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 |
|---|---|---|---|---|---|---|
Std.Tactic.BVDecide.Normalize.Bool.bne_to_beq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/Canonicalize.lean | theorem Bool.bne_to_beq (a b : Bool) : (a != b) = (!(a == b)) | a b : Bool
⊢ (a != b) = !a == b | simp [bne] | no goals | ab1087d2a1d4f388 |
lp.norm_le_of_tendsto | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem norm_le_of_tendsto {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C) {f : lp E p}
(hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) : ‖f‖ ≤ C | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : ↥(lp E p)
hf : Tendsto (id fun i => ↑(F i)) l (𝓝 ↑f)
⊢ ‖f‖ ≤ C | obtain ⟨i, hi⟩ := hCF.exists | case intro
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : ↥(lp E p)
hf : Tendsto (id fun i => ↑(F i)) l (𝓝 ↑f)
i : ι
hi : ‖F i‖ ≤ C
⊢ ‖f‖ ≤ C | d28404a6ed578ee2 |
DedekindDomain.ProdAdicCompletions.IsFiniteAdele.add | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | theorem add {x y : K_hat R K} (hx : x.IsFiniteAdele) (hy : y.IsFiniteAdele) :
(x + y).IsFiniteAdele | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : {x_1 | x x_1 ∉ adicCompletionIntegers K x_1}.Finite
hy : {x | y x ∉ adicCompletionIntegers K x}.Finite
v : HeightOneSpectrum R
hv : v ∈ {v | (x + y) v ∉ adicCom... | rw [mem_setOf] at hv | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : {x_1 | x x_1 ∉ adicCompletionIntegers K x_1}.Finite
hy : {x | y x ∉ adicCompletionIntegers K x}.Finite
v : HeightOneSpectrum R
hv : (x + y) v ∉ adicCompletionIn... | 82c2f2870b8e321c |
SimpleGraph.card_edgeFinset_le_card_choose_two | Mathlib/Combinatorics/SimpleGraph/Finite.lean | theorem card_edgeFinset_le_card_choose_two : #G.edgeFinset ≤ (Fintype.card V).choose 2 | V : Type u_1
G : SimpleGraph V
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype V
⊢ #G.edgeFinset ≤ (Fintype.card V).choose 2 | classical
rw [← card_edgeFinset_top_eq_card_choose_two]
exact card_le_card (edgeFinset_mono le_top) | no goals | ddbcc0b0091ad234 |
LieSubmodule.lie_baseChange | Mathlib/Algebra/Lie/BaseChange.lean | lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆ | R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ ... | rw [← toSubmodule_inj, coe_baseChange, lieIdeal_oper_eq_linear_span',
Submodule.baseChange_span, this, lieIdeal_oper_eq_linear_span'] | R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ ... | c0e93f202362c69d |
MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2 | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_le_lintegral_abs_det_fderiv_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
h's : μ s ≠ ⊤
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
⊢ Tendsto (... | refine ENNReal.Tendsto.mul_const ?_ (Or.inr h's) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
h's : μ s ≠ ⊤
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
⊢ Tendsto (... | b9f5a9f0ee656fbd |
SimpleGraph.hasse_preconnected_of_succ | Mathlib/Combinatorics/SimpleGraph/Hasse.lean | theorem hasse_preconnected_of_succ [SuccOrder α] [IsSuccArchimedean α] : (hasse α).Preconnected :=
fun a b => by
rw [reachable_iff_reflTransGen]
exact
reflTransGen_of_succ _ (fun c hc => Or.inl <| covBy_succ_of_not_isMax hc.2.not_isMax)
fun c hc => Or.inr <| covBy_succ_of_not_isMax hc.2.not_isMax
| α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
a b : α
⊢ (hasse α).Reachable a b | rw [reachable_iff_reflTransGen] | α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
a b : α
⊢ ReflTransGen (hasse α).Adj a b | d2571d725bafeabe |
CoxeterSystem.prod_alternatingWord_eq_prod_alternatingWord_sub | Mathlib/GroupTheory/Coxeter/Basic.lean | theorem prod_alternatingWord_eq_prod_alternatingWord_sub (i i' : B) (m : ℕ) (hm : m ≤ M i i' * 2) :
π (alternatingWord i i' m) = π (alternatingWord i' i (M i i' * 2 - m)) | case h
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 2 * k = k + k | ring | no goals | 8b276f2d43ad82bd |
SzemerediRegularity.card_increment | Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean | theorem card_increment (hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPG : ¬P.IsUniform G ε) :
#(increment hP G ε).parts = stepBound #P.parts | α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
hPα' : stepBound #P.parts ≤ Fintype.card α
hPpos : 0 < stepBound #P.parts
⊢ #(increment hP G ε).pa... | rw [increment, card_bind] | α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
hPα' : stepBound #P.parts ≤ Fintype.card α
hPpos : 0 < stepBound #P.parts
⊢ ∑ A ∈ P.parts.attach, ... | 76029e7fb47cdde8 |
CategoryTheory.Limits.hasLimitsOfShape_of_equivalence | Mathlib/CategoryTheory/Limits/HasLimits.lean | theorem hasLimitsOfShape_of_equivalence {J' : Type u₂} [Category.{v₂} J'] (e : J ≌ J')
[HasLimitsOfShape J C] : HasLimitsOfShape J' C | J : Type u₁
inst✝³ : Category.{v₁, u₁} J
C : Type u
inst✝² : Category.{v, u} C
J' : Type u₂
inst✝¹ : Category.{v₂, u₂} J'
e : J ≌ J'
inst✝ : HasLimitsOfShape J C
⊢ HasLimitsOfShape J' C | constructor | case has_limit
J : Type u₁
inst✝³ : Category.{v₁, u₁} J
C : Type u
inst✝² : Category.{v, u} C
J' : Type u₂
inst✝¹ : Category.{v₂, u₂} J'
e : J ≌ J'
inst✝ : HasLimitsOfShape J C
⊢ autoParam (∀ (F : J' ⥤ C), HasLimit F) _auto✝ | ecc4ff6321bd1dd3 |
NumberField.Embeddings.pow_eq_one_of_norm_eq_one | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ φ : K →+* A, ‖φ x‖ = 1) :
∃ (n : ℕ) (_ : 0 < n), x ^ n = 1 | K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ = 1
a : ℕ
x✝ : a ∈ univ
φ : K →+* A
⊢ ‖φ ((fun x_1 => x ^ x_1) a)‖ ≤ 1 | simp [hx φ] | no goals | 2db53376910a5986 |
EuclideanGeometry.dist_sq_add_dist_sq_eq_two_mul_dist_midpoint_sq_add_half_dist_sq | Mathlib/Geometry/Euclidean/Triangle.lean | theorem dist_sq_add_dist_sq_eq_two_mul_dist_midpoint_sq_add_half_dist_sq (a b c : P) :
dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2) | case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
hbc : b = c
⊢ dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2) | simp [hbc, midpoint_self, dist_self, two_mul] | no goals | 3df995f2b369cfd1 |
exists_lt_subset_ball | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' | case inl
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hs : IsClosed s
h : s ⊆ ball x r
hr : r ≤ 0
⊢ ∃ r' < r, s ⊆ ball x r' | rw [ball_eq_empty.2 hr, subset_empty_iff] at h | case inl
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hs : IsClosed s
h : s = ∅
hr : r ≤ 0
⊢ ∃ r' < r, s ⊆ ball x r' | 3e325a0ec1663f1a |
mul_div_cancel_right | Mathlib/Algebra/Group/Defs.lean | theorem mul_div_cancel_right (a b : G) : a * b / b = a | G : Type u_1
inst✝ : Group G
a b : G
⊢ a * b / b = a | rw [div_eq_mul_inv, mul_inv_cancel_right a b] | no goals | 20d41b8ab2b565ff |
