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 |
|---|---|---|---|---|---|---|
Polynomial.leadingCoeff_multiset_prod' | Mathlib/Algebra/Polynomial/BigOperators.lean | theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod ≠ 0) :
t.prod.leadingCoeff = (t.map leadingCoeff).prod | case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ a.leadingCoeff * t.prod.leadingCoeff = a.leadingCoeff * (Multiset... | rw [ih] | case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (Multiset.map leadingCoeff t).prod ≠ 0 | 08d45fcd3d906d49 |
ONote.repr_opow_aux₂ | Mathlib/SetTheory/Ordinal/Notation.lean | theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω ∣ repr a')
(e0 : repr a0 ≠ 0) (h : repr a' + m < (ω ^ repr a0)) (n : ℕ+) (k : ℕ) :
let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
(k ≠ 0 → R < ((ω ^ repr a0) ^ succ (k : Ordinal))) ∧
((ω ^ repr a0) ^ (k : Ordinal)) * (... | a0 a' : ONote
N0 : a0.NF
Na' : a'.NF
m : ℕ
d : ω ∣ a'.repr
e0 : a0.repr ≠ 0
h : a'.repr + ↑m < ω ^ a0.repr
n : ℕ+
k : ℕ
⊢ let R := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr;
(k ≠ 0 → R < (ω ^ a0.repr) ^ succ ↑k) ∧
(ω ^ a0.repr) ^ ↑k * (ω ^ a0.repr * ↑↑n + a'.repr) + R = (ω ^ a0.repr * ↑↑n + a'.repr + ↑m) ^ succ... | intro R' | a0 a' : ONote
N0 : a0.NF
Na' : a'.NF
m : ℕ
d : ω ∣ a'.repr
e0 : a0.repr ≠ 0
h : a'.repr + ↑m < ω ^ a0.repr
n : ℕ+
k : ℕ
R' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr
⊢ (k ≠ 0 → R' < (ω ^ a0.repr) ^ succ ↑k) ∧
(ω ^ a0.repr) ^ ↑k * (ω ^ a0.repr * ↑↑n + a'.repr) + R' = (ω ^ a0.repr * ↑↑n + a'.repr + ... | 5c92dbccef953ffc |
SimpleGraph.edist_eq_one_iff_adj | Mathlib/Combinatorics/SimpleGraph/Metric.lean | theorem edist_eq_one_iff_adj : G.edist u v = 1 ↔ G.Adj u v | V : Type u_1
G : SimpleGraph V
u v : V
⊢ G.edist u v = 1 ↔ G.Adj u v | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩ | case refine_1
V : Type u_1
G : SimpleGraph V
u v : V
h : G.edist u v = 1
⊢ G.Adj u v
case refine_2
V : Type u_1
G : SimpleGraph V
u v : V
h : G.Adj u v
⊢ G.edist u v = 1 | 8b97c4192f62d7c8 |
Monoid.CoprodI.lift_word_prod_nontrivial_of_not_empty | Mathlib/GroupTheory/CoprodI.lean | theorem lift_word_prod_nontrivial_of_not_empty {i j} (w : NeWord H i j) :
lift f w.prod ≠ 1 | case inl.intro.intro
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆... | exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w h1 h2 | no goals | 3e8798a67334cdb3 |
CategoryTheory.GrothendieckTopology.Plus.exists_of_sep | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
(hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
(X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
∃ t : ToType ((J.plusObj P).obj (op X)), Meq.mk S t = s | case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShap... | dsimp [Meq.mk] | case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShap... | adc5a5926d46b74e |
CategoryTheory.MonoidalCategory.associator_monoidal | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
tensorμ (X₁ ⊗ X₂) X₃ (Y₁ ⊗ Y₂) Y₃ ≫
(tensorμ X₁ X₂ Y₁ Y₂ ▷ (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
((α_ X₁ X₂ X₃).hom ⊗ (α_ Y₁ Y₂ Y₃).hom) ≫
tensorμ X₁ (X₂ ⊗ X₃) Y₁ (Y₂ ⊗ Y₃) ≫ ((X₁ ⊗ Y₁) ◁ tensorμ X₂ X₃ Y₂ Y₃) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
⊢ tensorμ (X₁ ⊗ X₂) X₃ (Y₁ ⊗ Y₂) Y₃ ≫ tensorμ X₁ X₂ Y₁ Y₂ ▷ (X₃ ⊗ Y₃) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
((α_ X₁ X₂ X₃).hom ⊗ (α_ Y₁ Y₂ Y₃).hom) ≫ tensorμ X₁ (X₂ ⊗ X₃) Y₁ (Y₂ ⊗ Y₃) ≫ (X₁ ⊗ Y₁... | dsimp only [tensorμ] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
⊢ ((α_ (X₁ ⊗ X₂) X₃ ((Y₁ ⊗ Y₂) ⊗ Y₃)).hom ≫
(X₁ ⊗ X₂) ◁ (α_ X₃ (Y₁ ⊗ Y₂) Y₃).inv ≫
(X₁ ⊗ X₂) ◁ (β_ X₃ (Y₁ ⊗ Y₂)).hom ▷ Y₃ ≫
