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 |
|---|---|---|---|---|---|---|
MonoidHom.transfer_eq_pow_aux | Mathlib/GroupTheory/Transfer.lean | theorem transfer_eq_pow_aux (g : G)
(key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k) :
g ^ H.index ∈ H | case neg
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun x => g • x) q ∈ H
f : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
fun q => ⟨g, ⋯⟩ ^ Function.minimalPeriod (fun x => g • ... | replace key :=
Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q | case neg
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
f : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
fun q => ⟨g, ⋯⟩ ^ Function.minimalPeriod (fun x => g • x) q.out
hf : ∀ (q : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H))), f q ∈... | 5eab6138ecfa4c55 |
Module.finitePresentation_of_free_of_surjective | Mathlib/Algebra/Module/FinitePresentation.lean | lemma Module.finitePresentation_of_free_of_surjective [Module.Free R M] [Module.Finite R M]
(l : M →ₗ[R] N)
(hl : Function.Surjective l) (hl' : (LinearMap.ker l).FG) :
Module.FinitePresentation R N | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : (LinearMap.ker l).FG
b : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R ... | refine ⟨(Set.finite_range (l ∘ b)).toFinset,
by simpa [Set.range_comp, LinearMap.range_eq_top], ?_⟩ | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : (LinearMap.ker l).FG
b : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R ... | d1abb803b7fe5bcc |
HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re | Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | /-- Formula for `hurwitzZeta s` as a Dirichlet series in the convergence range. We
restrict to `a ∈ Icc 0 1` to simplify the statement. -/
lemma hasSum_hurwitzZeta_of_one_lt_re {a : ℝ} (ha : a ∈ Icc 0 1) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℕ ↦ 1 / (n + a : ℂ) ^ s) (hurwitzZeta a s) | a : ℝ
ha : a ∈ Icc 0 1
s : ℂ
hs : 1 < s.re
n : ℕ
x✝ y✝ : ℂ
⊢ x✝ = (x✝ + y✝) / 2 + (x✝ - y✝) / 2 | ring | no goals | 24721f02c899b6bd |
Nat.repr_length | Mathlib/Data/Nat/Digits.lean | /-- The core implementation of `Nat.repr` returns a String with length less than or equal to the
number of digits in the decimal number (represented by `e`). For example, the decimal string
representation of any number less than 1000 (10 ^ 3) has a length less than or equal to 3. -/
lemma repr_length (n e : Nat) : 0 < ... | e n : ℕ
e0 : 0 < e
he : n + 1 < 10 ^ e
hterm : n.succ / 10 = 0
⊢ (toDigitsCore 10 (n + 1 + 1) (n + 1) []).length ≤ e | simp only [hterm, Nat.toDigitsCore] | e n : ℕ
e0 : 0 < e
he : n + 1 < 10 ^ e
hterm : n.succ / 10 = 0
⊢ (if True then [((n + 1) % 10).digitChar]
else
if True then [(0 % 10).digitChar, ((n + 1) % 10).digitChar]
else toDigitsCore 10 n (0 / 10) [(0 % 10).digitChar, ((n + 1) % 10).digitChar]).length ≤
e | d511f3017bbee59f |
Real.integrable_of_summable_norm_Icc | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | theorem Real.integrable_of_summable_norm_Icc {E : Type*} [NormedAddCommGroup E] {f : C(ℝ, E)}
(hf : Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)‖) :
Integrable f | E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
hf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
⊢ Integrable (⇑f) volume | refine integrable_of_summable_norm_restrict (.of_nonneg_of_le
(fun n : ℤ => mul_nonneg (norm_nonneg
(f.restrict (⟨Icc (n : ℝ) ((n : ℝ) + 1), isCompact_Icc⟩ : Compacts ℝ)))
ENNReal.toReal_nonneg) (fun n => ?_) hf) ?_ | case refine_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
hf : Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖
n : ℤ
⊢ ‖ContinuousMap.restrict (↑{ carrier := Icc (↑n) (↑n + 1), isCompact' := ⋯ }) f‖ *
(volume ↑{ carrier := Icc (↑n) (↑n + 1), isCompact' := ⋯ }).t... | 9f13b30eccc2ee44 |
SimpleGraph.reachable_deleteEdges_iff_exists_cycle.aux | Mathlib/Combinatorics/SimpleGraph/Path.lean | theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
(hb : ∀ p : G.Walk v w, s(v, w) ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
(he : s(v, w) ∈ c.edges)
(hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False | V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges
c : G.Walk u u
hc : c.IsTrail
he : s(v, w) ∈ c.edges
hw : w ∈ (c.takeUntil v ⋯).support
hv : v ∈ c.support
puw : G.Walk u w := (c.takeUntil v hv).takeUntil w hw
pwv : G.Walk w v := (c.takeUntil v hv).dropUntil w hw
p... | simp [puw, pwv, pvu] | no goals | df7e8865a04a9975 |
CochainComplex.HomComplex.Cochain.comp_zero | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | @[simp]
protected lemma comp_zero {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁)
(h : n₁ + n₂ = n₁₂) : z₁.comp (0 : Cochain G K n₂) h = 0 | case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp 0 h).v p q hpq = v 0 p q hpq | simp only [comp_v _ _ h p _ q rfl (by omega), zero_v, comp_zero] | no goals | 22eb973e1b7c6ddc |
Finsupp.neLocus_self_sub_left | Mathlib/Data/Finsupp/NeLocus.lean | theorem neLocus_self_sub_left : neLocus (f - g) f = g.support | α : Type u_1
N : Type u_3
inst✝² : DecidableEq α
inst✝¹ : DecidableEq N
inst✝ : AddGroup N
f g : α →₀ N
⊢ (f - g).neLocus f = g.support | rw [neLocus_comm, neLocus_self_sub_right] | no goals | cedddf5628405435 |
Polynomial.reflect_C_mul_X_pow | Mathlib/Algebra/Polynomial/Reverse.lean | theorem reflect_C_mul_X_pow (N n : ℕ) {c : R} : reflect N (C c * X ^ n) = C c * X ^ revAt N n | case a
R : Type u_1
inst✝ : Semiring R
N n : ℕ
c : R
n✝ : ℕ
⊢ (reflect N (C c * X ^ n)).coeff n✝ = (C c * X ^ (revAt N) n).coeff n✝ | rw [reflect_C_mul, coeff_C_mul, coeff_C_mul, coeff_X_pow, coeff_reflect] | case a
R : Type u_1
inst✝ : Semiring R
N n : ℕ
c : R
n✝ : ℕ
⊢ c * (X ^ n).coeff ((revAt N) n✝) = c * if n✝ = (revAt N) n then 1 else 0 | a042f8126a450bcc |
bernsteinPolynomial.sum_mul_smul | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem sum_mul_smul (n : ℕ) :
(∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν) =
(n * (n - 1)) • X ^ 2 | R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • X ^ 2 | let y : MvPolynomial Bool R := MvPolynomial.X false | R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • X ^ 2 | 5a1ea5f98d0a949e |
AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_fac | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | theorem desc_fac (F : J ⥤ PresheafedSpace.{_, _, v} C) (s : Cocone F) (j : J) :
(colimitCocone F).ι.app j ≫ desc F s = s.ι.app j | case h.w
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
j : J
U : Opens ↑↑s.pt
⊢ limit.lift
({ obj := fun X => u... | simp | no goals | f70d14ec7bfa8f34 |
NNReal.summable_coe | Mathlib/Topology/Instances/NNReal/Lemmas.lean | theorem summable_coe {f : α → ℝ≥0} : (Summable fun a => (f a : ℝ)) ↔ Summable f | α : Type u_1
f : α → ℝ≥0
⊢ (Summable fun a => ↑(f a)) ↔ Summable f | constructor | case mp
α : Type u_1
f : α → ℝ≥0
⊢ (Summable fun a => ↑(f a)) → Summable f
case mpr
α : Type u_1
f : α → ℝ≥0
⊢ Summable f → Summable fun a => ↑(f a) | 15f7fdaedb4d0401 |
List.getElem_eq_getElem_rotate | Mathlib/Data/List/Rotate.lean | theorem getElem_eq_getElem_rotate (l : List α) (n : ℕ) (k : Nat) (hk : k < l.length) :
