : CommSemiring R
s : σ
a : R
n : ℕ
⊢ C a * X s ^ n = (monomial (Finsupp.single s n)) a | rw [← zero_add (Finsupp.single s n), monomial_add_single, C_apply] | no goals | e507014d7f3c0aad |
List.getLast_tail | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getLast_tail (l : List α) (h : l.tail ≠ []) :
(tail l).getLast h = l.getLast (ne_nil_of_tail_ne_nil h) | case e_i
α : Type u_1
l : List α
h : l.tail ≠ []
⊢ l.length - 1 - 1 + 1 = l.length - 1 | match l with
| _ :: _ :: l => simp | no goals | 26ad402197e3f6fa |
SeparatingDual.exists_separating_of_ne | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | theorem exists_separating_of_ne {x y : V} (h : x ≠ y) :
∃ f : V →L[R] R, f x ≠ f y | case intro
R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace V
inst✝² : TopologicalSpace R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x y : V
h : x ≠ y
f : V →L[R] R
hf : f (x - y) ≠ 0
⊢ ∃ f, f x ≠ f y | exact ⟨f, by simpa [sub_ne_zero] using hf⟩ | no goals | d6044cc2572abdd9 |
Vector.vector₂_induction | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem vector₂_induction (P : Vector (Vector α n) m → Prop)
(of : ∀ (xss : Array (Array α)) (h₁ : xss.size = m) (h₂ : ∀ xs ∈ xss, xs.size = n),
P (mk (xss.attach.map (fun ⟨xs, m⟩ => mk xs (h₂ xs m))) (by simpa using h₁)))
(ass : Vector (Vector α n) m) : P ass | α : Type u_1
n m : Nat
P : Vector (Vector α n) m → Prop
ass : Vector (Vector α n) m
of :
P
{
toArray :=
Array.map
(fun x =>
match x with
| ⟨xs, m_1⟩ => { toArray := xs, size_toArray := ⋯ })
(map toArray ass).attach,
size_toArray := ⋯ }
⊢ P ass | simpa [Array.map_attach, Array.pmap_map] using of | no goals | ee0e1c9bcba1b731 |
RootPairing.root_sub_root_mem_of_pairingIn_pos | Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean | lemma root_sub_root_mem_of_pairingIn_pos (h : 0 < P.pairingIn ℤ i j) (h' : i ≠ j) :
α i - α j ∈ Φ | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : Finite ι
inst✝⁴ : CharZero R
inst✝³ : P.IsCrystallographic
i j : ι
inst✝² : NoZeroDivisors R
inst✝¹ : NoZeroSMulDivisors R M
i... | aesop | no goals | c7055659e7d25d4e |
WithTop.isLUB_sSup' | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | theorem isLUB_sSup' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
(hs : s.Nonempty) : IsLUB s (sSup s) | case pos.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∈ s
a : β
ha : Option.some a ∈ upperBounds s
⊢ ⊤ ≤ Option.some a | exact False.elim (not_top_le_coe a (ha h₁)) | no goals | 8c5594b3d704c24c |
isCompact_univ_pi | Mathlib/Topology/Compactness/Compact.lean | theorem isCompact_univ_pi {s : ∀ i, Set (X i)} (h : ∀ i, IsCompact (s i)) :
IsCompact (pi univ s) | ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : (i : ι) → Set (X i)
h : ∀ (i : ι), IsCompact (s i)
⊢ IsCompact (univ.pi s) | convert isCompact_pi_infinite h | case h.e'_3
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : (i : ι) → Set (X i)
h : ∀ (i : ι), IsCompact (s i)
⊢ univ.pi s = {x | ∀ (i : ι), x i ∈ s i} | c2f01e8a3d06382e |
Std.DHashMap.Internal.Raw₀.size_alter_le_size | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem size_alter_le_size [LawfulBEq α] (h : m.1.WF) {k : α} {f : Option (β k) → Option (β k)} :
(m.alter k f).1.size ≤ m.1.size + 1 | case isTrue
α : Type u
β : α → Type v
m : Raw₀ α β
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
h : m.val.WF
k : α
f : Option (β k) → Option (β k)
h✝ : (m.contains k && (f (m.get? k)).isNone) = true
⊢ m.val.size - 1 ≤ m.val.size + 1
case isFalse.isTrue
α : Type u
β : α → Type v
m : Raw₀ α β
inst✝² : BEq α
in... | all_goals omega | no goals | 6dbd8bba7fde94c6 |
CategoryTheory.Localization.Construction.uniq | Mathlib/CategoryTheory/Localization/Construction.lean | theorem uniq (G₁ G₂ : W.Localization ⥤ D) (h : W.Q ⋙ G₁ = W.Q ⋙ G₂) : G₁ = G₂ | case refine_2.mk.mk.inr.mk
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
X Y : C
w : { obj := Y }.obj ⟶ { obj := X }.obj
hw✝ : W w
hw : W.Q.map w = (wIso w hw✝).hom
⊢ (Quotient.functor (relations W) ⋙ G₁).map ... | have hw' := Functor.congr_hom h w | case refine_2.mk.mk.inr.mk
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
