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 |
|---|---|---|---|---|---|---|
Finset.preimage_inv | Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | theorem preimage_inv (s : Finset α) : s.preimage (·⁻¹) inv_injective.injOn = s⁻¹ :=
coe_injective <| by rw [coe_preimage, Set.inv_preimage, coe_inv]
| α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : InvolutiveInv α
s : Finset α
⊢ ↑(s.preimage (fun x => x⁻¹) ⋯) = ↑s⁻¹ | rw [coe_preimage, Set.inv_preimage, coe_inv] | no goals | 08856ec67b583d0d |
Mathlib.Tactic.Ring.add_pf_add_gt | Mathlib/Tactic/Ring/Basic.lean | theorem add_pf_add_gt (b₁ : R) (_ : a + b₂ = c) : a + (b₁ + b₂) = b₁ + c | R : Type u_1
inst✝ : CommSemiring R
a b₂ b₁ : R
⊢ a + (b₁ + b₂) = b₁ + (a + b₂) | simp [add_left_comm] | no goals | 3dce76cb2798260c |
BitVec.shiftLeft_ushiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem shiftLeft_ushiftRight {x : BitVec w} {n : Nat}:
x >>> n <<< n = x &&& BitVec.allOnes w <<< n | w n : Nat
ih : ∀ {x : BitVec w}, x >>> n <<< n = x &&& allOnes w <<< n
x : BitVec w
i : Nat
h : i < w
hw : ¬w = 0
hi₂ : ¬i = 0
⊢ 1 + (i - 1) = i | omega | no goals | d74277fca2e60b6a |
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 intro.inl
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
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (1 * (cs.simple i' * cs.simple i)⁻¹ ^ k) | simp | no goals | 8b276f2d43ad82bd |
EReal.exists_lt_mul_left_of_nonneg | Mathlib/Data/Real/EReal.lean | private lemma exists_lt_mul_left_of_nonneg (ha : 0 ≤ a) (hc : 0 ≤ c) (h : c < a * b) :
∃ a' ∈ Ico 0 a, c < a' * b | case inl
a c : EReal
ha : 0 ≤ a
hc : 0 ≤ c
h : c < a * ⊤
⊢ ∃ a' ∈ Ico 0 a, c < a' * ⊤ | rcases eq_or_lt_of_le ha with rfl | ha | case inl.inl
c : EReal
hc : 0 ≤ c
ha : 0 ≤ 0
h : c < 0 * ⊤
⊢ ∃ a' ∈ Ico 0 0, c < a' * ⊤
case inl.inr
a c : EReal
ha✝ : 0 ≤ a
hc : 0 ≤ c
h : c < a * ⊤
ha : 0 < a
⊢ ∃ a' ∈ Ico 0 a, c < a' * ⊤ | c6bb344d64a6ce34 |
WeierstrassCurve.Projective.equiv_iff_eq_of_Z_eq' | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma equiv_iff_eq_of_Z_eq' {P Q : Fin 3 → R} (hz : P z = Q z) (mem : Q z ∈ nonZeroDivisors R) :
P ≈ Q ↔ P = Q | R : Type r
inst✝ : CommRing R
P : Fin 3 → R
hz : P z = P z
mem : P z ∈ nonZeroDivisors R
⊢ P ≈ P | exact Setoid.refl _ | no goals | 6bc90cbf1590dd35 |
Matrix.det_eq_of_forall_row_eq_smul_add_pred_aux | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | case pos
R : Type v
inst✝ : CommRing R
n : ℕ
k✝ : Fin (n + 1)
k : Fin n
ih :
∀ (c : Fin n → R),
(∀ (i : Fin n), k.castSucc < i.succ → c i = 0) →
∀ {M N : Matrix (Fin n.succ) (Fin n.succ) R},
(∀ (j : Fin n.succ), M 0 j = N 0 j) →
(∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c ... | rfl | no goals | 8b96005e0c0eb652 |
BitVec.mul_comm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem mul_comm (x y : BitVec w) : x * y = y * x | case a
w : Nat
x y : BitVec w
⊢ (x * y).toFin = (y * x).toFin | simpa using Fin.mul_comm .. | no goals | c258f5a99a4638fb |
zpow_add₀ | Mathlib/Algebra/GroupWithZero/Basic.lean | lemma zpow_add₀ (ha : a ≠ 0) (m n : ℤ) : a ^ (m + n) = a ^ m * a ^ n | case hn
G₀ : Type u_2
inst✝ : GroupWithZero G₀
a : G₀
ha : a ≠ 0
m : ℤ
n : ℕ
ihn : a ^ (m + -↑n) = a ^ m * a ^ (-↑n)
⊢ a ^ (m + (-↑n - 1)) = a ^ m * a ^ (-↑n - 1) | rw [zpow_sub_one₀ ha, ← mul_assoc, ← ihn, ← zpow_sub_one₀ ha, Int.add_sub_assoc] | no goals | 56384098f6f089c9 |
Matrix.Represents.mul | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | theorem Matrix.Represents.mul {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A * A').Represents b (f * f') | ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A A' : Matrix ι ι R
f f' : Module.End R M
h : Represents b A f
h' : Represents b A' f'
⊢ ((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.linearCombination R R) b))
... | ext | case h.h
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A A' : Matrix ι ι R
f f' : Module.End R M
h : Represents b A f
h' : Represents b A' f'
i✝ : ι
⊢ (((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.linearComb... | 1051dbaedee97977 |
symmDiff_sdiff_inf | Mathlib/Order/SymmDiff.lean | theorem symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b | α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a \ (b ⊔ a ⊓ b) ⊔ b \ (a ⊔ a ⊓ b) = a ∆ b | simp [symmDiff] | no goals | dc2a903733928f81 |
exists_sum_eq_one_iff_pairwise_coprime | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i) | case h.e_a.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a... | rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat] | no goals | 8a28cdbc75233b6c |
AnalyticAt.eventually_constant_or_nhds_le_map_nhds | Mathlib/Analysis/Complex/OpenMapping.lean | theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : AnalyticAt ℂ g z₀) :
(∀ᶠ z in 𝓝 z₀, g z = g z₀) ∨ 𝓝 (g z₀) ≤ map g (𝓝 z₀) | case pos.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t => z₀ + t • z
gray : E → ℂ → ℂ := fun z => g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
h1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ (gray z) (ball 0 r)
h : ∀... | have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz | case pos.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t => z₀ + t • z
gray : E → ℂ → ℂ := fun z => g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
h1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ (gray z) (ball 0 r)
h : ∀... | bbddb65fce2e7708 |
RingHom.finitePresentation_ofLocalizationSpanTarget | Mathlib/RingTheory/RingHom/FinitePresentation.lean | theorem finitePresentation_ofLocalizationSpanTarget :
OfLocalizationSpanTarget @FinitePresentation | case h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjec... | rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem] | case h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjec... | 1e270402a60647b4 |
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 hbx : b * x ≤ x := calc
b * x ≤ 1 * x := by gcongr; exact le_of_lt hb.2
_ = x := by ring | 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 |
MeasureTheory.L1.integral_smul | Mathlib/MeasureTheory/Integral/BochnerL1.lean | theorem integral_smul (c : 𝕜) (f : α →₁[μ] E) : integral (c • f) = c • integral f | α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁵ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : CompleteSpace E
c : 𝕜
f : ↥(Lp E 1 μ)
⊢ integral (c • f) = c • integral f | simp only [integral] | α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁵ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : CompleteSpace E
c : 𝕜
f : ↥(Lp E 1 μ)
⊢ integralCLM (c • f) = c • integralCLM f | dd57e275448d5aed |
Complex.sinh_three_mul | Mathlib/Data/Complex/Trigonometric.lean | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | x : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh x * cosh (2 * x) + cosh x * sinh (2 * x) = 4 * sinh x ^ 3 + 3 * sinh x | simp only [cosh_two_mul, sinh_two_mul] | x : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh x * (cosh x ^ 2 + sinh x ^ 2) + cosh x * (2 * sinh x * cosh x) = 4 * sinh x ^ 3 + 3 * sinh x | 737e6f979e70c403 |