l[k] = ((l.rotate n)[(l.length - n % l.length + k) % l.length]'
((Nat.mod_lt _ (k.zero_le.trans_lt hk)).trans_eq (length_rotate _ _).symm)) | α : Type u
l : List α
n k : ℕ
hk : k < l.length
⊢ l[k] = l[((l.length - n % l.length + k) % l.length + n) % l.length] | refine congr_arg l.get (Fin.eq_of_val_eq ?_) | α : Type u
l : List α
n k : ℕ
hk : k < l.length
⊢ ↑⟨k, hk⟩ = ↑⟨((l.length - n % l.length + k) % l.length + n) % l.length, ⋯⟩ | 2583ca89c80a2b98 |
Set.Ioc_union_Ioc' | Mathlib/Order/Interval/Set/Basic.lean | theorem Ioc_union_Ioc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) | case neg
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : c ≤ b
h₂ : a ≤ d
x : α
hc : c < x
hd : ¬x ≤ d
⊢ a < x ∧ x ≤ b ∨ c < x ∧ x ≤ d ↔ (a < x ∨ c < x) ∧ (x ≤ b ∨ x ≤ d) | have hax : a < x := h₂.trans_lt (lt_of_not_ge hd) | case neg
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : c ≤ b
h₂ : a ≤ d
x : α
hc : c < x
hd : ¬x ≤ d
hax : a < x
⊢ a < x ∧ x ≤ b ∨ c < x ∧ x ≤ d ↔ (a < x ∨ c < x) ∧ (x ≤ b ∨ x ≤ d) | fd3d63edf0b5b459 |
Subgroup.isComplement_iff_existsUnique_mul_inv_mem | Mathlib/GroupTheory/Complement.lean | @[to_additive]
lemma isComplement_iff_existsUnique_mul_inv_mem :
IsComplement S T ↔ ∀ g, ∃! t : T, g * (t : G)⁻¹ ∈ S | G : Type u_1
inst✝ : Group G
S T : Set G
g : G
x : ↑S × ↑T
hx : ↑x.1 * ↑x.2 = g
hx' : ∀ (y : ↑S × ↑T), (fun x => ↑x.1 * ↑x.2 = g) y → y = x
⊢ (fun t => g * (↑t)⁻¹ ∈ S) x.2 | simp [← hx] | no goals | 4b235bce33546e1f |
List.max?_mem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean | theorem max?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
{xs : List α} → xs.max? = some a → a ∈ xs
| nil => by simp
| cons x xs => by
rw [max?]; rintro ⟨⟩
induction xs generalizing x with simp at *
| cons y xs ih =>
rcases ih (max x y) with h | h <;> simp [h]
simp [←... | α : Type u_1
a : α
inst✝ : Max α
min_eq_or : ∀ (a b : α), max a b = a ∨ max a b = b
x : α
xs : List α
⊢ some (foldl max x xs) = some a → a ∈ x :: xs | rintro ⟨⟩ | case refl
α : Type u_1
inst✝ : Max α
min_eq_or : ∀ (a b : α), max a b = a ∨ max a b = b
x : α
xs : List α
⊢ foldl max x xs ∈ x :: xs | 7317c10ebce96da2 |
Matrix.isRepresentation.toEnd_exists_mem_ideal | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | theorem Matrix.isRepresentation.toEnd_exists_mem_ideal (f : Module.End R M) (I : Ideal R)
(hI : LinearMap.range f ≤ I • ⊤) :
∃ M, Matrix.isRepresentation.toEnd R b hb M = f ∧ ∀ i j, M.1 i j ∈ I | 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
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
⊢ ∃ M_1, (toEnd R b hb) M_1 = f ∧ ∀ (i j : ι), ↑M_1 i j ∈ I | have : ∀ x, f x ∈ LinearMap.range (Ideal.finsuppTotal ι M I b) := by
rw [Ideal.range_finsuppTotal, hb]
exact fun x => hI (LinearMap.mem_range_self f x) | 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
hb : Submodule.span R (Set.range b) = ⊤
f : Module.End R M
I : Ideal R
hI : LinearMap.range f ≤ I • ⊤
this : ∀ (x : M), f x ∈ LinearMap.range (Ideal.finsuppTotal ι M ... | b3ecbfc9063659e9 |
Nat.stabilises_of_monotone | Mathlib/Order/Monotone/Basic.lean | /-- A monotone function `f : ℕ → ℕ` bounded by `b`, which is constant after stabilising for the
first time, stabilises in at most `b` steps. -/
lemma Nat.stabilises_of_monotone {f : ℕ → ℕ} {b n : ℕ} (hfmono : Monotone f) (hfb : ∀ m, f m ≤ b)
(hfstab : ∀ m, f m = f (m + 1) → f (m + 1) = f (m + 2)) (hbn : b ≤ n) : f ... | case intro.intro
f : ℕ → ℕ
b n : ℕ
hfmono : Monotone f
hfb : ∀ (m : ℕ), f m ≤ b
hfstab : ∀ (m : ℕ), f m = f (m + 1) → f (m + 1) = f (m + 2)
hbn : b ≤ n
m : ℕ
hmb : m ≤ b
hm : f m = f (m + 1)
key : ∀ (k : ℕ), k ≥ m → f k = f m
⊢ f n = f b | exact (key n (hmb.trans hbn)).trans (key b hmb).symm | no goals | d87475cd710a5b6c |
Polynomial.div_tendsto_atBot_of_degree_gt' | Mathlib/Analysis/Polynomial/Basic.lean | theorem div_tendsto_atBot_of_degree_gt' (hdeg : Q.degree < P.degree)
(hneg : P.leadingCoeff / Q.leadingCoeff < 0) :
Tendsto (fun x => eval x P / eval x Q) atTop atBot | 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hneg : P.leadingCoeff / Q.leadingCoeff < 0
hQ : Q ≠ 0
⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop | apply tendsto_zpow_atTop_atTop | case hn
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hneg : P.leadingCoeff / Q.leadingCoeff < 0
hQ : Q ≠ 0
⊢ 0 < ↑P.natDegree - ↑Q.natDegree | 3c76e2aec579866a |
Prod.mk_covBy_mk_iff_left | Mathlib/Order/Cover.lean | theorem mk_covBy_mk_iff_left : (a₁, b) ⋖ (a₂, b) ↔ a₁ ⋖ a₂ | α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
a₁ a₂ : α
b : β
⊢ (a₁, b) ⋖ (a₂, b) ↔ a₁ ⋖ a₂ | simp_rw [covBy_iff_wcovBy_and_lt, mk_wcovBy_mk_iff_left, mk_lt_mk_iff_left] | no goals | f751c7ce333358d7 |
Nat.choose_eq_asc_factorial_div_factorial | Mathlib/Data/Nat/Choose/Basic.lean | theorem choose_eq_asc_factorial_div_factorial (n k : ℕ) :
(n + k).choose k = (n + 1).ascFactorial k / k ! | n k : ℕ
⊢ k ! * (n + k).choose k = k ! * ((n + 1).ascFactorial k / k !) | rw [← ascFactorial_eq_factorial_mul_choose] | n k : ℕ
⊢ (n + 1).ascFactorial k = k ! * ((n + 1).ascFactorial k / k !) | e4c9bdb8704b9c5b |
CategoryTheory.Localization.LeftBousfield.W_adj_unit_app | Mathlib/CategoryTheory/Localization/Bousfield.lean | lemma W_adj_unit_app (X : D) : W (· ∈ Set.range F.obj) (adj.unit.app X) | case h.e'_3.h
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : G ⊣ F
inst✝¹ : F.Full
inst✝ : F.Faithful
X : D
Y : C
e_1✝ : ((G ⋙ F).obj X ⟶ F.obj Y) = (F.obj (G.obj X) ⟶ F.obj Y)
e_2✝ : ((𝟭 D).obj X ⟶ F.obj Y) = (X ⟶ F.obj Y)
⊢ (fun g => adj.unit.app X ≫... | aesop | no goals | ba82d5aab73d9d64 |
FormalMultilinearSeries.apply_eq_prod_smul_coeff | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | theorem apply_eq_prod_smul_coeff : p n y = (∏ i, y i) • p.coeff n | case h.e'_2.h.e'_1.h
𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
y : Fin n → 𝕜
x✝ : Fin n
⊢ y x✝ = y x✝ • 1 x✝ | simp only [Pi.one_apply, Algebra.id.smul_eq_mul, mul_one] | no goals | f6c046f6e19a80f3 |
Complex.sin_surjective | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | theorem sin_surjective : Function.Surjective sin | case intro
z : ℂ
⊢ ∃ a, sin a = cos z | exact ⟨z + π / 2, sin_add_pi_div_two z⟩ | no goals | e46b5a492c5b808d |
UniqueFactorizationMonoid.induction_on_prime_power | Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicative.lean | theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
(is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
(h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
(hcp : ∀ {x y}, IsRelPrime x y → P x → P y → P (x * y)) :
P (∏ p ∈ s, p ^ i p)... | case insert
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf'... | exact
hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)
(hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))
(ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>
is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq')) | no goals | e0665738e1c66545 |
Std.Tactic.BVDecide.Reflect.BitVec.cond_false | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Reflect.lean | theorem cond_false (discr : Bool) (lhs rhs : BitVec w) :
(discr || ((bif discr then lhs else rhs) == rhs)) = true | w : Nat
discr : Bool
lhs rhs : BitVec w