X Y : C
w : { obj := Y }.obj ⟶ { obj := X }.obj
hw✝ : W w
hw : W.Q.map w = (wIso w hw✝).hom
hw' : (W.Q ⋙ G₁).map w = eqToHom ⋯ ≫ (W.Q ⋙ ... | 62b80593df5a38d8 |
Nat.modCore_eq_mod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean | theorem modCore_eq_mod (n m : Nat) : Nat.modCore n m = n % m | case false
n m n✝ x✝ : Nat
h : ¬x✝ ≤ n✝ + 1
⊢ (n✝ + 1).modCore x✝ = n✝ + 1 | rw [Nat.modCore] | case false
n m n✝ x✝ : Nat
h : ¬x✝ ≤ n✝ + 1
⊢ ite (0 < x✝ ∧ x✝ ≤ n✝ + 1) ((n✝ + 1 - x✝).modCore x✝) (n✝ + 1) = n✝ + 1 | 092d7f99e558f524 |
finAddFlip_apply_natAdd | Mathlib/Logic/Equiv/Fin.lean | theorem finAddFlip_apply_natAdd (k : Fin n) (m : ℕ) :
finAddFlip (Fin.natAdd m k) = Fin.castAdd m k | n : ℕ
k : Fin n
m : ℕ
⊢ finAddFlip (Fin.natAdd m k) = Fin.castAdd m k | simp [finAddFlip] | no goals | 7244e2dac92d7bf0 |
ENNReal.inv_mul_cancel_right | Mathlib/Data/ENNReal/Inv.lean | /-- See `ENNReal.inv_mul_cancel_right'` for a stronger version. -/
protected lemma inv_mul_cancel_right (hb₀ : b ≠ 0) (hb : b ≠ ∞) : a * b⁻¹ * b = a :=
ENNReal.inv_mul_cancel_right' (by simp [hb₀]) (by simp [hb])
| a b : ℝ≥0∞
hb₀ : b ≠ 0
hb : b ≠ ⊤
⊢ b = 0 → a = 0 | simp [hb₀] | no goals | d50b291c25555585 |
MeasureTheory.AEContinuous.hasBoxIntegral | Mathlib/Analysis/BoxIntegral/Integrability.lean | theorem AEContinuous.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} (μ : Measure (ι → ℝ))
[IsLocallyFiniteMeasure μ] {I : Box ι} (hb : ∃ C : ℝ, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C)
(hc : ∀ᵐ x ∂μ, ContinuousAt f x) (l : IntegrationParams) :
HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul (∫ x in I, f x ∂μ) | case h.e'_13.right
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x
l : IntegrationParams
y... | obtain ⟨C, hC⟩ := hb | case h.e'_13.right.intro
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x
l : IntegrationParams
y : E
hy : HasIntegral I l f μ.t... | 1e4b7a409f71d4b9 |
seminormFromBounded_nonzero | Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean | theorem seminormFromBounded_nonzero (f_ne_zero : f ≠ 0) (f_nonneg : 0 ≤ f)
(f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) :
seminormFromBounded' f ≠ 0 | R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ seminormFromBounded' f ≠ 0 | obtain ⟨x, hx⟩ := Function.ne_iff.mp f_ne_zero | case intro
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : f x ≠ 0 x
⊢ seminormFromBounded' f ≠ 0 | 8820e77db939b97e |
summable_schlomilch_iff_of_nonneg | Mathlib/Analysis/PSeries.lean | theorem summable_schlomilch_iff_of_nonneg {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
(hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
(Summable fun k : ℕ => (u (k + 1) - (u k : ℝ)) * f (u k))... | C : ℕ
u : ℕ → ℕ
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
k : ℕ
⊢ ↑(u (k + 1)) - ↑(u k) = ↑(↑(u (k + 1)) - ↑(u k)) | have := Nat.cast_le (α := ℝ≥0).mpr <| (hu_strict k.lt_succ_self).le | C : ℕ
u : ℕ → ℕ
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
k : ℕ
this : ↑(u k) ≤ ↑(u k.succ)
⊢ ↑(u (k + 1)) - ↑(u k) = ↑(↑(u (k + 1)) - ↑(u k)) | 847a42aca13ee025 |
Cardinal.prime_of_aleph0_le | Mathlib/SetTheory/Cardinal/Divisibility.lean | theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a | case refine_2.inr.inr
a : Cardinal.{u}
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this : ∀ {a : Cardinal.{u}}, ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
⊢ a ∣ b ∨ a ∣ c | cases le_total c b <;> [solve_by_elim; rw [or_comm]] | case refine_2.inr.inr.inr
a : Cardinal.{u}
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this : ∀ {a : Cardinal.{u}}, ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
⊢ a ∣ c ∨ a ∣ b | 05a70ed30972affc |
CategoryTheory.toNerve₂.mk_naturality_δ1i | Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean | lemma toNerve₂.mk_naturality_δ1i (i : Fin 3) : toNerve₂.mk.naturalityProperty F (δ₂ i) | case h.«2».hx
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ X.map (δ 1).op (X.map (δ 2).op x) = X.map (δ 1 ≫ δ... | simp only [← FunctorToTypes.map_comp_apply, ← op_comp] | case h.«2».hx