(X₁ ⊗ X₂) ◁ (α_ (Y₁ ⊗ Y₂) X₃ Y₃).hom ≫ (α_ (X₁ ⊗ ... | 6bf761a2ac8fdbf4 |
Finset.prod_Ico_reflect | Mathlib/Algebra/BigOperators/Intervals.lean | theorem prod_Ico_reflect (f : ℕ → M) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :
(∏ j ∈ Ico k m, f (n - j)) = ∏ j ∈ Ico (n + 1 - m) (n + 1 - k), f j | case inr
M : Type u_2
inst✝ : CommMonoid M
f : ℕ → M
k m n : ℕ
h : m ≤ n + 1
this : ∀ i < m, i ≤ n
hkm : m ≤ k
⊢ ∏ j ∈ Ico k m, f (n - j) = ∏ j ∈ Ico (n + 1 - m) (n + 1 - k), f j | have : n + 1 - k ≤ n + 1 - m := by
rw [tsub_le_tsub_iff_left h]
exact hkm | case inr
M : Type u_2
inst✝ : CommMonoid M
f : ℕ → M
k m n : ℕ
h : m ≤ n + 1
this✝ : ∀ i < m, i ≤ n
hkm : m ≤ k
this : n + 1 - k ≤ n + 1 - m
⊢ ∏ j ∈ Ico k m, f (n - j) = ∏ j ∈ Ico (n + 1 - m) (n + 1 - k), f j | 83e148f6e39dd634 |
Measurable.simpleFunc_add | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem _root_.Measurable.simpleFunc_add
{E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddGroup E] [MeasurableAdd E]
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
Measurable ((f : α → E) + g) | case h_add
α : Type u_1
E : Type u_5
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : AddGroup E
inst✝ : MeasurableAdd E
g : α → E
hg : Measurable g
f f' : α →ₛ E
hff' : Disjoint (support ⇑f) (support ⇑f')
hf : Measurable (⇑f + g)
hf' : Measurable (⇑f' + g)
this : ⇑(f + f') + g = (support ⇑f).piecewise (⇑f + ... | exact Measurable.piecewise f.measurableSet_support hf hf' | no goals | 48fcdcf52a8216af |
isBigO_norm_Icc_restrict_atBot | Mathlib/Analysis/Fourier/PoissonSummation.lean | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atBot] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atBot] fun x : ℝ => |x| ^ (-b) | case h.e'_8
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
⊢ (fun x => |x| ^ (-b)) = (fun x => |x| ^ (-b)) ∘ Neg.neg | ext1 x | case h.e'_8.h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S x : ℝ
⊢ |x| ^ (-b) = ((fun x => |x| ^ (-b)) ∘ Neg.neg) x | 04ef8503390ca3ea |
WeierstrassCurve.Projective.addMap_of_Y_eq | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma addMap_of_Y_eq {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
(hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy' : P y * Q z = W.negY Q * P z) :
W.addMap ⟦P⟧ ⟦Q⟧ = ⟦![0, 1, 0]⟧ | case pos
F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy' : P y * Q z = W.negY Q * P z
hy : P y * Q z = Q y * P z
⊢ W.addMap ⟦P⟧ ⟦Q⟧ = ⟦![0, 1, 0]⟧ | rw [addMap_eq, add_of_Y_eq hP.left hPz hQz hx hy hy',
smul_eq _ <| isUnit_dblU_of_Y_eq hP hPz hQz hx hy hy'] | no goals | 548e3b0253acdae1 |
Ring.choose_smul_choose | Mathlib/RingTheory/Binomial.lean | theorem choose_smul_choose [NatPowAssoc R] (r : R) {n k : ℕ} (hkn : k ≤ n) :
(Nat.choose n k) • choose r n = choose r k * choose (r - k) (n - k) | R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ (n - k + k)).smeval r =
n.choose k • ((descPochhammer ℤ k).smeval r * (descPochhammer ℤ (n - k)).smeval (r - ↑k)) | rw [add_comm, ← descPochhammer_mul, smeval_mul, smeval_comp, smeval_sub, smeval_X,
← C_eq_natCast, smeval_C, npow_one, npow_zero, zsmul_one, Int.cast_natCast, nsmul_eq_mul] | no goals | 86e2c7c4fd4c502a |
ENNReal.exists_nat_pos_inv_mul_lt | Mathlib/Data/ENNReal/Inv.lean | theorem exists_nat_pos_inv_mul_lt (ha : a ≠ ∞) (hb : b ≠ 0) :
∃ n > 0, ((n : ℕ) : ℝ≥0∞)⁻¹ * a < b | a b : ℝ≥0∞
ha : a ≠ ⊤
hb : b ≠ 0
⊢ ∃ n > 0, (↑n)⁻¹ * a < b | rcases exists_nat_pos_mul_gt hb ha with ⟨n, npos, hn⟩ | case intro.intro
a b : ℝ≥0∞
ha : a ≠ ⊤
hb : b ≠ 0
n : ℕ
npos : n > 0
hn : a < ↑n * b
⊢ ∃ n > 0, (↑n)⁻¹ * a < b | d0e03ba40751de58 |
NNReal.IsConjExponent.conj_eq | Mathlib/Data/Real/ConjExponents.lean | lemma conj_eq : q = p / (p - 1) | p q : ℝ≥0
