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 |
|---|---|---|---|---|---|---|
NonUnitalAlgebra.map_bot | Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | theorem map_bot [IsScalarTower R B B] [SMulCommClass R B B]
(f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=
SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]
| R : Type u
A : Type v
B : Type w
inst✝⁸ : CommSemiring R
inst✝⁷ : NonUnitalNonAssocSemiring A
inst✝⁶ : Module R A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R B B
inst✝ : SMulCommClass R B B
f : A →ₙₐ[R] B
⊢ ↑(NonUnitalSubalg... | simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map] | no goals | d1056e7e4a75e18f |
Matroid.IsBasis.eq_of_closure_subset | Mathlib/Data/Matroid/Closure.lean | lemma IsBasis.eq_of_closure_subset (hI : M.IsBasis I X) (hJI : J ⊆ I) (hJ : X ⊆ M.closure J) :
J = I | α : Type u_2
M : Matroid α
X I J : Set α
hI : M.IsBasis I X
hJI : J ⊆ I
hJ : X ⊆ M.closure J
⊢ I ⊆ M.closure J | exact hI.subset.trans hJ | no goals | 05de755c7ba1211d |
Ideal.range_finsuppTotal | Mathlib/RingTheory/Ideal/Operations.lean | theorem range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) | case h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
x✝ : M
⊢ x✝ ∈ LinearMap.range (finsuppTotal ι M I v) ↔ x✝ ∈ I • Submodule.span R (Set.range v) | rw [Submodule.mem_ideal_smul_span_iff_exists_sum] | case h
ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
x✝ : M
⊢ x✝ ∈ LinearMap.range (finsuppTotal ι M I v) ↔ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (a.sum fun i c => c • v i) = x✝ | 458493e0f4c9f62a |
Submodule.mem_singleton_set_smul | Mathlib/Algebra/Module/Submodule/Pointwise.lean | lemma mem_singleton_set_smul [SMulCommClass R S M] (r : S) (x : M) :
x ∈ ({r} : Set S) • N ↔ ∃ (m : M), m ∈ N ∧ x = r • m | case mp.smul₀
R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
S : Type u_4
inst✝² : Monoid S
inst✝¹ : DistribMulAction S M
N : Submodule R M
inst✝ : SMulCommClass R S M
r : S
x : M
t : S
n : M
memₜ : t ∈ {r}
memₙ : n ∈ N
⊢ ∃ m ∈ N, t • n = r • m | aesop | no goals | 4bdd4b4b4b2d7d5a |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | case inr.inr.intro.intro.intro.intro.intro.intro.intro.right.right.right
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
... | constructor | case inr.inr.intro.intro.intro.intro.intro.intro.intro.right.right.right.left
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : F... | 4e9908103f3f6d5f |
Lean.Omega.IntList.neg_get | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean | theorem neg_get (xs : IntList) (i : Nat) : (- xs).get i = - xs.get i | xs : IntList
i : Nat
⊢ (Option.map (fun x => -x) xs[i]?).getD 0 = -xs[i]?.getD 0 | cases xs[i]? <;> simp | no goals | 20faeec2ffa34695 |
HurwitzKernelBounds.F_int_eq_of_mem_Icc | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | lemma F_int_eq_of_mem_Icc (k : ℕ) {a : ℝ} (ha : a ∈ Icc 0 1) {t : ℝ} (ht : 0 < t) :
F_int k a t = (F_nat k a t) + (F_nat k (1 - a) t) | case h.e'_2.h.e'_5.h
k : ℕ
a : ℝ
ha : a ∈ Icc 0 1
t : ℝ
ht : 0 < t
n : ℤ
⊢ f_int k a t n = Int.rec (f_nat k a t) (f_nat k (1 - a) t) n | cases n | case h.e'_2.h.e'_5.h.ofNat
k : ℕ
a : ℝ
ha : a ∈ Icc 0 1
t : ℝ
ht : 0 < t
a✝ : ℕ
⊢ f_int k a t (Int.ofNat a✝) = Int.rec (f_nat k a t) (f_nat k (1 - a) t) (Int.ofNat a✝)
case h.e'_2.h.e'_5.h.negSucc
k : ℕ
a : ℝ
ha : a ∈ Icc 0 1
t : ℝ
ht : 0 < t
a✝ : ℕ
⊢ f_int k a t (Int.negSucc a✝) = Int.rec (f_nat k a t) (f_nat k (1 - ... | 4a4bc0fd1e245f65 |
DFinsupp.filter_ne_eq_erase' | Mathlib/Data/DFinsupp/Defs.lean | theorem filter_ne_eq_erase' (f : Π₀ i, β i) (i : ι) : f.filter (i ≠ ·) = f.erase i | ι : Type u
β : ι → Type v
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : DecidableEq ι
f : Π₀ (i : ι), β i
i : ι
⊢ filter (fun x => i ≠ x) f = filter (fun x => x ≠ i) f | congr with j | case e_p.h.a
ι : Type u
β : ι → Type v
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : DecidableEq ι
f : Π₀ (i : ι), β i
i j : ι
⊢ i ≠ j ↔ j ≠ i | dc7d97d663e59d0e |
GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem integral_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
∫ x : ℝ, cexp (-b * (x + c * I) ^ 2) = (π / b) ^ (1 / 2 : ℂ) | b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
⊢ Tendsto (fun i... | let I₅ := fun T : ℝ => ∫ y : ℝ in (0 : ℝ)..c, cexp (-b * (-T + y * I) ^ 2) | b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ℂ := fu... | 8f46f2d622ac75ef |
Matrix.mul_eq_one_comm | Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean | theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A✝ B✝ A B : Matrix n n R
hAB : A * B = 1
h0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) = detp 1 A * detp 1 B + detp (-1) A * detp (-1) B
h :
B * A + (detp 1 A • (B * adjp (-1) B) + detp (-1) A • (B * adjp 1 B)) =
... | replace h := congr($h + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) • 1) | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A✝ B✝ A B : Matrix n n R
hAB : A * B = 1
h0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) = detp 1 A * detp 1 B + detp (-1) A * detp (-1) B
h :
B * A + (detp 1 A • (B * adjp (-1) B) + detp (-1) A • (B * adjp 1 B)) +
... | 7b3f481849b56cef |
MeasureTheory.Lp.dist_def | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem dist_def (f g : Lp E p μ) : dist f g = (eLpNorm (⇑f - ⇑g) p μ).toReal | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : ↥(Lp E p μ)
⊢ eLpNorm (↑↑(f - g)) p μ = eLpNorm (↑↑f - ↑↑g) p μ | apply eLpNorm_congr_ae (coeFn_sub _ _) | no goals | 95e354d102df0f62 |
Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective | Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
(hf : Function.Injective f) (p) (H : p ∈ minimalPrimes R) :
∃ p' : Ideal S, p'.IsPrime ∧ p'.comap f = p | R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
hf : Function.Injective ⇑f
p : Ideal R
H : p ∈ minimalPrimes R
⊢ ∃ p', p'.IsPrime ∧ comap f p' = p | have := H.1.1 | R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
hf : Function.Injective ⇑f
p : Ideal R
H : p ∈ minimalPrimes R
this : p.IsPrime
⊢ ∃ p', p'.IsPrime ∧ comap f p' = p | b40e77945c06e931 |
List.next_getElem | Mathlib/Data/List/Cycle.lean | theorem next_getElem (l : List α) (h : Nodup l) (i : Nat) (hi : i < l.length) :
next l l[i] (get_mem _ _) =
(l[(i + 1) % l.length]'(Nat.mod_lt _ (i.zero_le.trans_lt hi))) :=
match l, h, i, hi with
| [], _, i, hi => by simp at hi
| [_], _, _, _ => by simp
| x::y::l, _h, 0, h0 => by
have h₁ : (x :: ... | case inl.hx
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
h : l✝.Nodup
i : ℕ
hi✝ : i < l✝.length