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ X.map (δ 1 ≫ δ 2).op x = X.map (δ 1 ≫ δ 1).op x | 4128d2c817244193 |
Real.compact_inner_le_weight_mul_Lp_of_nonneg | Mathlib/Analysis/MeanInequalities.lean | /-- **Weighted Hölder inequality** in terms of `Finset.expect`. -/
lemma compact_inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) {w f : ι → ℝ}
(hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 ≤ f i) :
𝔼 i ∈ s, w i * f i ≤ (𝔼 i ∈ s, w i) ^ (1 - p⁻¹) * (𝔼 i ∈ s, w i * f i ^ p) ^ p⁻¹ | case hx
ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ 0 ≤ ∑ i ∈ s, w i * f i ^ p | exact sum_nonneg fun i _ ↦ by have := hw i; have := hf i; positivity | no goals | 03820a3c7b910519 |
HasDerivAt.lhopital_zero_atBot | Mathlib/Analysis/Calculus/LHopital.lean | theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
(hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
(hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l | case intro.intro.intro.intro.intro.intro.intro.refine_1.a
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atBot l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atBot
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atBot
hgg' : ∀ y ∈ s₂, HasDerivA... | exact (hl' hx).1.1 | no goals | 29d48f134b3a710b |
Relation.cutExpand_iff | Mathlib/Logic/Hydra.lean | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s : Multiset α
⊢ (∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = (s + t).erase a) ↔
∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | refine exists₂_congr fun t a ↦ ⟨?_, ?_⟩ | case refine_1
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s t : Multiset α
a : α
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = (s + t).erase a → (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t
case refine_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s t : Mu... | f1b635a3d8830f03 |
OrderBot.ext_bot | Mathlib/Order/BoundedOrder/Basic.lean | theorem OrderBot.ext_bot {α} {hA : PartialOrder α} (A : OrderBot α) {hB : PartialOrder α}
(B : OrderBot α) (H : ∀ x y : α, (haveI := hA; x ≤ y) ↔ x ≤ y) :
(@Bot.bot α (@OrderBot.toBot α hA.toLE A)) = (@Bot.bot α (@OrderBot.toBot α hB.toLE B)) | α : Type u_1
hA : PartialOrder α
A : OrderBot α
hB : PartialOrder α
B : OrderBot α
H : ∀ (x y : α), x ≤ y ↔ x ≤ y
⊢ ⊥ = ⊥ | cases PartialOrder.ext H | case refl
α : Type u_1
hA : PartialOrder α
A B : OrderBot α
H : ∀ (x y : α), x ≤ y ↔ x ≤ y
⊢ ⊥ = ⊥ | f8744f7c498d5843 |
mellin_inversion | Mathlib/Analysis/MellinInversion.lean | theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
(hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
mellinInv σ (mellin f) x = f x | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
h2π : 2 * π ≠ 0
this : Integrable (𝓕 fun u => rex... | exact this | no goals | eb76f337444b2548 |
LocalizedModule.subsingleton_iff_support_subset | Mathlib/RingTheory/Support.lean | lemma LocalizedModule.subsingleton_iff_support_subset {f : R} :
Subsingleton (LocalizedModule (.powers f) M) ↔
Module.support R M ⊆ PrimeSpectrum.zeroLocus {f} | case mp.intro
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : PrimeSpectrum R
hx' : x ∈ Module.support R M
f : R
H : ∀ (m : M), ∃ r ∈ Submonoid.powers f, r • m = 0
m : M
hm : (Submodule.span R {m}).annihilator ≤ x.asIdeal
⊢ f ∈ ↑x.asIdeal | obtain ⟨_, ⟨n, rfl⟩, e⟩ := H m | case mp.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : PrimeSpectrum R
hx' : x ∈ Module.support R M
f : R
H : ∀ (m : M), ∃ r ∈ Submonoid.powers f, r • m = 0
m : M
hm : (Submodule.span R {m}).annihilator ≤ x.asIdeal
n : ℕ
e : (fun x => f ^ x) n • m = ... | 75188a38276e1985 |
Std.Sat.AIG.mkAtomCached_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean | theorem mkAtomCached_decl_eq (aig : AIG α) (var : α) (idx : Nat) {h : idx < aig.decls.size}
{hbound} :
(aig.mkAtomCached var).aig.decls[idx]'hbound = aig.decls[idx] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
var : α
idx : Nat
h : idx < aig.decls.size
hbound : idx < (aig.mkAtomCached var).aig.decls.size
hcache : aig.cache.get? (Decl.atom var) = none