h : p.IsConjExponent q
⊢ q = p / (p - 1) | simpa only [← coe_one, ← NNReal.coe_sub h.one_le, ← NNReal.coe_div, coe_inj] using h.coe.conj_eq | no goals | de3eeb99028110c5 |
RootPairing.coroot'_reflection_perm | Mathlib/LinearAlgebra/RootSystem/Defs.lean | lemma coroot'_reflection_perm {i j : ι} :
P.coroot' (P.reflection_perm i j) = P.coroot' j ∘ₗ P.reflection i | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j : ι
⊢ P.coroot' ((P.reflection_perm i) j) = P.coroot' j ∘ₗ ↑(P.reflection i) | ext y | case h
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j : ι
y : M
⊢ (P.coroot' ((P.reflection_perm i) j)) y = (P.coroot' j ∘ₗ ↑(P.reflection i)) y | 54d8d76b4a0f012f |
Int.image_Ico_emod | Mathlib/Data/Int/Interval.lean | theorem image_Ico_emod (n a : ℤ) (h : 0 ≤ a) : (Ico n (n + a)).image (· % a) = Ico 0 a | case inr.h.mpr.inr.refine_2
n a : ℤ
h : 0 ≤ a
ha : 0 < a
i : ℤ
hia : 0 ≤ i ∧ i < a
hn : n % a + a * (n / a) = n
hi : n % a ≤ i
⊢ a * (n / a) ≤ n % a + a * (n / a) | simp only [Nat.zero_le, le_add_iff_nonneg_left] | case inr.h.mpr.inr.refine_2
n a : ℤ
h : 0 ≤ a
ha : 0 < a
i : ℤ
hia : 0 ≤ i ∧ i < a
hn : n % a + a * (n / a) = n
hi : n % a ≤ i
⊢ 0 ≤ n % a | 9d26ae2005c6604e |
Batteries.RBNode.Ordered.unique | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem Ordered.unique [@TransCmp α cmp] (ht : Ordered cmp t)
(hx : x ∈ t) (hy : y ∈ t) (e : cmp x y = .eq) : x = y | case node.inr.inr.inl
α : Type u_1
cmp : α → α → Ordering
x y : α
inst✝ : TransCmp cmp
e : cmp x y = Ordering.eq
c✝ : RBColor
l r : RBNode α
ihl : Ordered cmp l → x ∈ l → y ∈ l → x = y
ihr : Ordered cmp r → x ∈ r → y ∈ r → x = y
hl : Ordered cmp l
hr : Ordered cmp r
hx : Any (fun x_1 => x = x_1) r
ht : Ordered cmp (nod... | cases e.symm.trans <| OrientedCmp.cmp_eq_gt.2 (All_def.1 xr _ hx).1 | no goals | 77450b4176612ab8 |
Algebra.SubmersivePresentation.basisDeriv_apply | Mathlib/RingTheory/Smooth/StandardSmooth.lean | @[simp]
lemma basisDeriv_apply (i j : P.rels) :
P.basisDeriv i j = (aeval P.val) (pderiv (P.map j) (P.relation i)) | R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : SubmersivePresentation R S
i j : P.rels
⊢ P.basisDeriv i j = (aeval P.val) ((pderiv (P.map j)) (P.relation i)) | classical
simp [basisDeriv] | no goals | 894ddc991cebbc22 |
Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd' | Mathlib/Algebra/Module/Injective.lean | theorem ExtensionOfMaxAdjoin.extendIdealTo_wd' (h : Module.Baer R Q) {y : N} (r : R)
(eq1 : r • y = 0) : ExtensionOfMaxAdjoin.extendIdealTo i f h y r = 0 | R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
r : R
eq1 : r • y = 0
this : r ∈ ideal i... | rw [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h y r this] | R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
r : R
eq1 : r • y = 0
this : r ∈ ideal i... | ba5bece53f8b95be |
IsDenseInducing.extend_Z_bilin_aux | Mathlib/Topology/Algebra/UniformGroup/Defs.lean | theorem extend_Z_bilin_aux (x₀ : α) (y₁ : δ) : ∃ U₂ ∈ comap e (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂,
(fun p : β × δ => φ p.1 p.2) (x' - x, y₁) ∈ W' | α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ... | rw [nhds_prod_eq, prod_comap_comap_eq, ← nhds_prod_eq] | α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ... | 6766ff934c6fdbf1 |
Nat.emultiplicity_two_factorial_lt | Mathlib/Data/Nat/Multiplicity.lean | theorem emultiplicity_two_factorial_lt : ∀ {n : ℕ} (_ : n ≠ 0), emultiplicity 2 n ! < n | ⊢ ∀ {n : ℕ}, n ≠ 0 → emultiplicity 2 n ! < ↑n | have h2 := prime_two.prime | h2 : _root_.Prime 2
⊢ ∀ {n : ℕ}, n ≠ 0 → emultiplicity 2 n ! < ↑n | ce05fac28659a542 |
PhragmenLindelof.quadrant_IV | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem quadrant_IV (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