Filter.hasBasis_coclosedCompact | Mathlib/Topology/Compactness/Compact.lean | theorem hasBasis_coclosedCompact :
(Filter.coclosedCompact X).HasBasis (fun s => IsClosed s ∧ IsCompact s) compl | X : Type u
inst✝ : TopologicalSpace X
⊢ ∀ (i : Set X),
IsClosed i ∧ IsCompact i →
∀ (j : Set X), IsClosed j ∧ IsCompact j → ∃ k, (IsClosed k ∧ IsCompact k) ∧ kᶜ ⊆ iᶜ ∧ kᶜ ⊆ jᶜ | rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩ | case intro.intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs₁ : IsClosed s
hs₂ : IsCompact s
t : Set X
ht₁ : IsClosed t
ht₂ : IsCompact t
⊢ ∃ k, (IsClosed k ∧ IsCompact k) ∧ kᶜ ⊆ sᶜ ∧ kᶜ ⊆ tᶜ | e4c7c8b48b10e2ee |
MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
[NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
{L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
(h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ fi... | E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex... | norm_num | no goals | 375ae7d55500792c |
tendsto_div_of_monotone_of_exists_subseq_tendsto_div | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
(hmono : Monotone u)
(hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) | u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) *... | have A : a ≤ N - 1 := by
apply @Nat.le_of_add_le_add_right a 1 (N - 1)
rw [Nat.sub_add_cancel Npos]
exact aN | u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) *... | 2aab76659c6dcba8 |
HasCompactSupport.smul_right | Mathlib/Topology/Algebra/Support.lean | theorem HasCompactSupport.smul_right (hf : HasCompactSupport f) : HasCompactSupport (f • f') | α : Type u_2
M : Type u_7
R : Type u_8
inst✝³ : TopologicalSpace α
inst✝² : Zero R
inst✝¹ : Zero M
inst✝ : SMulWithZero R M
f : α → R
f' : α → M
hf : f =ᶠ[coclosedCompact α] 0
⊢ f • f' =ᶠ[coclosedCompact α] 0 | exact hf.mono fun x hx => by simp_rw [Pi.smul_apply', hx, Pi.zero_apply, zero_smul] | no goals | 097ecd58c57ed57f |
Int.cast_le | Mathlib/Algebra/Order/Ring/Cast.lean | @[simp, norm_cast] lemma cast_le : (m : R) ≤ n ↔ m ≤ n | R : Type u_1
inst✝⁴ : AddCommGroupWithOne R
inst✝³ : PartialOrder R
inst✝² : AddLeftMono R
inst✝¹ : ZeroLEOneClass R
inst✝ : NeZero 1
m n : ℤ
⊢ ↑m ≤ ↑n ↔ m ≤ n | rw [← sub_nonneg, ← cast_sub, cast_nonneg, sub_nonneg] | no goals | ec6509e4be5b2634 |
NNReal.inner_le_Lp_mul_Lp_of_norm_le_one | Mathlib/Analysis/MeanInequalities.lean | theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.IsConjExponent q) (hf : ∑ i ∈ s, f i ^ p ≤ 1) (hg : ∑ i ∈ s, g i ^ q ≤ 1) :
∑ i ∈ s, f i * g i ≤ 1 | ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : ∑ i ∈ s, f i ^ p ≤ 1
hg : ∑ i ∈ s, g i ^ q ≤ 1
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ ∑ i ∈ s, (f i ^ p / p.toNNReal + g i ^ q / q.toNNReal) =
(∑ i ∈ s, f i ^ p) / p.toNNReal + (∑ i ∈ s, g i ^ q) / q.toNNReal | rw [sum_add_distrib, sum_div, sum_div] | no goals | 8cfe6b868dff4ae8 |
ProbabilityTheory.integrable_cexp_mul_of_re_mem_integrableExpSet | Mathlib/Probability/Moments/IntegrableExpMul.lean | lemma integrable_cexp_mul_of_re_mem_integrableExpSet (hX : AEMeasurable X μ)
(hz : z.re ∈ integrableExpSet X μ) :
Integrable (fun ω ↦ cexp (z * X ω)) μ | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hX : AEMeasurable X μ
hz : z.re ∈ integrableExpSet X μ
⊢ Integrable (fun a => ‖cexp (z * ↑(X a))‖) μ | simpa [Complex.norm_exp] using hz | no goals | 668b6589bfab70ba |
CategoryTheory.Sheaf.isConstant_of_forget | Mathlib/CategoryTheory/Sites/ConstantSheaf.lean | lemma Sheaf.isConstant_of_forget [constantSheaf J D |>.Faithful] [constantSheaf J D |>.Full]
[constantSheaf J B |>.Faithful] [constantSheaf J B |>.Full]
[(sheafCompose J U).ReflectsIsomorphisms] [((sheafCompose J U).obj F).IsConstant J]
{T : C} (hT : IsTerminal T) : F.IsConstant J | C : Type u_1
inst✝¹² : Category.{u_5, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝¹¹ : Category.{u_4, u_2} D
inst✝¹⁰ : HasWeakSheafify J D
B : Type u_3
inst✝⁹ : Category.{u_6, u_3} B
U : D ⥤ B
inst✝⁸ : HasWeakSheafify J B
inst✝⁷ : J.PreservesSheafification U
inst✝⁶ : J.HasSheafCompose U
F : Sheaf J D
inst✝⁵ : (... | have : IsIso ((sheafCompose J U).map ((constantSheafAdj J D hT).counit.app F)) := by
rw [← constantSheafAdj_counit_w]
infer_instance | C : Type u_1
inst✝¹² : Category.{u_5, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝¹¹ : Category.{u_4, u_2} D
inst✝¹⁰ : HasWeakSheafify J D
B : Type u_3
inst✝⁹ : Category.{u_6, u_3} B
U : D ⥤ B
inst✝⁸ : HasWeakSheafify J B
inst✝⁷ : J.PreservesSheafification U
inst✝⁶ : J.HasSheafCompose U
F : Sheaf J D
inst✝⁵ : (... | 653af90f9dd3cfc4 |
SimpleGraph.edist_comm | Mathlib/Combinatorics/SimpleGraph/Metric.lean | theorem edist_comm : G.edist u v = G.edist v u | V : Type u_1
G : SimpleGraph V
u v : V
⊢ G.edist u v = G.edist v u | rw [edist_eq_sInf, ← Set.image_univ, ← Set.image_univ_of_surjective Walk.reverse_surjective,
← Set.image_comp, Set.image_univ, Function.comp_def] | V : Type u_1
G : SimpleGraph V
u v : V
⊢ sInf (Set.range fun x => ↑x.reverse.length) = G.edist v u | c53b18212a808f1f |
Cycle.length_nontrivial | Mathlib/Data/List/Cycle.lean | theorem length_nontrivial {s : Cycle α} (h : Nontrivial s) : 2 ≤ length s | case intro.intro.intro.intro.h
α : Type u_1
x y : α
hxy : x ≠ y
l : List α
hx : x ∈ Quot.mk (⇑(IsRotated.setoid α)) l
hy : y ∈ Quot.mk (⇑(IsRotated.setoid α)) l
⊢ 2 ≤ length (Quot.mk (⇑(IsRotated.setoid α)) l) | rcases l with (_ | ⟨hd, _ | ⟨hd', tl⟩⟩) | case intro.intro.intro.intro.h.nil
α : Type u_1
x y : α
hxy : x ≠ y
hx : x ∈ Quot.mk ⇑(IsRotated.setoid α) []
hy : y ∈ Quot.mk ⇑(IsRotated.setoid α) []
⊢ 2 ≤ length (Quot.mk ⇑(IsRotated.setoid α) [])
case intro.intro.intro.intro.h.cons.nil
α : Type u_1
x y : α
hxy : x ≠ y
hd : α
hx : x ∈ Quot.mk ⇑(IsRotated.setoid α) ... | 3276c3f816636265 |
MeasureTheory.SimpleFunc.map_setToSimpleFunc | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G}
(hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) :
(f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x) | case neg
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f... | rw [h_left_eq] | case neg
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f... | f53dabe35f13a020 |
MeasureTheory.IsSetSemiring.diff_sUnion_eq_sUnion_disjointOfDiffUnion | Mathlib/MeasureTheory/SetSemiring.lean | lemma diff_sUnion_eq_sUnion_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
(hI : ↑I ⊆ C) : s \ ⋃₀ I = ⋃₀ hC.disjointOfDiffUnion hs hI | α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ s \ ⋃₀ ↑I = ⋃₀ ↑(hC.disjointOfDiffUnion hs hI) | classical
rw [(hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.2]
simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
rw [sUnion_diff_singleton_empty] | no goals | d807fa02244180ba |