⊢ (aig.mkAtomCached var).aig.decls[idx] = aig.decls[idx] | have := mkAtomCached_miss_aig aig hcache | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
var : α
idx : Nat
h : idx < aig.decls.size
hbound : idx < (aig.mkAtomCached var).aig.decls.size
hcache : aig.cache.get? (Decl.atom var) = none
this : (aig.mkAtomCached var).aig.decls = aig.decls.push (Decl.atom var)
⊢ (aig.mkAtomCached var).aig.decls[idx] =... | 02f029f479e2f29c |
List.all_flatten | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem all_flatten {l : List (List α)} : l.flatten.all f = l.all (all · f) | α : Type u_1
f : α → Bool
l : List (List α)
⊢ l.flatten.all f = l.all fun x => x.all f | induction l <;> simp_all | no goals | bec3d372a87c3e02 |
ContMDiff.codRestrict_sphere | Mathlib/Geometry/Manifold/Instances/Sphere.lean | theorem ContMDiff.codRestrict_sphere {n : ℕ} [Fact (finrank ℝ E = n + 1)] {f : M → E}
(hf : ContMDiff I 𝓘(ℝ, E) m f) (hf' : ∀ x, f x ∈ sphere (0 : E) 1) :
ContMDiff I (𝓡 n) m (Set.codRestrict _ _ hf' : M → sphere (0 : E) 1) | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace ℝ E
m : WithTop ℕ∞
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ F H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I m M
n ... | intro v | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace ℝ E
m : WithTop ℕ∞
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ F H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I m M
n ... | 8dbeec34c7921ef6 |
DirichletCharacter.LFunctionTrivChar_residue_one | Mathlib/NumberTheory/LSeries/DirichletContinuation.lean | /-- The L function of the trivial Dirichlet character mod `N` has a simple pole with
residue `∏ p ∈ N.primeFactors, (1 - p⁻¹)` at `s = 1`. -/
lemma LFunctionTrivChar_residue_one :
Tendsto (fun s ↦ (s - 1) * LFunctionTrivChar N s) (𝓝[≠] 1)
(𝓝 <| ∏ p ∈ N.primeFactors, (1 - (p : ℂ)⁻¹)) | N : ℕ
inst✝ : NeZero N
H :
(fun s => (s - 1) * LFunctionTrivChar N s) =ᶠ[𝓝[≠] 1] fun s =>
(∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)
⊢ Tendsto (fun s => (s - 1) * LFunctionTrivChar N s) (𝓝[≠] 1) (𝓝 (∏ p ∈ N.primeFactors, (1 - (↑p)⁻¹))) | rw [tendsto_congr' H] | N : ℕ
inst✝ : NeZero N
H :
(fun s => (s - 1) * LFunctionTrivChar N s) =ᶠ[𝓝[≠] 1] fun s =>
(∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)
⊢ Tendsto (fun s => (∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)) (𝓝[≠] 1)
(𝓝 (∏ p ∈ N.primeFactors, (1 - (↑p)⁻¹))) | b015c88a63308f99 |
ENNReal.tendsto_toNNReal_iff | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tendsto_toNNReal_iff {f : α → ℝ≥0∞} {u : Filter α} (ha : a ≠ ∞) (hf : ∀ x, f x ≠ ∞) :
Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 (a.toNNReal)) ↔ Tendsto f u (𝓝 a) | α : Type u_1
a : ℝ≥0∞
f : α → ℝ≥0∞
u : Filter α
ha : a ≠ ⊤
hf : ∀ (x : α), f x ≠ ⊤
h : Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 a.toNNReal)
⊢ Tendsto ((fun x => ↑x) ∘ ENNReal.toNNReal ∘ f) u (𝓝 a) | exact (tendsto_coe_toNNReal ha).comp h | no goals | 0eb0792c000eed0b |
CategoryTheory.Limits.pullback_fst_map_snd_isPullback | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | theorem pullback_fst_map_snd_isPullback :
IsPullback (fst _ _ ≫ i₁ ≫ fst _ _)
(map i₁ i₂ (i₁ ≫ snd _ _) (i₂ ≫ snd _ _) _ _ _
(Category.id_comp _).symm (Category.id_comp _).symm)
(diagonal f)
(map (i₁ ≫ snd _ _) (i₂ ≫ snd _ _) f f (i₁ ≫ fst _ _) (i₂ ≫ fst _ _) i (by simp [condition])
... | C : Type u_1
inst✝² : Category.{?u.739784, u_1} C
X Y Z : C
inst✝¹ : HasPullbacks C
U V₁ V₂ : C
f : X ⟶ Y
i : U ⟶ Y
i₁ : V₁ ⟶ pullback f i
i₂ : V₂ ⟶ pullback f i
inst✝ : HasPullback i₁ i₂
⊢ (i₁ ≫ snd f i) ≫ i = (i₁ ≫ fst f i) ≫ f | simp [condition] | no goals | 3c998a8a2b31ce50 |
Std.DHashMap.Internal.List.getValueCast?_eraseKey | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast?_eraseKey [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α}
(hl : DistinctKeys l) :
getValueCast? a (eraseKey k l) = if k == a then none else getValueCast? a l | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k a : α
hl : DistinctKeys l
⊢ (Std.DHashMap.Internal.List.Option.dmap (if (k == a) = true then none else getEntry? a l) fun a_1 h' =>
cast ⋯ a_1.snd) =
