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 |
|---|---|---|---|---|---|---|
Set.Iic_union_Ioc' | Mathlib/Order/Interval/Set/Basic.lean | theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) | case neg
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c < b
x : α
hc : ¬c < x
⊢ x ≤ b ∨ c < x ∧ x ≤ d ↔ x ≤ b ∨ x ≤ d | have hxb : x ≤ b := (le_of_not_gt hc).trans h₁.le | case neg
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c < b
x : α
hc : ¬c < x
hxb : x ≤ b
⊢ x ≤ b ∨ c < x ∧ x ≤ d ↔ x ≤ b ∨ x ≤ d | 0e6589ca6df459fe |
Algebra.mem_adjoin_of_map_mul | Mathlib/RingTheory/Adjoin/Basic.lean | theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
(h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) | R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s
⊢ f x ∈ adjoin R (⇑f '' (s ∪ {1})) | induction h using adjoin_induction with
| mem a ha => exact subset_adjoin ⟨a, ⟨Set.subset_union_left ha, rfl⟩⟩
| algebraMap r =>
have : f 1 ∈ adjoin R (f '' (s ∪ {1})) :=
subset_adjoin ⟨1, ⟨Set.subset_union_right <| Set.mem_singleton 1, rfl⟩⟩
convert Subalgebra.smul_mem (adjoin R (f '' (s ∪ {1}))) this r
rw [... | no goals | 9ffa2155beb6f53e |
cfc_smul | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean | lemma cfc_smul {S : Type*} [SMul S R] [ContinuousConstSMul S R]
[SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
(s : S) (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) | R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
S : Type u_3
inst✝⁴ : S... | by_cases ha : p a | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
S : Type u_3
i... | c2d4183a5f441f58 |
SimpleGraph.odd_card_odd_degree_vertices_ne | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | theorem odd_card_odd_degree_vertices_ne [Fintype V] [DecidableEq V] [DecidableRel G.Adj] (v : V)
(h : Odd (G.degree v)) : Odd #{w | w ≠ v ∧ Odd (G.degree w)} | case intro
V : Type u
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableEq V
inst✝ : DecidableRel G.Adj
v : V
h : Odd (G.degree v)
k : ℕ
hg : #(filter (fun v => Odd (G.degree v)) univ) = k + k
hk : 0 < k
hc : (fun w => w ≠ v ∧ Odd (G.degree w)) = fun w => Odd (G.degree w) ∧ w ≠ v
⊢ Odd #(filter (fun w => Odd (G.de... | rw [← filter_filter, filter_ne', card_erase_of_mem] | case intro
V : Type u
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableEq V
inst✝ : DecidableRel G.Adj
v : V
h : Odd (G.degree v)
k : ℕ
hg : #(filter (fun v => Odd (G.degree v)) univ) = k + k
hk : 0 < k
hc : (fun w => w ≠ v ∧ Odd (G.degree w)) = fun w => Odd (G.degree w) ∧ w ≠ v
⊢ Odd (#(filter (fun w => Odd (G.d... | 1914419bacbd4e95 |
CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_monotone | Mathlib/CategoryTheory/MorphismProperty/TransfiniteComposition.lean | lemma transfiniteCompositionsOfShape_monotone :
Monotone (transfiniteCompositionsOfShape (C := C) (J := J)) | case intro
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
a✝ b✝ : MorphismProperty C
h : a✝ ≤ b✝
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
t : a✝.TransfiniteCompositionOfShape J f✝
⊢ b✝.transfiniteCompositionsOfShape J f✝ | exact ⟨t.ofLE h⟩ | no goals | 531354768a3a2a4b |
PSet.lt_rank_iff | Mathlib/SetTheory/ZFC/Rank.lean | theorem lt_rank_iff {o : Ordinal} {x : PSet} : o < rank x ↔ ∃ y ∈ x, o ≤ rank y | o : Ordinal.{u_1}
x : PSet.{u_1}
⊢ (∀ ⦃y : PSet.{u_1}⦄, y ∈ x → y.rank < o) ↔ ¬∃ y ∈ x, o ≤ y.rank | simp | no goals | afb72204928c1ab5 |
Relation.comp_eq_fun | Mathlib/Logic/Relation.lean | theorem comp_eq_fun (f : γ → β) : r ∘r (· = f ·) = (r · <| f ·) | case h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → β → Prop
f : γ → β
x : α
y : γ
⊢ (r ∘r fun x1 x2 => x1 = f x2) x y ↔ r x (f y) | simp [Comp] | no goals | b64652d28625c59f |
MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (B... | case intro.intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)... | have hI_tendsto :
Tendsto (fun k => ∫ x in Box.Icc (J k), ∑ i, f' x (e i) i) atTop
(𝓝 (∫ x in Box.Icc I, ∑ i, f' x (e i) i)) := by
simp only [IntegrableOn, ← Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi ⊢
rw [← Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi ⊢
exact tendsto_setIntegral_of_mo... | case intro.intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)... | d25a0c4e5a28a1f3 |
Part.le_total_of_le_of_le | Mathlib/Data/Part.lean | theorem le_total_of_le_of_le {x y : Part α} (z : Part α) (hx : x ≤ z) (hy : y ≤ z) :
x ≤ y ∨ y ≤ x | case inl
α : Type u_1
x y z : Part α
hx : x ≤ z
hy : y ≤ z
h : x = none
⊢ x ≤ y ∨ y ≤ x | rw [h] | case inl
α : Type u_1
x y z : Part α
hx : x ≤ z
hy : y ≤ z
h : x = none
⊢ none ≤ y ∨ y ≤ none | 9194d5f3a32aaef7 |
ModularGroup.exists_smul_mem_fd | Mathlib/NumberTheory/Modular.lean | theorem exists_smul_mem_fd (z : ℍ) : ∃ g : SL(2, ℤ), g • z ∈ 𝒟 | case intro.intro.intro.right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ -(1 / 2) ≤ (g • z).re ∧ (g • z).re ≤ 1 / 2 | constructor | case intro.intro.intro.right.left
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ -(1 / 2) ≤ (g • z).re
case intro.intro.intro.right.right
z : ... | 378a4bef3059a80f |
ruzsaSzemerediNumberNat_asymptotic_lower_bound | Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean | theorem ruzsaSzemerediNumberNat_asymptotic_lower_bound :
(fun n ↦ n ^ 2 * exp (-4 * sqrt (log n)) : ℕ → ℝ) =O[atTop]
fun n ↦ (ruzsaSzemerediNumberNat n : ℝ) | case refine_2
⊢ Nat.cast =O[atTop] fun n => ↑((n - 3) / 6) | rw [IsBigO_def] | case refine_2
⊢ ∃ c, IsBigOWith c atTop Nat.cast fun n => ↑((n - 3) / 6) | 72990b3a12cc99c5 |
WellFounded.prod_lex_of_wellFoundedOn_fiber | Mathlib/Order/WellFoundedSet.lean | theorem WellFounded.prod_lex_of_wellFoundedOn_fiber (hα : WellFounded (rα on f))
(hβ : ∀ a, (f ⁻¹' {a}).WellFoundedOn (rβ on g)) :
WellFounded (Prod.Lex rα rβ on fun c => (f c, g c)) | case inr.convert_4
α : Type u_2
β : Type u_3
γ : Type u_4
rα : α → α → Prop
rβ : β → β → Prop
f : γ → α
g : γ → β
hα : WellFounded (rα on f)
hβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)
c c' : γ
h : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'
h' : f c = f c' ∧ rβ (g c) (g c')
⊢ ↑(f ⁻¹' {↑⟨f c', ⋯⟩})
case h.e'_... | exacts [⟨c, h'.1⟩, PSigma.subtype_ext (Subtype.ext h'.1) rfl, h'.2] | no goals | 56092b86f3486016 |
Equiv.Perm.closure_cycle_adjacent_swap | Mathlib/GroupTheory/Perm/Closure.lean | theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.support = univ) (x : α) :
closure ({σ, swap x (σ x)} : Set (Perm α)) = ⊤ | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
⊢ closure {σ, swap x (σ x)} = ⊤ | let H := closure ({σ, swap x (σ x)} : Set (Perm α)) | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
h1 : σ.IsCycle
h2 : σ.support = univ
x : α
H : Subgroup (Perm α) := closure {σ, swap x (σ x)}
⊢ closure {σ, swap x (σ x)} = ⊤ | bc95869a76b17513 |
goldConj_irrational | Mathlib/Data/Real/GoldenRatio.lean | theorem goldConj_irrational : Irrational ψ | this✝ : Irrational √↑5