EReal.liminf_add_top_of_ne_bot | Mathlib/Topology/Instances/EReal/Lemmas.lean | lemma liminf_add_top_of_ne_bot (h : liminf u f = ⊤) (h' : liminf v f ≠ ⊥) :
liminf (u + v) f = ⊤ | α : Type u_3
f : Filter α
u v : α → EReal
h : liminf u f = ⊤
h' : liminf v f ≠ ⊥
⊢ ⊤ ≤ liminf u f + liminf v f | rw [h, top_add_of_ne_bot h'] | no goals | 09d39bee35675333 |
IsPathConnected.exists_path_through_family | Mathlib/Topology/Connected/PathConnected.lean | theorem IsPathConnected.exists_path_through_family {n : ℕ}
{s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ | case succ
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ n + 1, p' i ∈ s
⊢ ∃ γ, (∀ i ≤ n + 1, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s | rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩ | case succ.intro
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ n + 1, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
⊢ ∃ γ, (∀ i ≤ n + 1, p' i ∈ range ⇑γ) ∧... | b08141240b56f283 |
List.toFinsupp_append | Mathlib/Data/List/ToFinsupp.lean | theorem toFinsupp_append {R : Type*} [AddZeroClass R] (l₁ l₂ : List R)
[DecidablePred (getD (l₁ ++ l₂) · 0 ≠ 0)] [DecidablePred (getD l₁ · 0 ≠ 0)]
[DecidablePred (getD l₂ · 0 ≠ 0)] :
toFinsupp (l₁ ++ l₂) =
toFinsupp l₁ + (toFinsupp l₂).embDomain (addLeftEmbedding l₁.length) | case h.inl
R : Type u_2
inst✝³ : AddZeroClass R
l₁ l₂ : List R
inst✝² : DecidablePred fun x => (l₁ ++ l₂).getD x 0 ≠ 0
inst✝¹ : DecidablePred fun x => l₁.getD x 0 ≠ 0
inst✝ : DecidablePred fun x => l₂.getD x 0 ≠ 0
n : ℕ
h : n < l₁.length
⊢ (l₁ ++ l₂).getD n 0 = l₁.getD n 0 + (Finsupp.embDomain (addLeftEmbedding l₁.leng... | rw [getD_append _ _ _ _ h, Finsupp.embDomain_notin_range, add_zero] | case h.inl.h
R : Type u_2
inst✝³ : AddZeroClass R
l₁ l₂ : List R
inst✝² : DecidablePred fun x => (l₁ ++ l₂).getD x 0 ≠ 0
inst✝¹ : DecidablePred fun x => l₁.getD x 0 ≠ 0
inst✝ : DecidablePred fun x => l₂.getD x 0 ≠ 0
n : ℕ
h : n < l₁.length
⊢ n ∉ Set.range ⇑(addLeftEmbedding l₁.length) | 827e4cf140b9a24e |
Polynomial.leadingCoeff_C_mul_X_pow | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem leadingCoeff_C_mul_X_pow (a : R) (n : ℕ) : leadingCoeff (C a * X ^ n) = a | R : Type u
inst✝ : Semiring R
a : R
n : ℕ
⊢ (C a * X ^ n).leadingCoeff = a | rw [C_mul_X_pow_eq_monomial, leadingCoeff_monomial] | no goals | a4d0b4c40bf38547 |
zero_pow_eq_zero | Mathlib/Algebra/GroupWithZero/Basic.lean | @[simp]
lemma zero_pow_eq_zero [Nontrivial M₀] : (0 : M₀) ^ n = 0 ↔ n ≠ 0 :=
⟨by rintro h rfl; simp at h, zero_pow⟩
| M₀ : Type u_1
inst✝¹ : MonoidWithZero M₀
n : ℕ
inst✝ : Nontrivial M₀
⊢ 0 ^ n = 0 → n ≠ 0 | rintro h rfl | M₀ : Type u_1
inst✝¹ : MonoidWithZero M₀
inst✝ : Nontrivial M₀
h : 0 ^ 0 = 0
⊢ False | 69960446e0c3515a |
mul_pow_mul | Mathlib/Algebra/Group/Defs.lean | @[to_additive] lemma mul_pow_mul (a b : M) (n : ℕ) :
(a * b) ^ n * a = a * (b * a) ^ n | case zero
M : Type u_2
inst✝ : Monoid M
a b : M
⊢ (a * b) ^ 0 * a = a * (b * a) ^ 0 | simp | no goals | 027f8d0eb371a2ca |
AlgebraicGeometry.Scheme.Pullback.range_map | Mathlib/AlgebraicGeometry/PullbackCarrier.lean | lemma range_map {X' Y' S' : Scheme.{u}} (f' : X' ⟶ S') (g' : Y' ⟶ S') (i₁ : X ⟶ X')
(i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')
(e₂ : g ≫ i₃ = i₂ ≫ g') [Mono i₃] :
Set.range (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂).base =
(pullback.fst f' g').base ⁻¹' Set.range i₁.base ∩
(pullback.snd f... | case h
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
X' Y' S' : Scheme
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
t : ↑↑(pullback f g).toPresheafedSpace
⊢ (ConcreteCategory.hom (pullback.snd f g ≫ i₂).base) t =
(ConcreteCategory.hom (pullback.map f... | simp | no goals | 6ca8171f1040fa44 |
ProbabilityTheory.Kernel.comap_prod_swap | Mathlib/Probability/Kernel/Composition/Prod.lean | lemma comap_prod_swap (κ : Kernel α β) (η : Kernel γ δ) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
comap (prodMkRight α η ×ₖ prodMkLeft γ κ) Prod.swap measurable_swap
= map (prodMkRight γ κ ×ₖ prodMkLeft α η) Prod.swap | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel γ δ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
x : α × γ
f : δ × β → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (b : δ), ∫⁻ (c : β), f (b, c) ∂(prodMkLeft ... | swap | case hg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel γ δ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
x : α × γ
f : δ × β → ℝ≥0∞
hf : Measurable f
⊢ Measurable fun a => f a.swap
α : Type... | 35c706ed7361c3f8 |
MulRingNorm.equiv_symm | Mathlib/Analysis/Normed/Ring/Seminorm.lean | /-- Equivalence of multiplicative ring norms is symmetric. -/
@[deprecated "Use AbsoluteValue.isEquiv_symm instead" (since := "2025-01-07")]
lemma equiv_symm {f g : MulRingNorm R} (hfg : equiv f g) : equiv g f | case h
R : Type u_2
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g
⊢ 0 < 1 / c ∧ (fun x => g x ^ (1 / c)) = ⇑f | constructor | case h.left
R : Type u_2
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g
⊢ 0 < 1 / c
case h.right
R : Type u_2
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g
⊢ (fun x => g x ^ (1 / c)) = ⇑f | 3801cdabfb5ef1ad |
EulerSine.integral_cos_mul_cos_pow_even | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | theorem integral_cos_mul_cos_pow_even (n : ℕ) (hz : z ≠ 0) :
(((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2) *
∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n + 2)) =
(2 * n + 1 : ℂ) / (2 * n + 2) *
∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n) | z : ℂ
n : ℕ
hz : z ≠ 0
⊢ 2 ≤ 2 * n + 2 | omega | no goals | c42a63462e1474f3 |
Algebra.norm_eq_matrix_det | Mathlib/RingTheory/Norm/Defs.lean | theorem norm_eq_matrix_det [Fintype ι] [DecidableEq ι] (b : Basis ι R S) (s : S) :
norm R s = Matrix.det (Algebra.leftMulMatrix b s) | R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
ι : Type w
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
s : S
⊢ (norm R) s = ((leftMulMatrix b) s).det | rw [norm_apply, ← LinearMap.det_toMatrix b, ← toMatrix_lmul_eq] | R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
ι : Type w
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
s : S
⊢ ((toMatrix b b) ((lmul R S) s)).det = ((toMatrix b b) (mulLeft R s)).det | 263db825084bf49e |
isEquivalent_descFactorial | Mathlib/Analysis/SpecialFunctions/Choose.lean | /-- `n.descFactorial k` is asymptotically equivalent to `n^k`. -/
lemma isEquivalent_descFactorial (k : ℕ) :
(fun (n : ℕ) ↦ (n.descFactorial k : ℝ)) ~[atTop] (fun (n : ℕ) ↦ (n^k : ℝ)) | case succ
k : ℕ
h : (fun n => ↑(n.descFactorial k)) ~[atTop] fun n => ↑n ^ k
⊢ (fun n => ↑(n - k)) ~[atTop] Nat.cast | have hz : ∀ᶠ (x : ℕ) in atTop, (x : ℝ) ≠ 0 :=
eventually_atTop.mpr ⟨1, fun n hn ↦ ne_of_gt (mod_cast hn)⟩ | case succ
k : ℕ