if (k == a) = true then none else getValueCast? a l | by_cases h : k == a | case pos
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k a : α
hl : DistinctKeys l
h : (k == a) = true
⊢ (Std.DHashMap.Internal.List.Option.dmap (if (k == a) = true then none else getEntry? a l) fun a_1 h' =>
cast ⋯ a_1.snd) =
if (k == a) = true then none else getValueC... | dcb5eec118dbc7bf |
MeasureTheory.upcrossingsBefore_eq_sum | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 | rw [Set.indicator_of_mem] | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ 1 k = 1
case h
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ k ∈ {n | upperCrossingTime a b f N n ω < N} | 50c14e6e447fb83a |
minpoly.unique | Mathlib/FieldTheory/Minpoly/Field.lean | theorem unique {p : A[X]} (pmonic : p.Monic) (hp : Polynomial.aeval x p = 0)
(pmin : ∀ q : A[X], q.Monic → Polynomial.aeval x q = 0 → degree p ≤ degree q) :
p = minpoly A x | A : Type u_1
B : Type u_2
inst✝² : Field A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
pmonic : p.Monic
hp : (Polynomial.aeval x) p = 0
pmin : ∀ (q : A[X]), q.Monic → (Polynomial.aeval x) q = 0 → p.degree ≤ q.degree
hx : IsIntegral A x
⊢ minpoly A x = p | apply eq_of_sub_eq_zero | case h
A : Type u_1
B : Type u_2
inst✝² : Field A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
pmonic : p.Monic
hp : (Polynomial.aeval x) p = 0
pmin : ∀ (q : A[X]), q.Monic → (Polynomial.aeval x) q = 0 → p.degree ≤ q.degree
hx : IsIntegral A x
⊢ minpoly A x - p = 0 | bd66e2017cd29967 |
ofMul_image_zpowers_eq_zmultiples_ofMul | Mathlib/Algebra/Group/Subgroup/ZPowers/Basic.lean | theorem ofMul_image_zpowers_eq_zmultiples_ofMul {x : G} :
Additive.ofMul '' (Subgroup.zpowers x : Set G) = AddSubgroup.zmultiples (Additive.ofMul x) | case h.mp
G : Type u_1
inst✝ : Group G
x : G
y : Additive G
⊢ y ∈ ⇑Additive.ofMul '' ↑(Subgroup.zpowers x) → y ∈ ↑(AddSubgroup.zmultiples (Additive.ofMul x)) | rintro ⟨z, ⟨m, hm⟩, hz2⟩ | case h.mp.intro.intro.intro
G : Type u_1
inst✝ : Group G
x : G
y : Additive G
z : G
hz2 : Additive.ofMul z = y
m : ℤ
hm : (fun x_1 => x ^ x_1) m = z
⊢ y ∈ ↑(AddSubgroup.zmultiples (Additive.ofMul x)) | 866f88a37996c431 |
ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | theorem surjOn_closedBall_of_nonlinearRightInverse
(hf : ApproximatesLinearOn f f' s c)
(f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInv... | field_simp | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInv... | dd30cb34cb471f42 |
RegularExpression.mul_rmatch_iff | Mathlib/Computability/RegularExpressions.lean | theorem mul_rmatch_iff (P Q : RegularExpression α) (x : List α) :
(P * Q).rmatch x ↔ ∃ t u : List α, x = t ++ u ∧ P.rmatch t ∧ Q.rmatch u | α : Type u_1
inst✝ : DecidableEq α
a : α
x : List α
ih : ∀ (P Q : RegularExpression α), (P * Q).rmatch x = true ↔ ∃ t u, x = t ++ u ∧ P.rmatch t = true ∧ Q.rmatch u = true
P Q : RegularExpression α
hepsilon : ¬P.matchEpsilon = true
t u : List α
h : x = t ++ u
hP : (P.deriv a).rmatch t = true
hQ : Q.rmatch u = true
⊢ a ... | tauto | no goals | eb4ea08dacfd0649 |
MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ}
(hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
(Hd : ∀ x ∈ Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' volume a b) :
∫ x in a..b, f' x = f b - f a | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
hle : a ≤ b
s : Set ℝ
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hi : IntervalIntegrable f' volume a b
e : ℝ ≃L[ℝ] Fin 1 → ℝ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm
e_symm : ∀ (x : Fin 1 ... | refine
integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e ?_ ?_ (fun _ => f)
(fun _ => F') s hs a b hle (fun _ => Hc) (fun x hx _ => Hd x hx) _ ?_ ?_ | case refine_1
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
hle : a ≤ b
s : Set ℝ
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hi : IntervalIntegrable f' volume a b
e : ℝ ≃L[ℝ] Fin 1 → ℝ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm
e_symm :... | c234c86f513f79f8 |
ProbabilityTheory.Kernel.withDensity_tsum | Mathlib/Probability/Kernel/WithDensity.lean | theorem withDensity_tsum [Countable ι] (κ : Kernel α β) [IsSFiniteKernel κ] {f : ι → α → β → ℝ≥0∞}