x y : α
l : List α
hn : (x :: y :: l).Nodup
hi : l.length + 1 < (x :: y :: l).length
hx' : (x :: y :: l)[l.length + 1] ≠ x
hi' : l.length ≤ l.length
⊢ (x :: y :: l)[l.length + 1] = (x :: y :: l).getLast ⋯ | simp [getLast_eq_getElem] | no goals | 51201737832680f8 |
Finsupp.indicator_injective | Mathlib/Data/Finsupp/Indicator.lean | theorem indicator_injective : Injective fun f : ∀ i ∈ s, α => indicator s f | ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
a b : (i : ι) → i ∈ s → α
h : (fun f => indicator s f) a = (fun f => indicator s f) b
⊢ a = b | ext i hi | case h.h
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
a b : (i : ι) → i ∈ s → α
h : (fun f => indicator s f) a = (fun f => indicator s f) b
i : ι
hi : i ∈ s
⊢ a i hi = b i hi | 5cf50d46838d4703 |
LSeriesSummable.neg | Mathlib/NumberTheory/LSeries/Linearity.lean | lemma LSeriesSummable.neg {f : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) :
LSeriesSummable (-f) s | f : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
⊢ LSeriesSummable (-f) s | simpa [LSeriesSummable, term_neg] using Summable.neg hf | no goals | 1340e41f3ad4c867 |
det_reflection | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem det_reflection : LinearMap.det (reflection K).toLinearMap = (-1) ^ finrank 𝕜 Kᗮ | case pos
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 ↥K
hK : FiniteDimensional 𝕜 ↥Kᗮ
⊢ LinearMap.det ↑(reflection K).toLinearEquiv = (-1) ^ finrank 𝕜 ↥Kᗮ | let e := K.prodEquivOfIsCompl _ K.isCompl_orthogonal_of_completeSpace | case pos
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 ↥K
hK : FiniteDimensional 𝕜 ↥Kᗮ
e : (↥K × ↥Kᗮ) ≃ₗ[𝕜] E := K.prodEquivOfIsCompl Kᗮ ⋯
⊢ LinearMap.det ↑(reflection K).toLinearEquiv = (-1) ^ finrank 𝕜 ↥Kᗮ | ecc23853ea0269cb |
ContinuousMap.toNNReal_neg_algebraMap | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.lean | @[simp]
lemma toNNReal_neg_algebraMap (r : ℝ≥0) : (- algebraMap ℝ C(X, ℝ) r).toNNReal = 0 | case h.a
X : Type u_1
inst✝ : TopologicalSpace X
r : ℝ≥0
a✝ : X
⊢ ↑((-(algebraMap ℝ C(X, ℝ)) ↑r).toNNReal a✝) = ↑(0 a✝) | simp | no goals | c96f2070abad50c7 |
ProbabilityTheory.sum_meas_smul_cond_fiber | Mathlib/Probability/ConditionalProbability.lean | /-- The **law of total probability** for a random variable taking finitely many values: a measure
`μ` can be expressed as a linear combination of its conditional measures `μ[|X ← x]` on fibers of a
random variable `X` valued in a fintype. -/
lemma sum_meas_smul_cond_fiber {X : Ω → α} (hX : Measurable X) (μ : Measure Ω)... | case h
Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
⊢ (∑ x : α, μ (X ⁻¹' {x}) • μ[|X ⁻¹' {x}]) E = μ E | calc
_ = ∑ x, μ (X ⁻¹' {x} ∩ E) := by
simp only [Measure.coe_finset_sum, Measure.coe_smul, Finset.sum_apply,
Pi.smul_apply, smul_eq_mul]
simp_rw [mul_comm (μ _), cond_mul_eq_inter (hX (.singleton _))]
_ = _ := by
have : ⋃ x ∈ Finset.univ, X ⁻¹' {x} ∩ E = E := by ext; simp
rw [← measure_biUnion... | no goals | 184ad315f536fdf8 |
Nat.Primrec'.head | Mathlib/Computability/Primrec.lean | theorem head {n : ℕ} : @Primrec' n.succ head :=
(get 0).of_eq fun v => by simp [get_zero]
| n : ℕ
v : List.Vector ℕ n.succ
⊢ v.get 0 = v.head | simp [get_zero] | no goals | 14498809a9b87bba |
CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap' | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | lemma leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'
(h : S.HomologyData) (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) :
leftRightHomologyComparison' h₁ h₂ =
leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂ | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ leftRightHomologyComparison' h₁ h₂ = leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂ | simpa only [h.leftRightHomologyComparison'_eq] using
leftRightHomologyComparison'_compatibility h₁ h.left h₂ h.right | no goals | bbc20750fa6c89cb |
TrivSqZeroExt.isUnit_inv_iff | Mathlib/Algebra/TrivSqZeroExt.lean | theorem isUnit_inv_iff {x : tsze R M} : IsUnit x⁻¹ ↔ IsUnit x | R : Type u
M : Type v
inst✝⁴ : DivisionSemiring R
inst✝³ : AddCommGroup M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : Module R M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
x : tsze R M
⊢ IsUnit x⁻¹ ↔ IsUnit x | simp_rw [isUnit_iff_isUnit_fst, fst_inv, isUnit_iff_ne_zero, ne_eq, inv_eq_zero] | no goals | 7eb8553a16c07fcb |
Sbtw.angle₁₂₃_eq_pi | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | theorem _root_.Sbtw.angle₁₂₃_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∠ p₁ p₂ p₃ = π | case intro.intro.intro.intro.intro
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
p₁ p₂ p₃ : P
r : ℝ
hp₂ : (AffineMap.lineMap p₁ p₃) r = p₂
hr0 : 0 ≤ r
hr1 : r ≤ 1
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
⊢ -(1 - r) / r < 0 ∧ p₃ -ᵥ p₂ = (... | have hr0' : r ≠ 0 := by
rintro rfl
rw [← hp₂] at hp₂p₁
simp at hp₂p₁ | case intro.intro.intro.intro.intro
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
p₁ p₂ p₃ : P
r : ℝ
hp₂ : (AffineMap.lineMap p₁ p₃) r = p₂
hr0 : 0 ≤ r
hr1 : r ≤ 1
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
hr0' : r ≠ 0
⊢ -(1 - r) / r < 0 ∧... | 407371ce3952246e |
hasConstantSpeedOnWith_of_subsingleton | Mathlib/Analysis/ConstantSpeed.lean | theorem hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
(l : ℝ≥0) : HasConstantSpeedOnWith f s l | E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
hs : s.Subsingleton
l : ℝ≥0
x : ℝ
hx : x ∈ s
y : ℝ
hy : y ∈ s
⊢ eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x)) | cases hs hx hy | case refl
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
hs : s.Subsingleton
l : ℝ≥0
x : ℝ
hx hy : x ∈ s
⊢ eVariationOn f (s ∩ Icc x x) = ENNReal.ofReal (↑l * (x - x)) | 71fa565d21c0d47f |
CategoryTheory.Presieve.Arrows.Compatible.exists_familyOfElements | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | theorem exists_familyOfElements (hx : Compatible P π x) :
∃ (x' : FamilyOfElements P (ofArrows X π)), ∀ (i : I), x' _ (ofArrows.mk i) = x i | C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
x : (i : I) → P.obj (op (X i))
hx : Compatible P π x
⊢ ∃ x', ∀ (i : I), x' (π i) ⋯ = x i | choose i h h' using @ofArrows_surj _ _ _ _ _ π | C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
x : (i : I) → P.obj (op (X i))
hx : Compatible P π x
i : {Z : C} → (g : Z ⟶ B) → ofArrows X π g → I
h : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), X (i g hg) = Z
h' : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g)... | 6b25a2c28e17290e |