⊢ (discr || (bif discr then lhs else rhs) == rhs) = true | cases discr <;> simp | no goals | 2faca5387ca1854c |
TopCat.range_pullback_map | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | theorem range_pullback_map {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T)
(g₂ : Z ⟶ T) (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
(eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
Set.range (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) =
(pullback.fst g₁ g₂) ⁻¹' Set.range i... | case a
W X Y Z S T : TopCat
f₁ : W ⟶ S
f₂ : X ⟶ S
g₁ : Y ⟶ T
g₂ : Z ⟶ T
i₁ : W ⟶ Y
i₂ : X ⟶ Z
i₃ : S ⟶ T
H₃ : Mono i₃
eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁
eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂
x✝ : ↑(pullback g₁ g₂)
x₁ : ↑W
hx₁ : (ConcreteCategory.hom i₁) x₁ = (ConcreteCategory.hom (pullback.fst g₁ g₂)) x✝
x₂ : ↑X
hx₂ : (ConcreteCategory.hom i₂)... | rw [← ConcreteCategory.comp_apply, eq₁, ← ConcreteCategory.comp_apply, eq₂,
ConcreteCategory.comp_apply, ConcreteCategory.comp_apply, hx₁, hx₂,
← ConcreteCategory.comp_apply, pullback.condition, ConcreteCategory.comp_apply] | no goals | b99279db8a744b09 |
Std.Sat.CNF.mem_append | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean | theorem mem_append {v : α} {f1 f2 : CNF α} : Mem v (f1 ++ f2) ↔ Mem v f1 ∨ Mem v f2 | case mp.intro.intro.inl.h
α : Type u_1
v : α
f1 f2 : CNF α
c : Clause α
mc : Clause.Mem v c
mf1 : c ∈ f1
⊢ ∃ c, c ∈ f1 ∧ Clause.Mem v c | exact ⟨c, mf1, mc⟩ | no goals | 6ac2c3fbfebc18cc |
List.prod_eq_zero | Mathlib/Algebra/BigOperators/Ring/List.lean | /-- If zero is an element of a list `l`, then `List.prod l = 0`. If the domain is a nontrivial
monoid with zero with no divisors, then this implication becomes an `iff`, see
`List.prod_eq_zero_iff`. -/
lemma prod_eq_zero : ∀ {l : List M₀}, (0 : M₀) ∈ l → l.prod = 0
-- | absurd h (not_mem_nil _)
| a :: l, h => by
... | M₀ : Type u_4
inst✝ : MonoidWithZero M₀
a : M₀
l : List M₀
h : 0 ∈ a :: l
⊢ a * l.prod = 0 | rcases mem_cons.1 h with ha | hl | case inl
M₀ : Type u_4
inst✝ : MonoidWithZero M₀
a : M₀
l : List M₀
h : 0 ∈ a :: l
ha : 0 = a
⊢ a * l.prod = 0
case inr
M₀ : Type u_4
inst✝ : MonoidWithZero M₀
a : M₀
l : List M₀
h : 0 ∈ a :: l
hl : 0 ∈ l
⊢ a * l.prod = 0 | 5b137a5869a279e4 |
MeasureTheory.Measure.pi_eq_generateFrom | Mathlib/MeasureTheory/Constructions/Pi.lean | theorem pi_eq_generateFrom {C : ∀ i, Set (Set (α i))}
(hC : ∀ i, generateFrom (C i) = by apply_assumption) (h2C : ∀ i, IsPiSystem (C i))
(h3C : ∀ i, (μ i).FiniteSpanningSetsIn (C i)) {μν : Measure (∀ i, α i)}
(h₁ : ∀ s : ∀ i, Set (α i), (∀ i, s i ∈ C i) → μν (pi univ s) = ∏ i, μ i (s i)) :
Measure.pi μ ... | ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)
h₁ : ∀ (s : (i :... | rintro _ ⟨s, hs, rfl⟩ | case intro.intro
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)... | 6d99627d812669c7 |
isCyclic_of_subgroup_isDomain | Mathlib/RingTheory/IntegralDomain.lean | theorem isCyclic_of_subgroup_isDomain [Finite G] (f : G →* R) (hf : Injective f) : IsCyclic G | R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Finite G
f : G →* R
hf : Injective ⇑f
⊢ IsCyclic G | cases nonempty_fintype G | case intro
R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Finite G
f : G →* R
hf : Injective ⇑f
val✝ : Fintype G
⊢ IsCyclic G | 7493f2b4daf22b69 |
Real.mulExpNegMulSq_eq_sqrt_mul_mulExpNegMulSq_one | Mathlib/Analysis/SpecialFunctions/MulExpNegMulSq.lean | theorem mulExpNegMulSq_eq_sqrt_mul_mulExpNegMulSq_one (hε : 0 < ε) (x : ℝ) :
mulExpNegMulSq ε x = (sqrt ε)⁻¹ * mulExpNegMulSq 1 (sqrt ε * x) | ε : ℝ
hε : 0 < ε
x : ℝ
h✝ : √ε * x * (√ε * x) = ε * x * x
h : √ε = 0
⊢ False | rw [← pow_eq_zero_iff (Ne.symm (Nat.zero_ne_add_one 1)), sq_sqrt hε.le] at h | ε : ℝ
hε : 0 < ε
x : ℝ
h✝ : √ε * x * (√ε * x) = ε * x * x
h : ε = 0
⊢ False | ab3cedda73e3bb98 |
CFC.rpow_neg_one_eq_inv | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Basic.lean | lemma rpow_neg_one_eq_inv (a : Aˣ) (ha : (0 : A) ≤ a | A : Type u_1
inst✝⁵ : PartialOrder A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ≥0 fun a => 0 ≤ a
a : Aˣ
ha : autoParam (0 ≤ ↑a) _auto✝
⊢ ↑a ^ (-1) = ↑a⁻¹ | refine a.inv_eq_of_mul_eq_one_left ?_ |>.symm | A : Type u_1
inst✝⁵ : PartialOrder A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ≥0 fun a => 0 ≤ a
a : Aˣ
ha : autoParam (0 ≤ ↑a) _auto✝
⊢ ↑a ^ (-1) * ↑a = 1 | 134551fd52cd3afe |
Ideal.IsMaximal.comap_piEvalRingHom | Mathlib/RingTheory/Ideal/Maps.lean | theorem IsMaximal.comap_piEvalRingHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
{i : ι} {I : Ideal (R i)} (h : I.IsMaximal) : (I.comap <| Pi.evalRingHom R i).IsMaximal | case h.e'_5.h.inr
ι : Type u_4
R : ι → Type u_5
inst✝ : (i : ι) → Semiring (R i)
i : ι
I : Ideal (R i)
h : I.IsMaximal
J : Ideal ((i : ι) → R i)
x : (i : ι) → R i
le : comap (Pi.evalRingHom R i) I ≤ J
hxI : x ∉ comap (Pi.evalRingHom R i) I
hxJ : x ∈ J
r y : R i
hy : y ∈ I
eq : r * (Pi.evalRingHom R i) x + y = 1
j : ι
n... | simp [update_of_ne ne] | no goals | f7c735e2e46e1091 |
NumberField.mixedEmbedding.convexBodyLT_volume | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | theorem convexBodyLT_volume :
volume (convexBodyLT K f) = (convexBodyLTFactor K) * ∏ w, (f w) ^ (mult w) | K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
⊢ (↑2 ^ nrRealPlaces K * ∏ x : { w // w.IsReal }, ENNReal.ofReal ↑(f ↑x)) *
((∏ x : { w // w.IsComplex }, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑pi ^ nrComplexPlaces K) =
↑(convexBodyLTFactor K) *
((∏ x : { w // w.IsReal }, ENNReal.of... | simp_rw [convexBodyLTFactor, coe_mul, ENNReal.coe_pow] | K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
⊢ (↑2 ^ nrRealPlaces K * ∏ x : { w // w.IsReal }, ENNReal.ofReal ↑(f ↑x)) *
((∏ x : { w // w.IsComplex }, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑pi ^ nrComplexPlaces K) =
↑2 ^ nrRealPlaces K * ↑pi ^ nrComplexPlaces K *
((∏ x : { w // ... | b181101078b0e79b |
MeasureTheory.sigmaFinite_bot_iff | Mathlib/MeasureTheory/Measure/Typeclasses.lean | theorem sigmaFinite_bot_iff (μ : @Measure α ⊥) : SigmaFinite μ ↔ IsFiniteMeasure μ | α : Type u_1
μ : Measure α
h this : SigmaFinite μ
s : ℕ → Set α := spanningSets μ
hs_univ : ⋃ i, s i = univ
hs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ
⊢ μ univ < ⊤ | by_cases h_univ_empty : (Set.univ : Set α) = ∅ | case pos
α : Type u_1
μ : Measure α
h this : SigmaFinite μ
s : ℕ → Set α := spanningSets μ
hs_univ : ⋃ i, s i = univ
hs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ
h_univ_empty : univ = ∅
⊢ μ univ < ⊤
case neg
α : Type u_1
μ : Measure α
h this : SigmaFinite μ
s : ℕ → Set α := spanningSets μ
hs_univ : ⋃ i, s i = univ
hs_mea... | 46d89df9c3a056ab |
Matroid.IsBasis.isBasis_union_of_subset | Mathlib/Data/Matroid/Basic.lean | theorem IsBasis.isBasis_union_of_subset (hI : M.IsBasis I X) (hJ : M.Indep J) (hIJ : I ⊆ J) :
M.IsBasis J (J ∪ X) | α : Type u_1
M : Matroid α
I J X : Set α
hI : M.IsBasis I X
hJ : M.Indep J
hIJ : I ⊆ J
⊢ M.Indep J | assumption | no goals | 396e46a17248b608 |
CategoryTheory.Sheaf.isLocallyBijective_iff_isIso' | Mathlib/CategoryTheory/Sites/LocallyBijective.lean | /-- A morphism of sheaves of types is locally bijective iff it is an isomorphism.