(hf : ∀ i, Measurable (Function.uncurry (f i))) :
withDensity κ (∑' n, f n) = Kernel.sum fun n => withDensity κ (f n) | case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs... | exact Pi.summable.mpr fun p => ENNReal.summable | no goals | 4ff375ce758fb3f0 |
BitVec.shiftLeftRec_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem shiftLeftRec_eq {x : BitVec w₁} {y : BitVec w₂} {n : Nat} :
shiftLeftRec x y n = x <<< (y.setWidth (n + 1)).setWidth w₂ | w₁ w₂ : Nat
x : BitVec w₁
y : BitVec w₂
⊢ x.shiftLeftRec y 0 = x <<< setWidth w₂ (setWidth (0 + 1) y) | ext i | case pred
w₁ w₂ : Nat
x : BitVec w₁
y : BitVec w₂
i : Nat
a✝ : i < w₁
⊢ (x.shiftLeftRec y 0).getLsbD i = (x <<< setWidth w₂ (setWidth (0 + 1) y)).getLsbD i | 009cabe7d6272db5 |
OrdinalApprox.lfpApprox_mono_mid | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | theorem lfpApprox_mono_mid : Monotone (lfpApprox f) | case h.a.left
α : Type u
inst✝ : CompleteLattice α
f : α →o α
x₁ x₂ : α
h : x₁ ≤ x₂
i : Ordinal.{u}
ih : ∀ k < i, lfpApprox f x₁ k ≤ lfpApprox f x₂ k
⊢ x₁ ≤ x₂ ⊔ sSup {x | ∃ b < i, f (lfpApprox f x₂ b) = x} | exact le_sup_of_le_left h | no goals | 6ab22d20ceb2ddd3 |
MeasureTheory.DominatedFinMeasAdditive.of_measure_le | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem of_measure_le {μ' : Measure α} (h : μ ≤ μ') (hT : DominatedFinMeasAdditive μ T C)
(hC : 0 ≤ C) : DominatedFinMeasAdditive μ' T C | case h.hb
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ⊤ → μ' s = ⊤
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ⊤
hμs : μ s < ⊤
⊢ μ' s ≠ ⊤
case h.h
α... | exacts [hμ's.ne, h _] | no goals | 6e7946b9810c2855 |
integrableOn_rpow_mul_exp_neg_mul_rpow | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | theorem integrableOn_rpow_mul_exp_neg_mul_rpow {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b) :
IntegrableOn (fun x : ℝ => x ^ s * exp (- b * x ^ p)) (Ioi 0) | p s b : ℝ
hs : -1 < s
hp : 1 ≤ p
hb : 0 < b
⊢ IntegrableOn (fun x => x ^ s * rexp (-b * x ^ p)) (Ioi 0) volume | have hib : 0 < b ^ (-p⁻¹) := rpow_pos_of_pos hb _ | p s b : ℝ
hs : -1 < s
hp : 1 ≤ p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
⊢ IntegrableOn (fun x => x ^ s * rexp (-b * x ^ p)) (Ioi 0) volume | 4b843e74d032791b |
PythagoreanTriple.isPrimitiveClassified_of_coprime | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem isPrimitiveClassified_of_coprime (hc : Int.gcd x y = 1) : h.IsPrimitiveClassified | case neg
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hz : ¬0 < z
h' : PythagoreanTriple x y (-z)
⊢ 0 < -z | apply lt_of_le_of_ne _ (h'.ne_zero_of_coprime hc).symm | x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hz : ¬0 < z
h' : PythagoreanTriple x y (-z)
⊢ 0 ≤ -z | 1ef010cf94f0bf49 |
Matrix.detp_smul_adjp | Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean | theorem detp_smul_adjp (hAB : A * B = 1) :
A + (detp 1 A • adjp (-1) B + detp (-1) A • adjp 1 B) =
detp 1 A • adjp 1 B + detp (-1) A • adjp (-1) B | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
h0 :
detp 1 (A * B) + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) =
detp (-1) (A * B) + (detp 1 A * detp 1 B + detp (-1) A * detp (-1) B)
⊢ A + (detp 1 A • adjp (-1) B + detp (-1) A ... | rw [hAB, detp_one_one, detp_neg_one_one, zero_add] at h0 | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
h0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) = detp 1 A * detp 1 B + detp (-1) A * detp (-1) B
⊢ A + (detp 1 A • adjp (-1) B + detp (-1) A • adjp 1 B) = detp 1 A • adjp 1 B + detp ... | b531bbc77169e2ce |
CategoryTheory.regularTopology.mapToEqualizer_eq_comp | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | lemma mapToEqualizer_eq_comp (P : Cᵒᵖ ⥤ Type*) {X B : C} (π : X ⟶ B) [HasPullback π π] :
MapToEqualizer P π (pullback.fst π π) (pullback.snd π π) pullback.condition =
equalizer.lift (P.map π.op) (equalizerCondition_w' P π) ≫
(Types.equalizerIso _ _).hom | case h
C : Type u_1
inst✝¹ : Category.{u_5, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : HasPullback π π
⊢ (MapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯ ≫