this : Irrational (↑1 - √↑5)
⊢ Irrational ψ | convert this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num) | case h.e'_1
this✝ : Irrational √↑5
this : Irrational (↑1 - √↑5)
⊢ ψ = ↑0.5 * (↑1 - √↑5) | 7f7853f9e091a280 |
LieAlgebra.IsKilling.orthogonal_span_coroot_eq_ker | Mathlib/Algebra/Lie/Weights/Killing.lean | @[simp] lemma orthogonal_span_coroot_eq_ker (α : Weight K H L) :
(traceForm K H L).orthogonal (K ∙ coroot α) = α.ker | case refine_1
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
α : Weight K (↥H) L
hα : ¬α.IsZero
x : ↥H
hx : ∀ n ∈ span ... | specialize hx (coroot α) (Submodule.mem_span_singleton_self _) | case refine_1
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
α : Weight K (↥H) L
hα : ¬α.IsZero
x : ↥H
hx : (traceForm ... | 2fd7396025608b3a |
Matrix.det_nonsing_inv_mul_det | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | theorem det_nonsing_inv_mul_det (h : IsUnit A.det) : A⁻¹.det * A.det = 1 | n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h : IsUnit A.det
⊢ A⁻¹.det * A.det = 1 | rw [← det_mul, A.nonsing_inv_mul h, det_one] | no goals | 209f06c517eda8b4 |
PrimeSpectrum.zeroLocus_ideal_mem_irreducibleComponents | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | lemma zeroLocus_ideal_mem_irreducibleComponents {I : Ideal R} :
zeroLocus I ∈ irreducibleComponents (PrimeSpectrum R) ↔ I.radical ∈ minimalPrimes R | R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ zeroLocus ↑I ∈ irreducibleComponents (PrimeSpectrum R) ↔ vanishingIdeal (zeroLocus ↑I) ∈ minimalPrimes R | conv_lhs => rw [← (isClosed_zeroLocus _).closure_eq] | R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ closure (zeroLocus ↑I) ∈ irreducibleComponents (PrimeSpectrum R) ↔ vanishingIdeal (zeroLocus ↑I) ∈ minimalPrimes R | a2240dc4d96222c4 |
Set.zero_mem_smul_iff | Mathlib/Data/Set/Pointwise/SMul.lean | theorem zero_mem_smul_iff :
(0 : β) ∈ s • t ↔ (0 : α) ∈ s ∧ t.Nonempty ∨ (0 : β) ∈ t ∧ s.Nonempty | case mpr.inl.intro.intro
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMulWithZero α β
s : Set α
t : Set β
inst✝ : NoZeroSMulDivisors α β
hs : 0 ∈ s
b : β
hb : b ∈ t
⊢ 0 ∈ s • t | exact ⟨0, hs, b, hb, zero_smul _ _⟩ | no goals | 2eb057cfa2edc7c8 |
List.new_def_eq_old_def | Mathlib/Data/List/Indexes.lean | theorem new_def_eq_old_def :
∀ (f : ℕ → α → β) (l : List α), l.mapIdx f = List.oldMapIdx f l | α : Type u
β : Type v
⊢ ∀ (f : ℕ → α → β) (l : List α), mapIdx f l = List.oldMapIdx f l | intro f | α : Type u
β : Type v
f : ℕ → α → β
⊢ ∀ (l : List α), mapIdx f l = List.oldMapIdx f l | 9955599b4c417bfe |
NormedDivisionRing.norm_le_one_of_discrete | Mathlib/Analysis/Normed/Field/Basic.lean | @[simp]
lemma norm_le_one_of_discrete
(x : 𝕜) : ‖x‖ ≤ 1 | 𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
⊢ ‖x‖ ≤ 1 | rcases eq_or_ne x 0 with rfl|hx | case inl
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
⊢ ‖0‖ ≤ 1
case inr
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
⊢ ‖x‖ ≤ 1 | 89fedcb465594c43 |
Lean.Omega.IntList.cons_add_cons | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean | theorem cons_add_cons (x) (xs : IntList) (y) (ys : IntList) :
(x :: xs) + (y :: ys) = (x + y) :: (xs + ys) | x : Int
xs : IntList
y : Int
ys : IntList
⊢ x :: xs + y :: ys = (x + y) :: (xs + ys) | simp [add_def] | no goals | 3b33bea7c9067401 |
Real.arccos_eq_pi_div_two | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | theorem arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 | x : ℝ
⊢ arccos x = π / 2 ↔ x = 0 | simp [arccos] | no goals | 9f938bf572cccab2 |
ENNReal.rpow_natCast_mul | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | lemma rpow_natCast_mul (x : ℝ≥0∞) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z | x : ℝ≥0∞
n : ℕ
z : ℝ
⊢ x ^ (↑n * z) = (x ^ n) ^ z | rw [rpow_mul, rpow_natCast] | no goals | f5bec7f16d872074 |
Cardinal.ciSup_mul | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem ciSup_mul (c : Cardinal.{v}) : (⨆ i, f i) * c = ⨆ i, f i * c | case pos.inr.intro
ι : Type u
f : ι → Cardinal.{v}
c : Cardinal.{v}
h✝ : Nonempty ι
h0 : c ≠ 0
hf : BddAbove (range f)
this : ∀ (i : ι), f i * c ≤ (⨆ i, f i) * c
bdd : BddAbove (range fun x => f x * c)
hs : ℵ₀ ≤ ⨆ i, f i
i : ι
hi : 1 < f i
⊢ ⨆ i, f i ≤ ⨆ i, f i * c ∧ c ≤ ⨆ i, f i * c | exact ⟨ciSup_mono bdd fun i ↦ le_mul_right h0,
(le_mul_left (zero_lt_one.trans hi).ne').trans (le_ciSup bdd i)⟩ | no goals | 079631a7d791f82b |
List.mem_merge | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean | theorem mem_merge {a : α} {xs ys : List α} : a ∈ merge xs ys le ↔ a ∈ xs ∨ a ∈ ys | case cons.cons.isFalse
α : Type u_1
le : α → α → Bool
a x : α
xs : List α
ih✝ : ∀ {ys : List α}, a ∈ xs.merge ys le ↔ a ∈ xs ∨ a ∈ ys
y : α
ys : List α
ih : a ∈ (x :: xs).merge ys le ↔ a ∈ x :: xs ∨ a ∈ ys
h : ¬le x y = true
⊢ a ∈ y :: (x :: xs).merge ys le ↔ a ∈ x :: xs ∨ a ∈ y :: ys | simp only [mem_cons, or_assoc, Bool.not_eq_true, ih, ← or_assoc] | case cons.cons.isFalse
α : Type u_1
le : α → α → Bool
a x : α
xs : List α
ih✝ : ∀ {ys : List α}, a ∈ xs.merge ys le ↔ a ∈ xs ∨ a ∈ ys
y : α
ys : List α
ih : a ∈ (x :: xs).merge ys le ↔ a ∈ x :: xs ∨ a ∈ ys
h : ¬le x y = true
⊢ ((a = y ∨ a = x) ∨ a ∈ xs) ∨ a ∈ ys ↔ ((a = x ∨ a ∈ xs) ∨ a = y) ∨ a ∈ ys | ca0cd88629a52bfb |
solvableByRad.induction | Mathlib/FieldTheory/AbelRuffini.lean | theorem induction (P : solvableByRad F E → Prop)
(base : ∀ α : F, P (algebraMap F (solvableByRad F E) α))
(add : ∀ α β : solvableByRad F E, P α → P β → P (α + β))
(neg : ∀ α : solvableByRad F E, P α → P (-α))
(mul : ∀ α β : solvableByRad F E, P α → P β → P (α * β))
(inv : ∀ α : solvableByRad F E, P ... | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | revert α | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | 856cf0182c98d000 |
MeasureTheory.integral_le_measure | Mathlib/MeasureTheory/Integral/SetIntegral.lean | lemma integral_le_measure {f : X → ℝ} {s : Set X}
(hs : ∀ x ∈ s, f x ≤ 1) (h's : ∀ x ∈ sᶜ, f x ≤ 0) :
ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ s | case h
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x => f x ⊔ 0
g_int : Integrable g μ
⊢ ∫ (x : X), f x ∂μ ≤ ∫ (x : X), g x ∂μ | exact integral_mono H g_int (fun x ↦ le_max_left _ _) | no goals | 3c1b6cc50ba267ee |
SzemerediRegularity.le_sum_distinctPairs_edgeDensity_sq | Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean | lemma le_sum_distinctPairs_edgeDensity_sq (x : {i // i ∈ P.parts.offDiag}) (hε₁ : ε ≤ 1)
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5) :
(G.edgeDensity x.1.1 x.1.2 : ℝ) ^ 2 +
((if G.IsUniform ε x.1.1 x.1.2 then 0 else ε ^ 4 / 3) - ε ^ 5 / 25) ≤
(∑ i ∈ distinctPairs hP ... | case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
x : { i // i ∈ P.parts.offDiag }
hε₁ : ε ≤ 1
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
h : ¬G.IsUnifo... | exact edgeDensity_chunk_not_uniform hPα hPε hε₁ (mem_offDiag.1 x.2).2.2 h | no goals | 287f013480af0bef |