FiniteField.card | Mathlib/FieldTheory/Finite/Basic.lean | theorem card (p : ℕ) [CharP K p] : ∃ n : ℕ+, Nat.Prime p ∧ q = p ^ (n : ℕ) | case intro
K : Type u_1
inst✝² : Field K
inst✝¹ : Fintype K
p : ℕ
inst✝ : CharP K p
hp : Fact (Nat.Prime p)
this : Module (ZMod p) K :=
let __src := (ZMod.castHom ⋯ K).toModule;
Module.mk ⋯ ⋯
n : ℕ
h : q = p ^ n
⊢ ∃ n, Nat.Prime p ∧ q = p ^ ↑n | refine ⟨⟨n, ?_⟩, hp.1, h⟩ | case intro
K : Type u_1
inst✝² : Field K
inst✝¹ : Fintype K
p : ℕ
inst✝ : CharP K p
hp : Fact (Nat.Prime p)
this : Module (ZMod p) K :=
let __src := (ZMod.castHom ⋯ K).toModule;
Module.mk ⋯ ⋯
n : ℕ
h : q = p ^ n
⊢ 0 < n | a57ab5c16b101ae3 |
MeasureTheory.L1.norm_eq_integral_norm | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem L1.norm_eq_integral_norm (f : α →₁[μ] H) : ‖f‖ = ∫ a, ‖f a‖ ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_7
inst✝ : NormedAddCommGroup H
f : ↥(Lp H 1 μ)
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ).toReal = ∫ (a : α), ‖↑↑f a‖ ∂μ | rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall (by simp [norm_nonneg]))
(Lp.aestronglyMeasurable f).norm] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_7
inst✝ : NormedAddCommGroup H
f : ↥(Lp H 1 μ)
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ).toReal = (∫⁻ (a : α), ENNReal.ofReal ‖↑↑f a‖ ∂μ).toReal | 141cb4e78cb786f4 |
ContDiffAt.neg | Mathlib/Analysis/Calculus/ContDiff/Operations.lean | theorem ContDiffAt.neg {f : E → F} (hf : ContDiffAt 𝕜 n f x) :
ContDiffAt 𝕜 n (fun x => -f x) x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
n : WithTop ℕ∞
f : E → F
hf : ContDiffWithinAt 𝕜 n f univ x
⊢ ContDiffWithinAt 𝕜 n (fun x => -f x) univ x | exact hf.neg | no goals | b56c0d8babe331b9 |
List.filterTR_loop_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean | theorem filterTR_loop_eq (p : α → Bool) (as bs : List α) :
filterTR.loop p as bs = bs.reverse ++ filter p as | case nil
α : Type u
p : α → Bool
bs : List α
⊢ filterTR.loop p nil bs = bs.reverse ++ filter p nil | simp [filterTR.loop, filter] | no goals | 55649c6ec6544060 |
ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_atBot_zero | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma IsRatCondKernelCDFAux.tendsto_atBot_zero (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
(a : α) :
∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atBot (𝓝 0) | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n => f (a, t) (-↑n)) atTop (𝓝 0)
t : β
ht : Tendsto (fun n => f (a, t) (-↑n)) atTop (... | have h_anti : Antitone (fun q ↦ f (a, t) (-q)) := h_mono.comp_antitone monotone_id.neg | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n => f (a, t) (-↑n)) atTop (𝓝 0)
t : β
ht : Tendsto (fun n => f (a, t) (-↑n)) atTop (... | 54c74c7255804de4 |
Int.le_of_sub_nonpos | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem le_of_sub_nonpos {a b : Int} (h : a - b ≤ 0) : a ≤ b | a b : Int
h : a - b ≤ 0
⊢ a ≤ b | have h := Int.add_le_add_right h b | a b : Int
h✝ : a - b ≤ 0
h : a - b + b ≤ 0 + b
⊢ a ≤ b | 098b431f16e647af |
Profinite.exists_locallyConstant | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant {α : Type*} (hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom | case inr
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : Nonempty (Quotient S.toSetoid)
⊢ ∃ j g, f = LocallyConstant... | let f' : LocallyConstant C.pt S := ⟨S.proj, S.proj_isLocallyConstant⟩ | case inr
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : Nonempty (Quotient S.toSetoid)
f' : LocallyConstant (↑C.pt.... | 72c7aeebc62ad404 |
MeasureTheory.setIntegral_dirac | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem setIntegral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
(s : Set α) [Decidable (a ∈ s)] :
∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 | case pos
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
hE : CompleteSpace E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a | exact integral_dirac _ _ | no goals | fe372c35ee38024f |
CovBy.Ioi_eq | Mathlib/Order/Cover.lean | theorem CovBy.Ioi_eq (h : a ⋖ b) : Ioi a = Ici b | α : Type u_1
inst✝ : LinearOrder α
a b : α
h : a ⋖ b
⊢ Ioi a = Ici b | rw [← Ioo_union_Ici_eq_Ioi h.lt, h.Ioo_eq, empty_union] | no goals | 87b8b1cf4fa0e9e6 |
MeasureTheory.Measure.prod_swap | Mathlib/MeasureTheory/Measure/Prod.lean | theorem prod_swap : map Prod.swap (μ.prod ν) = ν.prod μ | case e_f.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
this :
(sum fun i => map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =
sum fun i => map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteSeq ν i.1))
i :... | refine (prod_eq ?_).symm | case e_f.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
this :
(sum fun i => map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =
sum fun i => map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteSeq ν i.1))
i :... | 604d2c93da35b541 |
IsPreconnected.subset_of_closure_inter_subset | Mathlib/Topology/Connected/Basic.lean | theorem IsPreconnected.subset_of_closure_inter_subset (hs : IsPreconnected s) (hu : IsOpen u)
(h'u : (s ∩ u).Nonempty) (h : closure u ∩ s ⊆ u) : s ⊆ u | α : Type u
inst✝ : TopologicalSpace α
s u : Set α
hs : IsPreconnected s
hu : IsOpen u
h'u : (s ∩ u).Nonempty
h : closure u ∩ s ⊆ u
A : s ⊆ u ∪ (closure u)ᶜ
⊢ s ⊆ u | apply hs.subset_left_of_subset_union hu isClosed_closure.isOpen_compl _ A h'u | α : Type u
inst✝ : TopologicalSpace α
s u : Set α
hs : IsPreconnected s
hu : IsOpen u
h'u : (s ∩ u).Nonempty
h : closure u ∩ s ⊆ u
A : s ⊆ u ∪ (closure u)ᶜ
⊢ Disjoint u (closure u)ᶜ | 643e8ab38b886c11 |
FermatLastTheoremForThreeGen.lambda_sq_dvd_c | Mathlib/NumberTheory/FLT/Three.lean | /-- Given `S' : Solution'`, we have that `λ ^ 2` divides `S'.c`. -/
lemma lambda_sq_dvd_c : λ ^ 2 ∣ S'.c | case h.e'_3.h.e'_5.h.e'_6
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hm : FiniteMultiplicity λ S'.c
this : 2 ≤ multiplicity λ S'.c
x : 𝓞 K
hx : S'.c = λ ^ multiplicity λ S'.c * x
⊢ 2 + (multiplicity λ S'.c - 2) = multipli... | simp [this] | no goals | c4ebb8ae9468f9d9 |
AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | lemma toSpec_preimage_basicOpen {f}
(t : NumDenSameDeg 𝒜 (.powers f)) :
(Opens.map (toSpec 𝒜 f).base).obj (sbo (HomogeneousLocalization.mk t)) =
Opens.comap ⟨_, continuous_subtype_val⟩ (pbo t.num.1) :=
Opens.ext <| Opens.map_coe _ _ ▸ by
convert (ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen f t... | case h.e'_2.h.e'_3
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
t : NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ ⇑(ConcreteCategory.hom (toSpec 𝒜 f).base) = ProjIsoSpecTopComponent.ToSpec.toFun f | exact funext fun _ => toSpec_base_apply_eq _ _ | no goals | 4e482b32b91f0901 |