one_le_gauge_of_not_mem | Mathlib/Analysis/Convex/Gauge.lean | theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
1 ≤ gauge s x :=
le_gauge_of_not_mem hs₁ hs₂ <| by rwa [one_smul]
| E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
hs₁ : StarConvex ℝ 0 s
hs₂ : Absorbs ℝ s {x}
hx : x ∉ s
⊢ x ∉ 1 • s | rwa [one_smul] | no goals | c71c0bf2f1d5db5a |
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.mpr
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
z : ↑↑(pullback f' g').toPresheafedSpace
x : ↑↑X.toPresheafedSpace
hx : (ConcreteCategory.hom i₁.base) x = (ConcreteCategory.hom (pull... | let T₁ : Triplet (pullback.fst f' g') i₁ := Triplet.mk' z x hx.symm | case h.mpr
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
z : ↑↑(pullback f' g').toPresheafedSpace
x : ↑↑X.toPresheafedSpace
hx : (ConcreteCategory.hom i₁.base) x = (ConcreteCategory.hom (pull... | 6ca8171f1040fa44 |
Nat.testBit_two_pow_sub_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem testBit_two_pow_sub_succ (h₂ : x < 2 ^ n) (i : Nat) :
testBit (2^n - (x + 1)) i = (decide (i < n) && ! testBit x i) | i : Nat
ih : ∀ {x n : Nat}, x < 2 ^ n → (2 ^ n - (x + 1)).testBit i = (decide (i < n) && !x.testBit i)
x n✝ n : Nat
h₂ : x < 2 ^ (n + 1)
⊢ ((2 ^ (n + 1) - (x + 1)) / 2).testBit i = (decide (i + 1 < n + 1) && !(x / 2).testBit i) | rw [Nat.two_pow_succ_sub_succ_div_two, ih] | i : Nat
ih : ∀ {x n : Nat}, x < 2 ^ n → (2 ^ n - (x + 1)).testBit i = (decide (i < n) && !x.testBit i)
x n✝ n : Nat
h₂ : x < 2 ^ (n + 1)
⊢ (decide (i < n) && !(x / 2).testBit i) = (decide (i + 1 < n + 1) && !(x / 2).testBit i)
i : Nat
ih : ∀ {x n : Nat}, x < 2 ^ n → (2 ^ n - (x + 1)).testBit i = (decide (i < n) && !x.... | 09c576eb1a89925c |
differentiableWithinAt_localInvariantProp | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | theorem differentiableWithinAt_localInvariantProp :
(contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
(DifferentiableWithinAtProp I I') :=
{ is_local | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | convert (this.differentiableWithinAt le_rfl).comp _ h _ | case h.e'_11
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ... | 3d1405d37db22f10 |
Subsingleton.helim | Mathlib/.lake/packages/lean4/src/lean/Init/Core.lean | theorem Subsingleton.helim {α β : Sort u} [h₁ : Subsingleton α] (h₂ : α = β) (a : α) (b : β) : HEq a b | α β : Sort u
h₁ : Subsingleton α
h₂ : α = β
a : α
b : β
⊢ HEq a b | subst h₂ | α : Sort u
h₁ : Subsingleton α
a b : α
⊢ HEq a b | 44744a880a1d9952 |
List.Perm.sizeOf_eq_sizeOf | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem Perm.sizeOf_eq_sizeOf [SizeOf α] {l₁ l₂ : List α} (h : l₁ ~ l₂) :
sizeOf l₁ = sizeOf l₂ | case swap
α : Type u_1
inst✝ : SizeOf α
l₁ l₂ : List α
x✝ y✝ : α
l✝ : List α
⊢ sizeOf (y✝ :: x✝ :: l✝) = sizeOf (x✝ :: y✝ :: l✝) | simp [Nat.add_left_comm] | no goals | 3d77188ce1f7e9a5 |
CStarAlgebra.span_nonneg_inter_ball | Mathlib/Analysis/CStarAlgebra/SpecialFunctions/PosPart.lean | /-- A C⋆-algebra is spanned by nonnegative elements of norm less than `r`. -/
lemma span_nonneg_inter_ball {r : ℝ} (hr : 0 < r) :
span ℂ ({x : A | 0 ≤ x} ∩ Metric.ball 0 r) = ⊤ | case h.H
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ Metric.closedBall 0 (r / 2) ⊆ Metric.ball 0 r | exact Metric.closedBall_subset_ball <| half_lt_self hr | no goals | eed7d32d179fb6f5 |
Nat.ordProj_pos | Mathlib/Data/Nat/Factorization/Basic.lean | theorem ordProj_pos (n p : ℕ) : 0 < ordProj[p] n | n p : ℕ
⊢ 0 < p ^ n.factorization p | if pp : p.Prime then simp [pow_pos pp.pos] else simp [pp] | no goals | 1096efa6604cd9d5 |
SimpleGraph.Walk.darts_dropUntil_subset | Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean | theorem darts_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) :
(p.dropUntil u h).darts ⊆ p.darts := fun x hx => by
rw [← take_spec p h, darts_append, List.mem_append]
exact Or.inr hx
| V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w : V
p : G.Walk v w
h : u ∈ p.support
x : G.Dart
hx : x ∈ (p.dropUntil u h).darts
⊢ x ∈ p.darts | rw [← take_spec p h, darts_append, List.mem_append] | V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w : V
p : G.Walk v w
h : u ∈ p.support
x : G.Dart
hx : x ∈ (p.dropUntil u h).darts
⊢ x ∈ (p.takeUntil u h).darts ∨ x ∈ (p.dropUntil u h).darts | b4133a6e267ddcc5 |
Profinite.NobelingProof.swapTrue_mem_C1 | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem swapTrue_mem_C1 (f : π (C1 C ho) (ord I · < o)) :
SwapTrue o f.val ∈ C1 C ho | I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
f : ↑(π (C1 C ho) fun x => ord I x < o)
⊢ SwapTrue o ↑f ∈ C1 C ho | obtain ⟨f, g, hg, rfl⟩ := f | case mk.intro.intro
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
g : I → Bool
hg : g ∈ C1 C ho
⊢ SwapTrue o ↑⟨Proj (fun x => ord I x < o) g, ⋯⟩ ∈ C1 C ho | 399ed2381c83746d |
MeasureTheory.unifTight_of_tendsto_Lp | Mathlib/MeasureTheory/Function/UnifTight.lean | theorem unifTight_of_tendsto_Lp (hp' : p ≠ ∞) (hf : ∀ n, MemLp (f n) p μ)
(hg : MemLp g p μ) (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)) :
UnifTight f p μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
⊢ UnifTight f p μ | have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp | α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x => g) + fun n => f n - g
⊢ UnifTight f p μ | e9a49ff753d29c27 |
sub_inv_antitoneOn_Icc_left | Mathlib/Algebra/Order/Field/Basic.lean | theorem sub_inv_antitoneOn_Icc_left (ha : b < c) :
AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Icc a b) | case neg
α : Type u_2
inst✝ : LinearOrderedField α
a b c : α
ha : b < c
hab : ¬a ≤ b
⊢ AntitoneOn (fun x => (x - c)⁻¹) (Set.Icc a b) | simp [hab, Set.Subsingleton.antitoneOn] | no goals | 4c600f8928948d7c |
Submodule.quotDualCoannihilatorToDual_nondegenerate | Mathlib/LinearAlgebra/Dual.lean | theorem quotDualCoannihilatorToDual_nondegenerate (W : Submodule R (Dual R M)) :
W.quotDualCoannihilatorToDual.Nondegenerate | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
W : Submodule R (Dual R M)
this : AddCommGroup ↥W := inferInstance
⊢ Function.Injective ⇑W.quotDualCoannihilatorToDual ∧ Function.Injective ⇑W.quotDualCoannihilatorToDual.flip | exact ⟨W.quotDualCoannihilatorToDual_injective, W.flip_quotDualCoannihilatorToDual_injective⟩ | no goals | 64b0bcaa076efee4 |
LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace | Mathlib/Algebra/Lie/Weights/Killing.lean | /-- This is Proposition 4.18 from [carter2005] except that we use
`LieModule.exists_forall_lie_eq_smul` instead of Lie's theorem (and so avoid
assuming `K` has characteristic zero). -/
lemma cartanEquivDual_symm_apply_mem_corootSpace (α : Weight K H L) :
(cartanEquivDual H).symm α ∈ corootSpace α | K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
