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
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|---|---|---|---|---|---|---|
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 = det N
|
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 i * M i.castSucc j) → M.det = N.det
c : Fin n → R
hc : ∀ (i : Fin n), k.succ < i.succ → c i = 0
M N : Matrix (Fin n.succ) (Fin n.succ) R
h0 : ∀ (j : Fin n.succ), M 0 j = N 0 j
hsucc : ∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c i * M i.castSucc j
M' : Matrix (Fin n.succ) (Fin n.succ) R := M.updateRow k.succ (N k.succ)
hM' : M' = M.updateRow k.succ (N k.succ)
hM : M = M'.updateRow k.succ (M' k.succ + c k • M k.castSucc)
k_ne_succ : k.castSucc ≠ k.succ
M_k : M k.castSucc = M' k.castSucc
i : Fin n
hi : ↑k ≤ ↑i
hik : i = k
⊢ 0 = 0
|
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))
(algEquivMatrix'.symm A * algEquivMatrix'.symm A') =
(PiToModule.fromEnd R b) (f * f')
|
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.linearCombination R R) b))
(algEquivMatrix'.symm A * algEquivMatrix'.symm A') ∘ₗ
LinearMap.single R (fun i => R) i✝)
1 =
((PiToModule.fromEnd R b) (f * f') ∘ₗ LinearMap.single R (fun i => R) i✝) 1
|
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 => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (insert a t \ {x}) \ {a} = t \ {x}
|
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 : ∀ z ∈ sphere 0 1, ∀ᶠ (t : ℂ) in 𝓝 0, gray z t = gray z 0
z : E
hz : z ∈ ball z₀ r
h' : ‖z - z₀‖ ≠ 0
w : E := ‖z - z₀‖⁻¹ • (z - z₀)
h3 : ∀ t ∈ ball 0 r, gray w t = g z₀
⊢ g z = g z₀
|
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 : ∀ z ∈ sphere 0 1, ∀ᶠ (t : ℂ) in 𝓝 0, gray z t = gray z 0
z : E
hz : z ∈ ball z₀ r
h' : ‖z - z₀‖ ≠ 0
w : E := ‖z - z₀‖⁻¹ • (z - z₀)
h3 : ∀ t ∈ ball 0 r, gray w t = g z₀
h4 : ‖z - z₀‖ < r
⊢ g z = g z₀
|
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.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
⊢ (Ideal.Quotient.mk I) (∑ g : { x // x ∈ s }, g' g * h' g) - 1 = 0
|
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.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
⊢ ∑ g : { x // x ∈ s }, g' g * h' g - 1 ∈ I
|
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 > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u + g u ∈ Set.Icc ((c₁ ⊓ c₃) * (f x + g x)) ((c₂ ⊔ c₄) * (f x + g x))
|
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 > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
⊢ f u + g u ∈ Set.Icc ((c₁ ⊓ c₃) * (f x + g x)) ((c₂ ⊔ c₄) * (f x + g x))
|
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' g').base ⁻¹' Set.range i₂.base
|
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 (pullback.fst f' g').base) z
y : ↑↑Y.toPresheafedSpace
hy : (ConcreteCategory.hom i₂.base) y = (ConcreteCategory.hom (pullback.snd f' g').base) z
⊢ z ∈ Set.range ⇑(ConcreteCategory.hom (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂).base)
|
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 (pullback.fst f' g').base) z
y : ↑↑Y.toPresheafedSpace
hy : (ConcreteCategory.hom i₂.base) y = (ConcreteCategory.hom (pullback.snd f' g').base) z
T₁ : Triplet (pullback.fst f' g') i₁ := Triplet.mk' z x ⋯
⊢ z ∈ Set.range ⇑(ConcreteCategory.hom (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂).base)
|
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.testBit i)
x n✝ n : Nat
h₂ : x < 2 ^ (n + 1)
⊢ x / 2 < 2 ^ n
|
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' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
|
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' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ ↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm = (↑I' ∘ ↑e' ∘ ↑I'.symm) ∘ ↑I' ∘ f ∘ ↑I.symm