List.take_isPrefix_take | Mathlib/Data/List/Infix.lean | @[simp] lemma take_isPrefix_take {m n : ℕ} : l.take m <+: l.take n ↔ m ≤ n ∨ l.length ≤ n | α : Type u_1
l : List α
m n : ℕ
⊢ take m l <+: take n l ↔ m ≤ n ∨ l.length ≤ n | simp [prefix_take_iff, take_prefix] | α : Type u_1
l : List α
m n : ℕ
⊢ m ⊓ l.length ≤ n ↔ m ≤ n ∨ l.length ≤ n | d3c718b96ade8919 |
Algebra.FormallyUnramified.isField_quotient_map_maximalIdeal | Mathlib/RingTheory/Unramified/LocalRing.lean | lemma FormallyUnramified.isField_quotient_map_maximalIdeal [FormallyUnramified R S] :
IsField (S ⧸ (maximalIdeal R).map (algebraMap R S)) | R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : FormallyUnramified R S
mR : Ideal S := Ideal.map (algebraMap R S) (maximalIdeal R)
hmR : mR ≤ maximalIdeal S
this✝⁵... | have : Nontrivial (S ⧸ mR) := Ideal.Quotient.nontrivial fun e ↦
(maximalIdeal.isMaximal S).ne_top (top_le_iff.mp <| e.symm.trans_le hmR) | R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : FormallyUnramified R S
mR : Ideal S := Ideal.map (algebraMap R S) (maximalIdeal R)
hmR : mR ≤ maximalIdeal S
this✝⁶... | 347d7077afd8552b |
List.Sorted.orderedInsert | Mathlib/Data/List/Sort.lean | theorem Sorted.orderedInsert (a : α) : ∀ l, Sorted r l → Sorted r (orderedInsert r a l)
| [], _ => sorted_singleton a
| b :: l, h => by
by_cases h' : a ≼ b
· -- Porting note: was
-- `simpa [orderedInsert, h', h] using fun b' bm => trans h' (rel_of_sorted_cons h _ bm)`
rw [List.orderedInsert, if_... | case neg
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : IsTotal α r
inst✝ : IsTrans α r
a b : α
l : List α
h : Sorted r (b :: l)
h' : ¬r a b
⊢ ∀ b' ∈ List.orderedInsert r a l, r b b' | intro b' bm | case neg
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : IsTotal α r
inst✝ : IsTrans α r
a b : α
l : List α
h : Sorted r (b :: l)
h' : ¬r a b
b' : α
bm : b' ∈ List.orderedInsert r a l
⊢ r b b' | da6a875b540ed7f6 |
LSeries.iteratedDeriv_alternating | Mathlib/NumberTheory/LSeries/Positivity.lean | /-- If all values of a `ℂ`-valued arithmetic function are nonnegative reals and `x` is a
real number in the domain of absolute convergence, then the `n`th iterated derivative
of the associated L-series is nonnegative real when `n` is even and nonpositive real
when `n` is odd. -/
lemma iteratedDeriv_alternating {a : ℕ →... | case isFalse
a : ℕ → ℂ
hn : 0 ≤ a
x : ℝ
h : abscissaOfAbsConv a < ↑x
n k : ℕ
h✝ : ¬k = 0
⊢ 0 ≤ logMul^[n] a k | induction n with
| zero => simpa only [Function.iterate_zero, id_eq] using hn k
| succ n IH =>
rw [Function.iterate_succ_apply']
refine mul_nonneg ?_ IH
simp only [← natCast_log, zero_le_real, Real.log_natCast_nonneg] | no goals | 9bfcd53e06a4912a |
Set.ordConnected_inter_Icc_iff | Mathlib/Order/Interval/Set/OrdConnected.lean | lemma ordConnected_inter_Icc_iff (hx : x ∈ s) (hy : y ∈ s) :
OrdConnected (s ∩ Icc x y) ↔ Ioo x y ⊆ s | α : Type u_1
inst✝ : PartialOrder α
s : Set α
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ (s ∩ Icc x y).OrdConnected ↔ Ioo x y ⊆ s | refine ⟨fun h ↦ Ioo_subset_Icc_self.trans fun z hz ↦ ?_, ordConnected_inter_Icc_of_subset⟩ | α : Type u_1
inst✝ : PartialOrder α
s : Set α
x y : α
hx : x ∈ s
hy : y ∈ s
h : (s ∩ Icc x y).OrdConnected
z : α
hz : z ∈ Icc x y
⊢ z ∈ s | f6769b380aecdfe3 |
Ordinal.accPt_subtype | Mathlib/SetTheory/Ordinal/Topology.lean | theorem accPt_subtype {p o : Ordinal} (S : Set Ordinal) (hpo : p < o) :
AccPt p (𝓟 S) ↔ AccPt ⟨p, hpo⟩ (𝓟 (Iio o ↓∩ S)) | case mpr
p o : Ordinal.{u_1}
S : Set Ordinal.{u_1}
hpo : p < o
h : ∀ U ∈ 𝓝 ⟨p, hpo⟩, ∃ y ∈ U ∩ Iio o ↓∩ S, y ≠ ⟨p, hpo⟩
⊢ ∀ U ∈ 𝓝 p, ∃ y ∈ U ∩ S, y ≠ p | intro u hu | case mpr
p o : Ordinal.{u_1}
S : Set Ordinal.{u_1}
hpo : p < o
h : ∀ U ∈ 𝓝 ⟨p, hpo⟩, ∃ y ∈ U ∩ Iio o ↓∩ S, y ≠ ⟨p, hpo⟩
u : Set Ordinal.{u_1}
hu : u ∈ 𝓝 p
⊢ ∃ y ∈ u ∩ S, y ≠ p | 830d905e8e28c141 |
Metric.inductiveLimitDist_eq_dist | Mathlib/Topology/MetricSpace/Gluing.lean | theorem inductiveLimitDist_eq_dist (I : ∀ n, Isometry (f n)) (x y : Σn, X n) :
∀ m (hx : x.1 ≤ m) (hy : y.1 ≤ m), inductiveLimitDist f x y =
dist (leRecOn hx (f _) x.2 : X m) (leRecOn hy (f _) y.2 : X m)
| 0, hx, hy => by
obtain ⟨i, x⟩ := x; obtain ⟨j, y⟩ := y
obtain rfl : i = 0 := nonpos_iff_eq_zer... | case pos
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
x y : (n : ℕ) × X n
m : ℕ
hx : x.fst ≤ x.fst ⊔ y.fst
hy : y.fst ≤ x.fst ⊔ y.fst
⊢ inductiveLimitDist f x y = dist (leRecOn hx (fun {k} => f k) x.snd) (leRecOn hy (fun {k} => f k) y.snd) | rfl | no goals | 35212214dc2b5acb |
Finset.Ico_eq_image_ssubsets | Mathlib/Data/Finset/Interval.lean | theorem Ico_eq_image_ssubsets (h : s ⊆ t) : Ico s t = (t \ s).ssubsets.image (s ∪ ·) | case h
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
⊢ s ≤ u ∧ u < t ↔ ∃ a ⊂ t \ s, s ∪ a = u | constructor | case h.mp
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
⊢ s ≤ u ∧ u < t → ∃ a ⊂ t \ s, s ∪ a = u
case h.mpr
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
⊢ (∃ a ⊂ t \ s, s ∪ a = u) → s ≤ u ∧ u < t | 1b047fbe71cc03e1 |
MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
(haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty | case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ ∀ (i : Compacts G), IsCompact (Icc 0 ↑(index ↑i ↑K₀)) | intro K | case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
⊢ IsCompact (Icc 0 ↑(index ↑K ↑K₀)) | 25c220e55fac0398 |
List.ofFn_get | Mathlib/Data/List/OfFn.lean | theorem ofFn_get : ∀ l : List α, (ofFn (get l)) = l
| [] => by rw [ofFn_zero]
| a :: l => by
rw [ofFn_succ]
congr
exact ofFn_get l
| α : Type u
a : α
l : List α
⊢ ((a :: l).get 0 :: ofFn fun i => (a :: l).get i.succ) = a :: l | congr | case e_tail
α : Type u
a : α
l : List α
⊢ (ofFn fun i => (a :: l).get i.succ) = l | d40ad03eedddb1a3 |
LSeries_eq_mul_integral_aux | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem LSeries_eq_mul_integral_aux {f : ℕ → ℂ} (hf : f 0 = 0) {r : ℝ} (hr : 0 ≤ r) {s : ℂ}
(hs : r < s.re) (hS : LSeriesSummable f s)
(hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ (n : ℝ) ^ r) :
LSeries f s = s * ∫ t in Set.Ioi (1 : ℝ), (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (- (s + 1)) | f : ℕ → ℂ
hf : f 0 = 0
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable f s
h₁ : (-s - 1).re + r < -1
h₂ : s ≠ 0
h₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x => ↑x ^ (-s)) t
h₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k
hO : (fun n => ∑ k ∈ Icc 0 n, f k) =O[atTop] fun n => ↑n ^ r