α : Weight K (↥H) L
e : L
he₀ : e ≠ 0
he : ∀ (x : ↥H), ⁅x, e⁆ = α x • e
heα : e ∈ rootSpace ... | simpa [inv_smul_eq_iff₀ hf] | no goals | 71b75475a1107604 |
UpperHalfPlane.tendsto_coe_atImInfty | Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean | lemma tendsto_coe_atImInfty :
Tendsto UpperHalfPlane.coe atImInfty (comap Complex.im atTop) | ⊢ Tendsto UpperHalfPlane.coe atImInfty (comap Complex.im atTop) | simpa only [atImInfty, tendsto_comap_iff, Function.comp_def,
funext UpperHalfPlane.coe_im] using tendsto_comap | no goals | 9026dcafb42e7013 |
List.eraseP_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem eraseP_filter (f : α → Bool) (l : List α) :
(filter f l).eraseP p = filter f (l.eraseP (fun x => p x && f x)) | α : Type u_1
p f : α → Bool
l : List α
⊢ eraseP p (filter f l) = filter f (eraseP (fun x => p x && f x) l) | rw [← filterMap_eq_filter, eraseP_filterMap] | α : Type u_1
p f : α → Bool
l : List α
⊢ filterMap (Option.guard fun x => f x = true)
(eraseP
(fun x =>
match Option.guard (fun x => f x = true) x with
| some y => p y
| none => false)
l) =
filterMap (Option.guard fun x => f x = true) (eraseP (fun x => p x && f x)... | 4bc6b7ef8c097500 |
PerfectClosure.mk_eq_iff | Mathlib/FieldTheory/PerfectClosure.lean | theorem mk_eq_iff (x y : ℕ × K) :
mk K p x = mk K p y ↔ ∃ z, (frobenius K p)^[y.1 + z] x.2 = (frobenius K p)^[x.1 + z] y.2 | case mp.trans.intro.intro
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝ y✝ x y z : ℕ × K
H1 : Relation.EqvGen (R K p) x y
H2 : Relation.EqvGen (R K p) y z
z1 : ℕ
ih1 : (⇑(frobenius K p))^[y.1 + z1] x.2 = (⇑(frobenius K p))^[x.1 + z1] y.2
z2 : ℕ
ih2 : (⇑(frobenius K p))^[z.1 + z2]... | rw [← iterate_add_apply] | case mp.trans.intro.intro
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝ y✝ x y z : ℕ × K
H1 : Relation.EqvGen (R K p) x y
H2 : Relation.EqvGen (R K p) y z
z1 : ℕ
ih1 : (⇑(frobenius K p))^[y.1 + z1] x.2 = (⇑(frobenius K p))^[x.1 + z1] y.2
z2 : ℕ
ih2 : (⇑(frobenius K p))^[z.1 + z2]... | 50fcb88ed45b5508 |
cauchySeq_shift | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u | case mpr
α : Type u
uniformSpace : UniformSpace α
u : ℕ → α
k : ℕ
h : CauchySeq u
⊢ CauchySeq fun n => u (n + k) | exact h.comp_tendsto (tendsto_add_atTop_nat k) | no goals | ea02bac41604f3b7 |
EReal.add_ne_top_iff_of_ne_bot_of_ne_top | Mathlib/Data/Real/EReal.lean | lemma add_ne_top_iff_of_ne_bot_of_ne_top {x y : EReal} (hy : y ≠ ⊥) (hy' : y ≠ ⊤) :
x + y ≠ ⊤ ↔ x ≠ ⊤ | x y : EReal
hy : y ≠ ⊥
hy' : y ≠ ⊤
⊢ x + y ≠ ⊤ ↔ x ≠ ⊤ | induction x <;> simp [add_ne_top_iff_of_ne_bot, hy, hy'] | no goals | c3c2f04cde960708 |
Int.clog_natCast | Mathlib/Data/Int/Log.lean | theorem clog_natCast (b : ℕ) (n : ℕ) : clog b (n : R) = Nat.clog b n | R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b n : ℕ
⊢ clog b ↑n = ↑(Nat.clog b n) | rcases n with - | n | case zero
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
⊢ clog b ↑0 = ↑(Nat.clog b 0)
case succ
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b n : ℕ
⊢ clog b ↑(n + 1) = ↑(Nat.clog b (n + 1)) | b2299eaf3f2b279a |
LieSubalgebra.isCartanSubalgebra_iff_isUcsLimit | Mathlib/Algebra/Lie/CartanSubalgebra.lean | theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit | R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = H.toLieSubmodule
hk' : LieSubmodule.ucs (k + 1) ⊥ = H.toLieSubmodule
⊢ H.normalizer = H | rw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk' | R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = H.toLieSubmodule
hk' : H.toLieSubmodule.normalizer = H.toLieSubmodule
⊢ H.normalizer = H | 3bb80ea66da3d89e |
StarConvex.add_left | Mathlib/Analysis/Convex/Star.lean | theorem StarConvex.add_left (hs : StarConvex 𝕜 x s) (z : E) :
StarConvex 𝕜 (z + x) ((fun x => z + x) '' s) | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x : E
s : Set E
hs : StarConvex 𝕜 x s
z : E
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
y' : E
hy' : y' ∈ s
⊢ a • (z + x) + b • (fun x => z + x) y' ∈ (fun x => z + x) '' s | refine ⟨a • x + b • y', hs hy' ha hb hab, ?_⟩ | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
x : E
s : Set E
hs : StarConvex 𝕜 x s
z : E
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
y' : E
hy' : y' ∈ s
⊢ (fun x => z + x) (a • x + b • y') = a • (z + x) + b • (fun x => z + x) y' | 194aa148110f99fc |
List.nil_union | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem nil_union (l : List α) : nil ∪ l = l | α : Type u_1
inst✝ : BEq α
l : List α
⊢ [] ∪ l = l | simp [List.union_def, foldr] | no goals | 42464ed6456290b9 |
isQuasiregular_iff_isUnit' | Mathlib/Algebra/Algebra/Quasispectrum.lean | lemma isQuasiregular_iff_isUnit' (R : Type*) {A : Type*} [CommSemiring R] [NonUnitalSemiring A]
[Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : A} :
IsQuasiregular x ↔ IsUnit (1 + x : Unitization R A) | R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalSemiring A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
x : A
hx : IsUnit (1 + ↑x)
⊢ PreQuasiregular.equiv.symm ↑((Unitization.unitsFstOne_mulEquiv_quasiregular R) ⟨hx.unit, ⋯⟩) = x | simp | no goals | 2d0099aa35b6727f |
BoundedContinuousFunction.arzela_ascoli₁ | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | theorem arzela_ascoli₁ [CompactSpace β] (A : Set (α →ᵇ β)) (closed : IsClosed A)
(H : Equicontinuous ((↑) : A → α → β)) : IsCompact A | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : CompactSpace β
A : Set (α →ᵇ β)
closed : IsClosed A
H : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x... | obtain ⟨x', x'tα, hx'⟩ := mem_iUnion₂.1 (htα (mem_univ x)) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro.intro
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : CompactSpace β
A : Set (α →ᵇ β)
closed : IsClosed A
H : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist ... | 57af0dfb992088dd |
MeasureTheory.integral_prod_symm | Mathlib/MeasureTheory/Integral/Prod.lean | theorem integral_prod_symm (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
∫ z, f z ∂μ.prod ν = ∫ y, ∫ x, f (x, y) ∂μ ∂ν | α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α × β → E
hf : Integrable f (μ.prod ν)
⊢ ∫ (z : β × α), f z.swap ∂ν.prod μ = ∫ (y : β), ∫ (x : α), f (x... | exact integral_prod _ hf.swap | no goals | 0410ab3469493d09 |
jacobiSum_mul_nontrivial | Mathlib/NumberTheory/JacobiSum/Basic.lean | theorem jacobiSum_mul_nontrivial {χ φ : MulChar F R} (h : χ * φ ≠ 1) (ψ : AddChar F R) :
gaussSum (χ * φ) ψ * jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ | F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
⊢ ∀ t ∈ univ \ {0}, (∑ x : F, χ x * φ (t - x)) * ψ t = (∑ y : F, χ (t * y) * φ (t - t * y)) * ψ t | intro t ht | F : Type u_1
R : Type u_2
inst✝³ : Field F
inst✝² : Fintype F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ φ : MulChar F R
h : χ * φ ≠ 1
ψ : AddChar F R
t : F
ht : t ∈ univ \ {0}
⊢ (∑ x : F, χ x * φ (t - x)) * ψ t = (∑ y : F, χ (t * y) * φ (t - t * y)) * ψ t | 6cf53dcbd81a1b9e |