(This is generalized below as `isLocallyBijective_iff_isIso`.) -/
private lemma isLocallyBijective_iff_isIso' :
IsLocallyInjective f ∧ IsLocallySurjective f ↔ IsIso f | case a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.val.app X))
right✝ : IsLocallySurjective f
X : Cᵒᵖ
s : G.val.obj X
H : Presieve.IsSheafFor F.val (Presheaf.imageSieve f.val s).arrows
t : Presieve.Family... | have eq₄ := congr_arg (G.val.map g₂.op) (Presheaf.app_localPreimage f.val s _ hf₂) | case a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.val.app X))
right✝ : IsLocallySurjective f
X : Cᵒᵖ
s : G.val.obj X
H : Presieve.IsSheafFor F.val (Presheaf.imageSieve f.val s).arrows
t : Presieve.Family... | 10840b64f67d3098 |
Ideal.sup_eq_top_iff_isCoprime | Mathlib/RingTheory/Ideal/Operations.lean | theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) :
span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y | R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ span {x} ⊔ span {y} = ⊤ ↔ IsCoprime x y | rw [eq_top_iff_one, Submodule.mem_sup] | R : Type u_2
inst✝ : CommSemiring R
x y : R
⊢ (∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1) ↔ IsCoprime x y | 251c1440396e8f0b |
Submodule.basis_of_pid_aux | Mathlib/LinearAlgebra/FreeModule/PID.lean | theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z... | case neg.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N <... | have ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0 := by
intro c z zN' hc
refine (mul_eq_zero.mp (y'_ortho_M' (a * c) z (N'_le_M' zN') ?_)).resolve_left a_zero
rw [mul_comm, mul_smul, hc] | case neg.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N <... | 48ee91e21f4317c3 |
IsFreeGroupoid.SpanningTree.endIsFree | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | /-- Given a free groupoid and an arborescence of its generating quiver, the vertex
group at the root is freely generated by loops coming from generating arrows
in the complement of the tree. -/
lemma endIsFree : IsFreeGroup (End (root' T)) :=
IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubqui... | case mk.mk
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
... | erw [Functor.mapEnd_apply, this, hF'] | case mk.mk
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
... | 4742e81f46ca4ae4 |
CategoryTheory.Abelian.Pseudoelement.pseudoZero_aux | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | theorem pseudoZero_aux {P : C} (Q : C) (f : Over P) : f ≈ (0 : Q ⟶ P) ↔ f.hom = 0 :=
⟨fun ⟨R, p, q, _, _, comm⟩ => zero_of_epi_comp p (by simp [comm]), fun hf =>
⟨biprod f.1 Q, biprod.fst, biprod.snd, inferInstance, inferInstance, by
rw [hf, Over.coe_hom, HasZeroMorphisms.comp_zero, HasZeroMorphisms.comp_ze... | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
P Q : C
f : Over P
x✝ : f ≈ Over.mk 0
R : C
p : R ⟶ f.left
q : R ⟶ (Over.mk 0).left
w✝¹ : Epi p
w✝ : Epi q
comm : p ≫ f.hom = q ≫ (Over.mk 0).hom
⊢ p ≫ f.hom = 0 | simp [comm] | no goals | 3427423e71788486 |
BitVec.getMsbD_sshiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getMsbD_sshiftRight {x : BitVec w} {i n : Nat} :
getMsbD (x.sshiftRight n) i = (decide (i < w) && if i < n then x.msb else getMsbD x (i - n)) | w : Nat
x : BitVec w
i n : Nat
⊢ (decide (i < w) && (!decide (w ≤ w - 1 - i) && if n + (w - 1 - i) < w then x.getLsbD (n + (w - 1 - i)) else x.msb)) =
(decide (i < w) && if i < n then x.msb else decide (i - n < w) && x.getLsbD (w - 1 - (i - n))) | by_cases h : i < w | case pos
w : Nat
x : BitVec w
i n : Nat
h : i < w
⊢ (decide (i < w) && (!decide (w ≤ w - 1 - i) && if n + (w - 1 - i) < w then x.getLsbD (n + (w - 1 - i)) else x.msb)) =
(decide (i < w) && if i < n then x.msb else decide (i - n < w) && x.getLsbD (w - 1 - (i - n)))
case neg
w : Nat
x : BitVec w
i n : Nat
h : ¬i < w... | 4692a33de500d022 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id) | n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
heq : f.clauses[id]! = none
hb : hasAs... | exact hf i b hb | no goals | 0ab20a88767ce69e |
ContMDiffWithinAt.mfderivWithin | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | theorem ContMDiffWithinAt.mfderivWithin {x₀ : N} {f : N → M → M'} {g : N → M}
{t : Set N} {u : Set M}
(hf : ContMDiffWithinAt (J.prod I) I' n (Function.uncurry f) (t ×ˢ u) (x₀, g x₀))
(hg : ContMDiffWithinAt J I m g t x₀) (hx₀ : x₀ ∈ t)
(hu : MapsTo g t u) (hmn : m + 1 ≤ n) (h'u : UniqueMDiffOn I u) :
... | case hst
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddC... | intro a ha | case hst
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddC... | b011e1f26afdb375 |
AddCircle.norm_coe_mul | Mathlib/Analysis/Normed/Group/AddCircle.lean | theorem norm_coe_mul (x : ℝ) (t : ℝ) :
‖(↑(t * x) : AddCircle (t * p))‖ = |t| * ‖(x : AddCircle p)‖ | case inr.e_a.h.mp
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) → ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z | rintro ⟨y, hy, rfl⟩ | case inr.e_a.h.mp.intro.intro
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
y : ℝ
hy : y - t * x ∈ zmultiples (t * p)
⊢ ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * |y| | 1dd8380c8d5f2dbd |
MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
(hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
(∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) | case intro.intro.intro.intro.intro.refine_2
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen u
s... | obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx | case intro.intro.intro.intro.intro.refine_2.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T... | 7d50e417c306b9b4 |
Cardinal.nat_is_prime_iff | Mathlib/SetTheory/Cardinal/Divisibility.lean | theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime | n : ℕ
h✝ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : b = 0 ∨ c = 0
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
h : ↑n = 0
⊢ False | cases hbc <;> contradiction | no goals | 0a33226042e64149 |
hasFDerivWithinAt_closure_of_tendsto_fderiv | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | theorem hasFDerivWithinAt_closure_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
(f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
(h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
HasFDerivWithinA... | case neg.intro.intro.mk.refine_2.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, Continuou... | apply ContinuousWithinAt.mono _ this | E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y
h : Tendsto (fun y => f... | b73104c22d482296 |
LSeries.term_congr | Mathlib/NumberTheory/LSeries/Basic.lean | lemma term_congr {f g : ℕ → ℂ} (h : ∀ {n}, n ≠ 0 → f n = g n) (s : ℂ) (n : ℕ) :
term f s n = term g s n | f g : ℕ → ℂ
h : ∀ {n : ℕ}, n ≠ 0 → f n = g n
s : ℂ
n : ℕ
⊢ term f s n = term g s n | rcases eq_or_ne n 0 with hn | hn <;> simp [hn, h] | no goals | bfa42cc3af7d977c |
Order.height_add_one_le | Mathlib/Order/KrullDimension.lean | lemma height_add_one_le {a b : α} (hab : a < b) : height a + 1 ≤ height b | case top
α : Type u_1
inst✝ : Preorder α
a b : α
hab : a < b
hfin : height a = ⊤
⊢ ⊤ + 1 ≤ height b | have : ⊤ ≤ height b := by
rw [← hfin]
gcongr | case top
α : Type u_1
inst✝ : Preorder α
a b : α
hab : a < b
hfin : height a = ⊤
this : ⊤ ≤ height b
⊢ ⊤ + 1 ≤ height b | fa171423c2de0314 |