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' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ MapsTo (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I'.symm ⁻¹' e'.source ∩ range ↑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 H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
hf : ((killingForm K L) e) f ≠ 0
this : ⁅e, f⁆ = ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))
⊢ ⁅(((killingForm K L) e) f)⁻¹ • e, f⁆ = ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))
|
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) l)
|
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] y.2 = (⇑(frobenius K p))^[y.1 + z2] z.2
⊢ (⇑(frobenius K p))^[x.1 + z1] ((⇑(frobenius K p))^[y.1 + z2] z.2) = (⇑(frobenius K p))^[x.1 + (z2 + (y.1 + z1))] z.2
|
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] y.2 = (⇑(frobenius K p))^[y.1 + z2] z.2
⊢ (⇑(frobenius K p))^[x.1 + z1 + (y.1 + z2)] z.2 = (⇑(frobenius K p))^[x.1 + (z2 + (y.1 + z1))] z.2
|
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') < ε
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
εε₁ : ε₁ < ε
ε₂ : ℝ := ε₁ / 2 / 2
ε₂0 : ε₂ > 0
U : α → Set α
hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ y ∈ U x, ∀ z ∈ U x, ∀ {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂
tα : Set α
left✝¹ : tα ⊆ univ
hfin✝ : tα.Finite
htα : univ ⊆ ⋃ x ∈ tα, U x
val✝¹ : Fintype ↑tα
tβ : Set β
left✝ : tβ ⊆ univ
hfin : tβ.Finite
htβ : univ ⊆ ⋃ y ∈ tβ, ball y ε₂
val✝ : Fintype ↑tβ
F : β → β
hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂
f : α →ᵇ β
hf : f ∈ A
g : α →ᵇ β
hg : g ∈ A
f_eq_g : (fun f a => ⟨F (↑f ↑a), ⋯⟩) ⟨f, hf⟩ = (fun f a => ⟨F (↑f ↑a), ⋯⟩) ⟨g, hg⟩
x : α
⊢ dist (↑⟨f, hf⟩ x) (↑⟨g, hg⟩ x) ≤ ε₁
|
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 (↑i x) (↑i x') < ε
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
εε₁ : ε₁ < ε
ε₂ : ℝ := ε₁ / 2 / 2
ε₂0 : ε₂ > 0
U : α → Set α
hU : ∀ (x : α), x ∈ U x ∧ IsOpen (U x) ∧ ∀ y ∈ U x, ∀ z ∈ U x, ∀ {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂
tα : Set α
left✝¹ : tα ⊆ univ
hfin✝ : tα.Finite
htα : univ ⊆ ⋃ x ∈ tα, U x
val✝¹ : Fintype ↑tα
tβ : Set β
left✝ : tβ ⊆ univ
hfin : tβ.Finite
htβ : univ ⊆ ⋃ y ∈ tβ, ball y ε₂
val✝ : Fintype ↑tβ
F : β → β
hF : ∀ (y : β), F y ∈ tβ ∧ dist y (F y) < ε₂
f : α →ᵇ β
hf : f ∈ A
g : α →ᵇ β
hg : g ∈ A
f_eq_g : (fun f a => ⟨F (↑f ↑a), ⋯⟩) ⟨f, hf⟩ = (fun f a => ⟨F (↑f ↑a), ⋯⟩) ⟨g, hg⟩
x x' : α
x'tα : x' ∈ tα
hx' : x ∈ U x'
⊢ dist (↑⟨f, hf⟩ x) (↑⟨g, hg⟩ x) ≤ ε₁
|
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, y) ∂μ ∂ν
|
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.partialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.partialHomeomorphSubtypeCoe hV).symm (↑e.symm y) = i (↑(U.partialHomeomorphSubtypeCoe hU).symm (↑e.symm y))
|
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.partialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.partialHomeomorphSubtypeCoe hV).symm (↑e.symm y) ∈ (V.partialHomeomorphSubtypeCoe hV).source
case refine_2
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.partialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.partialHomeomorphSubtypeCoe hV) (↑(V.partialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
↑(V.partialHomeomorphSubtypeCoe hV) (i (↑(U.partialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))
|
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 (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
h2 : ∃ a, ι a = gg ((ConcreteCategory.hom (C.π.app j)) x)
⊢ f x = if h : ∃ a, ι a = gg ((TopCat.Hom.hom (C.π.app j)) x) then h.choose else default
|
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 (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
h2 : ∃ a, ι a = gg ((ConcreteCategory.hom (C.π.app j)) x)
⊢ f x = h2.choose
|