(hB : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * ‖z‖ ^ c))
(hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C) (hz_re : 0 ≤ z.re)
(hz_im : z.im ≤ 0) : ‖f z‖ ≤ C | case intro.refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : z.re ≤ 0
... | exact him (-x) (neg_nonpos.2 hx) | no goals | f7b839a4c412b6d6 |
ProbabilityTheory.Kernel.iIndepFun.comp | Mathlib/Probability/Independence/Kernel.lean | lemma iIndepFun.comp {β γ : ι → Type*} {mβ : ∀ i, MeasurableSpace (β i)}
{mγ : ∀ i, MeasurableSpace (γ i)} {f : ∀ i, Ω → β i}
(h : iIndepFun mβ f κ μ) (g : ∀ i, β i → γ i) (hg : ∀ i, Measurable (g i)) :
iIndepFun mγ (fun i ↦ g i ∘ f i) κ μ | case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
γ : ι → Type u_9
mβ : (i : ι) → MeasurableSpace (β i)
mγ : (i : ι) → MeasurableSpace (γ i)
f : (i : ι) → Ω → β i
h :
∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
(∀ i ∈ S, ... | simp_rw [Set.preimage_comp] | case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
γ : ι → Type u_9
mβ : (i : ι) → MeasurableSpace (β i)
mγ : (i : ι) → MeasurableSpace (γ i)
f : (i : ι) → Ω → β i
h :
∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
(∀ i ∈ S, ... | dd1fe9dd37508bb1 |
LinearMap.BilinForm.iIsOrtho.not_isOrtho_basis_self_of_nondegenerate | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | theorem iIsOrtho.not_isOrtho_basis_self_of_nondegenerate {n : Type w} [Nontrivial R]
{B : BilinForm R M} {v : Basis n R M} (h : B.iIsOrtho v) (hB : B.Nondegenerate) (i : n) :
¬B.IsOrtho (v i) (v i) | R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
n : Type w
inst✝ : Nontrivial R
B : BilinForm R M
v : Basis n R M
h : B.iIsOrtho ⇑v
hB : B.Nondegenerate
i : n
ho : B.IsOrtho (v i) (v i)
m : M
⊢ (B (v i)) m = 0 | obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m | case intro
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
n : Type w
inst✝ : Nontrivial R
B : BilinForm R M
v : Basis n R M
h : B.iIsOrtho ⇑v
hB : B.Nondegenerate
i : n
ho : B.IsOrtho (v i) (v i)
vi : n →₀ R
⊢ (B (v i)) (v.repr.symm vi) = 0 | 1580a7998a36ecc0 |
LaurentPolynomial.trunc_C_mul_T | Mathlib/Algebra/Polynomial/Laurent.lean | theorem trunc_C_mul_T (n : ℤ) (r : R) : trunc (C r * T n) = ite (0 ≤ n) (monomial n.toNat r) 0 | case a
R : Type u_1
inst✝ : Semiring R
n : ℤ
r : R
⊢ (toFinsuppIso R) ((toFinsuppIso R).symm.toAddMonoidHom ((comapDomain.addMonoidHom ⋯) (Finsupp.single n r))) =
(toFinsuppIso R) (if 0 ≤ n then (monomial n.toNat) r else 0) | erw [comapDomain.addMonoidHom_apply Int.ofNat_injective] | case a
R : Type u_1
inst✝ : Semiring R
n : ℤ
r : R
⊢ comapDomain Int.ofNat (Finsupp.single n r) ⋯ = (toFinsuppIso R) (if 0 ≤ n then (monomial n.toNat) r else 0) | 77ffcc97fc60df9c |
Vitali.exists_disjoint_covering_ae | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈... | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | refine (hB a hat).trans ?_ | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | 4466d2fc3e80cb3f |
FractionalIdeal.isPrincipal.of_finite_maximals_of_inv | Mathlib/RingTheory/DedekindDomain/PID.lean | theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
[Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S ≤ R⁰)
(hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
Submodule.IsPrincipal (I : Submodule R A) | R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ ... | intro M hM | R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ ... | 739ab9d4c311b5e5 |
Dynamics.coverEntropyInf_eq_coverEntropy | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | lemma coverEntropyInf_eq_coverEntropy (T : X → X) {F : Set X} (h : MapsTo T F F) :