WithSeminorms.isVonNBounded_iff_seminorm_bounded | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem WithSeminorms.isVonNBounded_iff_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
Bornology.IsVonNBounded 𝕜 s ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i x < r | 𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
⊢ (∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r) ↔ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x ... | constructor | case mp
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
⊢ (∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r) → ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, ... | 24b13a155e892387 |
Finset.mulEnergy_eq_sum_sq' | Mathlib/Combinatorics/Additive/Energy.lean | @[to_additive] lemma mulEnergy_eq_sum_sq' (s t : Finset α) :
Eₘ[s, t] = ∑ a ∈ s * t, ((s ×ˢ t).filter fun (x, y) ↦ x * y = a).card ^ 2 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s t : Finset α
⊢ #(filter (fun x => x.1.1 * x.1.2 = x.2.1 * x.2.2) ((s ×ˢ t) ×ˢ s ×ˢ t)) =
∑ x ∈ s * t, #(filter (fun x_1 => x_1.1 * x_1.2 = x) (s ×ˢ t) ×ˢ filter (fun x_1 => x_1.1 * x_1.2 = x) (s ×ˢ t)) | rw [← card_disjiUnion] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s t : Finset α
⊢ #(filter (fun x => x.1.1 * x.1.2 = x.2.1 * x.2.2) ((s ×ˢ t) ×ˢ s ×ˢ t)) =
#((s * t).disjiUnion
(fun x => filter (fun x_1 => x_1.1 * x_1.2 = x) (s ×ˢ t) ×ˢ filter (fun x_1 => x_1.1 * x_1.2 = x) (s ×ˢ t)) ?h)
case h
α : Type u_1
inst✝¹ : Deci... | e47fe218b2118abc |
StrictConvex.add | Mathlib/Analysis/Convex/Strict.lean | theorem StrictConvex.add (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s + t) | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : TopologicalSpace E
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousAdd E
s t : Set E
hs : StrictConvex 𝕜 s
ht : StrictConvex 𝕜 t
v : E
hv : v ∈ s
w : E
hw : w ∈ t
x : E
hx : x ∈ s
y : E
... | rw [smul_add, smul_add, add_add_add_comm] | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : TopologicalSpace E
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousAdd E
s t : Set E
hs : StrictConvex 𝕜 s
ht : StrictConvex 𝕜 t
v : E
hv : v ∈ s
w : E
hw : w ∈ t
x : E
hx : x ∈ s
y : E
... | 6b28b716c7687ba5 |
Set.mem_accumulate | Mathlib/Data/Set/Accumulate.lean | theorem mem_accumulate [LE α] {x : α} {z : β} : z ∈ Accumulate s x ↔ ∃ y ≤ x, z ∈ s y | α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : LE α
x : α
z : β
⊢ z ∈ Accumulate s x ↔ ∃ y ≤ x, z ∈ s y | simp_rw [accumulate_def, mem_iUnion₂, exists_prop] | no goals | 699379f5632b978e |
Antitone.map_limsSup_of_continuousAt | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (· ≤ ·) | case a
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
... | apply le_of_forall_lt | case a.H
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSu... | 3b8bbb0ba829f542 |
AlgebraicGeometry.Scheme.Hom.app_invApp' | Mathlib/AlgebraicGeometry/OpenImmersion.lean | theorem app_invApp' (U) (hU : U ≤ f.opensRange) :
f.app U ≫ (f.appIso (f ⁻¹ᵁ U)).inv =
Y.presheaf.map (eqToHom (Opens.ext <| by simpa [Set.image_preimage_eq_inter_range])).op :=
PresheafedSpace.IsOpenImmersion.app_invApp _ _
| C : Type u
inst✝ : Category.{v, u} C
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f
U : Y.Opens
hU : U ≤ f.opensRange
⊢ ↑(f ''ᵁ f ⁻¹ᵁ U) = ↑U | simpa [Set.image_preimage_eq_inter_range] | no goals | b9a819cf2d2d108d |
Matrix.eval_charpolyRev | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | @[simp] lemma eval_charpolyRev :
eval 0 M.charpolyRev = 1 | R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ((evalRingHom 0).mapMatrix (1 - X • M.map ⇑C)).det = det 1 | have : (1 - (X : R[X]) • M.map C).map (eval 0) = 1 := by
ext i j; rcases eq_or_ne i j with hij | hij <;> simp [hij, one_apply] | R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
this : (1 - X • M.map ⇑C).map (eval 0) = 1
⊢ ((evalRingHom 0).mapMatrix (1 - X • M.map ⇑C)).det = det 1 | decfd8e0a3a0337f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.contradiction_of_insertUnit_success | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem contradiction_of_insertUnit_success {n : Nat} (assignments : Array Assignment)
(assignments_size : assignments.size = n)
(units : Array (Literal (PosFin n))) (foundContradiction : Bool) (l : Literal (PosFin n)) :
let insertUnit_res := insertUnit (units, assignments, foundContradiction) l
(foundC... | case inl.intro
n : Nat
assignments : Array Assignment
assignments_size : assignments.size = n
units : Array (Literal (PosFin n))
foundContradiction : Bool
l : Literal (PosFin n)
insertUnit_res : Array (Literal (PosFin n)) × Array Assignment × Bool :=
insertUnit (units, assignments, foundContradiction) l
h✝ : foundCon... | apply Exists.intro i | case inl.intro
n : Nat
assignments : Array Assignment
assignments_size : assignments.size = n
units : Array (Literal (PosFin n))
foundContradiction : Bool
l : Literal (PosFin n)
insertUnit_res : Array (Literal (PosFin n)) × Array Assignment × Bool :=
insertUnit (units, assignments, foundContradiction) l
h✝ : foundCon... | 7d5ecc4c40b25bd1 |
HomologicalComplex.truncLE'Map_f_eq | Mathlib/Algebra/Homology/Embedding/TruncLE.lean | lemma truncLE'Map_f_eq {i : ι} (hi : ¬ e.BoundaryLE i) {i' : ι'} (h : e.f i = i') :
(truncLE'Map φ e).f i =
(K.truncLE'XIso e h hi).hom ≫ φ.f i' ≫ (L.truncLE'XIso e h hi).inv :=
Quiver.Hom.op_inj
(by simpa using truncGE'Map_f_eq ((opFunctor C c').map φ.op) e.op (by simpa) h)
| ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncLE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : ¬e.BoundaryLE... | simpa using truncGE'Map_f_eq ((opFunctor C c').map φ.op) e.op (by simpa) h | no goals | 4c7e00fc23550068 |
ProbabilityTheory.measurableSet_isRatStieltjesPoint | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma measurableSet_isRatStieltjesPoint [MeasurableSpace α] (hf : Measurable f) :
MeasurableSet {a | IsRatStieltjesPoint f a} | α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
⊢ MeasurableSet {a | IsRatStieltjesPoint f a} | suffices {a | IsRatStieltjesPoint f a}
= ({a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)}
∩ {a | ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t}) by
rw [this]
exact (((h1.inter h2).inter h3).inter h4) | α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
⊢ {a | IsRatStieltjesPoint f a} =
{a | Monoto... | 234ff3cf974e672a |
Rat.fract_inv_num_lt_num_of_pos | Mathlib/Data/Rat/Floor.lean | theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
coprime_q_denom_q_num : q.den.Coprime q.num.natAbs
⊢ (↑q.den).natAbs = q.den | simp | no goals | d2870a8c7e295972 |
HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn
(hr : HasFPowerSeriesWithinOnBall f p s x r)
(h : AnalyticOn 𝕜 f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r | case hf'.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticO... | exact h.differentiableOn _ hy | no goals | d39a108ffb050c55 |
List.MergeSort.Internal.splitRevInTwo_snd | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Impl.lean | theorem splitRevInTwo_snd (l : { l : List α // l.length = n }) :
(splitRevInTwo l).2 = ⟨(splitInTwo l).2.1, by simp; omega⟩ | α : Type ?u.45246
n : Nat
l : { l // l.length = n }
⊢ (splitInTwo l).snd.val.length = n / 2 | simp | α : Type ?u.45246
n : Nat
l : { l // l.length = n }
⊢ l.val.length - (n + 1) / 2 = n / 2 | 26798ff4580edd69 |