PartialHomeomorph.subtypeRestr_symm_eqOn_of_le | Mathlib/Topology/PartialHomeomorph.lean | theorem subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Nonempty V)
(hUV : U ≤ V) : EqOn (e.subtypeRestr hV).symm (Set.inclusion hUV ∘ (e.subtypeRestr hU).symm)
(e.subtypeRestr hU).target | X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.partialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.partialHomeomorphSubt... | refine (V.partialHomeomorphSubtypeCoe hV).injOn ?_ trivial ?_ | case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.partialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.partial... | 914ca04e5b54b50a |
groupCohomology.H0Map_id | Mathlib/RepresentationTheory/GroupCohomology/Functoriality.lean | theorem H0Map_id : H0Map (MonoidHom.id _) (𝟙 A) = 𝟙 _ | k H : Type u
inst✝¹ : CommRing k
inst✝ : Group H
A : Rep k H
⊢ H0Map (MonoidHom.id H) (𝟙 A) = 𝟙 (H0 A) | rfl | no goals | f7fe638b41e0f450 |
comp_equiv_symm_dotProduct | Mathlib/Data/Matrix/Mul.lean | theorem comp_equiv_symm_dotProduct (e : m ≃ n) : u ∘ e.symm ⬝ᵥ x = u ⬝ᵥ x ∘ e :=
(e.sum_comp _).symm.trans <|
Finset.sum_congr rfl fun _ _ => by simp only [Function.comp, Equiv.symm_apply_apply]
| m : Type u_2
n : Type u_3
α : Type v
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalNonAssocSemiring α
u : m → α
x : n → α
e : m ≃ n
x✝¹ : m
x✝ : x✝¹ ∈ Finset.univ
⊢ (u ∘ ⇑e.symm) (e x✝¹) * x (e x✝¹) = u x✝¹ * (x ∘ ⇑e) x✝¹ | simp only [Function.comp, Equiv.symm_apply_apply] | no goals | be7d1f6c11a84e99 |
Polynomial.preHilbertPoly_eq_choose_sub_add | Mathlib/RingTheory/Polynomial/HilbertPoly.lean | lemma preHilbertPoly_eq_choose_sub_add [CharZero F] (d : ℕ) {k n : ℕ} (hkn : k ≤ n):
(preHilbertPoly F d k).eval (n : F) = (n - k + d).choose d | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
d k n : ℕ
hkn : k ≤ n
⊢ eval (↑n) (preHilbertPoly F d k) = ↑((n - k + d).choose d) | have : (d ! : F) ≠ 0 := by norm_cast; positivity | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
d k n : ℕ
hkn : k ≤ n
this : ↑d ! ≠ 0
⊢ eval (↑n) (preHilbertPoly F d k) = ↑((n - k + d).choose d) | abed0e0ae54a68db |
Profinite.exists_locallyConstant_finite_nonempty | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant_finite_nonempty {α : Type*} [Finite α] [Nonempty α]
(hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom | case intro.intro.h
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fu... | erw [dif_pos h2] | case intro.intro.h
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fu... | a1c1fd2fab6330f3 |
MeasureTheory.exists_measurable_le_forall_setLIntegral_eq | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem exists_measurable_le_forall_setLIntegral_eq [SFinite μ] (f : α → ℝ≥0∞) :
∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∀ s, ∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ | case intro
α : Type u_1
m : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), f a ⊓ ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x =... | exact ⟨⌈C⌉₊, fun x ↦ (hC <| ψ.mem_range_self x).trans (Nat.le_ceil _)⟩ | no goals | f381f21756fe17db |
Matrix.PosDef.intCast | Mathlib/LinearAlgebra/Matrix/PosDef.lean | theorem intCast [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R]
(d : ℤ) (hd : 0 < d) :
PosDef (d : Matrix n n R) :=
⟨isHermitian_intCast _, fun x hx => by
simp only [intCast_mulVec, dotProduct_smul]
rw [Int.cast_smul_eq_zsmul]
exact zsmul_pos (dotProduct_star_self_pos_iff.mpr hx) hd⟩
| n : Type u_2
R : Type u_3
inst✝⁶ : Fintype n
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : DecidableEq n
inst✝ : NoZeroDivisors R
d : ℤ
hd : 0 < d
x : n → R
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ ↑d *ᵥ x | simp only [intCast_mulVec, dotProduct_smul] | n : Type u_2
R : Type u_3
inst✝⁶ : Fintype n
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : DecidableEq n
inst✝ : NoZeroDivisors R
d : ℤ
hd : 0 < d
x : n → R
hx : x ≠ 0
⊢ 0 < ↑d • (star x ⬝ᵥ x) | 28da97a3d0cf32ce |
lowerCentralSeries.map | Mathlib/GroupTheory/Nilpotent.lean | theorem lowerCentralSeries.map {H : Type*} [Group H] (f : G →* H) (n : ℕ) :
Subgroup.map f (lowerCentralSeries G n) ≤ lowerCentralSeries H n | case succ
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
f : G →* H
d : ℕ
hd : Subgroup.map f (lowerCentralSeries G d) ≤ lowerCentralSeries H d
⊢ Subgroup.map f (lowerCentralSeries G (d + 1)) ≤ lowerCentralSeries H (d + 1) | rintro a ⟨x, hx : x ∈ lowerCentralSeries G d.succ, rfl⟩ | case succ.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
f : G →* H
d : ℕ
hd : Subgroup.map f (lowerCentralSeries G d) ≤ lowerCentralSeries H d
x : G
hx : x ∈ lowerCentralSeries G d.succ
⊢ f x ∈ lowerCentralSeries H (d + 1) | a402c9a39a4d66f9 |
Monoid.PushoutI.NormalWord.rcons_injective | Mathlib/GroupTheory/PushoutI.lean | theorem rcons_injective {i : ι} : Function.Injective (rcons (d := d) i) | case mk.mk.mk.mk
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
i : ι
head₁ : G i
tail₁ : Word G
fstIdx_ne✝¹ : tail₁.fstIdx ≠ some i
normalized✝¹ :
∀ (i_1 : ι) (g : G i_1),
... | simp only [rcons, NormalWord.mk.injEq, EmbeddingLike.apply_eq_iff_eq,
Word.Pair.mk.injEq, Pair.mk.injEq, and_imp] | case mk.mk.mk.mk
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
i : ι
head₁ : G i
tail₁ : Word G
fstIdx_ne✝¹ : tail₁.fstIdx ≠ some i
normalized✝¹ :
∀ (i_1 : ι) (g : G i_1),
... | 8eb9cc30ff7f40e1 |
Mathlib.Meta.NormNum.isInt_eq_false | Mathlib/Tactic/NormNum/Eq.lean | theorem isInt_eq_false [Ring α] [CharZero α] : {a b : α} → {a' b' : ℤ} →
IsInt a a' → IsInt b b' → decide (a' = b') = false → ¬a = b
| _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simpa using of_decide_eq_false h
| α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
n✝¹ n✝ : ℤ
h : decide (n✝¹ = n✝) = false
⊢ ¬↑n✝¹ = ↑n✝ | simpa using of_decide_eq_false h | no goals | 3398fa8d9db915c4 |
directedOn_image | Mathlib/Order/Directed.lean | theorem directedOn_image {s : Set β} {f : β → α} :
DirectedOn r (f '' s) ↔ DirectedOn (f ⁻¹'o r) s | α : Type u
β : Type v
r : α → α → Prop
s : Set β
f : β → α
⊢ DirectedOn r (f '' s) ↔ DirectedOn (f ⁻¹'o r) s | simp only [DirectedOn, Set.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂, Order.Preimage] | no goals | 819f3339eebcd917 |
Module.support_subset_of_surjective | Mathlib/RingTheory/Support.lean | lemma Module.support_subset_of_surjective (hf : Function.Surjective f) :
Module.support R N ⊆ Module.support R M | case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Surjective ⇑f
x : PrimeSpectrum R
m : M
hm : ∀ r ∉ x.asIdeal, r • f m ≠ 0
⊢ ∃ m, ∀ r ∉ x.asIdeal, r • m ≠ 0 | exact ⟨m, fun r hr e ↦ hm r hr (by simpa using congr(f $e))⟩ | no goals | aeacf4d0367f2dda |
InnerProductSpace.Core.inner_add_right | Mathlib/Analysis/InnerProductSpace/Defs.lean | theorem inner_add_right (x y z : F) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y z : F