coverEntropyInf T F = coverEntropy T F | X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
h : MapsTo T F F
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ coverEntropyEntourage T F U ≤ coverEntropyInf T F | rcases comp_symm_mem_uniformity_sets U_uni with ⟨V, V_uni, V_symm, V_U⟩ | case intro.intro.intro
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
h : MapsTo T F F
U : Set (X × X)
U_uni : U ∈ 𝓤 X
V : Set (X × X)
V_uni : V ∈ 𝓤 X
V_symm : SymmetricRel V
V_U : V ○ V ⊆ U
⊢ coverEntropyEntourage T F U ≤ coverEntropyInf T F | 5d992315baed6191 |
LinearMap.iterate_bijective | Mathlib/Algebra/Module/LinearMap/End.lean | theorem iterate_bijective (h : Bijective f') : ∀ n : ℕ, Bijective (f' ^ n)
| 0 => bijective_id
| n + 1 => by
rw [iterate_succ]
exact (iterate_bijective h n).comp h
| R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f' : M →ₗ[R] M
h : Bijective ⇑f'
n : ℕ
⊢ Bijective ⇑(f' ^ (n + 1)) | rw [iterate_succ] | R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f' : M →ₗ[R] M
h : Bijective ⇑f'
n : ℕ
⊢ Bijective ⇑((f' ^ n) ∘ₗ f') | de7cf4517f6aa767 |
Ordnode.Valid'.rotateL_lemma₄ | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.rotateL_lemma₄ {a b : ℕ} (H3 : 2 * b ≤ 9 * a + 3) : 3 * b ≤ 16 * a + 9 | a b : ℕ
H3 : 2 * b ≤ 9 * a + 3
⊢ 3 * b ≤ 16 * a + 9 | omega | no goals | c43ea1f5bee87a19 |
quadraticChar_card_card | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | theorem quadraticChar_card_card [DecidableEq F] (hF : ringChar F ≠ 2) {F' : Type*} [Field F']
[Fintype F'] [DecidableEq F'] (hF' : ringChar F' ≠ 2) (h : ringChar F' ≠ ringChar F) :
quadraticChar F (Fintype.card F') =
quadraticChar F' (quadraticChar F (-1) * Fintype.card F) | F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
⊢ (quadraticChar F) ↑(... | have h := Char.card_pow_card hχ₁ ((quadraticChar_isQuadratic F).comp _) h hF' | F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h✝ : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
h : (χ (-1) * ↑(Finty... | 0521617d4359a853 |
MeasureTheory.AECover.lintegral_tendsto_of_nat | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞}
(hfm : AEMeasurable f μ) : Tendsto (∫⁻ x in φ ·, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) | α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
φ : ℕ → Set α
hφ : AECover μ atTop φ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
lim₁ : Tendsto (fun i => ∫⁻ (x : α) in ⋂ k ∈ Ici i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
lim₂ : Tendsto (fun i => ∫⁻ (x : α) in ⋃ k ∈ Iic i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
... | refine tendsto_of_tendsto_of_tendsto_of_le_of_le lim₁ lim₂ (fun n ↦ ?_) fun n ↦ ?_ | case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
φ : ℕ → Set α
hφ : AECover μ atTop φ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
lim₁ : Tendsto (fun i => ∫⁻ (x : α) in ⋂ k ∈ Ici i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
lim₂ : Tendsto (fun i => ∫⁻ (x : α) in ⋃ k ∈ Iic i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x :... | 82c8c1c847dae885 |
List.head_takeWhile | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean | theorem head_takeWhile (p : α → Bool) (l : List α) (w) :
(l.takeWhile p).head w = l.head (by rintro rfl; simp_all) | α : Type u_1
p : α → Bool
l : List α
w : takeWhile p l ≠ []
⊢ (takeWhile p l).head w = l.head ⋯ | cases l with
| nil => rfl
| cons x xs =>
simp only [takeWhile_cons, head_cons]
simp only [takeWhile_cons] at w
split <;> simp_all | no goals | a309347b1cb9d239 |
Set.enumerate_mem | Mathlib/Data/Set/Enumerate.lean | theorem enumerate_mem (h_sel : ∀ s a, sel s = some a → a ∈ s) :
∀ {s n a}, enumerate sel s n = some a → a ∈ s
| s, 0, a => h_sel s a