(Types.equalizerIso (P.map (pullback.fst π π).op) (P.map (pullback.snd π π).op)).inv) ≫
equalizer.ι (P.map (pullback.fst π π).op) ... | aesop | no goals | ffb6c7fcd1536b07 |
Array.of_push_eq_push | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/BasicAux.lean | theorem Array.of_push_eq_push {as bs : Array α} (h : as.push a = bs.push b) : as = bs ∧ a = b | case mk
α : Type u_1
a b : α
bs : Array α
h₂ : a = b
toList✝ : List α
h : { toList := toList✝ }.toList.concat a = bs.toList.concat b
h₁ : { toList := toList✝ }.toList = bs.toList
⊢ { toList := toList✝ } = bs ∧ a = b | cases bs | case mk.mk
α : Type u_1
a b : α
h₂ : a = b
toList✝¹ toList✝ : List α
h : { toList := toList✝¹ }.toList.concat a = { toList := toList✝ }.toList.concat b
h₁ : { toList := toList✝¹ }.toList = { toList := toList✝ }.toList
⊢ { toList := toList✝¹ } = { toList := toList✝ } ∧ a = b | f5a946fc3697cf55 |
cbiSup_eq_of_not_forall | Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean | theorem cbiSup_eq_of_not_forall {p : ι → Prop} {f : Subtype p → α} (hp : ¬ (∀ i, p i)) :
⨆ (i) (h : p i), f ⟨i, h⟩ = iSup f ⊔ sSup ∅ | case pos.a.a.inl
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp✝ : ¬∀ (i : ι), p i
i₀ : ι
hi₀ : ¬p i₀
this : Nonempty ι
H : BddAbove (range f)
B : BddAbove (range fun i => if h : p i then f ⟨i, h⟩ else sSup ∅)
hp : IsEmpty (Subtype p)
⊢ sSup ∅ ≤ ⨆ i, if h : p i th... | convert le_ciSup B i₀ | case h.e'_3
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp✝ : ¬∀ (i : ι), p i
i₀ : ι
hi₀ : ¬p i₀
this : Nonempty ι
H : BddAbove (range f)
B : BddAbove (range fun i => if h : p i then f ⟨i, h⟩ else sSup ∅)
hp : IsEmpty (Subtype p)
⊢ sSup ∅ = if h : p i₀ then f ⟨i₀,... | f8e2dff97f22af64 |
BitVec.sshiftRight_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem sshiftRight_add {x : BitVec w} {m n : Nat} :
x.sshiftRight (m + n) = (x.sshiftRight m).sshiftRight n | case neg
w : Nat
x : BitVec w
m n i : Nat
a✝ : i < w
h₁ : ¬w ≤ i
⊢ (if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb) =
if n + i < w then !decide (w ≤ n + i) && if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb
else (x.sshiftRight m).msb | by_cases h₂ : n + ↑i < w | case pos
w : Nat
x : BitVec w
m n i : Nat
a✝ : i < w
h₁ : ¬w ≤ i
h₂ : n + i < w
⊢ (if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb) =
if n + i < w then !decide (w ≤ n + i) && if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb
else (x.sshiftRight m).msb
case neg
w : Nat
x : BitVec w
m n i : N... | af6792ce3f635bc7 |
Nat.Prime.even_iff | Mathlib/Data/Nat/Prime/Basic.lean | theorem Prime.even_iff {p : ℕ} (hp : Prime p) : Even p ↔ p = 2 | p : ℕ
hp : Prime p
⊢ Even p ↔ p = 2 | rw [even_iff_two_dvd, prime_dvd_prime_iff_eq prime_two hp, eq_comm] | no goals | 6c94cef2ca53b50c |
Equiv.Perm.card_fixedPoints | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem card_fixedPoints (σ : Equiv.Perm α) :
Fintype.card (Function.fixedPoints σ) = Fintype.card α - σ.cycleType.sum | case e_s
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ Finset.filter (Membership.mem (Function.fixedPoints ⇑σ)) Finset.univ = σ.supportᶜ | aesop | no goals | 868ccbc5bc12036b |
maximalIdeal_isPrincipal_of_isDedekindDomain | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [IsLocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal | R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : (Ideal.span {a}).radical = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
h... | obtain ⟨_, ⟨y, hy, rfl⟩, hy' : x * algebraMap R K y = algebraMap R K 1⟩ := this | case intro.intro.intro.intro
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^... | e801db6f819d1797 |
String.ltb_cons_addChar | Mathlib/Data/String/Basic.lean | theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos) :
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ = ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ | case base₂
c : Char
cs₁✝ cs₂✝ : List Char
i₁✝ i₂✝ : Pos
cs₁ cs₂ : List Char
i₁ i₂ : Pos
a✝ : ¬{ s := { data := cs₂ }, i := i₂ }.hasNext = true
⊢ ltb { s := { data := c :: { data := cs₁ }.data }, i := i₁ + c }