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 => ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : α →ₛ ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
C : ℝ≥0
hC : C ∈ upperBounds ↑ψ.range
⊢ ∃ n, ∀ (x : α), ψ x ≤ ↑n
|
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),
⟨i_1, g⟩ ∈ { head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹ }.tail.toList → g ∈ d.set i_1
head₂ : G i
tail₂ : Word G
fstIdx_ne✝ : tail₂.fstIdx ≠ some i
normalized✝ :
∀ (i_1 : ι) (g : G i_1),
⟨i_1, g⟩ ∈ { head := head₂, tail := tail₂, fstIdx_ne := fstIdx_ne✝ }.tail.toList → g ∈ d.set i_1
⊢ rcons i { head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹, normalized := normalized✝¹ } =
rcons i { head := head₂, tail := tail₂, fstIdx_ne := fstIdx_ne✝, normalized := normalized✝ } →
{ head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹, normalized := normalized✝¹ } =
{ head := head₂, tail := tail₂, fstIdx_ne := fstIdx_ne✝, normalized := normalized✝ }
|
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),
⟨i_1, g⟩ ∈ { head := head₁, tail := tail₁, fstIdx_ne := fstIdx_ne✝¹ }.tail.toList → g ∈ d.set i_1
head₂ : G i
tail₂ : Word G
fstIdx_ne✝ : tail₂.fstIdx ≠ some i
normalized✝ :
∀ (i_1 : ι) (g : G i_1),
⟨i_1, g⟩ ∈ { head := head₂, tail := tail₂, fstIdx_ne := fstIdx_ne✝ }.tail.toList → g ∈ d.set i_1
⊢ ↑(⋯.equiv head₁).2 = ↑(⋯.equiv head₂).2 →
tail₁ = tail₂ → (⋯.equiv head₁).1 = (⋯.equiv head₂).1 → head₁ = head₂ ∧ tail₁ = tail₂
|
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) - 1) * (w * z) * (w * z) = 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_transfiniteCompositionOfShape :
∃ (J : Type w) (_ : LinearOrder J) (_ : OrderBot J) (_ : SuccOrder J)
(_ : WellFoundedLT J),
Nonempty ((generatingMonomorphisms G).pushouts.TransfiniteCompositionOfShape J f)
|
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 : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
this✝¹ : IsIso (L.map β.s)
this✝ : IsIso (L.map γ.s)
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
hT₃ : T₃ ∈ distinguishedTriangles
hβγσ : W (β.s ≫ γ.s ≫ σ)
ψ₃ : T₂.obj₃ ⟶ T₃.obj₃
hψ₃ : W ψ₃
hψ₁ : T₂.mor₂ ≫ ψ₃ = (β.s ≫ γ.s ≫ σ) ≫ T₃.mor₂
hψ₂ : T₂.mor₃ ≫ (shiftFunctor C 1).map α.s = ψ₃ ≫ T₃.mor₃
ψ : T₂ ⟶ T₃ := T₂.homMk T₃ α.s (β.s ≫ γ.s ≫ σ) ψ₃ ⋯ hψ₁ hψ₂
this : IsIso (L.mapTriangle.map ψ)
⊢ (L.mapTriangle.map (completeDistinguishedTriangleMorphism T₁ T₃ hT₁ hT₃ α.f (γ.f ≫ σ) ⋯)).hom₂ =
b ≫ (L.mapTriangle.map ψ).hom₂
|
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 : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
this✝¹ : IsIso (L.map β.s)
this✝ : IsIso (L.map γ.s)
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
hT₃ : T₃ ∈ distinguishedTriangles
hβγσ : W (β.s ≫ γ.s ≫ σ)
ψ₃ : T₂.obj₃ ⟶ T₃.obj₃
hψ₃ : W ψ₃
hψ₁ : T₂.mor₂ ≫ ψ₃ = (β.s ≫ γ.s ≫ σ) ≫ T₃.mor₂
hψ₂ : T₂.mor₃ ≫ (shiftFunctor C 1).map α.s = ψ₃ ≫ T₃.mor₃
ψ : T₂ ⟶ T₃ := T₂.homMk T₃ α.s (β.s ≫ γ.s ≫ σ) ψ₃ ⋯ hψ₁ hψ₂
this : IsIso (L.mapTriangle.map ψ)
⊢ L.map (γ.f ≫ σ) = b ≫ L.map (β.s ≫ γ.s ≫ σ)
|
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
⊢ ∀ x ∈ closure s,
∀ y ∈ closure s,
(∃ t, x = t.sup fun x => ↑x.1 \ ↑x.2) →
(∃ t, y = t.sup fun x => ↑x.1 \ ↑x.2) → ∃ t, x ⊔ y = t.sup fun x => ↑x.1 \ ↑x.2
case refine_3
α : Type u_2
inst✝ : BooleanAlgebra α
s : Set α
isSublattice : IsSublattice s
bot_mem : ⊥ ∈ s
top_mem : ⊤ ∈ s
a : α
⊢ ∀ x ∈ closure s, (∃ t, x = t.sup fun x => ↑x.1 \ ↑x.2) → ∃ t, xᶜ = t.sup fun x => ↑x.1 \ ↑x.2
|
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 : ι
⊢ ∀ x_1 ∈ univ.erase i, Pi.mulSingle x_1 (x x_1) i = 1
|
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 exact where `B ⟶ C`
is epi, then `F(A) ⟶ F(B) ⟶ F(C) ⟶ 0` is exact and `F(B) ⟶ F(C)` is epi as well.
- `F` preserves cokernels.
- `F` preserves finite colimits.