| s, n + 1, a => by
cases h : sel s with
| none => simp [enumerate_eq_none_of_sel, h]
| some a' =>
simp only [enumerate, h, Nat.add_eq, add_zero]
exact fun... | case some
α : Type u_1
sel : Set α → Option α
h_sel : ∀ (s : Set α) (a : α), sel s = some a → a ∈ s
s : Set α
n : ℕ
a a' : α
h : sel s = some a'
⊢ (do
let a ← some a'
enumerate sel (s \ {a}) n) =
some a →
a ∈ s | exact fun h' : enumerate sel (s \ {a'}) n = some a ↦
have : a ∈ s \ {a'} := enumerate_mem h_sel h'
this.left | no goals | 6ee0b04deba0206d |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ... | constructor | case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst... | 02ca1706b397b254 |
Topology.IsClosedEmbedding.polishSpace | Mathlib/Topology/MetricSpace/Polish.lean | theorem _root_.Topology.IsClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β]
[PolishSpace β] {f : α → β} (hf : IsClosedEmbedding f) : PolishSpace α | α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
this✝¹ : UpgradedPolishSpace β := upgradePolishSpace β
this✝ : MetricSpace α := IsEmbedding.comapMetricSpace f ⋯
this : SecondCountableTopology α
⊢ PolishSpace α | have : CompleteSpace α := by
rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
exact hf.isClosed_range.isComplete | α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
this✝² : UpgradedPolishSpace β := upgradePolishSpace β
this✝¹ : MetricSpace α := IsEmbedding.comapMetricSpace f ⋯
this✝ : SecondCountableTopology α
this : CompleteSpace α
⊢ PolishSp... | 4a7835d35855f550 |
Int.dvd_of_dvd_mul_right_of_gcd_one | Mathlib/Data/Int/GCD.lean | theorem dvd_of_dvd_mul_right_of_gcd_one {a b c : ℤ} (habc : a ∣ b * c) (hab : gcd a b = 1) :
a ∣ c | a b c : ℤ
habc : a ∣ b * c
hab : a.gcd b = 1
⊢ a ∣ c | rw [mul_comm] at habc | a b c : ℤ
habc : a ∣ c * b
hab : a.gcd b = 1
⊢ a ∣ c | 1a586d02866ccac7 |
List.find?_range'_eq_some | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
(range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j | s n i : Nat
p : Nat → Bool
h : p i = true
h₃ : ∀ (j : Nat), s ≤ j → j < i → p j = false
h₁ : s ≤ i
h₂ : i < s + n
a : Nat
a₁ : s ≤ a
a₂ : a < s + (i - s)
⊢ a < i | omega | no goals | f6cd80357862798f |
Orientation.oangle_eq_pi_iff_sameRay_neg | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_eq_pi_iff_sameRay_neg {x y : V} :
o.oangle x y = π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y) | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y) | rw [← o.oangle_eq_zero_iff_sameRay] | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ o.oangle x (-y) = 0 | a3732c0f955bed29 |
Equiv.Perm.nodup_of_pairwise_disjoint | Mathlib/GroupTheory/Perm/Support.lean | theorem nodup_of_pairwise_disjoint {l : List (Perm α)} (h1 : (1 : Perm α) ∉ l)
(h2 : l.Pairwise Disjoint) : l.Nodup | α : Type u_1
l : List (Perm α)
h1 : 1 ∉ l
h2 : List.Pairwise Disjoint l
⊢ l.Nodup | refine List.Pairwise.imp_of_mem ?_ h2 | α : Type u_1
l : List (Perm α)
h1 : 1 ∉ l
h2 : List.Pairwise Disjoint l
⊢ ∀ {a b : Perm α}, a ∈ l → b ∈ l → a.Disjoint b → a ≠ b | d28a7770c33ffe6c |
isGδ_iff_eq_iInter_nat | Mathlib/Topology/GDelta/Basic.lean | lemma isGδ_iff_eq_iInter_nat {s : Set X} :
IsGδ s ↔ ∃ (f : ℕ → Set X), (∀ n, IsOpen (f n)) ∧ s = ⋂ n, f n | case refine_1.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT : ∀ t ∈ T, IsOpen t
T_count : T.Countable
⊢ ∃ f, (∀ (n : ℕ), IsOpen (f n)) ∧ ⋂₀ T = ⋂ n, f n | rcases Set.eq_empty_or_nonempty T with rfl|hT | case refine_1.intro.intro.intro.inl
X : Type u_1
inst✝ : TopologicalSpace X
hT : ∀ t ∈ ∅, IsOpen t
T_count : ∅.Countable
⊢ ∃ f, (∀ (n : ℕ), IsOpen (f n)) ∧ ⋂₀ ∅ = ⋂ n, f n
case refine_1.intro.intro.intro.inr
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT✝ : ∀ t ∈ T, IsOpen t