Batteries.RBNode.Path.insert_toList | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem insert_toList {p : Path α} :
(p.insert t v).toList = p.withList (t.setRoot v).toList | α : Type u_1
t : RBNode α
v : α
p : Path α
⊢ (match t with
| nil => p.insertNew v
| node c a v_1 b => p.fill (node c a v b)).toList =
p.listL ++ ((setRoot v t).toList ++ p.listR) | split <;> simp [setRoot] | no goals | cd568f15082ffe83 |
tangentCone_eq_univ | Mathlib/Analysis/Calculus/TangentCone.lean | theorem tangentCone_eq_univ {s : Set 𝕜} {x : 𝕜} (hx : (𝓝[s \ {x}] x).NeBot) :
tangentConeAt 𝕜 s x = univ | case inr.intro.intro.intro.refine_2
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
x : 𝕜
hx : (𝓝[s \ {x}] x).NeBot
y : 𝕜
hy : y ≠ 0
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → 𝕜
hv : ∀ (n : ℕ), v n ∈ s \ {x} ∩ Metric.ball x (u n)
d : ℕ → 𝕜 := fun n => v n - x
d_ne : ∀ (... | have B (n : ℕ) : ‖d n‖ ≤ u n := by
specialize hv n
simp only [mem_inter_iff, mem_diff, mem_singleton_iff, Metric.mem_ball, dist_eq_norm] at hv
simpa using hv.2.le | case inr.intro.intro.intro.refine_2
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
x : 𝕜
hx : (𝓝[s \ {x}] x).NeBot
y : 𝕜
hy : y ≠ 0
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → 𝕜
hv : ∀ (n : ℕ), v n ∈ s \ {x} ∩ Metric.ball x (u n)
d : ℕ → 𝕜 := fun n => v n - x
d_ne : ∀ (... | aeb42d5f7b21be66 |
not_summable_of_antitone_of_neg | Mathlib/Analysis/SumOverResidueClass.lean | /-- If `f : ℕ → ℝ` is decreasing and has a negative term, then `f` is not summable. -/
lemma not_summable_of_antitone_of_neg {f : ℕ → ℝ} (hf : Antitone f) {n : ℕ} (hn : f n < 0) :
¬ Summable f | case intro
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
N : ℕ
hN : |f (n ⊔ N)| < |f n|
⊢ False | contrapose! hN | case intro
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
N : ℕ
hN : ¬False
⊢ |f n| ≤ |f (n ⊔ N)| | bd42ef63684ae4ee |
List.dProd_monoid | Mathlib/Algebra/GradedMonoid.lean | theorem List.dProd_monoid {α} [AddMonoid ι] [Monoid R] (l : List α) (fι : α → ι) (fA : α → R) :
@List.dProd _ _ (fun _ : ι => R) _ _ l fι fA = (l.map fA).prod | ι : Type u_1
R : Type u_2
α : Type u_3
inst✝¹ : AddMonoid ι
inst✝ : Monoid R
l : List α
fι : α → ι
fA : α → R
⊢ [].dProd fι fA = (map fA []).prod | rw [List.dProd_nil, List.map_nil, List.prod_nil] | ι : Type u_1
R : Type u_2
α : Type u_3
inst✝¹ : AddMonoid ι
inst✝ : Monoid R
l : List α
fι : α → ι
fA : α → R
⊢ GradedMonoid.GOne.one = 1 | caf25bac3a8dd0b6 |
AlgebraicTopology.DoldKan.N₁Γ₀_inv_app_f_f | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | theorem N₁Γ₀_inv_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ ((toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).inv ≫ (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom).f.f
n =
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n | apply id_comp | no goals | e8b77721e7e81f14 |
CategoryTheory.MorphismProperty.multiplicativeClosure_le_iff | Mathlib/CategoryTheory/MorphismProperty/Composition.lean | /-- The multiplicative closure of `W` is the smallest multiplicative property greater than or equal
to `W`. -/
@[simp]
lemma multiplicativeClosure_le_iff (W' : MorphismProperty C) [W'.IsMultiplicative] :
multiplicativeClosure W ≤ W' ↔ W ≤ W' where
mp h := le_multiplicativeClosure W |>.trans h
mpr h | C : Type u
inst✝¹ : Category.{v, u} C
W W' : MorphismProperty C
inst✝ : W'.IsMultiplicative
h : W ≤ W'
⊢ W.multiplicativeClosure ≤ W' | intro _ _ _ hf | C : Type u
inst✝¹ : Category.{v, u} C
W W' : MorphismProperty C
inst✝ : W'.IsMultiplicative
h : W ≤ W'
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
hf : W.multiplicativeClosure f✝
⊢ W' f✝ | eaad7e68c7e5f5d0 |
MeasureTheory.UniformIntegrable.spec' | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, StronglyMeasurable (f i))
(hfu : UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε) :
∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofRea... | obtain ⟨-, hfu, M, hM⟩ := hfu | case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
⊢ ∃ C, ∀ (i : ι... | 489767214d928b49 |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | Mathlib/NumberTheory/KummerDedekind.lean | theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
(hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
normalizedFactors (I.map (algebraMap R S)) =
Multiset.map
(fun f =>
((normalizedFactorsMapEquivNormaliz... | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsI... | have := emultiplicity_factors_map_eq_emultiplicity hI hI' hx hx' hJ | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsI... | c49652504dc8313f |
Std.DHashMap.Raw.contains_insertIfNew_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem contains_insertIfNew_self [EquivBEq α] [LawfulHashable α] (h : m.WF) {k : α} {v : β k} :
(m.insertIfNew k v).contains k | α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
k : α
v : β k
⊢ (m.insertIfNew k v).contains k = true | simp_to_raw using Raw₀.contains_insertIfNew_self | no goals | 03dc4f81bcadf9cc |
Int.exists_mul_self | Mathlib/Data/Int/Sqrt.lean | theorem exists_mul_self (x : ℤ) : (∃ n, n * n = x) ↔ sqrt x * sqrt x = x :=
⟨fun ⟨n, hn⟩ => by rw [← hn, sqrt_eq, ← Int.ofNat_mul, natAbs_mul_self], fun h => ⟨sqrt x, h⟩⟩
| x : ℤ
x✝ : ∃ n, n * n = x
n : ℤ
hn : n * n = x
⊢ sqrt x * sqrt x = x | rw [← hn, sqrt_eq, ← Int.ofNat_mul, natAbs_mul_self] | no goals | 9b6c51ba9e0d245b |
ForInStep.bindList_append | Mathlib/.lake/packages/batteries/Batteries/Control/ForInStep/Lemmas.lean | theorem ForInStep.bindList_append [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (s : ForInStep β) (l₁ l₂) :
s.bindList f (l₁ ++ l₂) = s.bindList f l₁ >>= (·.bindList f l₂) | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
s : ForInStep β
l₁ l₂ : List α
⊢ bindList f (l₁ ++ l₂) s = do
let x ← bindList f l₁ s
bindList f l₂ x | induction l₁ generalizing s <;> simp [*] | no goals | e2eca0728b17d41f |
Nat.lt_of_testBit | Mathlib/Data/Nat/Bitwise.lean | theorem lt_of_testBit {n m : ℕ} (i : ℕ) (hn : testBit n i = false) (hm : testBit m i = true)
(hnm : ∀ j, i < j → testBit n j = testBit m j) : n < m | case f.z
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), n.testBit i = false → m.testBit i = true → (∀ (j : ℕ), i < j → n.testBit j = m.testBit j) → n < m
i : ℕ
hn : (bit b n).testBit i = false
hm : testBit 0 i = true
hnm : ∀ (j : ℕ), i < j → (bit b n).testBit j = testBit 0 j
⊢ bit b n < 0 | exact False.elim (Bool.false_ne_true ((zero_testBit i).symm.trans hm)) | no goals | 6ffde8c2e514a295 |
AddCircle.volume_closedBall | Mathlib/MeasureTheory/Integral/Periodic.lean | theorem volume_closedBall {x : AddCircle T} (ε : ℝ) :
volume (Metric.closedBall x ε) = ENNReal.ofReal (min T (2 * ε)) | T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hT' : |T| = T
I : Set ℝ := Ioc (-(T / 2)) (T / 2)
hε : ε < T / 2
⊢ Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε | rw [inter_eq_left, Real.closedBall_eq_Icc, zero_sub, zero_add] | T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hT' : |T| = T