-/
lemma preservesFiniteColimits_tfae : List.TFAE
[
∀ (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.map S.g),
∀ ⦃X Y : C⦄ (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F,
PreservesFiniteColimits F
]
|
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.map S.g)
hF : ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
X Y : C
f : X ⟶ Y
S : ShortComplex C := ShortComplex.mk f (cokernel.π f) ⋯
⊢ IsColimit (CokernelCofork.map (Cofork.ofπ (cokernel.π f) ⋯) 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.map S.g)
hF : ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
X Y : C
f : X ⟶ Y
S : ShortComplex C := ShortComplex.mk f (cokernel.π f) ⋯
hS : (S.map F).Exact ∧ Epi (F.map S.g) := hF S ⟨exact_cokernel f, inferInstance⟩
⊢ IsColimit (CokernelCofork.map (Cofork.ofπ (cokernel.π f) ⋯) 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⟩
rcases dvd with ⟨k2, pr⟩
have k2_nonzero : k2 ≠ 0 := fun k2_zero => by simp [k2_zero] at pr
rw [Nat.mul_div_cancel_left k (Nat.pos_of_ne_zero a_split), pr,
Nat.mul_div_cancel_left k2 (Nat.pos_of_ne_zero c_split), Nat.mul_comm ((c + 1) * k2) k, ←
Nat.mul_assoc k (c + 1) k2, Nat.mul_div_cancel _ (Nat.pos_of_ne_zero k2_nonzero),
Nat.mul_div_cancel _ (Nat.pos_of_ne_zero c_split)]
|
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 l = b :: L₁ ++ L₂
⊢ ∃ l₁ l₂, x :: l = l₁ ++ l₂ ∧ filterMap f l₁ = b :: L₁ ∧ filterMap f l₂ = L₂
|
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 = filterMap f l₂
h✝ : filterMap f (l₁ ++ l₂) = filterMap f l₁ ++ filterMap f l₂
h : b :: filterMap f (l₁ ++ l₂) = b :: filterMap f l₁ ++ filterMap f l₂
⊢ ∃ l₁_1 l₂_1,
x :: (l₁ ++ l₂) = l₁_1 ++ l₂_1 ∧ filterMap f l₁_1 = b :: filterMap f l₁ ∧ filterMap f l₂_1 = filterMap f l₂
|
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 ∈ closedBall (w j) (K * δ j)
ε : ℝ
hε : ε > 0
h : l.NeBot
δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0
δlim : Tendsto δ l (𝓝 0)
hK : 0 < K + 1
⊢ ∀ (x_1 : ι),
(dist (δ x_1) 0 < ε / (K + 1) ∧ δ x_1 ∈ Ioi 0) ∧ x ∈ closedBall (w x_1) (K * δ x_1) →
closedBall (w x_1) (δ x_1) ⊆ closedBall x ε
|
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, x ∈ closedBall (w j) (K * δ j)
ε : ℝ
hε : ε > 0
h : l.NeBot
δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0
δlim : Tendsto δ l (𝓝 0)
hK : 0 < K + 1
j : ι
hx : x ∈ closedBall (w j) (K * δ j)
hjε : dist (δ j) 0 < ε / (K + 1)
hj₀ : 0 < δ j
y : α
hy : y ∈ closedBall (w j) (δ j)
⊢ y ∈ closedBall x ε
|
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 n` tend to zero.
Then the function is a Cauchy sequence."]
lemma cauchySeq_of_tendsto_div_nhds_one {f : ℕ → G}
(hf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)) :
CauchySeq 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 (M, M + k).2 / f (M, M + k).1 ∈ ↑t
|
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.intro.intro.inr
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
hlt' : n < k + 1
⊢ ¬P n
|
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 = Algebra.adjoin F s
⊢ f.fieldRange = IntermediateField.adjoin F s
|
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 : g.range = Algebra.adjoin F s
⊢ f.fieldRange.toSubfield = (IntermediateField.adjoin F s).toSubfield
|
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.toAddCommGroup
he✝ : NormedSpace.toModule = NormedSpace.toModule
⊢ 2 • (innerSL ℝ) (T x₀) = (fderivInnerCLM ℝ (T x₀, id x₀)).comp (T.prod (ContinuousLinearMap.id ℝ F))
|
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
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
⊢ Tendsto (fun n => μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))
|
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
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
N : ∃ n, μ (t \ closedBall 0 ↑n) ≠ ⊤
⊢ Tendsto (fun n => μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))
|
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 =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
|
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), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
aux₂ :
(Function.uncurry fun x y => (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) y) =
{p | p.2 ∈ Ioc 0 (f p.1)}.indicator fun p => ENNReal.ofReal (g p.2)
⊢ MeasurableSet {p | p.2 ∈ Ioc 0 (f p.1)}
|
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
|
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