T_count : T.Countable
hT : T.No... | 5f43a0c95c012d1d |
Ideal.span_singleton_mul_right_inj | Mathlib/RingTheory/Ideal/Operations.lean | theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) :
span {x} * I = span {x} * J ↔ I = J | R : Type u
inst✝¹ : CommSemiring R
I J : Ideal R
inst✝ : IsDomain R
x : R
hx : x ≠ 0
⊢ span {x} * I = span {x} * J ↔ I = J | simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] | no goals | ea6e9aae94f661b3 |
BitVec.toInt_allOnes | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toInt_allOnes : (allOnes w).toInt = if 0 < w then -1 else 0 | case pos
w : Nat
h : w = 0
⊢ (allOnes w).toInt = if 0 < w then -1 else 0 | subst h | case pos
⊢ (allOnes 0).toInt = if 0 < 0 then -1 else 0 | f00205704d5f5dc2 |
Topology.CWComplex.cellFrontier_subset_finite_closedCell | Mathlib/Topology/CWComplex/Classical/Basic.lean | lemma CWComplex.cellFrontier_subset_finite_closedCell [CWComplex C] (n : ℕ) (i : cell C n) :
∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ ⋃ (m < n) (j ∈ I m), closedCell m j | case intro
X : Type u_1
t : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
n : ℕ
i : cell C n
I : (m : ℕ) → Finset (cell C m)
hI : MapsTo (↑(map n i)) (sphere 0 1) (∅ ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)
⊢ ∃ I, cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j | use I | case h
X : Type u_1
t : TopologicalSpace X
C : Set X
inst✝ : CWComplex C
n : ℕ
i : cell C n
I : (m : ℕ) → Finset (cell C m)
hI : MapsTo (↑(map n i)) (sphere 0 1) (∅ ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)
⊢ cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, closedCell m j | 2ef54141ee55975e |
εNFA.mem_εClosure_iff_exists_path | Mathlib/Computability/EpsilonNFA.lean | theorem mem_εClosure_iff_exists_path {s₁ s₂ : σ} :
s₂ ∈ M.εClosure {s₁} ↔ ∃ n, M.IsPath s₁ s₂ (.replicate n none) where
mp h | case h
α : Type u
σ : Type v
M : εNFA α σ
s₁ s₂ s✝ t✝ : σ
a✝¹ : t✝ ∈ M.step s✝ none
a✝ : M.εClosure {s₁} s✝
n : ℕ
h✝ : M.IsPath s₁ s✝ (List.replicate n none)
⊢ ∃ t, M.IsPath s₁ t (List.replicate n none) ∧ M.IsPath t t✝ (List.replicate 1 none) | tauto | no goals | a532dacc19d8227b |
Std.DHashMap.Raw.get_modify_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem get_modify_self [LawfulBEq α] {k : α} {f : β k → β k} (h : m.WF) {hc : k ∈ m.modify k f} :
haveI h' : k ∈ m := mem_modify h |>.mp hc
(m.modify k f).get k hc = f (m.get k h') | α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
m : Raw α β
inst✝ : LawfulBEq α
k : α
f : β k → β k
h : m.WF
hc : k ∈ m.modify k f
⊢ (m.modify k f).contains k = true → (m.modify k f).get k hc = f (m.get k ⋯) | simp_to_raw using Raw₀.get_modify_self | no goals | 090b34cbf68550b9 |
toIxxMod_cyclic_left | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIxxMod_cyclic_left {x₁ x₂ x₃ : α} (h : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃) :
toIcoMod hp x₂ x₃ ≤ toIocMod hp x₂ x₁ | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
x₁ x₂ x₃ : α
h✝ : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃
x₂' : α := toIcoMod hp x₁ x₂
x₃' : α := toIcoMod hp x₂' x₃
h : x₂' ≤ toIocMod hp x₁ x₃'
h₂₁ : x₂' < x₁ + p
⊢ toIcoMod hp x₂ x₃ ≤ toIocMod hp x₂ x₁ | have h₃₂ : x₃' - p < x₂' := sub_lt_iff_lt_add.2 (toIcoMod_lt_right _ _ _) | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
x₁ x₂ x₃ : α
h✝ : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃
x₂' : α := toIcoMod hp x₁ x₂
x₃' : α := toIcoMod hp x₂' x₃
h : x₂' ≤ toIocMod hp x₁ x₃'
h₂₁ : x₂' < x₁ + p
h₃₂ : x₃' - p < x₂'
⊢ toIcoMod hp x₂ x₃ ≤ toIocMod hp x₂ x₁ | e7f67c561e8fcc20 |
hasFDerivAt_polarCoord_symm | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