h : (fun n => ↑(n.descFactorial k)) ~[atTop] fun n => ↑n ^ k
hz : ∀ᶠ (x : ℕ) in atTop, ↑x ≠ 0
⊢ (fun n => ↑(n - k)) ~[atTop] Nat.cast | d9a0bc9f825f756b |
Liouville.frequently_exists_num | Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean | theorem frequently_exists_num (hx : Liouville x) (n : ℕ) :
∃ᶠ b : ℕ in atTop, ∃ a : ℤ, x ≠ a / b ∧ |x - a / b| < 1 / (b : ℝ) ^ n | case intro.intro.intro.intro.intro.intro.intro.intro
x : ℝ
hx : Liouville x
n N : ℕ
hN : ∀ b ≥ N, ∀ (x_1 : ℤ), x ≠ ↑x_1 / ↑b → 1 / ↑b ^ n ≤ |x - ↑x_1 / ↑b|
this✝ : ∀ b > 1, ∀ᶠ (m : ℕ) in atTop, ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
this : ∀ᶠ (m : ℕ) in atTop, ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
m : ℕ... | push_cast at hne hlt | case intro.intro.intro.intro.intro.intro.intro.intro
x : ℝ
hx : Liouville x
n N : ℕ
hN : ∀ b ≥ N, ∀ (x_1 : ℤ), x ≠ ↑x_1 / ↑b → 1 / ↑b ^ n ≤ |x - ↑x_1 / ↑b|
this✝ : ∀ b > 1, ∀ᶠ (m : ℕ) in atTop, ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
this : ∀ᶠ (m : ℕ) in atTop, ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
m : ℕ... | ebbdebce3ecf7dd1 |
CategoryTheory.Functor.Final.exists_coeq_of_locally_small | Mathlib/CategoryTheory/Filtered/Final.lean | theorem Functor.Final.exists_coeq_of_locally_small [IsFilteredOrEmpty C] [Final F] {d : D} {c : C}
(s s' : d ⟶ F.obj c) : ∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t | C : Type v₁
inst✝³ : Category.{v₁, v₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
F : C ⥤ D
inst✝¹ : IsFilteredOrEmpty C
inst✝ : F.Final
d : D
c : C
s s' : d ⟶ F.obj c
⊢ colimit.ι (F ⋙ coyoneda.obj (op d)) c s = colimit.ι (F ⋙ coyoneda.obj (op d)) c s' | apply (Final.colimitCompCoyonedaIso F d).toEquiv.injective | case a
C : Type v₁
inst✝³ : Category.{v₁, v₁} C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
F : C ⥤ D
inst✝¹ : IsFilteredOrEmpty C
inst✝ : F.Final
d : D
c : C
s s' : d ⟶ F.obj c
⊢ (colimitCompCoyonedaIso F d).toEquiv (colimit.ι (F ⋙ coyoneda.obj (op d)) c s) =
(colimitCompCoyonedaIso F d).toEquiv (colimit.ι (F ⋙ coyon... | d59ab7087435c805 |
UnitAddTorus.hasSum_mFourier_series_apply_of_summable | Mathlib/Analysis/Fourier/AddCircleMulti.lean | theorem hasSum_mFourier_series_apply_of_summable (h : Summable (mFourierCoeff f))
(x : UnitAddTorus d) : HasSum (fun i ↦ mFourierCoeff f i • mFourier i x) (f x) | d : Type u_1
inst✝ : Fintype d
f : C(UnitAddTorus d, ℂ)
h : Summable (mFourierCoeff ⇑f)
x : UnitAddTorus d
⊢ HasSum (fun i => mFourierCoeff (⇑f) i • (mFourier i) x) (f x) | simpa only [_root_.map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
(hasSum_mFourier_series_of_summable h) | no goals | 0d9aeb9f01cefe67 |
ENNReal.limsup_const_mul_of_ne_top | Mathlib/Order/Filter/ENNReal.lean | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
x : ℝ≥0∞
⊢ (fun x => a⁻¹ * x) (g x) = x | simp [g, ← mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top] | no goals | d30eb6c16ac314e0 |
Equiv.Perm.signAux_eq_signAux2 | Mathlib/GroupTheory/Perm/Sign.lean | theorem signAux_eq_signAux2 {n : ℕ} :
∀ (l : List α) (f : Perm α) (e : α ≃ Fin n) (_h : ∀ x, f x ≠ x → x ∈ l),
signAux ((e.symm.trans f).trans e) = signAux2 l f
| [], f, e, h => by
have : f = 1 := Equiv.ext fun y => Classical.not_not.1 (mt (h y) (List.not_mem_nil _))
rw [this, one_def, Equiv.trans_r... | α : Type u
inst✝ : DecidableEq α
n : ℕ
x : α
l : List α
f : Perm α
e : α ≃ Fin n
h : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l
hfx : ¬x = f x
hy : ∀ (y : α), (swap x (f x) * f) y ≠ y → y ∈ l
⊢ (e.symm.trans (swap x (f x) * f)).trans e = swap (e x) (e (f x)) * (e.symm.trans f).trans e | ext | case H.h
α : Type u
inst✝ : DecidableEq α
n : ℕ
x : α
l : List α
f : Perm α
e : α ≃ Fin n
h : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l
hfx : ¬x = f x
hy : ∀ (y : α), (swap x (f x) * f) y ≠ y → y ∈ l
x✝ : Fin n
⊢ ↑(((e.symm.trans (swap x (f x) * f)).trans e) x✝) = ↑((swap (e x) (e (f x)) * (e.symm.trans f).trans e) x✝) | f94966561d754325 |
ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | /-- The auxiliary function agrees on `re s > 1` with the L-series of the von Mangoldt function
restricted to the residue class `a : ZMod q` minus the principal part `(q.totient)⁻¹/(s-1)`
of its pole at `s = 1`. -/
lemma eqOn_LFunctionResidueClassAux (ha : IsUnit a) :
Set.EqOn (LFunctionResidueClassAux a)
(fun... | q : ℕ
a : ZMod q
inst✝ : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s + ∑ χ ∈ {1}ᶜ, χ a⁻¹ * deriv (LFunction χ) s / LFunction χ s =
∑ χ : DirichletCharacter ℂ q, χ a⁻¹ * (deriv (LFunction χ) s / LFunction χ s) + 1 / (s - 1) | classical rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
simp only [mul_div_assoc]
congrm (?_ + _)
have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
Function.update_of_ne hs₁, LFunctionTrivChar, add... | no goals | 86a79bf7dd9cebb1 |
CategoryTheory.ShortComplex.SnakeInput.L₁'_exact | Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | /-- Exactness of `L₀.X₂ ⟶ L₀.X₃ ⟶ L₃.X₁`. -/
lemma L₁'_exact : S.L₁'.Exact | case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A₀ : C
x₃ : A₀ ⟶ S.L₀.X₃
hx₃ : x₃ ≫ S.δ = 0
A₁ : C
π₁ : A₁ ⟶ A₀
hπ₁ : Epi π₁
p : A₁ ⟶ S.P
hp : π₁ ≫ x₃ = p ≫ pullback.snd S.L₁.g S.v₀₁.τ₃
hp' : (p ≫ S.φ₁) ≫ S.v₂₃.τ₁ = 0
A₂ : C
π₂ : A₂ ⟶ A... | exact ⟨A₂, π₂ ≫ π₁, epi_comp _ _, k₂, by simp only [assoc, L₁'_f, ← hk₂', hp]⟩ | no goals | a73b0907fae173cf |
BitVec.signExtend_eq_setWidth_of_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem signExtend_eq_setWidth_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w):
x.signExtend v = x.setWidth v | w : Nat
x : BitVec w
v : Nat
hv : v ≤ w
⊢ signExtend v x = setWidth v x | ext i h | case pred
w : Nat
x : BitVec w
v : Nat
hv : v ≤ w
i : Nat
h : i < v
⊢ (signExtend v x).getLsbD i = (setWidth v x).getLsbD i | ed542328457bdbfe |
Std.DHashMap.Internal.Raw₀.getD_alter_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getD_alter_self [LawfulBEq α] {k : α} {fallback : β k} (h : m.1.WF)
{f : Option (β k) → Option (β k)} :
(m.alter k f).getD k fallback = (f (m.get? k)).getD fallback | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
k : α
fallback : β k
h : m.val.WF
f : Option (β k) → Option (β k)
⊢ (m.alter k f).getD k fallback = (f (m.get? k)).getD fallback | simp only [getD_alter, h, beq_self_eq_true, reduceDIte, cast_eq_id, Option.map_id_fun, id_eq] | no goals | 1c6121e42940b96f |
StarAlgebra.elemental.starAlgHomClass_ext | Mathlib/Topology/Algebra/StarSubalgebra.lean | theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
[FunLike F (elemental R a) B] [AlgHomClass F R _ B] [StarHomClass F _ B]
{φ ψ : F} (hφ : Continuous φ)
(hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ | R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : StarRing A
inst✝¹¹ : IsTopologicalSemiring A