⊢ ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜 | rw [← inner_conj_symm, inner_add_left, RingHom.map_add] | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y z : F
⊢ (starRingEnd 𝕜) ⟪y, x⟫_𝕜 + (starRingEnd 𝕜) ⟪z, x⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜 | 7df8bd68cffbf670 |
Real.sinh_nonpos_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | theorem sinh_nonpos_iff : sinh x ≤ 0 ↔ x ≤ 0 | x : ℝ
⊢ sinh x ≤ 0 ↔ x ≤ 0 | simpa only [sinh_zero] using @sinh_le_sinh x 0 | no goals | 37d5273f32e1482a |
Multiset.lt_replicate_succ | Mathlib/Data/Multiset/Replicate.lean | theorem lt_replicate_succ {m : Multiset α} {x : α} {n : ℕ} :
m < replicate (n + 1) x ↔ m ≤ replicate n x | α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) ↔ m ≤ replicate n x | constructor | case mp
α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) → m ≤ replicate n x
case mpr
α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x | c4daba5a64d21b93 |
Pell.eq_pow_of_pell | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem eq_pow_of_pell {m n k} :
n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
2 * a * n = t + (n * n + 1) ∧ m < t ∧
n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) | case mp
n k : ℕ
⊢ k = 0 ∧ n ^ k = 1 ∨
0 < k ∧
(n = 0 ∧ n ^ k = 0 ∨
0 < n ∧
∃ w a t z,
∃ (a1 : 1 < a),
xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
2 * a * n = t + (n * n + 1) ∧
n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1)... | refine k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, ?_⟩ | case mp
n k : ℕ
hk : 0 < k
⊢ n = 0 ∧ n ^ k = 0 ∨
0 < n ∧
∃ w a t z,
∃ (a1 : 1 < a),
xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
2 * a * n = t + (n * n + 1) ∧
n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1 | a363d7dbe39afd6c |
CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_transfiniteCompositionOfShape | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/EnoughInjectives.lean | /-- Let `C` be a Grothendieck abelian category. Assume that `G : C` is a generator
of `C`. Then, any morphism in `C` is a transfinite composition of pushouts
of monomorphisms in the family `generatingMonomorphisms G` which consists
of the inclusions of the subobjects of `G`. -/
lemma exists_transfiniteCompositionOfShap... | C : Type u
inst✝³ : Category.{v, u} C
G : C
inst✝² : Abelian C
hG : IsSeparator G
X : C
inst✝¹ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝ : Mono f
o : Ordinal.{w}
j : o.toType
hj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤
⊢ o ≠ 0 | simpa only [← Ordinal.toType_nonempty_iff_ne_zero] using Nonempty.intro j | no goals | 628936178606867e |
CategoryTheory.Triangulated.Localization.complete_distinguished_triangle_morphism | Mathlib/CategoryTheory/Localization/Triangulated.lean | lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle D)
(hT₁ : T₁ ∈ L.essImageDistTriang) (hT₂ : T₂ ∈ L.essImageDistTriang)
(a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretria... | dsimp [ψ] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretria... | d504ba1cf51cb937 |
BooleanSubalgebra.mem_closure_iff_sup_sdiff | Mathlib/Order/BooleanSubalgebra.lean | theorem mem_closure_iff_sup_sdiff {a : α} :
a ∈ closure s ↔ ∃ t : Finset (s × s), a = t.sup fun x ↦ x.1.1 \ x.2.1 | α : Type u_2
inst✝ : BooleanAlgebra α
s : Set α
isSublattice : IsSublattice s
bot_mem : ⊥ ∈ s
top_mem : ⊤ ∈ s
a : α
⊢ a ∈ closure s ↔ ∃ t, a = t.sup fun x => ↑x.1 \ ↑x.2 | refine ⟨closure_bot_sup_induction
(fun x h ↦ ⟨{(⟨x, h⟩, ⟨⊥, bot_mem⟩)}, by simp⟩) ⟨∅, by simp⟩ ?_ ?_, ?_⟩ | case refine_1
α : Type u_2
inst✝ : BooleanAlgebra α
s : Set α
isSublattice : IsSublattice s
bot_mem : ⊥ ∈ s
top_mem : ⊤ ∈ s
a : α
⊢ (∃ t, a = t.sup fun x => ↑x.1 \ ↑x.2) → a ∈ closure s
case refine_2
α : Type u_2
inst✝ : BooleanAlgebra α
s : Set α
isSublattice : IsSublattice s
bot_mem : ⊥ ∈ s
top_mem : ⊤ ∈ s
a : α
⊢ ∀... | 764585926be33cc2 |
Array.findIdx_lt_size | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean | theorem findIdx_lt_size {p : α → Bool} {xs : Array α} :
xs.findIdx p < xs.size ↔ ∃ x ∈ xs, p x | α : Type u_1
p : α → Bool
xs : Array α
⊢ findIdx p xs < xs.size ↔ ∃ x, x ∈ xs ∧ p x = true | rcases xs with ⟨xs⟩ | case mk
α : Type u_1
p : α → Bool
xs : List α
⊢ findIdx p { toList := xs } < { toList := xs }.size ↔ ∃ x, x ∈ { toList := xs } ∧ p x = true | 4c708ef584b1a91e |
MeasureTheory.eLpNorm_one_le_of_le | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem eLpNorm_one_le_of_le {r : ℝ≥0} (hfint : Integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ)
(hf : ∀ᵐ ω ∂μ, f ω ≤ r) : eLpNorm f 1 μ ≤ 2 * μ Set.univ * r | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
⊢ ∫ (x : α), -f x ∂μ = 0 | rw [integral_neg, neg_eq_zero] | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
⊢ ∫ (a : α), f a ∂μ = 0 | f01ab8867e396603 |
Finset.noncommProd_mul_single | Mathlib/Data/Finset/NoncommProd.lean | theorem noncommProd_mul_single [Fintype ι] [DecidableEq ι] (x : ∀ i, M i) :
(univ.noncommProd (fun i => Pi.mulSingle i (x i)) fun i _ j _ _ =>
Pi.mulSingle_apply_commute x i j) = x | case h.convert_8
ι : Type u_2
M : ι → Type u_6
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : (i : ι) → M i
i : ι
⊢ (insert i (univ.erase i)).noncommProd (fun j => Pi.mulSingle j (x j) i) ⋯ = x i | rw [noncommProd_insert_of_not_mem _ _ _ _ (not_mem_erase _ _),
noncommProd_eq_pow_card (univ.erase i), one_pow, mul_one] | case h.convert_8
ι : Type u_2
M : ι → Type u_6
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : (i : ι) → M i
i : ι
⊢ Pi.mulSingle i (x i) i = x i
case h.convert_8.h
ι : Type u_2
M : ι → Type u_6
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : (i : ι) → M i
i : ... | 0aba06cc50ea81cb |
integrable_inv_one_add_sq | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | theorem integrable_inv_one_add_sq : Integrable fun (x : ℝ) ↦ (1 + x ^ 2)⁻¹ | this : Integrable (fun x => (1 + ‖x‖ ^ 2) ^ (-2 / 2)) volume
⊢ Integrable (fun x => (1 + x ^ 2)⁻¹) volume | simpa [rpow_neg_one] | no goals | 03fc0c858722fcae |
ModP.preVal_eq_zero | Mathlib/RingTheory/Perfection.lean | theorem preVal_eq_zero {x : ModP O p} : preVal K v O p x = 0 ↔ x = 0 :=
⟨fun hvx =>
by_contradiction fun hx0 : x ≠ 0 => by
rw [← v_p_lt_preVal (hv := hv), hvx] at hx0
exact not_lt_zero' hx0,
fun hx => hx.symm ▸ preVal_zero⟩
| K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
hvx : preVal K v O p x = 0
hx0 : x ≠ 0
⊢ False | rw [← v_p_lt_preVal (hv := hv), hvx] at hx0 | K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
hvx : preVal K v O p x = 0
hx0 : v ↑p < 0
⊢ False | d0945b2aa54346ef |
CategoryTheory.Functor.preservesFiniteColimits_tfae | Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean | /--
For an addivite functor `F : C ⥤ D` between abelian categories, the following are equivalent:
- `F` preserves short exact sequences on the right hand side, i.e. if `0 ⟶ A ⟶ B ⟶ C ⟶ 0` is
exact then `F(A) ⟶ F(B) ⟶ F(C) ⟶ 0` is exact.