Module.projective_of_localization_maximal' | Mathlib/RingTheory/LocalProperties/Projective.lean | theorem Module.projective_of_localization_maximal'
(H : ∀ (I : Ideal R) (_ : I.IsMaximal), Module.Projective (Rₚ I) (Mₚ I))
[Module.FinitePresentation R M] : Module.Projective R M | case H
R : Type u_1
M : Type uM
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [inst : P.IsMaximal] → Type u_4
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaxi... | intros P hP | case H
R : Type u_1
M : Type uM
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [inst : P.IsMaximal] → Type u_4
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaxi... | 60b47373a59430e6 |
support_subset_support_matPolyEquiv | Mathlib/RingTheory/PolynomialAlgebra.lean | theorem support_subset_support_matPolyEquiv (m : Matrix n n R[X]) (i j : n) :
support (m i j) ⊆ support (matPolyEquiv m) | R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
⊢ (m i j).support ⊆ (matPolyEquiv m).support | intro k | R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ
⊢ k ∈ (m i j).support → k ∈ (matPolyEquiv m).support | e123ec4fe2b71ce0 |
LucasLehmer.order_ω | Mathlib/NumberTheory/LucasLehmer.lean | theorem order_ω (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
orderOf (ωUnit (p' + 2)) = 2 ^ (p' + 2) | case h₂
p' : ℕ
h : lucasLehmerResidue (p' + 2) = 0
⊢ orderOf (ωUnit (p' + 2)) ∣ 2 ^ (p' + 1 + 1) | apply orderOf_dvd_iff_pow_eq_one.2 | case h₂
p' : ℕ
h : lucasLehmerResidue (p' + 2) = 0
⊢ ωUnit (p' + 2) ^ 2 ^ (p' + 1 + 1) = 1 | 9c324a727016e67e |
Nat.pow_le_pow_iff_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem pow_le_pow_iff_right {a n m : Nat} (h : 1 < a) :
a ^ n ≤ a ^ m ↔ n ≤ m | a n m : Nat
h : 1 < a
⊢ a ^ n ≤ a ^ m ↔ n ≤ m | constructor | case mp
a n m : Nat
h : 1 < a
⊢ a ^ n ≤ a ^ m → n ≤ m
case mpr
a n m : Nat
h : 1 < a
⊢ n ≤ m → a ^ n ≤ a ^ m | 1988d7c3f55269ea |
LieAlgebra.IsSemisimple.finitelyAtomistic | Mathlib/Algebra/Lie/Semisimple/Basic.lean | /--
In a semisimple Lie algebra,
Lie ideals that are contained in the supremum of a finite collection of atoms
are themselves the supremum of a finite subcollection of those atoms.
By a compactness argument, this statement can be extended to arbitrary sets of atoms.
See `atomistic`.
The proof is by induction on the f... | case inr.intro.intro.intro.intro.intro.intro.right
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieId... | rw [← hJI] | case inr.intro.intro.intro.intro.intro.intro.right
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieId... | 9feaa310a91f7482 |
Polynomial.exists_primitive_lcm_of_isPrimitive | Mathlib/RingTheory/Polynomial/Content.lean | theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : r.IsPrimitive
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧ (p ∣ s ∧ q ∣ ... | contrapose! rs | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : r.IsPrimitive
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧ (p ∣ s ∧ q ∣ ... | 806b283d42e7cf3d |
Nat.digits_zero | Mathlib/Data/Nat/Digits.lean | theorem digits_zero (b : ℕ) : digits b 0 = [] | b : ℕ
⊢ b.digits 0 = [] | rcases b with (_ | ⟨_ | ⟨_⟩⟩) <;> simp [digits, digitsAux0, digitsAux1] | no goals | a6638a93ed5ee15f |
Complex.neg_ofReal_mem_slitPlane | Mathlib/Analysis/Complex/Basic.lean | @[simp]
lemma neg_ofReal_mem_slitPlane {x : ℝ} : -↑x ∈ slitPlane ↔ x < 0 | x : ℝ
⊢ -↑x ∈ slitPlane ↔ x < 0 | simpa using ofReal_mem_slitPlane (x := -x) | no goals | fe6a681ee1a1a067 |
Int.ediv_eq_zero_of_lt_abs | Mathlib/Algebra/Order/Group/Unbundled/Int.lean | theorem ediv_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < |b|) : a / b = 0 :=
match b, |b|, abs_eq_natAbs b, H2 with
| (n : ℕ), _, rfl, H2 => ediv_eq_zero_of_lt H1 H2
| -[n+1], _, rfl, H2 => neg_injective <| by rw [← Int.ediv_neg]; exact ediv_eq_zero_of_lt H1 H2
| a b : ℤ
H1 : 0 ≤ a
H2✝ : a < |b|
n : ℕ
H2 : a < ↑-[n+1].natAbs
⊢ a / - -[n+1] = -0 | exact ediv_eq_zero_of_lt H1 H2 | no goals | 03b1768a6087094b |
Int.bodd_add_div2 | Mathlib/Data/Int/Bitwise.lean | theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| (n : ℕ) => by
rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl]
exact congr_arg ofNat n.bodd_add_div2
| -[n+1] => by
refine Eq.trans ?_ (congr_arg negSucc n.bodd_add_div2)
dsimp [bodd]; cases Nat.bodd... | n : ℕ
⊢ (bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n | rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl] | n : ℕ
⊢ ↑(bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n | b52aac4a92c4fde8 |
Real.natFloor_logb_natCast | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem natFloor_logb_natCast (b : ℕ) (n : ℕ) : ⌊logb b n⌋₊ = Nat.log b n | b n : ℕ
⊢ ⌊logb ↑b ↑n⌋₊ = Nat.log b n | obtain _ | _ | b := b | case zero
n : ℕ
⊢ ⌊logb ↑0 ↑n⌋₊ = Nat.log 0 n
case succ.zero
n : ℕ
⊢ ⌊logb ↑(0 + 1) ↑n⌋₊ = Nat.log (0 + 1) n
case succ.succ
n b : ℕ
⊢ ⌊logb ↑(b + 1 + 1) ↑n⌋₊ = Nat.log (b + 1 + 1) n | ea4e06a3d65b55dd |
List.Vector.insertIdx_comm | Mathlib/Data/Vector/Basic.lean | theorem insertIdx_comm (a b : α) (i j : Fin (n + 1)) (h : i ≤ j) :
∀ v : Vector α n,
(v.insertIdx a i).insertIdx b j.succ = (v.insertIdx b j).insertIdx a (Fin.castSucc i)
| ⟨l, hl⟩ => by
refine Subtype.eq ?_
simp only [insertIdx_val, Fin.val_succ, Fin.castSucc, Fin.coe_castAdd]
apply List.insert... | case x
α : Type u_1
n : ℕ
a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
⊢ ↑j ≤ n | exact Nat.le_of_succ_le_succ j.2 | no goals | e169c22cbc8ad2b0 |
Ordinal.sup_eq_zero_iff | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} :
sup.{_, v} f = 0 ↔ ∀ i, f i = 0 | ι : Type u
f : ι → Ordinal.{max u v}
⊢ sup f = 0 ↔ ∀ (i : ι), f i = 0 | refine
⟨fun h i => ?_, fun h =>
le_antisymm (sup_le fun i => Ordinal.le_zero.2 (h i)) (Ordinal.zero_le _)⟩ | ι : Type u
f : ι → Ordinal.{max u v}
h : sup f = 0
i : ι
⊢ f i = 0 | ea86e1aa2e7db71c |
Std.Sat.AIG.of_isConstant | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Lemmas.lean | theorem of_isConstant {aig : AIG α} {assign : α → Bool} {ref : Ref aig} {b : Bool} :
aig.isConstant ref b → ⟦aig, ref, assign⟧ = b | case mk.h_2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
assign : α → Bool
b : Bool
gate : Nat
hgate : gate < aig.decls.size
decl✝ : Decl α
x✝ : ∀ (val : Bool), aig.decls[gate] = Decl.const val → False
h : false = true
⊢ ⟦assign, { aig := aig, ref := { gate := gate, hgate := hgate } }⟧ = b | contradiction | no goals | 47238698698b790d |
IsGalois.of_separable_splitting_field | Mathlib/FieldTheory/Galois/Basic.lean | theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separable) :
IsGalois F E | case base
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
hFE : FiniteDimensional F E
this : DecidableEq E := Classical.decEq E
s : Set E := p.rootSet E
adjoin_root : adjoin F s = ⊤
P : IntermediateField F E → Prop := fun K... | apply @Fintype.card_congr _ _ _ (_) _ | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
hFE : FiniteDimensional F E
this : DecidableEq E := Classical.decEq E
s : Set E := p.rootSet E
adjoin_root : adjoin F s = ⊤
P : IntermediateField F E → Prop := fun K => Fintyp... | d4b369ecd5955422 |
ArithmeticFunction.vonMangoldt.summable_residueClass_non_primes_div | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | /-- The function `n ↦ Λ n / n`, restricted to non-primes in a residue class, is summable.