inst✝¹⁰ : ContinuousStar A
inst✝⁹ : Algebra R A
inst✝⁸ : StarModule R A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : S... | simp only [map_mul, hx, hy] | no goals | 5c269248ad380788 |
Cardinal.toENat_le_natCast_iff | Mathlib/SetTheory/Cardinal/Finite.lean | theorem _root_.Cardinal.toENat_le_natCast_iff {c : Cardinal} {n : ℕ} :
toENat c ≤ n ↔ c ≤ n | c : Cardinal.{u_3}
n : ℕ
⊢ toENat c ≤ ↑n ↔ c ≤ ↑n | simp | no goals | fcb7f5913131e7be |
Subgroup.cyclic_of_min | Mathlib/GroupTheory/Archimedean.lean | theorem Subgroup.cyclic_of_min {H : Subgroup G} {a : G}
(ha : IsLeast { g : G | g ∈ H ∧ 1 < g } a) : H = closure {a} | G : Type u_1
inst✝¹ : LinearOrderedCommGroup G
inst✝ : MulArchimedean G
H : Subgroup G
a : G
a_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right✝ : ∀ (y : ℤ), (fun k => a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
h✝ : ¬g / a ^ k = 1
h ... | simpa [zpow_add_one, div_lt_iff_lt_mul'] using lt | no goals | 3c057d2c8d17bd12 |
Stream'.WSeq.head_think | Mathlib/Data/Seq/WSeq.lean | theorem head_think (s : WSeq α) : head (think s) = (head s).think | α : Type u
s : WSeq α
⊢ s.think.head = s.head.think | simp [head] | no goals | 18d3ce8310e836e9 |
KaehlerDifferential.range_kerCotangentToTensor | Mathlib/RingTheory/Kaehler/Basic.lean | theorem KaehlerDifferential.range_kerCotangentToTensor
(h : Function.Surjective (algebraMap A B)) :
LinearMap.range (kerCotangentToTensor R A B) =
(LinearMap.ker (KaehlerDifferential.mapBaseChange R A B)).restrictScalars A | case h
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : B ⊗[A] Ω[A⁄R]
⊢ x ∈ LinearMap.range (kerCotangentToTensor R A B) ↔
x ∈ Subm... | constructor | case h.mp
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : B ⊗[A] Ω[A⁄R]
⊢ x ∈ LinearMap.range (kerCotangentToTensor R A B) →
x ∈ S... | 1495c43685b1e38d |
LTSeries.apply_add_index_le_apply_add_index_int | Mathlib/Order/RelSeries.lean | /--
In ℤ, two entries in an `LTSeries` differ by at least the difference of their indices.
(Expressed in a way that avoids subtraction).
-/
lemma apply_add_index_le_apply_add_index_int (p : LTSeries ℤ) (i j : Fin (p.length + 1))
(hij : i ≤ j) : p i + j ≤ p j + i | p : LTSeries ℤ
i✝ j : Fin (p.length + 1)
i : ℕ
hi : i < p.length + 1
hij : ⟨i, hi⟩ ≤ j
⊢ p.toFun ⟨i, hi⟩ + ↑↑j ≤ p.toFun j + ↑↑⟨i, hi⟩ | have ⟨j, hj⟩ := j | p : LTSeries ℤ
i✝ j✝ : Fin (p.length + 1)
i : ℕ
hi : i < p.length + 1
j : ℕ
hj : j < p.length + 1
hij : ⟨i, hi⟩ ≤ ⟨j, hj⟩
⊢ p.toFun ⟨i, hi⟩ + ↑↑⟨j, hj⟩ ≤ p.toFun ⟨j, hj⟩ + ↑↑⟨i, hi⟩ | 12eb8f68589a7c04 |
ProbabilityTheory.setLIntegral_condKernel_univ_right | Mathlib/Probability/Kernel/Disintegration/Integral.lean | lemma setLIntegral_condKernel_univ_right (hf : Measurable f) (a : α) {s : Set β}
(hs : MeasurableSet s) :
∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in s ×ˢ Set.univ, f x ∂(κ a) | α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
f : β × Ω → ℝ≥0∞
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet... | rw [← setLIntegral_condKernel hf a hs MeasurableSet.univ] | α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
f : β × Ω → ℝ≥0∞
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet... | 87a1f66eea7a7796 |
Real.log_eq_zero | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | theorem log_eq_zero {x : ℝ} : log x = 0 ↔ x = 0 ∨ x = 1 ∨ x = -1 | case mp
x : ℝ
h : log x = 0
⊢ x = 0 ∨ x = 1 ∨ x = -1 | rcases lt_trichotomy x 0 with (x_lt_zero | rfl | x_gt_zero) | case mp.inl
x : ℝ
h : log x = 0
x_lt_zero : x < 0
⊢ x = 0 ∨ x = 1 ∨ x = -1
case mp.inr.inl
h : log 0 = 0
⊢ 0 = 0 ∨ 0 = 1 ∨ 0 = -1
case mp.inr.inr
x : ℝ
h : log x = 0
x_gt_zero : 0 < x
⊢ x = 0 ∨ x = 1 ∨ x = -1 | 7bca46bca69a9763 |
SetTheory.PGame.birthday_zero | Mathlib/SetTheory/Game/Birthday.lean | theorem birthday_zero : birthday 0 = 0 | ⊢ birthday 0 = 0 | simp [inferInstanceAs (IsEmpty PEmpty)] | no goals | d61f66677aaf6621 |
HomologicalComplex.tensor_unit_d₂ | Mathlib/Algebra/Homology/Monoidal.lean | @[simp]
lemma tensor_unit_d₂ (i₁ i₂ j : I) :
mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0 | case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex... | by_cases h₂ : ComplexShape.π c c c (i₁, c.next i₂) = j | case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex... | d444fe28d5c10a8e |
ENNReal.lintegral_Lp_add_le_of_le_one | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_Lp_add_le_of_le_one {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hp0 : 0 ≤ p)
(hp1 : p ≤ 1) :
(∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
(2 : ℝ≥0∞) ^ (1 / p - 1) * ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) | case h₁.hfg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hp0 : 0 ≤ p
hp1 : p ≤ 1
hp : 0 < p
a : α
⊢ (f + g) a ^ p ≤ f a ^ p + g a ^ p | exact rpow_add_le_add_rpow _ _ hp0 hp1 | no goals | 38aa19c6d5135811 |
IsometryEquiv.toDilationEquiv_ratio | Mathlib/Topology/MetricSpace/DilationEquiv.lean | @[simp]
lemma _root_.IsometryEquiv.toDilationEquiv_ratio (e : X ≃ᵢ Y) : ratio e.toDilationEquiv = 1 | X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
e : X ≃ᵢ Y
⊢ ratio e.toDilationEquiv = 1 | rw [← ratio_toDilation, IsometryEquiv.toDilationEquiv_toDilation, Isometry.toDilation_ratio] | no goals | 18d5ff267a2f919a |
SimpleGraph.edgeSet_replaceVertex_of_adj | Mathlib/Combinatorics/SimpleGraph/Operations.lean | theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} | V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
ha : G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : ¬x✝ = t
h✝ : y✝ = t
⊢ G.Adj x✝ s ↔
(G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x => s(x, t)) '' G.neighborSet s) ∧
s(x✝, y✝) ∉ {s(t, t)} | rw [adj_comm] | V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
ha : G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : ¬x✝ = t
h✝ : y✝ = t
⊢ G.Adj s x✝ ↔
(G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x => s(x, t)) '' G.neighborSet s) ∧
s(x✝, y✝) ∉ {s(t, t)} | 5f238941ab198a03 |
IsUnit.mem_submonoid_iff | Mathlib/Algebra/Group/Submonoid/Basic.lean | theorem IsUnit.mem_submonoid_iff {M : Type*} [Monoid M] (a : M) :
a ∈ IsUnit.submonoid M ↔ IsUnit a | M : Type u_4
inst✝ : Monoid M
a : M
⊢ a ∈ submonoid M ↔ IsUnit a | change a ∈ setOf IsUnit ↔ IsUnit a | M : Type u_4
inst✝ : Monoid M
a : M
⊢ a ∈ setOf IsUnit ↔ IsUnit a | 43df31f7a27e9f16 |
MvPolynomial.schwartz_zippel_sup_sum | Mathlib/Algebra/MvPolynomial/SchwartzZippel.lean | /-- The **Schwartz-Zippel lemma**
For a nonzero multivariable polynomial `p` over an integral domain, the probability that `p`
evaluates to zero at points drawn at random from a product of finite subsets `S i` of the integral
domain is bounded by the supremum of `∑ i, degᵢ s / #(S i)` ranging over monomials `s` of `p`... | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R... | gcongr | case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (F... | 0aa93d268fe4b9d8 |