- `F` preserves exact sequences on the right hand side, i.e. if `A ⟶ B ⟶ C` is ex... | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
(∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F... | let hS := hF S ⟨exact_cokernel f, inferInstance⟩ | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
(∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F... | fda2dfddec035e24 |
Nat.div_div_div_eq_div | Mathlib/Data/Nat/Init.lean | @[simp] lemma div_div_div_eq_div (dvd : b ∣ a) (dvd2 : a ∣ c) : c / (a / b) / b = c / a :=
match a, b, c with
| 0, _, _ => by simp
| a + 1, 0, _ => by simp at dvd
| a + 1, c + 1, _ => by
have a_split : a + 1 ≠ 0 := succ_ne_zero a
have c_split : c + 1 ≠ 0 := succ_ne_zero c
rcases dvd2 with ⟨k, rfl⟩
... | case intro
a✝ b c✝ a c : ℕ
dvd : c + 1 ∣ a + 1
a_split : a + 1 ≠ 0
c_split : c + 1 ≠ 0
k : ℕ
⊢ (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1) | rcases dvd with ⟨k2, pr⟩ | case intro.intro
a✝ b c✝ a c : ℕ
a_split : a + 1 ≠ 0
c_split : c + 1 ≠ 0
k k2 : ℕ
pr : a + 1 = (c + 1) * k2
⊢ (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1) | eab95b5d7fe19e6b |
Finsupp.mem_toMultiset | Mathlib/Data/Finsupp/Multiset.lean | theorem mem_toMultiset (f : α →₀ ℕ) (i : α) : i ∈ toMultiset f ↔ i ∈ f.support | α : Type u_1
f : α →₀ ℕ
i : α
⊢ i ∈ toMultiset f ↔ i ∈ f.support | classical
rw [← Multiset.count_ne_zero, Finsupp.count_toMultiset, Finsupp.mem_support_iff] | no goals | 3b970731ab3a9f50 |
toAdd_list_sum | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem toAdd_list_sum (s : List (Multiplicative α)) : s.prod.toAdd = (s.map toAdd).sum | α : Type u_3
inst✝ : AddMonoid α
s : List (Multiplicative α)
⊢ s.prod = s.sum | rfl | no goals | c8980a13b1be89a2 |
MeasureTheory.MemLp.eLpNorm_indicator_norm_ge_le | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem MemLp.eLpNorm_indicator_norm_ge_le (hf : MemLp f p μ) (hmeas : StronglyMeasurable f) {ε : ℝ}
(hε : 0 < ε) : ∃ M : ℝ, eLpNorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε | case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε | refine ⟨1, hp_ne_zero.symm ▸ ?_⟩ | case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ eLpNorm ({x | 1 ≤ ↑‖f x‖₊}.indicator f) 0 μ ≤ ENNReal.ofReal ε | bfa6a8839e926f05 |
MvPolynomial.indicator_mem_restrictDegree | Mathlib/FieldTheory/Finite/Polynomial.lean | theorem indicator_mem_restrictDegree (c : σ → K) :
indicator c ∈ restrictDegree σ K (Fintype.card K - 1) | case refine_2
K : Type u_1
σ : Type u_2
inst✝² : Fintype K
inst✝¹ : Fintype σ
inst✝ : CommRing K
c : σ → K
n : σ
⊢ n ∉ Finset.univ → (Fintype.card K - 1) * Multiset.count n {n} = 0 | intro h | case refine_2
K : Type u_1
σ : Type u_2
inst✝² : Fintype K
inst✝¹ : Fintype σ
inst✝ : CommRing K
c : σ → K
n : σ
h : n ∉ Finset.univ
⊢ (Fintype.card K - 1) * Multiset.count n {n} = 0 | fbdd72868819cd4c |
CategoryTheory.IsIso.inv_comp | Mathlib/CategoryTheory/Iso.lean | theorem inv_comp [IsIso f] [IsIso h] : inv (f ≫ h) = inv h ≫ inv f | case hom_inv_id
C : Type u
inst✝² : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
h : Y ⟶ Z
inst✝¹ : IsIso f
inst✝ : IsIso h
⊢ (f ≫ h) ≫ inv h ≫ inv f = 𝟙 X | simp | no goals | 4c5858ea8a1326f3 |
List.filterMap_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filterMap_eq_append_iff {f : α → Option β} :
filterMap f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂ | case mp.cons.h_2.inr.intro.intro
α : Type u_1
β : Type u_2
L₂ : List β
f : α → Option β
x : α
l : List α
ih : ∀ {L₁ : List β}, filterMap f l = L₁ ++ L₂ → ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂
x✝ : Option β
b : β
w : f x = some b
L₁ : List β
h✝ : filterMap f l = L₁ ++ L₂
h : b :: filterMap f ... | obtain ⟨l₁, l₂, rfl, rfl, rfl⟩ := ih ‹_› | case mp.cons.h_2.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
x : α
x✝ : Option β
b : β
w : f x = some b
l₁ l₂ : List α
ih :
∀ {L₁ : List β},
filterMap f (l₁ ++ l₂) = L₁ ++ filterMap f l₂ →
∃ l₁_1 l₂_1, l₁ ++ l₂ = l₁_1 ++ l₂_1 ∧ filterMap f l₁_1 = L₁ ∧ filterMap f l₂_1 ... | e698eece5419b8d8 |
IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | theorem tendsto_closedBall_filterAt {K : ℝ} {x : α} {ι : Type*} {l : Filter ι} (w : ι → α)
(δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0)) (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)) :
Tendsto (fun j => closedBall (w j) (δ j)) l ((vitaliFamily μ K).filterAt x) | case refine_2.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x : α
ι : Type u_2
l : Filter ι
w : ι → α
δ : ι → ℝ
xmem : ∀ᶠ (j : ι) in l, x ∈ closedB... | rintro j ⟨⟨hjε, hj₀ : 0 < δ j⟩, hx⟩ y hy | case refine_2.inr.intro.intro
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x : α
ι : Type u_2
l : Filter ι
w : ι → α
δ : ι → ℝ
xmem : ∀ᶠ (j : ι) in l,... | 09e360431f9e0376 |
minpoly.unique' | Mathlib/FieldTheory/Minpoly/Basic.lean | theorem unique' {p : A[X]} (hm : p.Monic) (hp : Polynomial.aeval x p = 0)
(hl : ∀ q : A[X], degree q < degree p → q = 0 ∨ Polynomial.aeval x q ≠ 0) :
p = minpoly A x | A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
hm : p.Monic
hp : (Polynomial.aeval x) p = 0
hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0
⊢ p = minpoly A x | nontriviality A | A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
hm : p.Monic
hp : (Polynomial.aeval x) p = 0
hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0
a✝ : Nontrivial A
⊢ p = minpoly A x | 7aeb1fe2585c14a6 |
strictConcaveOn_log_Ioi | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log | x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < z - y