⊢ LSeries f s = ∫... | refine tendsto_nhds_unique ((tendsto_add_atTop_iff_nat 1).mpr hS.hasSum.tendsto_sum_nat) ?_ | f : ℕ → ℂ
hf : f 0 = 0
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable f s
h₁ : (-s - 1).re + r < -1
h₂ : s ≠ 0
h₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x => ↑x ^ (-s)) t
h₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k
hO : (fun n => ∑ k ∈ Icc 0 n, f k) =O[atTop] fun n => ↑n ^ r
⊢ Tendsto (fun n ... | 31197df722225f38 |
NonUnitalAlgebra.adjoin_induction_subtype | Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | /-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/
@[elab_as_elim, deprecated adjoin_induction (since := "2024-10-11")]
lemma adjoin_induction_subtype {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)
(mem : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)
(add : ∀ x y... | case zero
R : Type u
A : Type v
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
s : Set A
p : ↥(adjoin R s) → Prop
a : ↥(adjoin R s)
mem : ∀ (x : A) (h : x ∈ s), p ⟨x, ⋯⟩
add : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)
zero :... | exact zero | no goals | d2cdd471248160a8 |
Ring.coe_jacobson_quotient | Mathlib/RingTheory/Jacobson/Radical.lean | theorem coe_jacobson_quotient (I : Ideal R) [I.IsTwoSided] :
(jacobson (R ⧸ I) : Set (R ⧸ I)) = Module.jacobson R (R ⧸ I) | R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ 0 ∈ Module.jacobson R (R ⧸ I) | exact zero_mem _ | no goals | 0a5f67c64f9b114a |
Algebra.Presentation.differentialsSolution_isPresentation | Mathlib/Algebra/Module/Presentation/Differentials.lean | lemma differentialsSolution_isPresentation :
pres.differentialsSolution.IsPresentation | case right
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pres : Presentation R S
⊢ LinearMap.ker pres.differentialsSolution.π = Submodule.span S (Set.range pres.differentialsRelations.relation) | rw [← Module.Relations.range_map] | case right
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pres : Presentation R S
⊢ LinearMap.ker pres.differentialsSolution.π = LinearMap.range pres.differentialsRelations.map | d3c9201e4431d782 |
Ideal.LiesOver.of_eq_map_equiv | Mathlib/RingTheory/Ideal/Over.lean | theorem LiesOver.of_eq_map_equiv [P.LiesOver p] {E : Type*} [EquivLike E B C]
[AlgEquivClass E A B C] (σ : E) (h : Q = P.map σ) : Q.LiesOver p | A : Type u_2
inst✝⁷ : CommSemiring A
B : Type u_3
C : Type u_4
inst✝⁶ : Semiring B
inst✝⁵ : Semiring C
inst✝⁴ : Algebra A B
inst✝³ : Algebra A C
P : Ideal B
Q : Ideal C
p : Ideal A
inst✝² : P.LiesOver p
E : Type u_5
inst✝¹ : EquivLike E B C
inst✝ : AlgEquivClass E A B C
σ : E
h : Q = comap (↑σ).symm P
⊢ Q.LiesOver p | exact of_eq_comap p (σ : B ≃ₐ[A] C).symm h | no goals | 903c76e889b7a255 |
HomologicalComplex.mono_of_mono_f | Mathlib/Algebra/Homology/HomologicalComplex.lean | lemma mono_of_mono_f {K L : HomologicalComplex V c} (φ : K ⟶ L)
(hφ : ∀ i, Mono (φ.f i)) : Mono φ where
right_cancellation g h eq | ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K L : HomologicalComplex V c
φ : K ⟶ L
hφ : ∀ (i : ι), Mono (φ.f i)
Z✝ : HomologicalComplex V c
g h : Z✝ ⟶ K
eq : g ≫ φ = h ≫ φ
⊢ g = h | ext i | case h
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K L : HomologicalComplex V c
φ : K ⟶ L
hφ : ∀ (i : ι), Mono (φ.f i)
Z✝ : HomologicalComplex V c
g h : Z✝ ⟶ K
eq : g ≫ φ = h ≫ φ
i : ι
⊢ g.f i = h.f i | 6cbecd34f313dfa6 |
Finset.erdos_ko_rado | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | theorem erdos_ko_rado {𝒜 : Finset (Finset (Fin n))} {r : ℕ}
(h𝒜 : (𝒜 : Set (Finset (Fin n))).Intersecting) (h₂ : (𝒜 : Set (Finset (Fin n))).Sized r)
(h₃ : r ≤ n / 2) :
#𝒜 ≤ (n - 1).choose (r - 1) | n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < #𝒜
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < #𝒜ᶜˢ
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk ... | rw [card_union_of_disjoint ‹_›] | n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < #𝒜
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < #𝒜ᶜˢ
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk ... | f540a864ab99f6db |
exists_smooth_tsupport_subset | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s ∈ 𝓝 x) :
∃ f : E → ℝ,
tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 | case intro.intro.refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ... | rintro t ⟨y, rfl⟩ | case intro.intro.refine_3.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ ... | ae1451e3dfa82ee2 |
WithCStarModule.pi_norm | Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean | lemma pi_norm (x : C⋆ᵐᵒᵈ (Π i, E i)) : ‖x‖ = √‖∑ i, ⟪x i, x i⟫_A‖ | A : Type u_1
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → Module ℂ (E i)
inst✝¹ : (i : ι) → SMul Aᵐᵒᵖ (E i)
inst✝ : (i : ι) → CStarModule A (E i)
x : C⋆ᵐᵒᵈ ((i : ι) → E i)
⊢ ‖x‖ = √‖∑ i : ι, ⟪x i,... | rfl | no goals | 1925bb2d55913242 |
ContDiffWithinAt.comp_continuousLinearMap | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem ContDiffWithinAt.comp_continuousLinearMap {x : G} (g : G →L[𝕜] E)
(hf : ContDiffWithinAt 𝕜 n f s (g x)) : ContDiffWithinAt 𝕜 n (f ∘ g) (g ⁻¹' s) x | case intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n✝ : WithTop ℕ∞
x : G
g ... | refine g.continuous.continuousWithinAt.tendsto_nhdsWithin ?_ hu | case intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n✝ : WithTop ℕ∞
x : G
g ... | bd3e17e15121cc21 |
NormedAddGroupHom.ker_completion | Mathlib/Analysis/Normed/Group/HomCompletion.lean | theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
(h : f.SurjectiveOnWith f.range C) :
(f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range | case intro.intro
G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
⊢ ∃ b ∈ ↑(toCo... | rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩ | case intro.intro.intro.intro
G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ ... | f0c87529fe1bb1ed |
Fin.appendEquiv_eq_Homeomorph | Mathlib/Topology/Homeomorph.lean | theorem _root_.Fin.appendEquiv_eq_Homeomorph (m n : ℕ) : Fin.appendEquiv m n =
((sumArrowHomeomorphProdArrow).symm.trans
(piCongrLeft (Y := fun _ ↦ X) finSumFinEquiv)).toEquiv | case neg
X : Type u_1
inst✝ : TopologicalSpace X
m n : ℕ
x1 : Fin m → X
x2 : Fin n → X
l : Fin (m + n)
h : ¬↑l < m
⊢ (if h : ↑l < m then x1 (l.castLT ⋯) else ⋯ ▸ x2 (Fin.subNat m (Fin.cast ⋯ l) ⋯)) =
({ toFun := fun f => (f ∘ Sum.inl, f ∘ Sum.inr), invFun := fun p => Sum.elim p.1 p.2, left_inv := ⋯, right_inv := ⋯,... | simp [h] | no goals | 431174ae15665125 |
balancedHull_subset_convexHull_union_neg | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | lemma balancedHull_subset_convexHull_union_neg {s : Set E} :