MeasureTheory.ae_bdd_condExp_of_ae_bdd | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem ae_bdd_condExp_of_ae_bdd {R : ℝ≥0} {f : α → ℝ} (hbdd : ∀ᵐ x ∂μ, |f x| ≤ R) :
∀ᵐ x ∂μ, |(μ[f|m]) x| ≤ R | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R | by_cases hnm : m ≤ m0 | case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑... | 8f98e6efac06e3d2 |
WeierstrassCurve.toCharThreeNF_spec_of_b₂_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean | theorem toCharThreeNF_spec_of_b₂_eq_zero (hb₂ : W.b₂ = 0) :
(W.variableChange W.toCharThreeNF).IsShortNF | F : Type u_2
inst✝¹ : Field F
inst✝ : CharP F 3
W : WeierstrassCurve F
hb₂ : W.b₂ = 0
⊢ (W.variableChange W.toCharThreeNF).IsShortNF | rw [toCharThreeNF, toShortNFOfCharThree_a₂, hb₂, div_zero, ← VariableChange.id,
VariableChange.id_comp] | F : Type u_2
inst✝¹ : Field F
inst✝ : CharP F 3
W : WeierstrassCurve F
hb₂ : W.b₂ = 0
⊢ (W.variableChange W.toShortNFOfCharThree).IsShortNF | 4e48cf448d6c63fd |
GromovHausdorff.candidates_dist_bound | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | theorem candidates_dist_bound (fA : f ∈ candidates X Y) :
∀ {x y : X ⊕ Y}, f (x, y) ≤ maxVar X Y * dist x y
| inl x, inl y =>
calc
f (inl x, inl y) = dist x y := candidates_dist_inl fA x y
_ = dist (α := X ⊕ Y) (inl x) (inl y) | X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : GromovHausdorff.ProdSpaceFun X Y
fA : f ∈ candidates X Y
x y : X
⊢ 1 * dist (inl x) (inl y) ≤ ↑(GromovHausdorff.maxVar X Y) * dist (inl x) (inl y) | gcongr | case h
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : GromovHausdorff.ProdSpaceFun X Y
fA : f ∈ candidates X Y
x y : X
⊢ 1 ≤ ↑(GromovHausdorff.maxVar X Y) | f9e53a1d40273e71 |
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso | Mathlib/Algebra/Homology/BifunctorShift.lean | lemma mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso :
mapBifunctorShift₁Iso K₁ (K₂⟦y⟧) F x ≪≫
(CategoryTheory.shiftFunctor _ x).mapIso (mapBifunctorShift₂Iso K₁ K₂ F y) =
(x * y).negOnePow • (mapBifunctorShift₂Iso (K₁⟦x⟧) K₂ F y ≪≫
(CategoryTheory.shiftFunctor _ y).mapIso (mapBifunctorShift₁... | case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁)... | rw [Functor.map_comp, Functor.map_comp, assoc, assoc, assoc,
← HomologicalComplex₂.totalShift₁Iso_hom_naturality_assoc,
HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom,
← HomologicalComplex₂.totalShift₂Iso_hom_naturality_assoc,
Linear.comp_units_smul, Linear.comp_units_smul,
smul_left_cancel_iff,
... | case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁)... | 1e66381cd8e0c1b4 |
Complex.arg_mul_cos_add_sin_mul_I | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
⊢ θ < 0 | linarith | no goals | b22411895133f55c |
Nat.Linear.Poly.denote_eq_cancelAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Poly.denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly)
(h : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) | case pos.h
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ : List (Nat × Var)
k₂ : Nat
m₂ : List (Nat × Var)
hltv hgtv : ¬blt v₁ v₁ = true
hltk : k₁.blt k₂ = tr... | rw [Nat.mul_sub_right_distrib, ← Nat.add_assoc, ← Nat.add_sub_assoc haux] | case pos.h
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ : List (Nat × Var)
k₂ : Nat
m₂ : List (Nat × Var)
hltv hgtv : ¬blt v₁ v₁ = true
hltk : k₁.blt k₂ = tr... | 9bfcd466b3bdd638 |
CategoryTheory.Functor.preservesEqualizer_of_preservesKernels | Mathlib/CategoryTheory/Preadditive/LeftExact.lean | /-- A functor between preadditive categories preserves the equalizer of two
morphisms if it preserves all kernels. -/
lemma preservesEqualizer_of_preservesKernels
[∀ {X Y} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F]
{X Y : C} (f g : X ⟶ Y) : PreservesLimit (parallelPair f g) F | case preserves
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : HasBinaryBiproducts C
inst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
X Y : C
f g : X ⟶ Y
this✝ : Prese... | let c' := isLimitKernelForkOfFork (i.ofIsoLimit (Fork.isoForkOfι c)) | case preserves
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : HasBinaryBiproducts C
inst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
X Y : C
f g : X ⟶ Y
this✝ : Prese... | 6bda66db73d4b92f |
UpperHalfPlane.dist_triangle | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c | a b c : ℍ
⊢ ¬(↑b).im = 0 | exact b.im_ne_zero | no goals | ed6dc2cd0e48ba7b |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.readyForRupAdd_ofArray | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem readyForRupAdd_ofArray {n : Nat} (arr : Array (Option (DefaultClause n))) :
ReadyForRupAdd (ofArray arr) | n : Nat
arr : Array (Option (DefaultClause n))
hsize✝¹ : (ofArray arr).assignments.size = n
ModifiedAssignmentsInvariant : Array Assignment → Prop :=
fun assignments =>
∃ hsize, ∀ (i : PosFin n) (b : Bool), hasAssignment b assignments[i.val] = true → unit (i, b) ∈ (ofArray arr).toList
hb : ModifiedAssignmentsInva... | exact ih h | no goals | 0a237b5f7a9a7c15 |
Std.Sat.AIG.RefVec.map.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem map.go_decl_eq {aig : AIG α} (i) (hi)
(s : RefVec aig i) (input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
[LawfulOperator α Ref f] [LawfulMapOperator α f] :
∀ (idx : Nat) (h1) (h2), (go aig i hi s input f).1.decls[idx]'h2 = aig.decls[idx]'h1 | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
i : Nat
hi : i ≤ len
s : aig.RefVec i
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
res : RefVecEntry α len
h✝ : i < len
hgo :
(let res := f aig (inpu... | dsimp only at hgo | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
i : Nat
hi : i ≤ len
s : aig.RefVec i
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
res : RefVecEntry α len
h✝ : i < len
hgo : go (f aig (input.get i h✝... | 733cd3cf6f81bdb1 |
Array.count_pos_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Count.lean | theorem count_pos_iff {a : α} {l : Array α} : 0 < count a l ↔ a ∈ l | α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
l : Array α
⊢ 0 < count a l ↔ a ∈ l | simp only [count, countP_pos_iff, beq_iff_eq, exists_eq_right] | no goals | 709182ce4b9c62d1 |
affineSegment_comm | Mathlib/Analysis/Convex/Between.lean | theorem affineSegment_comm (x y : P) : affineSegment R x y = affineSegment R y x | case mpr.intro.intro.refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : OrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y z : P
t : R
ht : t ∈ Set.Icc 0 1
hxy : (lineMap y x) t = z
⊢ 1 - t ∈ Set.Icc 0 1 | rwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero] | no goals | 38c8af264e747923 |