This is used to convert results on `ArithmeticFunction.vonMangoldt.residueClass` to results
on primes in an arithmetic progression. -/
lemma summable_residueClass_non_primes_div :
Summable fun n : ℕ ↦ (if n.Prime then 0 else r... | case refine_1.refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ ArithmeticFunction.vonMangoldt.F₀ n
hF₀ : ∀ (p : Nat.Primes), ArithmeticFunction.vonMangoldt.F₀ ↑p = 0
hFF' : ArithmeticFunction.vonMango... | simp only [Set.range_prod_map, Set.range_id, Set.mem_prod, Set.mem_univ, Set.mem_range,
Nat.exists_add_one_eq, true_and, not_lt, nonpos_iff_eq_zero] at hu | case refine_1.refine_2
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ ArithmeticFunction.vonMangoldt.F₀ n
hF₀ : ∀ (p : Nat.Primes), ArithmeticFunction.vonMangoldt.F₀ ↑p = 0
hFF' : ArithmeticFunction.vonMango... | 6c8eee8a1fa8d5a9 |
Matrix.frobenius_norm_col | Mathlib/Analysis/Matrix.lean | theorem frobenius_norm_col (v : n → α) : ‖col ι v‖ = ‖(WithLp.equiv 2 _).symm v‖ | n : Type u_4
α : Type u_5
ι : Type u_7
inst✝² : Fintype n
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : n → α
⊢ (∑ x : n, ‖col ι v x default‖ ^ 2) ^ (1 / 2) = (∑ i : n, ‖(WithLp.equiv 2 (n → α)).symm v i‖ ^ 2) ^ (1 / 2) | simp only [col_apply, Real.rpow_two, WithLp.equiv_symm_pi_apply] | no goals | 5a15b318ee26225c |
Int.sq_of_gcd_eq_one | Mathlib/RingTheory/Int/Basic.lean | theorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :
∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 | a b c : ℤ
h : a.gcd b = 1
heq : a * b = c ^ 2
⊢ IsUnit 1 | exact isUnit_one | no goals | de99b93abef95ba2 |
Padic.padicNormE_lim_le | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
‖f.lim‖ ≤ a | case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(const norm f.lim - f) j‖ < a
⊢ ‖f.lim‖ ≤ a | calc
‖f.lim‖ = ‖f.lim - f N + f N‖ := by simp
_ ≤ max ‖f.lim - f N‖ ‖f N‖ := padicNormE.nonarchimedean _ _
_ ≤ a := max_le (le_of_lt (hN _ le_rfl)) (hf _) | no goals | 1cb646383844b024 |
Unitization.splitMul_injective_of_clm_mul_injective | Mathlib/Analysis/Normed/Algebra/Unitization.lean | theorem splitMul_injective_of_clm_mul_injective
(h : Function.Injective (mul 𝕜 A)) :
Function.Injective (splitMul 𝕜 A) | case inl_add_inr.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
h : Function.Injective ⇑(mul 𝕜 A)
a✝ : A
hx : (mul 𝕜 A) a✝ = (mul 𝕜 A) 0
⊢ ↑a✝ = 0 | rw [h hx, inr_zero] | no goals | 95669d928e59e42c |
measurable_to_prop | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurable_to_prop {f : α → Prop} (h : MeasurableSet (f ⁻¹' {True})) : Measurable f | α : Type u_1
inst✝ : MeasurableSpace α
f : α → Prop
h : MeasurableSet (f ⁻¹' {True})
x : Prop
⊢ MeasurableSet (f ⁻¹' {x}) | by_cases hx : x | case pos
α : Type u_1
inst✝ : MeasurableSpace α
f : α → Prop
h : MeasurableSet (f ⁻¹' {True})
x : Prop
hx : x
⊢ MeasurableSet (f ⁻¹' {x})
case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → Prop
h : MeasurableSet (f ⁻¹' {True})
x : Prop
hx : ¬x
⊢ MeasurableSet (f ⁻¹' {x}) | 827afcfaaaf16cca |
Std.Range.size_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Range/Lemmas.lean | theorem size_eq (r : Std.Range) (h : i < r.stop) :
(r.stop - i + r.step - 1) / r.step =
(r.stop - (i + r.step) + r.step - 1) / r.step + 1 | i : Nat
r : Range
h : i < r.stop
w : 0 < r.step
h✝ : i + r.step < r.stop
this✝ : (r.stop - (i + r.step) + r.step - 1) / r.step * r.step ≤ r.stop - (i + r.step) + r.step - 1
this : r.stop - (i + r.step) + r.step - 1 - r.step < (r.stop - (i + r.step) + r.step - 1) / r.step * r.step
⊢ r.stop - i + r.step - 1 ≤ (r.stop - (... | omega | no goals | 958938ce3486a63a |
ProbabilityTheory.Kernel.exists_ae_eq_isMarkovKernel | Mathlib/Probability/Kernel/Basic.lean | lemma exists_ae_eq_isMarkovKernel {μ : Measure α}
(h : ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a)) (h' : μ ≠ 0) :
∃ (η : Kernel α β), (κ =ᵐ[μ] η) ∧ IsMarkovKernel η | case intro.intro.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
⊢ ∃ η, ⇑κ =ᶠ[ae μ] ⇑η ∧ IsMarkovKernel η | obtain ⟨a, ha⟩ : sᶜ.Nonempty := by
contrapose! h'; simpa [μs, h'] using measure_univ_le_add_compl s (μ := μ) | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ ∃ η, ⇑κ =ᶠ[ae μ] ⇑η ∧ Is... | be621af5684ed1da |
List.Nodup.mem_diff_iff | Mathlib/Data/List/Nodup.lean | theorem Nodup.mem_diff_iff [DecidableEq α] (hl₁ : l₁.Nodup) : a ∈ l₁.diff l₂ ↔ a ∈ l₁ ∧ a ∉ l₂ | α : Type u
l₁ l₂ : List α
a : α
inst✝ : DecidableEq α
hl₁ : l₁.Nodup
⊢ a ∈ l₁.diff l₂ ↔ a ∈ l₁ ∧ a ∉ l₂ | rw [hl₁.diff_eq_filter, mem_filter, decide_eq_true_iff] | no goals | 88b203d17a366fa7 |
AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | theorem f_invApp_f_app (i j k : D.J) (U : Opens (D.V (i, j)).carrier) :
(D.f_open i j).invApp _ U ≫ (D.f i k).c.app _ =
(π₁ i, j, k).c.app (op U) ≫
(π₂⁻¹ i, j, k) (unop _) ≫
(D.V _).presheaf.map
(eqToHom
(by
delta IsOpenImmersion.opensFunctor
... | C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this :
∀ (U : (Opens ↑↑(D.U i))ᵒᵖ),
(D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) =
((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens... | simp_rw [Category.assoc] | C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this :
∀ (U : (Opens ↑↑(D.U i))ᵒᵖ),
(D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) =
((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens... | 12d445cb71884f59 |
Prod.gameAdd_iff | Mathlib/Order/GameAdd.lean | theorem gameAdd_iff {rα rβ} {x y : α × β} :