balancedHull ℝ s ⊆ convexHull ℝ (s ∪ -s) | case intro.intro.intro.intro.a
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x => r • x) y ∈ balancedHull ℝ s
this✝ : 0 ≤ 1 + r
this : 0 ≤ 1 - r
⊢ ((1 + r) / 2) • y + ((1 - r) / 2) • -y = (fun x => r • x) y | rw [smul_neg, ← sub_eq_add_neg, ← sub_smul] | case intro.intro.intro.intro.a
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x => r • x) y ∈ balancedHull ℝ s
this✝ : 0 ≤ 1 + r
this : 0 ≤ 1 - r
⊢ ((1 + r) / 2 - (1 - r) / 2) • y = (fun x => r • x) y | 4f6d0197db595fc3 |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall ... | simp only [H.ne, (hg n H).left.trans hi, if_false] | no goals | 70d4d5d8aa8a7de2 |
MvPolynomial.weightedDegree_eq_zero_iff | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | theorem weightedDegree_eq_zero_iff [CanonicallyOrderedAdd M]
(hw : NonTorsionWeight w) {m : σ →₀ ℕ} :
weight w m = 0 ↔ ∀ x : σ, m x = 0 | case h
M : Type u_2
σ : Type u_3
inst✝¹ : OrderedAddCommMonoid M
w : σ → M
inst✝ : CanonicallyOrderedAdd M
hw : NonTorsionWeight w
m : σ →₀ ℕ
x : σ
⊢ x ∈ m.support → m x • w x = 0 ↔ m x = 0 | rw [Finsupp.mem_support_iff] | case h
M : Type u_2
σ : Type u_3
inst✝¹ : OrderedAddCommMonoid M
w : σ → M
inst✝ : CanonicallyOrderedAdd M
hw : NonTorsionWeight w
m : σ →₀ ℕ
x : σ
⊢ m x ≠ 0 → m x • w x = 0 ↔ m x = 0 | 2175db668788ba02 |
Submodule.closure_neg | Mathlib/Algebra/Module/Submodule/Pointwise.lean | theorem closure_neg (s : Set M) : span R (-s) = -span R s | case a
R : Type u_2
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
s : Set M
⊢ span R (-s) ≤ -span R s | rw [span_le, coe_set_neg, ← Set.neg_subset, neg_neg] | case a
R : Type u_2
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
s : Set M
⊢ s ⊆ ↑(span R s) | a87cf00d9a094d82 |
Equiv.Perm.cycleType_extendDomain | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem cycleType_extendDomain {β : Type*} [Fintype β] [DecidableEq β] {p : β → Prop}
[DecidablePred p] (f : α ≃ Subtype p) {g : Perm α} :
cycleType (g.extendDomain f) = cycleType g | case induction_disjoint
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
β : Type u_2
inst✝² : Fintype β
inst✝¹ : DecidableEq β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : (σ.extendDomain f).cycleType = σ.cycleType
hτ : (τ.extendDomain f).cycleType = τ... | rw [hd.cycleType, ← extendDomain_mul, (hd.extendDomain f).cycleType, hσ, hτ] | no goals | a5e15d1fef0edcde |
Algebra.Generators.sq_ker_comp_le_ker_compLocalizationAwayAlgHom | Mathlib/RingTheory/CotangentLocalizationAway.lean | lemma sq_ker_comp_le_ker_compLocalizationAwayAlgHom :
((localizationAway g (S := T)).comp P).ker ^ 2 ≤
RingHom.ker (compLocalizationAwayAlgHom T g P) | case a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
∀ {x : ((localizationAway g).comp P).Ring},
x ∈ Ide... | apply sup_le | case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
∀ {x : ((localizationAway g).comp P).Ring},
x ∈ I... | 077970dcc2a9682d |
MvPolynomial.isNoetherianRing_fin_0 | Mathlib/RingTheory/Polynomial/Basic.lean | theorem isNoetherianRing_fin_0 [IsNoetherianRing R] :
IsNoetherianRing (MvPolynomial (Fin 0) R) | case f
R : Type u
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
⊢ MvPolynomial (Fin 0) R ≃+* R | apply MvPolynomial.isEmptyRingEquiv R (Fin 0) | no goals | ed587d1812a0114a |
contDiff_tsum_of_eventually | Mathlib/Analysis/Calculus/SmoothSeries.lean | theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
(hv : ∀ k : ℕ, k ≤ N → Summable (v k))
(h'f : ∀ k : ℕ, k ≤ N →
∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
ContDiff 𝕜 N fun x => ∑' i, f i x | α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 ... | ext1 x | case h
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContD... | 430bd82f35db2e88 |
BitVec.setWidth_cons | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem setWidth_cons {x : BitVec w} : (cons a x).setWidth w = x | w : Nat
a : Bool
x : BitVec w
⊢ setWidth w (cons a x) = x | simp [cons, setWidth_append] | no goals | 5cffe49d6dc4e45f |
contractLeft_assoc_coevaluation' | Mathlib/LinearAlgebra/Coevaluation.lean | theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ∘ₗ (TensorProduct.lid K _).toLinearMap | case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
(∑ x : ↑(Ba... | rw [map_sum] | case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),
(LinearMap.lT... | fa558da7fb2ef0c7 |
birkhoffAverage_apply_sub_birkhoffAverage | Mathlib/Dynamics/BirkhoffSum/Average.lean | theorem birkhoffAverage_apply_sub_birkhoffAverage {α M : Type*} (R : Type*) [DivisionRing R]
[AddCommGroup M] [Module R M] (f : α → α) (g : α → M) (n : ℕ) (x : α) :
birkhoffAverage R f g n (f x) - birkhoffAverage R f g n x =
(n : R)⁻¹ • (g (f^[n] x) - g x) | α : Type u_1
M : Type u_2
R : Type u_3
inst✝² : DivisionRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : α → α
g : α → M
n : ℕ
x : α
⊢ birkhoffAverage R f g n (f x) - birkhoffAverage R f g n x = (↑n)⁻¹ • (g (f^[n] x) - g x) | simp only [birkhoffAverage, birkhoffSum_apply_sub_birkhoffSum, ← smul_sub] | no goals | 8485044105a55052 |
Real.log_sqrt | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | theorem log_sqrt {x : ℝ} (hx : 0 ≤ x) : log (√x) = log x / 2 | x : ℝ
hx : 0 ≤ x
⊢ 2 ≠ 0 | exact two_ne_zero | no goals | 578e824fad84ecc5 |
Stirling.stirlingSeq'_bounded_by_pos_constant | Mathlib/Analysis/SpecialFunctions/Stirling.lean | theorem stirlingSeq'_bounded_by_pos_constant : ∃ a, 0 < a ∧ ∀ n : ℕ, a ≤ stirlingSeq (n + 1) | case intro
c : ℝ
h : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))
n : ℕ
⊢ c ≤ Real.log (stirlingSeq (n + 1)) | exact h n | no goals | 1c1410667e74f0cc |
Besicovitch.SatelliteConfig.exists_normalized | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
(lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ Pairwise fun i j => 1 - δ ≤ ‖c' i - c' j‖ | case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fi... | exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej | no goals | ef3e2825a59b79a7 |
DyckWord.semilength_outsidePart_lt | Mathlib/Combinatorics/Enumerative/DyckWord.lean | theorem semilength_outsidePart_lt : p.outsidePart.semilength < p.semilength | p : DyckWord
h : p ≠ 0
this : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
⊢ p.outsidePart.semilength < p.semilength | omega | no goals | 7cafc38442c8583f |
Topology.IsEmbedding.t5Space | Mathlib/Topology/Separation/Regular.lean | theorem Topology.IsEmbedding.t5Space [TopologicalSpace Y] [T5Space Y] {e : X → Y}