Submodule.LinearDisjoint.linearIndependent_right_of_flat | Mathlib/LinearAlgebra/LinearDisjoint.lean | theorem linearIndependent_right_of_flat (H : M.LinearDisjoint N) [Module.Flat R M]
{ι : Type*} {n : ι → N} (hn : LinearIndependent R n) : LinearMap.ker (mulRightMap M n) = ⊥ | R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥M
ι : Type u_1
n : ι → ↥N
hn : LinearIndependent R n
⊢ Function.Injective ⇑(M.mulRightMap n) | simp_rw [mulRightMap_eq_mulMap_comp, LinearMap.coe_comp, LinearEquiv.coe_coe,
← Function.comp_assoc, EquivLike.injective_comp] | R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥M
ι : Type u_1
n : ι → ↥N
hn : LinearIndependent R n
⊢ Function.Injective (⇑(M.mulMap N) ∘ ⇑(LinearMap.lTensor (↥M) (Finsupp.linearCombination R n))) | caecda1942908fed |
ProbabilityTheory.convex_integrableExpSet | Mathlib/Probability/Moments/IntegrableExpMul.lean | /-- `integrableExpSet X μ` is a convex subset of `ℝ` (it is an interval). -/
lemma convex_integrableExpSet : Convex ℝ (integrableExpSet X μ) | Ω✝ : Type u_1
m✝ : MeasurableSpace Ω✝
X✝ : Ω✝ → ℝ
μ✝ : Measure Ω✝
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t₁ : ℝ
ht₁ : t₁ ∈ integrableExpSet X μ
t₂ : ℝ
ht₂ : t₂ ∈ integrableExpSet X μ
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_le : t₁ ≤ t₂
⊢ t₁ = a * t₁ + b * t₁ | rw [← add_mul, hab, one_mul] | no goals | 9c405d0dd0f17582 |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case neg.intro
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible... | use ⟨Units.mk0 u hu0, 0, 0, 0⟩ | case h
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := in... | 5599f8482dc6841e |
Multiset.inter_add_distrib | Mathlib/Data/Multiset/UnionInter.lean | lemma inter_add_distrib (s t u : Multiset α) : s ∩ t + u = (s + u) ∩ (t + u) | case intro
α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
h : s ∩ t + u ≠ (s + u) ∩ (t + u)
a : α
ha : a ::ₘ (s ∩ t + u) ≤ (s + u) ∩ (t + u)
⊢ False | rw [← cons_add] at ha | case intro
α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
h : s ∩ t + u ≠ (s + u) ∩ (t + u)
a : α
ha : a ::ₘ s ∩ t + u ≤ (s + u) ∩ (t + u)
⊢ False | e2689f3d231690a7 |
InfiniteGalois.normal_iff_isGalois | Mathlib/FieldTheory/Galois/Infinite.lean | theorem normal_iff_isGalois (L : IntermediateField k K) [IsGalois k K] :
L.fixingSubgroup.Normal ↔ IsGalois k L | k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
h : L.fixingSubgroup.Normal
f : ↥L → IntermediateField k K :=
fun x =>
IntermediateField.lift
(IntermediateField.fixedField
(Subgroup.map (restrictNormalHom ↥(adjoin k {↑x}... | let _ := IsGalois.of_fixedField_normal_subgroup (Subgroup.map (restrictNormalHom
(adjoin k {l.1})) L.fixingSubgroup) | k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
h : L.fixingSubgroup.Normal
f : ↥L → IntermediateField k K :=
fun x =>
IntermediateField.lift
(IntermediateField.fixedField
(Subgroup.map (restrictNormalHom ↥(adjoin k {↑x}... | 96fb7648f7b24b91 |
WeierstrassCurve.φ_zero | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | @[simp]
lemma φ_zero : W.φ 0 = 1 | R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ W.φ 0 = 1 | rw [φ, ψ_zero, zero_pow two_ne_zero, mul_zero, zero_sub, zero_add, ψ_one, one_mul, zero_sub,
ψ_neg, neg_neg, ψ_one] | no goals | 546b025ad86b0fba |
VectorFourier.norm_fourierPowSMulRight_le | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | lemma norm_fourierPowSMulRight_le (f : V → E) (v : V) (n : ℕ) :
‖fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖ | case h.h.h1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
v : V
n : ℕ
m : Fin n → W
i : Fin n
_hi : i ∈ Finset.univ
⊢ |(L v) (m i)|... | exact L.le_opNorm₂ v (m i) | no goals | 6c68e758ca2b7452 |
Int.natAbs_eq_natAbs_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem natAbs_eq_natAbs_iff {a b : Int} : a.natAbs = b.natAbs ↔ a = b ∨ a = -b | case mpr.inl
a b : Int
h✝ : a = b
⊢ a.natAbs = b.natAbs | subst a | case mpr.inl
b : Int
⊢ b.natAbs = b.natAbs | e42c147e74a9944d |
CategoryTheory.ShortComplex.exact_iff_image_eq_kernel | Mathlib/CategoryTheory/Abelian/Exact.lean | theorem exact_iff_image_eq_kernel : S.Exact ↔ imageSubobject S.f = kernelSubobject S.g | case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : imageSubobject S.f = kernelSubobject S.g
⊢ ((kernelSubobject S.g).ofLE (imageSubobject S.f) ⋯ ≫ imageToKernel S.f S.g ⋯) ≫ (kernelSubobject S.g).arrow =
𝟙 (Subobject.underlying.obj (kernelSubobject S.g)) ≫ (kernelSubobject S.g... | simp | no goals | f041bfaa62f3cb75 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case hfalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = l... | rw [blastDivSubtractShift_denote_mem_prefix] | case hfalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = l... | 123e2e165ae62131 |
MeasureTheory.lintegral_nnnorm_condExpL2_le | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | theorem lintegral_nnnorm_condExpL2_le (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) (f : Lp ℝ 2 μ) :
∫⁻ x in s, ‖(condExpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ | case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
f : ↥(Lp ℝ 2 μ)
h_meas : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) f)) μ := lpMeas.aeStronglyMeasurable ((condExpL2 ℝ ℝ hm) f)
g : α → ℝ := Exists.choose h_meas
hg_meas : StronglyMeasurable g
hg... | exact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs | no goals | ee579ebf8eed737f |
HurwitzZeta.hurwitzZeta_neg_nat | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | theorem hurwitzZeta_neg_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZeta x (-k) =
-1 / (k + 1) * ((Polynomial.bernoulli (k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) | x : ℝ
hx : x ∈ Icc 0 1
n : ℕ
hk : 2 * n ≠ 0
⊢ n ≠ 0 | omega | no goals | fe75c9bfe836a847 |
Bool.eq_true_of_not_eq_false' | Mathlib/Data/Bool/Basic.lean | theorem eq_true_of_not_eq_false' {a : Bool} : !a = false → a = true | a : Bool
⊢ (!decide (a = false)) = true → a = true | cases a <;> decide | no goals | b6616827947f02bc |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
{F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : Tendsto (F · x₀) l (𝓝 (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀) :
ContinuousWithinAt f S x₀ | ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
⊢ ContinuousWithinAt f S x₀ | intro U hU | ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
⊢ ... | b1ad817ea502fd0e |
ExistsContDiffBumpBase.y_pos_of_mem_ball | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x | case right
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D -... | apply closedBall_subset_closedBall' _ (ball_subset_closedBall hy) | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 ... | 646f97b7cc0fbf51 |