I : Set ℝ := Ioc (-(T / 2)) (T / 2)
hε : ε < T / 2
⊢ Icc (-ε) ε ⊆ I | 7eec94214cffe2c7 |
MeasureTheory.llr_smul_right | Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | lemma llr_smul_right [IsFiniteMeasure μ] [Measure.HaveLebesgueDecomposition μ ν]
(hμν : μ ≪ ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ∞) :
llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log c.toReal | case h
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : μ.rnDeriv (c • ν) =ᶠ[ae ν] c⁻¹ • μ.rnDeriv ν
x : α
hx_eq : μ.rnDeriv (c • ν) x = (c⁻¹ • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : ... | rw [log_mul] | case h
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : μ.rnDeriv (c • ν) =ᶠ[ae ν] c⁻¹ • μ.rnDeriv ν
x : α
hx_eq : μ.rnDeriv (c • ν) x = (c⁻¹ • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : ... | f2c098759e8140a2 |
DirichletCharacter.IsPrimitive.completedLFunction_one_sub | Mathlib/NumberTheory/LSeries/DirichletContinuation.lean | theorem completedLFunction_one_sub {χ : DirichletCharacter ℂ N} (hχ : IsPrimitive χ) (s : ℂ) :
completedLFunction χ (1 - s) = N ^ (s - 1 / 2) * rootNumber χ * completedLFunction χ⁻¹ s | N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j : ZMod N, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
⊢ ↑N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (fun j => χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s =
↑N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedL... | rw [completedLFunction, completedLFunction_const_mul] | N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j : ZMod N, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
⊢ ↑N ^ (s - 1) * χ (-1) / ε * (χ⁻¹ (-1) * gaussSum χ stdAddChar * ZMod.completedLFunction (⇑χ⁻¹) s) =
↑N ^ (s - 1) / ε * gaussSum χ stdAddChar * ZMod.completedLFun... | ff38d9c55b1ef998 |
NormedSpace.invOf_exp_of_mem_ball | Mathlib/Analysis/Normed/Algebra/Exponential.lean | theorem invOf_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) [Invertible (exp 𝕂 x)] :
⅟ (exp 𝕂 x) = exp 𝕂 (-x) | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕂
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕂 𝔸
inst✝² : CompleteSpace 𝔸
inst✝¹ : CharZero 𝕂
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
inst✝ : Invertible (exp 𝕂 x)
this : Invertible (exp 𝕂 x) := invertibleExpOfMemBall hx
⊢ ⅟(exp 𝕂 x)... | convert (rfl : ⅟ (exp 𝕂 x) = _) | no goals | 708b111fb388522e |
ProbabilityTheory.IndepFun.variance_sum | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_sum [IsProbabilityMeasure μ] {ι : Type*} {X : ι → Ω → ℝ}
{s : Finset ι} (hs : ∀ i ∈ s, MemLp (X i) 2 μ)
(h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j) μ) :
variance (∑ i ∈ s, X i) μ = ∑ i ∈ s, variance (X i) μ | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) 2 μ
h : (↑(... | rw [mul_sum, sum_fn] | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) 2 μ
h : (↑(... | 1d5529d2c6cdc1ee |
Real.rpow_def_of_neg | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | theorem rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) | x : ℝ
hx : x < 0
y : ℝ
this : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I
⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = rexp (log x * y) * cos (y * π) | rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Complex.ofReal_cos, ←
Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ← mul_assoc, ← Complex.ofReal_mul,
Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.ofReal_im,
Real.log_neg_eq_log] | x : ℝ
hx : x < 0
y : ℝ
this : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I
⊢ rexp (log x * y) * cos (y * π) + ((↑(rexp (log x * y) * sin (y * π))).re * 0 - 0 * Complex.I.im) =
rexp (log x * y) * cos (y * π) | 22867cb721b4dcb8 |
Nat.bodd_add_div2 | Mathlib/Data/Nat/Bits.lean | lemma bodd_add_div2 : ∀ n, (bodd n).toNat + 2 * div2 n = n
| 0 => rfl
| succ n => by
simp only [bodd_succ, Bool.cond_not, div2_succ, Nat.mul_comm]
refine Eq.trans ?_ (congr_arg succ (bodd_add_div2 n))
cases bodd n
· simp
· simp; omega
| case false
n : ℕ
⊢ (!false).toNat + (bif false then n.div2.succ else n.div2) * 2 = (false.toNat + 2 * n.div2).succ | simp | no goals | cbe835fd2836b11c |
Matroid.IsRkFinite.iUnion | Mathlib/Data/Matroid/Rank/Finite.lean | /-- A union of finitely many `IsRkFinite` sets is `IsRkFinite`. -/
lemma IsRkFinite.iUnion {ι : Type*} [Finite ι] {Xs : ι → Set α} (h : ∀ i, M.IsRkFinite (Xs i)) :
M.IsRkFinite (⋃ i, Xs i) | α : Type u_1
M : Matroid α
ι : Type u_2
inst✝ : Finite ι
Xs : ι → Set α
h : ∀ (i : ι), M.IsRkFinite (Xs i)
Is : ι → Set α
hIs : ∀ (i : ι), M.IsBasis' (Is i) (Xs i)
hfin : (⋃ i, Is i).Finite
⊢ (⋃ i, Xs i) ∩ M.E ⊆ M.closure (⋃ i, Is i) | rw [iUnion_inter, iUnion_subset_iff] | α : Type u_1
M : Matroid α
ι : Type u_2
inst✝ : Finite ι
Xs : ι → Set α
h : ∀ (i : ι), M.IsRkFinite (Xs i)
Is : ι → Set α
hIs : ∀ (i : ι), M.IsBasis' (Is i) (Xs i)
hfin : (⋃ i, Is i).Finite
⊢ ∀ (i : ι), Xs i ∩ M.E ⊆ M.closure (⋃ i, Is i) | da1fd80baf9993d9 |
sum_div_pow_sq_le_div_sq | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
(∑ i ∈ range N with j < c ^ i, (1 : ℝ) / (c ^ i) ^ 2) ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 | N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
C : c⁻¹ ^ 2 < 1
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
⊢ 1 / j ^ 2 / (c⁻¹ ^ 2) ^ 1 / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2 | have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow₀ hc two_ne_zero)).ne' | N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
C : c⁻¹ ^ 2 < 1
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
this : c ^ 2 - 1 ≠ 0
⊢ 1 / j ^ 2 / (c⁻¹ ^ 2) ^ 1 / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2 | c250b05c7bec5514 |
norm_sup_sub_sup_le_add_norm | Mathlib/Analysis/Normed/Order/Lattice.lean | theorem norm_sup_sub_sup_le_add_norm (a b c d : α) : ‖a ⊔ b - c ⊔ d‖ ≤ ‖a - c‖ + ‖b - d‖ | α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
⊢ ‖a ⊔ b - c ⊔ d‖ ≤ ‖a - c‖ + ‖b - d‖ | rw [← norm_abs_eq_norm (a - c), ← norm_abs_eq_norm (b - d)] | α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
⊢ ‖a ⊔ b - c ⊔ d‖ ≤ ‖|a - c|‖ + ‖|b - d|‖ | c8b7a8b484c87394 |
eHolderNorm_lt_top | Mathlib/Topology/MetricSpace/HolderNorm.lean | @[simp] lemma eHolderNorm_lt_top : eHolderNorm r f < ∞ ↔ MemHolder r f | X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
⊢ eHolderNorm r f < ⊤ ↔ MemHolder r f | refine ⟨fun h => ?_,
fun hf => let ⟨C, hC⟩ := hf; iInf_lt_top.2 ⟨C, iInf_lt_top.2 ⟨hC, coe_lt_top⟩⟩⟩ | X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
h : eHolderNorm r f < ⊤
⊢ MemHolder r f | 1cc4a5c9bdcd78e2 |
IsDedekindDomainInv.dimensionLEOne | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem dimensionLEOne : DimensionLEOne A := ⟨by
-- We're going to show that `P` is maximal because any (maximal) ideal `M`
-- that is strictly larger would be `⊤`.