(he : IsEmbedding e) : T5Space X where
__ := he.t1Space
completely_normal | X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T5Space Y
e : X → Y
he : IsEmbedding e
this : CompletelyNormalSpace X
⊢ ∀ ⦃s t : Set X⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t) | exact completely_normal | no goals | 30c21ed92eb6fa83 |
ZFSet.sUnion_empty | Mathlib/SetTheory/ZFC/Basic.lean | theorem sUnion_empty : ⋃₀ (∅ : ZFSet.{u}) = ∅ | case a
z✝ : ZFSet.{u}
⊢ z✝ ∈ ⋃₀ ∅ ↔ z✝ ∈ ∅ | simp | no goals | ab36a3c52f4f0608 |
Finset.subset_map_symm | Mathlib/Data/Finset/Image.lean | theorem subset_map_symm {t : Finset β} {f : α ≃ β} : s ⊆ t.map f.symm ↔ s.map f ⊆ t | case mpr
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
f : α ≃ β
h : map f.toEmbedding s ⊆ t
x : α
hx : x ∈ s
⊢ x ∈ map f.symm.toEmbedding t | simp only [mem_map_equiv] | case mpr
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
f : α ≃ β
h : map f.toEmbedding s ⊆ t
x : α
hx : x ∈ s
⊢ f.symm.symm x ∈ t | 10bf1d1adae13662 |
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 | 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 ... | simp only [B.ne', Ne, not_false_iff, mul_one, field_simps] | 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 |
IsLocalization.injective_of_map_algebraMap_zero | Mathlib/RingTheory/Localization/Defs.lean | lemma injective_of_map_algebraMap_zero {T} [CommRing T] (f : S →+* T)
(h : ∀ x, f (algebraMap R S x) = 0 → algebraMap R S x = 0) :
Function.Injective f | R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
T : Type u_5
inst✝ : CommRing T
f : S →+* T
h : ∀ (x : R), f ((algebraMap R S) x) = 0 → (algebraMap R S) x = 0
x y : R
hz : f ((algebraMap R S) (x - y)) = 0
⊢ (algebraMap R S) x = (algebraM... | apply h at hz | R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
T : Type u_5
inst✝ : CommRing T
f : S →+* T
h : ∀ (x : R), f ((algebraMap R S) x) = 0 → (algebraMap R S) x = 0
x y : R
hz : (algebraMap R S) (x - y) = 0
⊢ (algebraMap R S) x = (algebraMap R... | 79e8190181b60f92 |
CategoryTheory.PreGaloisCategory.not_initial_iff_fiber_nonempty | Mathlib/CategoryTheory/Galois/Basic.lean | /-- An object is not initial if and only if its fiber is nonempty. -/
lemma not_initial_iff_fiber_nonempty (X : C) : (IsInitial X → False) ↔ Nonempty (F.obj X) | C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : PreGaloisCategory C
inst✝ : FiberFunctor F
X : C
h : IsInitial X → False
he : IsEmpty (F.obj X).carrier
⊢ False | exact Nonempty.elim ((initial_iff_fiber_empty F X).mpr he) h | no goals | 779726b45235097b |
Function.Periodic.sub_eq | Mathlib/Algebra/Ring/Periodic.lean | theorem Periodic.sub_eq [AddGroup α] (h : Periodic f c) (x : α) : f (x - c) = f x | α : Type u_1
β : Type u_2
f : α → β
c : α
inst✝ : AddGroup α
h : Periodic f c
x : α
⊢ f (x - c) = f x | simpa only [sub_add_cancel] using (h (x - c)).symm | no goals | f1cf9b006eda9a92 |
IsCoprime.mul_add_right_right | Mathlib/RingTheory/Coprime/Basic.lean | theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) | R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + z * x) | exact h.add_mul_right_right z | no goals | ccdc7fe9457e2b06 |
Ordinal.cof_type_lt | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem cof_type_lt [LinearOrder α] [IsWellOrder α (· < ·)] :
(@type α (· < ·) _).cof = @Order.cof α (· ≤ ·) | α : Type u
inst✝¹ : LinearOrder α
inst✝ : IsWellOrder α fun x1 x2 => x1 < x2
⊢ (type fun x1 x2 => x1 < x2).cof = Order.cof fun x1 x2 => x1 ≤ x2 | rw [cof_type, compl_lt, swap_ge] | no goals | 289c21badbb59060 |
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 refine_2
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
⊢ 0 < μ (support fun i => w D i * φ (x - i)) | have B : 0 < 1 + D := by linarith | case refine_2
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
⊢ 0 < μ (support fun i => w D i ... | 646f97b7cc0fbf51 |
extend_partialOrder | Mathlib/Order/Extension/Linear.lean | theorem extend_partialOrder {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] :
∃ s : α → α → Prop, IsLinearOrder α s ∧ r ≤ s | case refine_1.inl.inl
α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
S : Set (α → α → Prop) := {s | IsPartialOrder α s}
hS : ∀ c ⊆ S, IsChain (fun x1 x2 => x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub
s : α → α → Prop
hrs : r ≤ s
hs : Maximal (fun x => x ∈ S) s
this : IsPartialOrder α s
x y : α
h : ¬s x y ∧ ... | exact Or.inl (_root_.trans ab bc) | no goals | f4cf57fbd32a5c9a |
Matroid.Restriction.isBase_iff_of_spanning | Mathlib/Data/Matroid/Closure.lean | lemma Restriction.isBase_iff_of_spanning {N : Matroid α} (hR : N ≤r M) (hN : M.Spanning N.E) :
N.IsBase B ↔ (M.IsBase B ∧ B ⊆ N.E) | α : Type u_2
M : Matroid α
B : Set α
N : Matroid α
hR : N ≤r M
hN : M.Spanning N.E
⊢ N.IsBase B ↔ M.IsBase B ∧ B ⊆ N.E | obtain ⟨R, hR : R ⊆ M.E, rfl⟩ := hR | case intro.intro
α : Type u_2
M : Matroid α
B R : Set α
hR : R ⊆ M.E
hN : M.Spanning (M ↾ R).E
⊢ (M ↾ R).IsBase B ↔ M.IsBase B ∧ B ⊆ (M ↾ R).E | f0c9b183e76cd544 |
Topology.IsLower.isTopologicalSpace_basis | Mathlib/Topology/Order/LowerUpperTopology.lean | lemma isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U | case pos
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : U = univ
⊢ IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U | simp only [hU, isOpen_univ, compl_Ici, true_or] | no goals | f1c47d20246a5fbc |
FiniteField.sum_pow_units | Mathlib/FieldTheory/Finite/Basic.lean | theorem sum_pow_units [DecidableEq K] (i : ℕ) :
(∑ x : Kˣ, (x ^ i : K)) = if q - 1 ∣ i then -1 else 0 | case h
K : Type u_1
inst✝² : Field K
inst✝¹ : Fintype K
inst✝ : DecidableEq K
i : ℕ
φ : Kˣ →* K := { toFun := fun x => ↑x ^ i, map_one' := ⋯, map_mul' := ⋯ }
this : Decidable (φ = 1)
x : Kˣ
⊢ x ^ i = 1 ↔ φ x = 1 x | simp [φ, Units.ext_iff] | no goals | 682b7d68ce16f7cb |
Submodule.baseChange_top | Mathlib/LinearAlgebra/TensorProduct/Tower.lean | @[simp]
lemma baseChange_top : (⊤ : Submodule R M).baseChange A = ⊤ | R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ∀ (x : A ⊗[R] M), x ∈ span A ↑(LinearMap.range ((TensorProduct.mk R A M) 1)) | intro x | R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : A ⊗[R] M
⊢ x ∈ span A ↑(LinearMap.range ((TensorProduct.mk R A M) 1)) | 7a47417502811d97 |
Fin.coe_int_sub_eq_ite | Mathlib/Data/Fin/Basic.lean | theorem coe_int_sub_eq_ite {n : Nat} (u v : Fin n) :
((u - v : Fin n) : Int) = if v ≤ u then (u - v : Int) else (u - v : Int) + n | case isFalse
n : ℕ
u v : Fin n
h✝ : ¬v ≤ u
⊢ ↑↑⟨(n - ↑v + ↑u) % n, ⋯⟩ = ↑↑u - ↑↑v + ↑n | rw [ofNat_emod, Int.emod_eq_of_lt] <;> omega | no goals | b2f66d2e550d4eca |