{ s := { data := c :: { data := cs₂ }.data }, i := i₂ + c } =
ltb { s := { data := { data := cs₁ }.da... | conv => lhs; unfold ltb | case base₂
c : Char
cs₁✝ cs₂✝ : List Char
i₁✝ i₂✝ : Pos
cs₁ cs₂ : List Char
i₁ i₂ : Pos
a✝ : ¬{ s := { data := cs₂ }, i := i₂ }.hasNext = true
⊢ (if { s := { data := c :: { data := cs₂ }.data }, i := i₂ + c }.hasNext = true then
if { s := { data := c :: { data := cs₁ }.data }, i := i₁ + c }.hasNext = true then
... | 335ab19a00638e55 |
Polynomial.exists_approx_polynomial_aux | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | theorem exists_approx_polynomial_aux [Ring Fq] {d : ℕ} {m : ℕ} (hm : Fintype.card Fq ^ d ≤ m)
(b : Fq[X]) (A : Fin m.succ → Fq[X]) (hA : ∀ i, degree (A i) < degree b) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ degree (A i₁ - A i₀) < ↑(natDegree b - d) | Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Ring Fq
d m : ℕ
hm : Fintype.card Fq ^ d ≤ m
b : Fq[X]
A : Fin m.succ → Fq[X]
hA : ∀ (i : Fin m.succ), (A i).degree < b.degree
hb : b ≠ 0
f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff (b.natDegree - ↑j.succ)
this : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ)
i... | rw [← Nat.succ_sub hbj, Nat.succ_sub_succ, tsub_tsub_cancel_of_le hbj.le] | no goals | 98b1c3c064e383ea |
Int.dvd_of_dvd_mul_left_of_gcd_one | Mathlib/Data/Int/GCD.lean | theorem dvd_of_dvd_mul_left_of_gcd_one {a b c : ℤ} (habc : a ∣ b * c) (hab : gcd a c = 1) :
a ∣ b | a b c : ℤ
habc : a ∣ b * c
hab : a.gcd c = 1
this : ↑(a.gcd c) = a * a.gcdA c + c * a.gcdB c
⊢ a ∣ b | simp only [hab, Int.ofNat_zero, Int.ofNat_succ, zero_add] at this | a b c : ℤ
habc : a ∣ b * c
hab : a.gcd c = 1
this : 1 = a * a.gcdA c + c * a.gcdB c
⊢ a ∣ b | 54039853ecd11ddc |
MeasureTheory.hahn_decomposition | Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | theorem hahn_decomposition (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
∃ s, MeasurableSet s ∧ (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧
∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNRea... | simpa only [mul_zero, tsub_zero] | no goals | b74e26e9476d0e19 |
AnalyticAt.order_eq_nat_iff | Mathlib/Analysis/Analytic/Order.lean | /-- The order of an analytic function `f` at `z₀` equals a natural number `n` iff `f` can locally
be written as `f z = (z - z₀) ^ n • g z`, where `g` is analytic and does not vanish at `z₀`. -/
lemma order_eq_nat_iff (hf : AnalyticAt 𝕜 f z₀) (n : ℕ) : hf.order = ↑n ↔
∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧... | case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ¬∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z | contrapose! h | case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
⊢ ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0 | 54f3aa2f84cf69fe |
Lean.Omega.Constraint.div_sat | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/Constraint.lean | theorem div_sat (c : Constraint) (t : Int) (k : Nat) (n : k ≠ 0) (h : (k : Int) ∣ t) (w : c.sat t) :
(c.div k).sat (t / k) | case mk.none.some.h
t : Int
k : Nat
h : ↑k ∣ t
n : ↑k > 0
u : Int
w : t ≤ u
⊢ 0 ≤ u / ↑k - t / ↑k | rw [← Int.sub_ediv_of_dvd _ h, ← ge_iff_le, Int.div_nonneg_iff_of_pos n] | case mk.none.some.h
t : Int
k : Nat
h : ↑k ∣ t
n : ↑k > 0
u : Int
w : t ≤ u
⊢ u - t ≥ 0 | 3a7beadca86a93d3 |
EReal.le_div_iff_mul_le | Mathlib/Data/Real/EReal.lean | lemma le_div_iff_mul_le (h : b > 0) (h' : b ≠ ⊤) : a ≤ c / b ↔ a * b ≤ c | a b c : EReal
h : b > 0
h' : b ≠ ⊤
⊢ a ≤ c / b ↔ a * b ≤ c | nth_rw 1 [← @mul_div_cancel a b (ne_bot_of_gt h) h' (ne_of_gt h)] | a b c : EReal
h : b > 0
h' : b ≠ ⊤
⊢ b * (a / b) ≤ c / b ↔ a * b ≤ c | f00c095c28af760e |
uniqueDiffOn_Ioc | Mathlib/Analysis/Calculus/TangentCone.lean | theorem uniqueDiffOn_Ioc (a b : ℝ) : UniqueDiffOn ℝ (Ioc a b) :=
if hab : a < b then
uniqueDiffOn_convex (convex_Ioc a b) <| by simp only [interior_Ioc, nonempty_Ioo, hab]
else by simp only [Ioc_eq_empty hab, uniqueDiffOn_empty]
| a b : ℝ
hab : ¬a < b
⊢ UniqueDiffOn ℝ (Ioc a b) | simp only [Ioc_eq_empty hab, uniqueDiffOn_empty] | no goals | e87678dbc5b14295 |
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