rintro P P_ne hP
refine Ideal.isMaximal_def.mpr ⟨hP.ne_top, fun M hM => ?_⟩
-- We may assume `P` and `M` (as fractional ideals) are nonzero.
ha... | case intro.intro.intro.intro
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (fun a => ↑a) ((↑M)⁻¹ * ↑P)
z : A
hz... | have zy_mem := mul_mem_mul (mem_coeIdeal_of_mem A⁰ hzM) hx | case intro.intro.intro.intro
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (fun a => ↑a) ((↑M)⁻¹ * ↑P)
z : A
hz... | ec848de7999ea57e |
SSet.OneTruncation₂.homOfEq_edge | Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean | @[simp]
lemma OneTruncation₂.homOfEq_edge
{X : SSet.Truncated.{u} 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X}
(f : x₁ ⟶ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :
(Quiver.homOfEq f hx hy).edge = f.edge | X : Truncated 2
x₁ y₁ : OneTruncation₂ X
f : x₁ ⟶ y₁
⊢ (Quiver.homOfEq f ⋯ ⋯).edge = f.edge | rfl | no goals | 9d7b73d235e55a99 |
Std.Sat.AIG.mkGateCached.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean | theorem mkGateCached.go_decl_eq (aig : AIG α) (input : GateInput aig) :
∀ (idx : Nat) (h1) (h2), (go aig input).aig.decls[idx]'h1 = aig.decls[idx]'h2 | case h_2.h_4
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝⁴ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input... | rw [LawfulOperator.decl_eq (f := AIG.mkConstCached)] | no goals | f550d988487b2b3c |
Lean.Order.Array.monotone_foldrM | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_foldrM
(f : γ → α → β → m β) (init : β) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
monotone (fun x => xs.foldrM (f x) init start stop) | m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → β → m β
init : β
xs : Array α
start stop : Nat
hmono : monotone f
⊢ monotone fun x => Array.foldrM (f x) init xs start stop | unfold Array.foldrM | m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → β → m β
init : β
xs : Array α
start stop : Nat
hmono : monotone f
⊢ monotone fun x =>
if h : start ≤ xs.size then if stop < start then Array.foldrM.fold (f x... | c22e1149b4a24d85 |
List.Perm.prod_eq' | Mathlib/Algebra/BigOperators/Group/List/Lemmas.lean | /-- If elements of a list commute with each other, then their product does not
depend on the order of elements. -/
@[to_additive "If elements of a list additively commute with each other, then their sum does not
depend on the order of elements."]
lemma Perm.prod_eq' (h : l₁ ~ l₂) (hc : l₁.Pairwise Commute) : l₁.prod = ... | case H₁
M : Type u_4
inst✝ : Monoid M
l₁ l₂ : List M
h : l₁ ~ l₂
hc : Pairwise Commute l₁
x✝ : M
a✝ : x✝ ∈ l₁
z✝ : M
⊢ x✝ * (x✝ * z✝) = x✝ * (x✝ * z✝) | rfl | no goals | 782ddb106ff11709 |
WeierstrassCurve.ofJ_1728_of_two_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/ModelsWithJ.lean | lemma ofJ_1728_of_two_eq_zero (h2 : (2 : F) = 0) : ofJ 1728 = ofJ0 F | F : Type u_2
inst✝¹ : Field F
inst✝ : DecidableEq F
h2 : 2 = 0
⊢ ofJ 1728 = ofJ0 F | rw [ofJ, if_pos (by linear_combination 864 * h2),
if_neg ((show (3 : F) = 1 by linear_combination h2) ▸ one_ne_zero)] | no goals | 87d9097f856c22a6 |
MeasureTheory.setLIntegral_eq_of_support_subset | Mathlib/MeasureTheory/Integral/Lebesgue.lean | lemma setLIntegral_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
∫⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hsf : support f ⊆ s
⊢ ∫⁻ (x : α), f x ∂μ = ∫⁻ (a : α), s.indicator f a ∂μ | congr with x | case e_f.h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hsf : support f ⊆ s
x : α
⊢ f x = s.indicator f x | 768002bc29cf4dd9 |
Nat.pow_length_le_mul_ofDigits | Mathlib/Data/Nat/Digits.lean | theorem pow_length_le_mul_ofDigits {b : ℕ} {l : List ℕ} (hl : l ≠ []) (hl2 : l.getLast hl ≠ 0) :
(b + 2) ^ l.length ≤ (b + 2) * ofDigits (b + 2) l | b : ℕ
l : List ℕ
hl : l ≠ []
hl2 : l.getLast hl ≠ 0
⊢ 0 < l.getLast hl | rwa [pos_iff_ne_zero] | no goals | a56162e7f0190c1c |
continuousAt_clog | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | theorem continuousAt_clog {x : ℂ} (h : x ∈ slitPlane) : ContinuousAt log x | case refine_2
x : ℂ
h : x ∈ slitPlane
h_cont_mul : Continuous fun x => x * I
⊢ ContinuousAt (fun x => ↑x.arg * I) x | refine h_cont_mul.continuousAt.comp (continuous_ofReal.continuousAt.comp ?_) | case refine_2
x : ℂ
h : x ∈ slitPlane
h_cont_mul : Continuous fun x => x * I
⊢ ContinuousAt arg x | 3806bdd6f92461d6 |
Quiver.Path.eq_toPath_comp_of_length_eq_succ | Mathlib/Combinatorics/Quiver/Path.lean | lemma eq_toPath_comp_of_length_eq_succ (p : Path a b) {n : ℕ}
(hp : p.length = n + 1) :
∃ (c : V) (f : a ⟶ c) (q : Quiver.Path c b) (_ : q.length = n),
p = f.toPath.comp q | case cons
V : Type u
inst✝ : Quiver V
a b c d : V
p : Path a c
q : c ⟶ d
h : ∀ {n : ℕ}, p.length = n + 1 → ∃ c_1 f q x, p = f.toPath.comp q
n : ℕ
hp : (p.cons q).length = n + 1
⊢ ∃ c_1 f q_1 x, p.cons q = f.toPath.comp q_1 | cases n | case cons.zero
V : Type u
inst✝ : Quiver V
a b c d : V
p : Path a c
q : c ⟶ d
h : ∀ {n : ℕ}, p.length = n + 1 → ∃ c_1 f q x, p = f.toPath.comp q
hp : (p.cons q).length = 0 + 1
⊢ ∃ c_1 f q_1 x, p.cons q = f.toPath.comp q_1
case cons.succ
V : Type u
inst✝ : Quiver V
a b c d : V
p : Path a c
q : c ⟶ d
h : ∀ {n : ℕ}, p.le... | e85693f555647484 |
CauSeq.inf_limZero | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun _ H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
rw [abs_lt] at H₁ H₂ ⊢
exact ⟨lt_inf_iff.mpr ⟨H₁.1, H₂.1⟩, inf_lt_iff.mpr (Or.inl H₁.2)⟩
| α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : f.LimZero
hg : g.LimZero
ε : α
ε0 : ε > 0
x✝ : ℕ
H : ∀ j ≥ x✝, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ x✝
H₁ : |↑f j| < ε
H₂ : |↑g j| < ε
⊢ |↑(f ⊓ g) j| < ε | rw [abs_lt] at H₁ H₂ ⊢ | α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : f.LimZero
hg : g.LimZero
ε : α
ε0 : ε > 0
x✝ : ℕ
H : ∀ j ≥ x✝, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ x✝
H₁ : -ε < ↑f j ∧ ↑f j < ε
H₂ : -ε < ↑g j ∧ ↑g j < ε
⊢ -ε < ↑(f ⊓ g) j ∧ ↑(f ⊓ g) j < ε | 824af2fd8ca10b38 |
iSupIndep_of_dfinsupp_lsum_injective | Mathlib/LinearAlgebra/DFinsupp.lean | theorem iSupIndep_of_dfinsupp_lsum_injective (p : ι → Submodule R N)
(h : Function.Injective (lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype)) :
iSupIndep p | ι : Type u_1
R : Type u_2
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
i : ι
x : ↥(p i)
v : Π₀ (i : ι), ↥(p i)
hv : ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ↑x
⊢ ((lsum ℕ) fun i ... | simpa only [lsum_single] using hv | no goals | 85bd08e695cc9c88 |
QuadraticMap.polar_neg_right | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | theorem polar_neg_right (x y : M) : polar Q x (-y) = -polar Q x y | R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
x y : M