GameAdd rα rβ x y ↔ rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 | case mp
α : Type u_1
β : Type u_2
rα : α → α → Prop
rβ : β → β → Prop
x y : α × β
⊢ GameAdd rα rβ x y → rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 | rintro (@⟨a₁, a₂, b, h⟩ | @⟨a, b₁, b₂, h⟩) | case mp.fst
α : Type u_1
β : Type u_2
rα : α → α → Prop
rβ : β → β → Prop
a₁ a₂ : α
b : β
h : rα a₁ a₂
⊢ rα (a₁, b).1 (a₂, b).1 ∧ (a₁, b).2 = (a₂, b).2 ∨ rβ (a₁, b).2 (a₂, b).2 ∧ (a₁, b).1 = (a₂, b).1
case mp.snd
α : Type u_1
β : Type u_2
rα : α → α → Prop
rβ : β → β → Prop
a : α
b₁ b₂ : β
h : rβ b₁ b₂
⊢ rα (a, b₁).1 ... | 28d98cf43a79cccd |
SSet.const_comp | Mathlib/AlgebraicTopology/SimplicialSet/Basic.lean | @[simp]
lemma const_comp {X Y Z : SSet.{u}} (y : Y _⦋0⦌) (g : Y ⟶ Z) :
const (X := X) y ≫ g = const (g.app _ y) | X Y Z : SSet
y : Y _⦋0⦌
g : Y ⟶ Z
⊢ const y ≫ g = const (g.app (Opposite.op ⦋0⦌) y) | ext m x | case w.h
X Y Z : SSet
y : Y _⦋0⦌
g : Y ⟶ Z
m : SimplexCategoryᵒᵖ
x : X.obj m
⊢ (const y ≫ g).app m x = (const (g.app (Opposite.op ⦋0⦌) y)).app m x | a2cd29a5b6629e5d |
HurwitzZeta.continuousOn_oddKernel | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | /-- The odd kernel is continuous on `Ioi 0`. -/
lemma continuousOn_oddKernel (a : UnitAddCircle) : ContinuousOn (oddKernel a) (Ioi 0) | case H.refine_2.hg
a x : ℝ
hx : x ∈ Ioi 0
hf : Continuous fun u => (↑a * I * ↑u, I * ↑u)
⊢ ContinuousAt (fun x => ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)) x | exact continuousAt_const.mul <| (continuousAt_jacobiTheta₂ (a * I * x)
(by rwa [I_mul_im, ofReal_re])).comp (f := fun u : ℝ ↦ (a * I * u, I * u)) hf.continuousAt | no goals | ada121cff347bbb8 |
contMDiff_isOpenEmbedding | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | /-- If the `ChartedSpace` structure on a manifold `M` is given by an open embedding `e : M → H`,
then `e` is `C^n`. -/
lemma contMDiff_isOpenEmbedding [Nonempty M] :
haveI := h.singletonChartedSpace; ContMDiff I I n e | case right
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : WithTop ℕ∞
inst✝ : Nonempty M
this : IsManif... | rw [I.right_inv] | case right
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : WithTop ℕ∞
inst✝ : Nonempty M
this : IsManif... | d917ee07659392c5 |
HasDerivAt.comp_ofReal | Mathlib/Analysis/Complex/RealDeriv.lean | theorem HasDerivAt.comp_ofReal (hf : HasDerivAt e e' ↑z) : HasDerivAt (fun y : ℝ => e ↑y) e' z | e : ℂ → ℂ
e' : ℂ
z : ℝ
hf : HasDerivAt e e' ↑z
⊢ HasDerivAt (fun y => e ↑y) e' z | simpa only [ofRealCLM_apply, ofReal_one, mul_one] using hf.comp z ofRealCLM.hasDerivAt | no goals | 43927f10f66f83fd |
FirstOrder.Language.Embedding.subtype_substructureEquivMap | Mathlib/ModelTheory/Substructures.lean | theorem subtype_substructureEquivMap (f : M ↪[L] N) (s : L.Substructure M) :
(subtype _).comp (f.substructureEquivMap s).toEmbedding = f.comp (subtype _) | case h
L : Language
M : Type w
N : Type u_1
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ↪[L] N
s : L.Substructure M
x✝ : ↥s
⊢ ((Substructure.map f.toHom s).subtype.comp (f.substructureEquivMap s).toEmbedding) x✝ = (f.comp s.subtype) x✝ | rfl | no goals | bc35d1ab2a889613 |
CategoryTheory.pullbackShiftFunctorAdd'_inv_app | Mathlib/CategoryTheory/Shift/Pullback.lean | lemma pullbackShiftFunctorAdd'_inv_app :
(shiftFunctorAdd' _ a₁ a₂ a₃ h).inv.app X =
(shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).hom.app X) ≫
(pullbackShiftIso C φ a₂ b₂ h₂).hom.app _ ≫
(shiftFunctorAdd' C b₁ b₂ b₃ (by rw [h₁, h₂, h₃, ← h, φ.map_add])).inv.app X ... | C : Type u_1
inst✝³ : Category.{?u.25018, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ b₁ + b₂ = b₃ | rw [h₁, h₂, h₃, ← h, φ.map_add] | no goals | a1ceed23baed48f7 |
Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing | Mathlib/RingTheory/Unramified/Field.lean | theorem bijective_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] :
Function.Bijective (algebraMap K A) | case e_f.a.H
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijec... | intros r s | case e_f.a.H
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijec... | 5099a67fd835b73a |
Cardinal.isSuccLimit_aleph0 | Mathlib/SetTheory/Cardinal/Basic.lean | theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ | ⊢ ℵ₀ ≠ 0 ∧ IsSuccPrelimit ℵ₀ | exact ⟨aleph0_ne_zero, isSuccPrelimit_aleph0⟩ | no goals | ed43733052682f40 |
IsNilpotent.isUnit_quotient_mk_iff | Mathlib/RingTheory/Ideal/Quotient/Nilpotent.lean | theorem IsNilpotent.isUnit_quotient_mk_iff {R : Type*} [CommRing R] {I : Ideal R}
(hI : IsNilpotent I) {x : R} : IsUnit (Ideal.Quotient.mk I x) ↔ IsUnit x | R : Type u_2
inst✝ : CommRing R
I : Ideal R
hI : IsNilpotent I
x : R
⊢ IsUnit ((Ideal.Quotient.mk I) x) ↔ IsUnit x | refine ⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩ | R : Type u_2
inst✝ : CommRing R
I : Ideal R
hI : IsNilpotent I
x : R
⊢ IsUnit ((Ideal.Quotient.mk I) x) → IsUnit x | bd2278e113ece423 |
Setoid.eq_eqv_class_of_mem | Mathlib/Data/Setoid/Partition.lean | theorem eq_eqv_class_of_mem {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {s y}
(hs : s ∈ c) (hy : y ∈ s) : s = { x | mkClasses c H x y } | case h.mp
α : Type u_1
c : Set (Set α)
H : ∀ (a : α), ∃! b, b ∈ c ∧ a ∈ b
s : Set α
y : α
hs : s ∈ c
hy : y ∈ s
x : α
hx : x ∈ s
_s' : Set α
hs' : _s' ∈ c
hx' : x ∈ _s'
⊢ y ∈ _s' | rwa [eq_of_mem_eqv_class H hs' hx' hs hx] | no goals | 1601737e771c7bfb |
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