Equiv.succ_embeddingFinSucc_fst_symm_apply | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | /-- Technical lemma used in the proof of `hasFTaylorSeriesUpTo_iteratedFDeriv`, to compare sums
over embedding of `Fin k` and `Fin (k + 1)`. -/
private lemma _root_.Equiv.succ_embeddingFinSucc_fst_symm_apply {ι : Type*} [DecidableEq ι]
{n : ℕ} (e : Fin (n+1) ↪ ι) {k : ι}
(h'k : k ∈ Set.range (Equiv.embeddingFin... | case intro
ι : Type u_4
inst✝ : DecidableEq ι
n : ℕ
e : Fin (n + 1) ↪ ι
j : Fin (n + 1)
h'k : e j ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hj : j ≠ 0
this : e j = ((Equiv.embeddingFinSucc n ι) e).fst (j.pred hj)
⊢ (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨((Equiv.embeddingFinSucc n ι) e).fst (j.pred... | simp [-Equiv.embeddingFinSucc_fst] | no goals | 812eabf947ba927a |
LieAlgebra.IsKilling.chainBotCoeff_add_chainTopCoeff | Mathlib/Algebra/Lie/Weights/RootSystem.lean | lemma chainBotCoeff_add_chainTopCoeff :
chainBotCoeff α β + chainTopCoeff α β = chainLength α β | case neg.a
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : ¬α.IsZero
e : (chainLength α β - c... | exact rootSpace_neg_nsmul_add_chainTop_of_lt α β hα (Nat.lt_succ_self _) | no goals | 699e482ae04ad602 |
Equiv.Perm.IsCycle.isConj | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.isConj (hσ : IsCycle σ) (hτ : IsCycle τ) (h : #σ.support = #τ.support) :
IsConj σ τ | α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ τ : Perm α
hσ : σ.IsCycle
hτ : τ.IsCycle
h : #σ.support = #τ.support
⊢ IsConj σ τ | refine
isConj_of_support_equiv
(hσ.zpowersEquivSupport.symm.trans <|
(zpowersEquivZPowers <| by rw [hσ.orderOf, h, hτ.orderOf]).trans hτ.zpowersEquivSupport)
?_ | α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ τ : Perm α
hσ : σ.IsCycle
hτ : τ.IsCycle
h : #σ.support = #τ.support
⊢ ∀ (x : α) (hx : x ∈ ↑σ.support),
↑((hσ.zpowersEquivSupport.symm.trans ((zpowersEquivZPowers ⋯).trans hτ.zpowersEquivSupport)) ⟨σ x, ⋯⟩) =
τ ↑((hσ.zpowersEquivSupport.symm.trans ((zpow... | 05f090ec94345134 |
MeasureTheory.unifIntegrable_finite | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem unifIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, MemLp (f i) p μ) : UnifIntegrable f p μ | case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ⇑hn.some.symm
hg... | assumption | no goals | 64d4c0197b2312fa |
Submodule.goursat | Mathlib/LinearAlgebra/Goursat.lean | /-- **Goursat's lemma** for an arbitrary submodule of a product.
If `L` is a submodule of `M × N`, then there exist submodules `M'' ≤ M' ≤ M` and `N'' ≤ N' ≤ N` such
that `L ≤ M' × N'`, and `L` is (the image in `M × N` of) the preimage of the graph of an `R`-linear
isomorphism `M' ⧸ M'' ≃ N' ⧸ N''`. -/
lemma goursat :... | case intro
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
P : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M N).s... | use M', N', L'.goursatFst, L'.goursatSnd, e | case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
P : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M N).submo... | 8bafbf13c72f3711 |
toIcoDiv_neg | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIcoDiv_neg (a b : α) : toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1) | case h
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p) | obtain ⟨hc, ho⟩ := sub_toIcoDiv_zsmul_mem_Ico hp a (-b) | case h.intro
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
hc : a ≤ -b - toIcoDiv hp a (-b) • p
ho : -b - toIcoDiv hp a (-b) • p < a + p
⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p) | 3f17671b359fe1fd |
Order.exists_series_of_le_height | Mathlib/Order/KrullDimension.lean | /-- There exist a series ending in a element for any length up to the element’s height. -/
lemma exists_series_of_le_height (a : α) {n : ℕ} (h : n ≤ height a) :
∃ p : LTSeries α, p.last = a ∧ p.length = n | case h.intro.mk
α : Type u_1
inst✝ : Preorder α
n m : ℕ
p : LTSeries α
hne : Nonempty { p_1 // RelSeries.last p_1 = RelSeries.last p }
ha : ∀ (p_1 : LTSeries α), RelSeries.last p_1 = RelSeries.last p → p_1.length ≠ n
hp : (fun x => (↑x).length) ⟨p, ⋯⟩ = m
⊢ m ≤ n | simp only at hp | case h.intro.mk
α : Type u_1
inst✝ : Preorder α
n m : ℕ
p : LTSeries α
hne : Nonempty { p_1 // RelSeries.last p_1 = RelSeries.last p }
ha : ∀ (p_1 : LTSeries α), RelSeries.last p_1 = RelSeries.last p → p_1.length ≠ n
hp : p.length = m
⊢ m ≤ n | 0e15cb2199a1ee03 |
AkraBazziRecurrence.GrowsPolynomially.add | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | protected lemma GrowsPolynomially.add {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hg : GrowsPolynomially g) (hf' : 0 ≤ᶠ[atTop] f) (hg' : 0 ≤ᶠ[atTop] g) :
GrowsPolynomially fun x => f x + g x | case h
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ ... | have fx_nonneg : 0 ≤ f x := hf' x hbx | case h
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ ... | 23da4f580f793a06 |
Asymptotics.IsLittleO.smul_isBigO | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem IsLittleO.smul_isBigO (h₁ : k₁ =o[l] k₂) (h₂ : f' =O[l] g') :
(fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x | α : Type u_1
E' : Type u_6
F' : Type u_7
R : Type u_13
𝕜' : Type u_16
inst✝⁷ : SeminormedAddCommGroup E'
inst✝⁶ : SeminormedAddCommGroup F'
inst✝⁵ : SeminormedRing R
inst✝⁴ : NormedDivisionRing 𝕜'
f' : α → E'
g' : α → F'
l : Filter α
inst✝³ : Module R E'
inst✝² : BoundedSMul R E'
inst✝¹ : Module 𝕜' F'
inst✝ : Bounde... | rcases h₂.exists_pos with ⟨c', c'pos, hc'⟩ | case intro.intro
α : Type u_1
E' : Type u_6
F' : Type u_7
R : Type u_13
𝕜' : Type u_16
inst✝⁷ : SeminormedAddCommGroup E'
inst✝⁶ : SeminormedAddCommGroup F'
inst✝⁵ : SeminormedRing R
inst✝⁴ : NormedDivisionRing 𝕜'
f' : α → E'
g' : α → F'
l : Filter α
inst✝³ : Module R E'
inst✝² : BoundedSMul R E'
inst✝¹ : Module 𝕜' ... | 72c7be5113299dac |
Std.Sat.AIG.mkConstCached_eval_eq_mkConst_eval | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean | theorem mkConstCached_eval_eq_mkConst_eval {aig : AIG α} :
⟦aig.mkConstCached val, assign⟧ = ⟦aig.mkConst val, assign⟧ | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
val : Bool
aig : AIG α
⊢ ⟦assign, aig.mkConstCached val⟧ = ⟦assign, aig.mkConst val⟧ | simp only [mkConstCached] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
val : Bool
aig : AIG α
⊢ ⟦assign,
match aig.cache.get? (Decl.const val) with
| some hit =>
{ aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ }, ref := { gate := hit.idx, hgate := ⋯ } }
| none =>
{
... | 55155a61c814d3d1 |
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