hyz' : 0 < z / y
⊢ z / y ≠ 1 | contrapose! h | x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
hyz' : 0 < z / y
h : z / y = 1
⊢ z - y ≤ 0 | b669e8d780cc761c |
Module.finite_of_finrank_pos | Mathlib/LinearAlgebra/Dimension/Free.lean | theorem finite_of_finrank_pos (h : 0 < finrank R M) : Module.Finite R M | R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : StrongRankCondition R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
h : 0 < finrank R M
⊢ Module.Finite R M | contrapose h | R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : StrongRankCondition R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
h : ¬Module.Finite R M
⊢ ¬0 < finrank R M | 5b0cb550fa8d5395 |
NonarchimedeanGroup.cauchySeq_of_tendsto_div_nhds_one | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | /-- Let `G` be a nonarchimedean abelian group, and let `f : ℕ → G` be a function
such that the quotients `f (n + 1) / f n` tend to one. Then the function is a Cauchy sequence. -/
@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : ℕ → G` be a
function such that the differences `f (n + 1) - f... | case intro
G : Type u_2
inst✝³ : CommGroup G
inst✝² : UniformSpace G
inst✝¹ : UniformGroup G
inst✝ : NonarchimedeanGroup G
f : ℕ → G
hf : Tendsto (fun n => f (n + 1) / f n) atTop (𝓝 1)
s : Set G
hs : s ∈ 𝓝 1
t : OpenSubgroup G
ht : ↑t ⊆ s
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → f (b + 1) / f b ∈ t
M : ℕ
hMN : N ≤ M
k : ℕ
⊢ f (... | induction k with
| zero => simpa using one_mem t
| succ k ih => simpa using t.mul_mem (hN _ (by omega : N ≤ M + k)) ih | no goals | 95cdab31772386ea |
Finsupp.mapDomain_apply' | Mathlib/Data/Finsupp/Basic.lean | theorem mapDomain_apply' (S : Set α) {f : α → β} (x : α →₀ M) (hS : (x.support : Set α) ⊆ S)
(hf : Set.InjOn f S) {a : α} (ha : a ∈ S) : mapDomain f x (f a) = x a | case neg
α : Type u_1
β : Type u_2
M : Type u_5
inst✝ : AddCommMonoid M
S : Set α
f : α → β
x : α →₀ M
hS : ↑x.support ⊆ S
hf : Set.InjOn f S
a : α
ha : a ∈ S
hax : a ∉ x.support
i : α
hi : i ∈ x.support
⊢ ¬f i = f a | exact hf.ne (hS hi) ha (ne_of_mem_of_not_mem hi hax) | no goals | 7d55c7cd92e5faee |
Int.natCast_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Pow.lean | theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n | b n : Nat
⊢ ↑(b ^ n) = ↑b ^ n | match n with
| 0 => rfl
| n + 1 =>
simp only [Nat.pow_succ, Int.pow_succ, natCast_mul, natCast_pow _ n] | no goals | 22381b0d4e566b6c |
Polynomial.isIntegrallyClosed_iff' | Mathlib/RingTheory/Polynomial/GaussLemma.lean | theorem isIntegrallyClosed_iff' [IsDomain R] :
IsIntegrallyClosed R ↔
∀ p : R[X], p.Monic → (Irreducible p ↔ Irreducible (p.map <| algebraMap R K)) | case mpr
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
⊢ (∀ (p : R[X]), p.Monic → (Irreducible p ↔ Irreducible (map (algebraMap R K) p))) → IsIntegrallyClosed R | intro H | case mpr
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
H : ∀ (p : R[X]), p.Monic → (Irreducible p ↔ Irreducible (map (algebraMap R K) p))
⊢ IsIntegrallyClosed R | 399cb77d787c13b1 |
Nat.findGreatest_eq_iff | Mathlib/Data/Nat/Find.lean | lemma findGreatest_eq_iff :
Nat.findGreatest P k = m ↔ m ≤ k ∧ (m ≠ 0 → P m) ∧ ∀ ⦃n⦄, m < n → n ≤ k → ¬P n | case neg.mp.intro.intro
P : ℕ → Prop
inst✝ : DecidablePred P
k : ℕ
ihk : ∀ {m : ℕ}, findGreatest P k = m ↔ m ≤ k ∧ (m ≠ 0 → P m) ∧ ∀ ⦃n : ℕ⦄, m < n → n ≤ k → ¬P n
m : ℕ
hk : ¬P (k + 1)
hle✝ : m ≤ k
hP : m ≠ 0 → P m
hm : ∀ ⦃n : ℕ⦄, m < n → n ≤ k → ¬P n
n : ℕ
hlt : m < n
hle : n ≤ k + 1
⊢ ¬P n | rcases Decidable.eq_or_lt_of_le hle with (rfl | hlt') | case neg.mp.intro.intro.inl
P : ℕ → Prop
inst✝ : DecidablePred P
k : ℕ
ihk : ∀ {m : ℕ}, findGreatest P k = m ↔ m ≤ k ∧ (m ≠ 0 → P m) ∧ ∀ ⦃n : ℕ⦄, m < n → n ≤ k → ¬P n
m : ℕ
hk : ¬P (k + 1)
hle✝ : m ≤ k
hP : m ≠ 0 → P m
hm : ∀ ⦃n : ℕ⦄, m < n → n ≤ k → ¬P n
hlt : m < k + 1
hle : k + 1 ≤ k + 1
⊢ ¬P (k + 1)
case neg.mp.in... | 9905ebac396451ee |
Complex.HadamardThreeLines.diffContOnCl_interpStrip | Mathlib/Analysis/Complex/Hadamard.lean | lemma diffContOnCl_interpStrip :
DiffContOnCl ℂ (interpStrip f) (verticalStrip 0 1) | case h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ ¬sSupNormIm f 0 = 0 | rwa [eq_comm] | no goals | a658f86b7165f3fe |
padicNorm.int_eq_one_iff | Mathlib/NumberTheory/Padics/PadicNorm.lean | theorem int_eq_one_iff (m : ℤ) : padicNorm p m = 1 ↔ ¬(p : ℤ) ∣ m | case pos
p : ℕ
hp : Fact (Nat.Prime p)
m : ℤ
h✝ : ↑m = 0
⊢ (↑p)⁻¹ < 0 → 0 = 1 | rw [inv_lt_zero, ← Nat.cast_zero, Nat.cast_lt] | case pos
p : ℕ
hp : Fact (Nat.Prime p)
m : ℤ
h✝ : ↑m = 0
⊢ p < 0 → ↑0 = 1 | 5a294b98dd015827 |
List.Vector.scanl_get | Mathlib/Data/Vector/Basic.lean | theorem scanl_get (i : Fin n) :
(scanl f b v).get i.succ = f ((scanl f b v).get (Fin.castSucc i)) (v.get i) | case succ.zero
α : Type u_1
β : Type u_6
f : β → α → β
b : β
v : Vector α (0 + 1)
i : Fin (0 + 1)
⊢ (scanl f b v).get i.succ = f ((scanl f b v).get i.castSucc) (v.get i) | have i0 : i = 0 := Fin.eq_zero _ | case succ.zero
α : Type u_1
β : Type u_6
f : β → α → β
b : β
v : Vector α (0 + 1)
i : Fin (0 + 1)
i0 : i = 0
⊢ (scanl f b v).get i.succ = f ((scanl f b v).get i.castSucc) (v.get i) | d55723b163424dfc |
isApproximateSubgroup_one | Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean | /-- A `1`-approximate subgroup is the same thing as a subgroup. -/
@[to_additive (attr := simp)
"A `1`-approximate subgroup is the same thing as a subgroup."]