Denumerable.encode_ofNat | Mathlib/Logic/Denumerable.lean | theorem encode_ofNat (n) : encode (ofNat α n) = n | case intro.intro
α : Type u_1
inst✝ : Denumerable α
n : ℕ
a : α
h : a ∈ decode n
e : encode a = n
⊢ encode (ofNat α n) = n | rwa [ofNat_of_decode h] | no goals | 2b239cd57d797aca |
LinearMap.isProj_iff_isIdempotentElem | Mathlib/LinearAlgebra/Projection.lean | theorem isProj_iff_isIdempotentElem (f : M →ₗ[S] M) :
(∃ p : Submodule S M, IsProj p f) ↔ IsIdempotentElem f | case h.map_id
S : Type u_5
inst✝² : Semiring S
M : Type u_6
inst✝¹ : AddCommMonoid M
inst✝ : Module S M
f : M →ₗ[S] M
h : IsIdempotentElem f
x : M
hx : x ∈ range f
⊢ f x = x | obtain ⟨y, hy⟩ := mem_range.1 hx | case h.map_id.intro
S : Type u_5
inst✝² : Semiring S
M : Type u_6
inst✝¹ : AddCommMonoid M
inst✝ : Module S M
f : M →ₗ[S] M
h : IsIdempotentElem f
x : M
hx : x ∈ range f
y : M
hy : f y = x
⊢ f x = x | 449bde4cb58bd7c8 |
LindemannWeierstrass.exp_polynomial_approx | Mathlib/NumberTheory/Transcendental/Lindemann/Init/AnalyticalPart.lean | theorem exp_polynomial_approx (f : ℤ[X]) (hf : f.eval 0 ≠ 0) :
∃ c,
∀ p > (eval 0 f).natAbs, p.Prime →
∃ n : ℤ, ¬ ↑p ∣ n ∧ ∃ gp : ℤ[X], gp.natDegree ≤ p * f.natDegree - 1 ∧
∀ {r : ℂ}, r ∈ f.aroots ℂ →
‖n • exp r - p • aeval r gp‖ ≤ c ^ p / (p - 1)! | case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1)... | rw [mem_roots'] at hr | case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1)... | 69cf0983a5f08c5e |
hammingDist_zero_left | Mathlib/InformationTheory/Hamming.lean | theorem hammingDist_zero_left : hammingDist (0 : ∀ i, β i) = hammingNorm :=
funext fun x => by rw [hammingDist_comm, hammingDist_zero_right]
| ι : Type u_2
β : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → DecidableEq (β i)
inst✝ : (i : ι) → Zero (β i)
x : (i : ι) → β i
⊢ hammingDist 0 x = hammingNorm x | rw [hammingDist_comm, hammingDist_zero_right] | no goals | 4435c7e78d8e7f02 |
convexHull_eq | Mathlib/Analysis/Convex/Combination.lean | theorem convexHull_eq (s : Set E) : convexHull R s =
{ x : E | ∃ (ι : Type) (t : Finset ι) (w : ι → R) (z : ι → E), (∀ i ∈ t, 0 ≤ w i) ∧
∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x } | case refine_3
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Set E
⊢ {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x} ⊆ (convexHull R) s | rintro _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩ | case refine_3.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Set E
ι : Type
t : Finset ι
w : ι → R
z : ι → E
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
hz : ∀ i ∈ t, z i ∈ s
⊢ t.centerMass w z ∈ (convexHull R) s | 84416efa323f8fb0 |
intervalIntegral.continuous_parametric_primitive_of_continuous | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | theorem continuous_parametric_primitive_of_continuous
{a₀ : ℝ} (hf : Continuous f.uncurry) :
Continuous fun p : X × ℝ ↦ ∫ t in a₀..p.2, f p.1 t ∂μ | E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NoAtoms μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous (Function.uncurry f)
⊢ Continuous fun p => ∫ (t : ℝ) in a₀..p.2, f p.1 t ∂μ | rw [continuous_iff_continuousAt] | E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NoAtoms μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous (Function.uncurry f)
⊢ ∀ (x : X × ℝ), ContinuousAt (fun p => ∫ (t : ℝ) in a₀..p.2, f p.1 t ∂μ) x | cf605f961886ddc3 |
Std.DHashMap.Raw.getKey!_insertIfNew | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getKey!_insertIfNew [EquivBEq α] [LawfulHashable α] [Inhabited α] (h : m.WF) {k a : α}
{v : β k} :
getKey! (m.insertIfNew k v) a = if k == a ∧ ¬k ∈ m then k else getKey! m a | α : Type u
β : α → Type v
m : Raw α β
inst✝⁴ : BEq α
inst✝³ : Hashable α
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
h : m.WF
k a : α
v : β k
⊢ (m.insertIfNew k v).getKey! a = if (k == a) = true ∧ ¬k ∈ m then k else m.getKey! a | simp only [mem_iff_contains, Bool.not_eq_true] | α : Type u
β : α → Type v
m : Raw α β
inst✝⁴ : BEq α
inst✝³ : Hashable α
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
h : m.WF
k a : α
v : β k
⊢ (m.insertIfNew k v).getKey! a = if (k == a) = true ∧ m.contains k = false then k else m.getKey! a | 73565f263c0d691d |
MeasureTheory.Measure.inf_apply | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | lemma inf_apply {s : Set α} (hs : MeasurableSet s) :
(μ ⊓ ν) s = sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} | α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
⊢ (μ ⊓ ν) s = sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} | rw [← sInf_pair, Measure.sInf_apply hs, OuterMeasure.sInf_apply
(image_nonempty.2 <| insert_nonempty μ {ν})] | α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
⊢ ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ μ_1 ∈ toOuterMeasure '' {μ, ν}, μ_1 (t n) =
sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)} | 07170df6d33ce878 |
Basis.sumCoords_self_apply | Mathlib/LinearAlgebra/Basis/Defs.lean | theorem sumCoords_self_apply : b.sumCoords (b i) = 1 | ι : Type u_1
R : Type u_3
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
i : ι
⊢ b.sumCoords (b i) = 1 | simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self,
Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index] | no goals | 036ad117ae2237fb |
Finset.centroid_pair | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem centroid_pair [DecidableEq ι] [Invertible (2 : k)] (p : ι → P) (i₁ i₂ : ι) :
({i₁, i₂} : Finset ι).centroid k p = (2⁻¹ : k) • (p i₂ -ᵥ p i₁) +ᵥ p i₁ | case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Invertible 2
p : ι → P
i₁ i₂ : ι
h : ¬i₁ = i₂
⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁ | have hc : (#{i₁, i₂} : k) ≠ 0 := by
rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]
norm_num
exact Invertible.ne_zero _ | case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Invertible 2
p : ι → P
i₁ i₂ : ι
h : ¬i₁ = i₂
hc : ↑(#{i₁, i₂}) ≠ 0
⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁ | ad891ef500f95909 |
MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | theorem tendsto_zero_testAgainstNN_of_tendsto_zero_mass {γ : Type*} {F : Filter γ}
{μs : γ → FiniteMeasure Ω} (mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 0) | Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_3
F : Filter γ
μs : γ → FiniteMeasure Ω
mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)
f : Ω →ᵇ ℝ≥0
⊢ Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 0) | apply tendsto_iff_dist_tendsto_zero.mpr | Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_3
F : Filter γ
μs : γ → FiniteMeasure Ω
mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)
f : Ω →ᵇ ℝ≥0
⊢ Tendsto (fun b => dist ((μs b).testAgainstNN f) 0) F (𝓝 0) | 1379066a3f281830 |
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