⊢ polar (⇑Q) x (-y) = -polar (⇑Q) x y | rw [← neg_one_smul R y, polar_smul_right, neg_one_smul] | no goals | 499dbaaca5f091d5 |
FirstOrder.Language.exists_countable_is_age_of_iff | Mathlib/ModelTheory/Fraisse.lean | theorem exists_countable_is_age_of_iff [Countable (Σ l, L.Functions l)] :
(∃ M : Bundled.{w} L.Structure, Countable M ∧ L.age M = K) ↔
K.Nonempty ∧ (∀ M N : Bundled.{w} L.Structure, Nonempty (M ≃[L] N) → (M ∈ K ↔ N ∈ K)) ∧
(Quotient.mk' '' K).Countable ∧ (∀ M : Bundled.{w} L.Structure, M ∈ K → Structure... | case mpr
L : Language
K : Set (Bundled L.Structure)
inst✝ : Countable ((l : ℕ) × L.Functions l)
⊢ K.Nonempty ∧
(∀ (M N : Bundled L.Structure), Nonempty (↑M ≃[L] ↑N) → (M ∈ K ↔ N ∈ K)) ∧
(Quotient.mk' '' K).Countable ∧ (∀ M ∈ K, Structure.FG L ↑M) ∧ Hereditary K ∧ JointEmbedding K →
∃ M, Countable ↑M ∧... | rintro ⟨Kn, _, cq, hfg, hp, jep⟩ | case mpr.intro.intro.intro.intro.intro
L : Language
K : Set (Bundled L.Structure)
inst✝ : Countable ((l : ℕ) × L.Functions l)
Kn : K.Nonempty
left✝ : ∀ (M N : Bundled L.Structure), Nonempty (↑M ≃[L] ↑N) → (M ∈ K ↔ N ∈ K)
cq : (Quotient.mk' '' K).Countable
hfg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEm... | 3807c6e77bd58f4a |
multiplicity_ne_zero | Mathlib/RingTheory/Multiplicity.lean | theorem multiplicity_ne_zero :
multiplicity a b ≠ 0 ↔ a ∣ b | α : Type u_1
inst✝ : Monoid α
a b : α
⊢ multiplicity a b ≠ 0 ↔ a ∣ b | simp [multiplicity_eq_zero] | no goals | e02c26e2adc9a670 |
tangentMapWithin_eq_tangentMap | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem tangentMapWithin_eq_tangentMap {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.1)
(h : MDifferentiableAt I I' f p.1) : tangentMapWithin I I' f s p = tangentMap I I' f p | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | rw [← tangentMapWithin_univ] | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | d08fa7327e0a1275 |
MulChar.isQuadratic_iff_sq_eq_one | Mathlib/NumberTheory/MulChar/Basic.lean | /-- A multiplicative character `χ` into an integral domain is quadratic
if and only if `χ^2 = 1`. -/
lemma isQuadratic_iff_sq_eq_one {M R : Type*} [CommMonoid M] [CommRing R] [NoZeroDivisors R]
[Nontrivial R] {χ : MulChar M R} :
IsQuadratic χ ↔ χ ^ 2 = 1 | case refine_1
M : Type u_4
R : Type u_5
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
χ : MulChar M R
h : χ.IsQuadratic
x : Mˣ
⊢ (χ ^ 2) ↑x = 1 ↑x | rw [one_apply_coe, χ.pow_apply_coe] | case refine_1
M : Type u_4
R : Type u_5
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
χ : MulChar M R
h : χ.IsQuadratic
x : Mˣ
⊢ χ ↑x ^ 2 = 1 | 6577a1512c271721 |
CategoryTheory.Localization.homEquiv_comp | Mathlib/CategoryTheory/Localization/HomEquiv.lean | lemma homEquiv_comp (f : L₁.obj X ⟶ L₁.obj Y) (g : L₁.obj Y ⟶ L₁.obj Z) :
homEquiv W L₁ L₂ (f ≫ g) = homEquiv W L₁ L₂ f ≫ homEquiv W L₁ L₂ g | C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{u_9, u_1} C
inst✝³ : Category.{u_8, u_5} D₁
inst✝² : Category.{u_10, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y Z : C
f : L₁.obj X ⟶ L₁.obj Y
g : L₁.obj Y ⟶ L₁.obj Z
⊢ (homEquiv W L₁ L₂) ... | apply LocalizerMorphism.homMap_comp | no goals | 146c2c882e335dcd |
Polynomial.coeff_pow_mul_natDegree | Mathlib/Algebra/Polynomial/Degree/Operations.lean | theorem coeff_pow_mul_natDegree (p : R[X]) (n : ℕ) :
(p ^ n).coeff (n * p.natDegree) = p.leadingCoeff ^ n | case pos
R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hi : (p ^ i).coeff (i * p.natDegree) = p.leadingCoeff ^ i
hp1 : p.leadingCoeff ^ i = 0
⊢ (p ^ i * p).coeff (i * p.natDegree + p.natDegree) = 0 | by_cases hp2 : p ^ i = 0 | case pos
R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hi : (p ^ i).coeff (i * p.natDegree) = p.leadingCoeff ^ i
hp1 : p.leadingCoeff ^ i = 0
hp2 : p ^ i = 0
⊢ (p ^ i * p).coeff (i * p.natDegree + p.natDegree) = 0
case neg
R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hi : (p ^ i).coeff (i * p.natDegree) = p.leadingCoef... | 9ebde2d144b436bc |
norm_eq_iInf_iff_real_inner_le_zero | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u : F} {v : F}
(hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 | case mp
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ... | by_cases hq : q = 0 | case pos
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
... | 2a92713ac13db1c9 |
Ordnode.Valid'.glue_aux | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.glue_aux {l r o₁ o₂} (hl : Valid' o₁ l o₂) (hr : Valid' o₁ r o₂)
(sep : l.All fun x => r.All fun y => x < y) (bal : BalancedSz (size l) (size r)) :
Valid' o₁ (@glue α l r) o₂ ∧ size (glue l r) = size l + size r | case node.node
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx r... | dsimp [glue] | case node.node
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx r... | 8646db9152bba08b |
IsPrimitiveRoot.is_roots_of_minpoly | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | theorem is_roots_of_minpoly [DecidableEq K] :
primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset | n : ℕ
K : Type u_1
inst✝³ : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝² : IsDomain K
inst✝¹ : CharZero K
inst✝ : DecidableEq K
⊢ primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset | by_cases hn : n = 0 | case pos
n : ℕ
K : Type u_1
inst✝³ : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝² : IsDomain K
inst✝¹ : CharZero K
inst✝ : DecidableEq K
hn : n = 0
⊢ primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset
case neg
n : ℕ
K : Type u_1
inst✝³ : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝² : IsD... | 9ec698b90ad3017a |
Nat.Partrec.Code.evaln_mono | Mathlib/Computability/PartrecCode.lean | theorem evaln_mono : ∀ {k₁ k₂ c n x}, k₁ ≤ k₂ → x ∈ evaln k₁ c n → x ∈ evaln k₂ c n
| 0, k₂, c, n, x, _, h => by simp [evaln] at h
| k + 1, k₂ + 1, c, n, x, hl, h => by
have hl' := Nat.le_of_succ_le_succ hl
have :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ → (x ∈ o₁ → x ∈ o₂) →
x ∈ ... | case prec.succ
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x ... | exact fun y h₁ h₂ => ⟨y, evaln_mono hl' h₁, hg _ _ h₂⟩ | no goals | 231a1123011d0046 |
RingHom.finitePresentation_ofLocalizationSpanTarget | Mathlib/RingTheory/RingHom/FinitePresentation.lean | theorem finitePresentation_ofLocalizationSpanTarget :
OfLocalizationSpanTarget @FinitePresentation | case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Su... | rw [← hl] | case intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Su... | 1e270402a60647b4 |
norm_iteratedFDerivWithin_prod_le | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem norm_iteratedFDerivWithin_prod_le [DecidableEq ι] [NormOneClass A'] {u : Finset ι}
{f : ι → E → A'} {N : WithTop ℕ∞} (hf : ∀ i ∈ u, ContDiffOn 𝕜 N (f i) s)
(hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : n ≤ N) :
‖iteratedFDerivWithin 𝕜 n (∏ j ∈ u, f j ·) s x‖ ≤
∑ p ∈ u.sym n, (p... | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx ... | simp | no goals | 511530e6d71b1b7d |
IsSelfAdjoint.star_iff | Mathlib/Algebra/Star/SelfAdjoint.lean | theorem star_iff [InvolutiveStar R] {x : R} : IsSelfAdjoint (star x) ↔ IsSelfAdjoint x | R : Type u_1
inst✝ : InvolutiveStar R
x : R
⊢ IsSelfAdjoint (star x) ↔ IsSelfAdjoint x | simpa only [IsSelfAdjoint, star_star] using eq_comm | no goals | e6bd364a1830d040 |
FDerivMeasurableAux.norm_sub_le_of_mem_A | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem norm_sub_le_of_mem_A {c : 𝕜} (hc : 1 < ‖c‖) {r ε : ℝ} (hε : 0 < ε) (hr : 0 < r) {x : E}
{L₁ L₂ : E →L[𝕜] F} (h₁ : x ∈ A f L₁ r ε) (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ‖c‖ * ε | case h₂
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
... | apply le_of_mem_A h₁ | case h₂.hy
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r... | 2ee71d508cf2305e |
HomologicalComplex.liftCycles_homologyπ_eq_zero_of_boundary | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | @[reassoc]
lemma liftCycles_homologyπ_eq_zero_of_boundary {A : C} (k : A ⟶ K.X i) (j : ι)
(hj : c.next i = j) {i' : ι} (x : A ⟶ K.X i') (hx : k = x ≫ K.d i' i) :
K.liftCycles k j hj (by rw [hx, assoc, K.d_comp_d, comp_zero]) ≫ K.homologyπ i = 0 | case neg
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx : k = x ≫ K.d i' i
h : ¬c.Rel i' i
this : K.liftCycles k j hj ⋯ = 0
⊢ K.liftCycle... | rw [this, zero_comp] | no goals | 5f8a23e175db7b00 |
Convex.integral_mem | Mathlib/Analysis/Convex/Integral.lean | theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
(hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s | case intro.intro.intro.refine_1
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := bor... | exact fun _ _ => ENNReal.toReal_nonneg | no goals | eb893caa8bcc87be |
DihedralGroup.reciprocalFactors_even | Mathlib/GroupTheory/CommutingProbability.lean | lemma reciprocalFactors_even {n : ℕ} (h0 : n ≠ 0) (h2 : Even n) :
reciprocalFactors n = 3 :: reciprocalFactors (n / 2) | n : ℕ
h0 : n ≠ 0
h2 : Even n
⊢ n ≠ 1 | rintro rfl | h0 : 1 ≠ 0
h2 : Even 1
⊢ False | 17c06f8c80712055 |
IsPrimal.mul | Mathlib/Algebra/GroupWithZero/Divisibility.lean | theorem IsPrimal.mul {α} [CancelCommMonoidWithZero α] {m n : α}
(hm : IsPrimal m) (hn : IsPrimal n) : IsPrimal (m * n) | case inr.intro.intro.intro.intro.intro.intro
α : Type u_2
inst✝ : CancelCommMonoidWithZero α
n : α
hn : IsPrimal n
a₁ a₂ b c : α
hm : IsPrimal (a₁ * a₂)
h0 : a₁ * a₂ ≠ 0
h : n ∣ b * c
⊢ ∃ a₁_1 a₂_1, a₁_1 ∣ a₁ * b ∧ a₂_1 ∣ a₂ * c ∧ a₁ * a₂ * n = a₁_1 * a₂_1 | obtain ⟨a₁', a₂', h₁, h₂, rfl⟩ := hn h | case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_2
inst✝ : CancelCommMonoidWithZero α
a₁ a₂ b c : α
hm : IsPrimal (a₁ * a₂)
h0 : a₁ * a₂ ≠ 0
a₁' a₂' : α
h₁ : a₁' ∣ b
h₂ : a₂' ∣ c
hn : IsPrimal (a₁' * a₂')
h : a₁' * a₂' ∣ b * c
⊢ ∃ a₁_1 a₂_1, a₁_1 ∣ a₁ * b ∧ a₂_1 ∣ a₂ * c ∧ a₁ * a₂ * (a₁' ... | ab5031a4a2a30aa7 |
Filter.le_limsup_iff | Mathlib/Order/LiminfLimsup.lean | theorem le_limsup_iff {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α} {u : α → β} {x : β}
(h₁ : f.IsCoboundedUnder (· ≤ ·) u | α : Type u_6
β : Type u_7
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
x : β
h₁ : autoParam (IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h₂ : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h : ∀ y < x, ∃ᶠ (a : α) in f, y < u a
h' : ∃ y < x, ∀ (z : β), z ≤ y ∨ x ≤ z
⊢ ∃ᶠ (x_1 :... | rcases h' with ⟨z, z_x, hz⟩ | case intro.intro
α : Type u_6
β : Type u_7
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
x : β
h₁ : autoParam (IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h₂ : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h : ∀ y < x, ∃ᶠ (a : α) in f, y < u a
z : β
z_x : z < x
hz : ∀ (z_1 : β... | 672106753e3701a8 |
Surreal.Multiplication.ih24_neg | Mathlib/SetTheory/Surreal/Multiplication.lean | /-- Symmetry properties of `IH24`. -/
lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) | case refine_3
x₁ x₂ y : PGame
h : ∀ ⦃z : PGame⦄, (z.IsOption x₁ → P24 z x₂ y) ∧ (z.IsOption x₂ → P24 x₁ z y) ∧ (z.IsOption y → P24 x₁ x₂ z)
z : PGame
⊢ z.IsOption y → P24 x₁ x₂ z | first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2 | no goals | b6642f68aa213d25 |
measurableSet_eq_fun | Mathlib/MeasureTheory/Group/Arithmetic.lean | theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [MeasurableSpace E] [AddGroup E]
[MeasurableSingletonClass E] [MeasurableSub₂ E] {f g : α → E} (hf : Measurable f)
(hg : Measurable g) : MeasurableSet { x | f x = g x } | α : Type u_3
m : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
h_set_eq : {x | f x = g x} = {x | (f - g) x = 0}
⊢ MeasurableSet {x | f x = g x} | rw [h_set_eq] | α : Type u_3
m : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
h_set_eq : {x | f x = g x} = {x | (f - g) x = 0}
⊢ MeasurableSet {x | (f - g) x = 0} | 56b01b1a716a13bc |
AkraBazziRecurrence.rpow_p_mul_one_sub_smoothingFn_le | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma rpow_p_mul_one_sub_smoothingFn_le :
∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r i n) - ... | filter_upwards [eventually_gt_atTop ⌈(b i)⁻¹⌉₊, eventually_gt_atTop 1] with n hn hn' | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r ... | 75ad81c80333a79d |
Profinite.NobelingProof.Products.max_eq_eval | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem Products.max_eq_eval [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) :
Linear_CC' C hsC ho (l.eval C) = l.Tail.eval (C' C ho) | case h
I : Type u
C : Set (I → Bool)
inst✝² : LinearOrder I
inst✝¹ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
inst✝ : Inhabited I
l : Products I
hl : ↑l ≠ []
hlh : (↑l).head! = term I ho
hlc : List.Chain' (fun x1 x2 => x1 > x2) (term I ho :: ↑l.Tail)
x ... | have hi' : ∀ i, i ∈ l.Tail.val → (x.val i = SwapTrue o x.val i) := by
intro i hi
simp only [SwapTrue, @eq_comm _ (x.val i), ite_eq_right_iff, ord_term ho]
rintro rfl
exact ((List.Chain.rel hlc hi).ne rfl).elim | case h
I : Type u
C : Set (I → Bool)
inst✝² : LinearOrder I
inst✝¹ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
inst✝ : Inhabited I
l : Products I
hl : ↑l ≠ []
hlh : (↑l).head! = term I ho
hlc : List.Chain' (fun x1 x2 => x1 > x2) (term I ho :: ↑l.Tail)
x ... | 1e685f18c6783584 |
exists_continuous_one_zero_of_isCompact_of_isGδ | Mathlib/Topology/UrysohnsLemma.lean | theorem exists_continuous_one_zero_of_isCompact_of_isGδ [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), s = f ⁻¹' {1} ∧ EqOn f 0 t ∧ HasCompactSupport f
∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | case intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(... | simpa using this.ne | no goals | 06fdcc97382f69bb |
List.toFinset_eq_iff_perm_dedup | Mathlib/Data/Finset/Dedup.lean | theorem toFinset_eq_iff_perm_dedup : l.toFinset = l'.toFinset ↔ l.dedup ~ l'.dedup | α : Type u_1
inst✝ : DecidableEq α
l l' : List α
⊢ l.toFinset = l'.toFinset ↔ l.dedup ~ l'.dedup | simp [Finset.ext_iff, perm_ext_iff_of_nodup (nodup_dedup _) (nodup_dedup _)] | no goals | dcdaa6495ac81792 |
MeasureTheory.OuterMeasure.trim_binop | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | theorem trim_binop {m₁ m₂ m₃ : OuterMeasure α} {op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞}
(h : ∀ s, m₁ s = op (m₂ s) (m₃ s)) (s : Set α) : m₁.trim s = op (m₂.trim s) (m₃.trim s) | case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
m₁ m₂ m₃ : OuterMeasure α
op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞
h : ∀ (s : Set α), m₁ s = op (m₂ s) (m₃ s)
s t : Set α
_hst : s ⊆ t
_ht : MeasurableSet t
htm : m₁ t = m₁.trim s ∧ m₂ t = m₂.trim s ∧ m₃ t = m₃.trim s ∧ ∀ (i : Fin 0), (![] i) t = (![] i).trim s
⊢ m₁.trim s =... | rw [← htm.1, ← htm.2.1, ← htm.2.2.1, h] | no goals | 751141c35a7948ba |
MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_aux | Mathlib/Probability/Martingale/BorelCantelli.lean | theorem Submartingale.exists_tendsto_of_abs_bddAbove_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) | Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
ht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n => stoppedValue f (leastGE f (↑i) n) ω) atTop (𝓝 c)... | unfold hitting | Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
ht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n => stoppedValue f (leastGE f (↑i) n) ω) atTop (𝓝 c)... | 2d1bffccac8cf548 |
Primrec.fin_val_iff | Mathlib/Computability/Primrec.lean | theorem fin_val_iff {n} {f : α → Fin n} : (Primrec fun a => (f a).1) ↔ Primrec f | α : Type u_1
inst✝ : Primcodable α
n : ℕ
f : α → Fin n
this : Primcodable { a // id a < n } := Primcodable.subtype ⋯
⊢ (Primrec fun a => ↑(f a)) ↔ Primrec fun a => ↑(Fin.equivSubtype (f a)) | rfl | no goals | 92a5276ffe1a417c |
Int.abs_sign_of_nonzero | Mathlib/Algebra/Order/Group/Unbundled/Int.lean | theorem abs_sign_of_nonzero {z : ℤ} (hz : z ≠ 0) : |z.sign| = 1 | z : ℤ
hz : z ≠ 0
⊢ |z.sign| = 1 | rw [abs_eq_natAbs, natAbs_sign_of_nonzero hz, Int.ofNat_one] | no goals | 052d6b25d0cf7969 |
Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt | Mathlib/Analysis/Calculus/MeanValue.lean | theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s)
{f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y)
(hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
f' : E → E →L[𝕜] G
hs : Convex ℝ s
f : E → G
hder : ∀ᶠ (y :... | obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0,
ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } :=
mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) | case intro.intro
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
f' : E → E →L[𝕜] G
hs : Convex ℝ s
f : E →... | c2b1e3faf61ae726 |
BitVec.ofFin_intCast | Mathlib/Data/BitVec.lean | theorem ofFin_intCast (z : ℤ) : ofFin (z : Fin (2^w)) = ↑z | z : ℤ
w : ℕ
⊢ {
toFin :=
(fun x =>
match x with
| Int.ofNat n => ↑n
| Int.negSucc n => -↑(n + 1))
z } =
BitVec.ofInt (w + 1) z | rcases z with z | z | case ofNat
w z : ℕ
⊢ {
toFin :=
(fun x =>
match x with
| Int.ofNat n => ↑n
| Int.negSucc n => -↑(n + 1))
(Int.ofNat z) } =
BitVec.ofInt (w + 1) (Int.ofNat z)
case negSucc
w z : ℕ
⊢ {
toFin :=
(fun x =>
match x with
| ... | 4e8ad9400412075e |
List.MergeSort.Internal.splitRevInTwo'_snd | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Impl.lean | theorem splitRevInTwo'_snd (l : { l : List α // l.length = n }) :
(splitRevInTwo' l).2 = ⟨(splitInTwo ⟨l.1.reverse, by simpa using l.2⟩).1.1.reverse, by simp; omega⟩ | α : Type u_1
n : Nat
l : { l // l.length = n }
⊢ ⟨drop (n / 2) l.val, ⋯⟩ = ⟨drop (l.val.reverse.length - (n + 1) / 2) l.val, ⋯⟩ | congr 2 | case e_val.e_a
α : Type u_1
n : Nat
l : { l // l.length = n }
⊢ n / 2 = l.val.reverse.length - (n + 1) / 2 | c6c7aeefc787b40e |
schnirelmannDensity_eq_one_iff | Mathlib/Combinatorics/Schnirelmann.lean | /-- The Schnirelmann density of `A` is `1` if and only if `A` contains all the positive naturals. -/
lemma schnirelmannDensity_eq_one_iff : schnirelmannDensity A = 1 ↔ {0}ᶜ ⊆ A | case mp
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
x : ℕ
hx : x ∈ {0}ᶜ
hx' : x ∉ A
⊢ schnirelmannDensity A < 1 | apply (schnirelmannDensity_le_of_not_mem hx').trans_lt | case mp
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
x : ℕ
hx : x ∈ {0}ᶜ
hx' : x ∉ A
⊢ 1 - (↑x)⁻¹ < 1 | 77c035c561a694c5 |
MeasureTheory.Measure.pi_eq_generateFrom | Mathlib/MeasureTheory/Constructions/Pi.lean | theorem pi_eq_generateFrom {C : ∀ i, Set (Set (α i))}
(hC : ∀ i, generateFrom (C i) = by apply_assumption) (h2C : ∀ i, IsPiSystem (C i))
(h3C : ∀ i, (μ i).FiniteSpanningSetsIn (C i)) {μν : Measure (∀ i, α i)}
(h₁ : ∀ s : ∀ i, Set (α i), (∀ i, s i ∈ C i) → μν (pi univ s) = ∏ i, μ i (s i)) :
Measure.pi μ ... | ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)
h₁ : ∀ (s : (i :... | have h4C : ∀ (i) (s : Set (α i)), s ∈ C i → MeasurableSet s := by
intro i s hs; rw [← hC]; exact measurableSet_generateFrom hs | ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)
h₁ : ∀ (s : (i :... | 6d99627d812669c7 |
ComputablePred.computable_iff | Mathlib/Computability/Halting.lean | theorem computable_iff {p : α → Prop} :
ComputablePred p ↔ ∃ f : α → Bool, Computable f ∧ p = fun a => (f a : Prop) :=
⟨fun ⟨_, h⟩ => ⟨_, h, funext fun _ => propext (Bool.decide_iff _).symm⟩, by
rintro ⟨f, h, rfl⟩; exact ⟨by infer_instance, by simpa using h⟩⟩
| α : Type u_1
inst✝ : Primcodable α
f : α → Bool
h : Computable f
⊢ Computable fun a => decide ((fun a => f a = true) a) | simpa using h | no goals | 6a4ea15bc92dc591 |
IsLocalization.scaleRoots_commonDenom_mem_lifts | Mathlib/RingTheory/Localization/Integral.lean | theorem IsLocalization.scaleRoots_commonDenom_mem_lifts (p : Rₘ[X])
(hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range) :
p.scaleRoots (algebraMap R Rₘ <| IsLocalization.commonDenom M p.support p.coeff) ∈
Polynomial.lifts (algebraMap R Rₘ) | case neg
R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
Rₘ : Type u_3
inst✝² : CommRing Rₘ
inst✝¹ : Algebra R Rₘ
inst✝ : IsLocalization M Rₘ
p : Rₘ[X]
hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range
n : ℕ
h₁ : p.coeff n = 0
⊢ 0 ∈ Set.range ⇑(algebraMap R Rₘ) | exact zero_mem (algebraMap R Rₘ).range | no goals | 2bf54bf7debabe79 |
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