lemma isApproximateSubgroup_one {A : Set G} :
IsApproximateSubgroup 1 (A : Set G) ↔ ∃ H : Subgroup G, H = A where
mp hA | case intro.intro
G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
K : Finset G
hKA : A ^ 2 ⊆ ↑K • A
hK : ∃ x, K = ∅ ∨ K = {x}
⊢ ∃ x, A * A ⊆ x • A | obtain ⟨x, rfl | rfl⟩ := hK | case intro.intro.intro.inl
G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hKA : A ^ 2 ⊆ ↑∅ • A
⊢ ∃ x, A * A ⊆ x • A
case intro.intro.intro.inr
G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hKA : A ^ 2 ⊆ ↑{x} • A
⊢ ∃ x, A * A ⊆ x • A | 1d755e438e264b32 |
pow_unbounded_of_one_lt | Mathlib/Algebra/Order/Archimedean/Basic.lean | lemma pow_unbounded_of_one_lt [ExistsAddOfLE α] (x : α) (hy1 : 1 < y) : ∃ n : ℕ, x < y ^ n | α : Type u_1
inst✝² : StrictOrderedSemiring α
inst✝¹ : Archimedean α
y : α
inst✝ : ExistsAddOfLE α
x : α
hy1 : 1 < y
⊢ ∃ n, x < y ^ n | obtain ⟨z, hz, rfl⟩ := exists_pos_add_of_lt' hy1 | case intro.intro
α : Type u_1
inst✝² : StrictOrderedSemiring α
inst✝¹ : Archimedean α
inst✝ : ExistsAddOfLE α
x z : α
hz : 0 < z
hy1 : 1 < 1 + z
⊢ ∃ n, x < (1 + z) ^ n | ea14394363994ad4 |
Filter.Tendsto.exists_within_forall_le | Mathlib/Order/Filter/Cofinite.lean | theorem Filter.Tendsto.exists_within_forall_le {α β : Type*} [LinearOrder β] {s : Set α}
(hs : s.Nonempty) {f : α → β} (hf : Filter.Tendsto f Filter.cofinite Filter.atTop) :
∃ a₀ ∈ s, ∀ a ∈ s, f a₀ ≤ f a | case inl.intro.intro.intro
α : Type u_4
β : Type u_5
inst✝ : LinearOrder β
s : Set α
hs : s.Nonempty
f : α → β
hf : Tendsto f cofinite atTop
y : α
hys : y ∈ s
x : β
hx : f y < x
this : {y | ¬x ≤ f y}.Finite
⊢ ∃ a₀ ∈ s, ∀ a ∈ s, f a₀ ≤ f a | simp only [not_le] at this | case inl.intro.intro.intro
α : Type u_4
β : Type u_5
inst✝ : LinearOrder β
s : Set α
hs : s.Nonempty
f : α → β
hf : Tendsto f cofinite atTop
y : α
hys : y ∈ s
x : β
hx : f y < x
this : {y | f y < x}.Finite
⊢ ∃ a₀ ∈ s, ∀ a ∈ s, f a₀ ≤ f a | ca1090b0ea7bdd91 |
IsFractionRing.algHom_fieldRange_eq_of_comp_eq_of_range_eq | Mathlib/FieldTheory/IntermediateField/Adjoin/Algebra.lean | theorem algHom_fieldRange_eq_of_comp_eq_of_range_eq
(h : RingHom.comp f (algebraMap A K) = (g : A →+* L))
{s : Set L} (hs : g.range = Algebra.adjoin F s) :
f.fieldRange = IntermediateField.adjoin F s | F : Type u_1
A : Type u_2
K : Type u_3
L : Type u_4
inst✝⁸ : Field F
inst✝⁷ : CommRing A
inst✝⁶ : Algebra F A
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : Field L
inst✝ : Algebra F L
g : A →ₐ[F] L
f : K →ₐ[F] L
h : (↑f).comp (algebraMap A K) = ↑g
s : Set L
hs : g.range... | apply IntermediateField.toSubfield_injective | case a
F : Type u_1
A : Type u_2
K : Type u_3
L : Type u_4
inst✝⁸ : Field F
inst✝⁷ : CommRing A
inst✝⁶ : Algebra F A
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : Field L
inst✝ : Algebra F L
g : A →ₐ[F] L
f : K →ₐ[F] L
h : (↑f).comp (algebraMap A K) = ↑g
s : Set L
hs : ... | 685c99ec804c2a47 |
LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
(hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ (T x₀))) x₀ | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
T : F →L[ℝ] F
hT : (↑T).IsSymmetric
x₀ : F
⊢ HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ) (T x₀)) x₀ | convert T.hasStrictFDerivAt.inner ℝ (hasStrictFDerivAt_id x₀) using 1 | case h.e'_12.h.h.h
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
T : F →L[ℝ] F
hT : (↑T).IsSymmetric
x₀ : F
e_4✝ : NormedAddCommGroup.toAddCommGroup = SeminormedAddCommGroup.toAddCommGroup
he✝¹ : NormedSpace.toModule = NormedSpace.toModule
e_8✝ : Real.instAddCommGroup = SeminormedAddCommGroup... | e5f6c554ea5c2120 |
Nat.pos_of_mem_divisors | Mathlib/NumberTheory/Divisors.lean | theorem pos_of_mem_divisors {m : ℕ} (h : m ∈ n.divisors) : 0 < m | n m : ℕ
h : m ∈ n.divisors
⊢ 0 < m | cases m | case zero
n : ℕ
h : 0 ∈ n.divisors
⊢ 0 < 0
case succ
n n✝ : ℕ
h : n✝ + 1 ∈ n.divisors
⊢ 0 < n✝ + 1 | f81a5b51eb4f3d13 |
Array.size_extract_loop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem size_extract_loop (as bs : Array α) (size start : Nat) :
(extract.loop as size start bs).size = bs.size + min size (as.size - start) | α : Type u_1
as : Array α
size : Nat
ih : ∀ (bs : Array α) (start : Nat), (extract.loop as size start bs).size = bs.size + min size (as.size - start)
bs : Array α
start : Nat
h : ¬start < as.size
⊢ (extract.loop as (size + 1) start bs).size = bs.size + min (size + 1) (as.size - start) | have h := Nat.le_of_not_gt h | α : Type u_1
as : Array α
size : Nat
ih : ∀ (bs : Array α) (start : Nat), (extract.loop as size start bs).size = bs.size + min size (as.size - start)
bs : Array α
start : Nat
h✝ : ¬start < as.size
h : as.size ≤ start
⊢ (extract.loop as (size + 1) start bs).size = bs.size + min (size + 1) (as.size - start) | 33a6be218948d9fd |
MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
(h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
(ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
... | have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
⟨0, ((measure_mono diff_subset).trans_lt h''t.lt_top).ne⟩ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
... | 8e4b11ebb89db50e |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
(μ : Measure α) [SFinite μ]
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
... | α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNRea... | simpa only [mem_univ, Pi.zero_apply, true_and] using
measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ | no goals | 028c71a79fc18349 |
Real.sin_int_mul_pi_sub | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem sin_int_mul_pi_sub (x : ℝ) (n : ℤ) : sin (n * π - x) = -((-1) ^ n * sin x) | x : ℝ
n : ℤ
⊢ sin (↑n * π - x) = -((-1) ^ n * sin x) | simpa only [sin_neg, mul_neg, Int.cast_negOnePow] using sin_antiperiodic.int_mul_sub_eq n | no goals | 66c8a0c13694c32a |
DFinsupp.single_eq_single_iff | Mathlib/Data/DFinsupp/Defs.lean | theorem single_eq_single_iff (i j : ι) (xi : β i) (xj : β j) :
DFinsupp.single i xi = DFinsupp.single j xj ↔ i = j ∧ HEq xi xj ∨ xi = 0 ∧ xj = 0 | case neg
ι : Type u
β : ι → Type v
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : DecidableEq ι
i j : ι
xi : β i
xj : β j
h : single i xi = single j xj
hij : ¬i = j
⊢ i = j ∧ HEq xi xj ∨ xi = 0 ∧ xj = 0 | have h_coe : ⇑(DFinsupp.single i xi) = DFinsupp.single j xj := congr_arg (⇑) h | case neg
ι : Type u
β : ι → Type v
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : DecidableEq ι
i j : ι
xi : β i
xj : β j
h : single i xi = single j xj
hij : ¬i = j
h_coe : ⇑(single i xi) = ⇑(single j xj)
⊢ i = j ∧ HEq xi xj ∨ xi = 0 ∧ xj = 0 | 1a074302bda6952c |
legendreSym.eq_zero_mod_of_eq_neg_one | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | theorem eq_zero_mod_of_eq_neg_one {p : ℕ} [Fact p.Prime] {a : ℤ} (h : legendreSym p a = -1)
{x y : ZMod p} (hxy : x ^ 2 - a * y ^ 2 = 0) : x = 0 ∧ y = 0 | p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
h : legendreSym p a = -1
x y : ZMod p
hxy : x ^ 2 - ↑a * y ^ 2 = 0
hf : ↑a = 0
⊢ False | rw [(eq_zero_iff p a).mpr hf] at h | p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℤ
h : 0 = -1
x y : ZMod p
hxy : x ^ 2 - ↑a * y ^ 2 = 0
hf : ↑a = 0
⊢ False | 627576b8706b1158 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.