HasFDerivAt polarCoord.symm (fderivPolarCoordSymm p) p | p : ℝ × ℝ
⊢ HasFDerivAt (↑polarCoord.symm)
((cos p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (-p.1 * sin p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ).prod
(sin p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 * cos p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ))
p | convert HasFDerivAt.prod (𝕜 := ℝ)
(hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
(hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
simp [smul_smul, add_comm, neg_mul, smul_neg, neg_smul _ (ContinuousLinearMap.snd ℝ ℝ ℝ)] | no goals | b0f6064757e906eb |
Polynomial.natDegree_mod_lt | Mathlib/Algebra/Polynomial/FieldDivision.lean | lemma natDegree_mod_lt [Field k] (p : k[X]) {q : k[X]} (hq : q.natDegree ≠ 0) :
(p % q).natDegree < q.natDegree | case refine_2
k : Type y
inst✝ : Field k
p q : k[X]
hq : q.natDegree ≠ 0
hq' : q.leadingCoeff ≠ 0
⊢ q * C q.leadingCoeff⁻¹ ≠ 1 | contrapose! hq | case refine_2
k : Type y
inst✝ : Field k
p q : k[X]
hq' : q.leadingCoeff ≠ 0
hq : q * C q.leadingCoeff⁻¹ = 1
⊢ q.natDegree = 0 | c1edaa831c1aaa7d |
Orientation.volumeForm_robust_neg | Mathlib/Analysis/InnerProductSpace/Orientation.lean | theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation ≠ o) :
o.volumeForm = -b.toBasis.det | case h.e'_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ e.toBasis.or... | exact o.finOrthonormalBasis_orientation _ _ | no goals | b3c42ea7b4a2a624 |
hasDerivAt_ofReal_cpow_const' | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | theorem hasDerivAt_ofReal_cpow_const' {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1) :
HasDerivAt (fun y : ℝ => (y : ℂ) ^ (r + 1) / (r + 1)) (x ^ r) x | case h.e'_9
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ (r + 1) * ↑x ^ r = (r + 1) * ↑x ^ r * ?inl.convert_4.convert_2 | exact (mul_one _).symm | no goals | e70c1aa0a6bd69b4 |
LieAlgebra.IsKilling.isReduced_rootSystem | Mathlib/Algebra/Lie/Weights/RootSystem.lean | theorem isReduced_rootSystem : (rootSystem H).IsReduced | case intro.inl.h
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α : Weight K (↥H) L
hα : α ∈ LieSubalgebra.root
β : Wei... | ext x | case intro.inl.h.h
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α : Weight K (↥H) L
hα : α ∈ LieSubalgebra.root
β : W... | 19d8de96cdfca1d4 |
recursion' | Mathlib/Data/Real/Pi/Irrational.lean | /--
Auxiliary for the proof that `π` is irrational.
While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction... | case convert_2.convert_2
θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 *... | have (x) : u₂' x = (2 * n + 1) * f x ^ n - 2 * n * f x ^ (n - 1) := by
cases n with
| zero => simp [u₂']
| succ n => ring! | case convert_2.convert_2
θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 *... | 834ac76d1ec60c71 |
CategoryTheory.ObjectProperty.prop_X₂_of_exact | Mathlib/CategoryTheory/Abelian/SerreClass/Basic.lean | lemma prop_X₂_of_exact {S : ShortComplex C} (hS : S.Exact)
(h₁ : P S.X₁) (h₃ : P S.X₃) : P S.X₂ | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
S : ShortComplex C
hS : S.Exact
h₁ : P S.X₁
h₃ : P S.X₃
⊢ P S.X₂ | let d := S.homologyData | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
S : ShortComplex C
hS : S.Exact
h₁ : P S.X₁
h₃ : P S.X₃
d : S.HomologyData := S.homologyData
⊢ P S.X₂ | 01c6d97250e3eb76 |
analyticAt_clog | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z | case h
z : ℂ
m : z ∈ slitPlane
⊢ ∀ a ∈ slitPlane, DifferentiableAt ℂ log a | intro z m | case h
z✝ : ℂ
m✝ : z✝ ∈ slitPlane
z : ℂ
m : z ∈ slitPlane
⊢ DifferentiableAt ℂ log z | 057972827dcb7615 |
le_gronwallBound_of_liminf_deriv_right_le | Mathlib/Analysis/ODE/Gronwall.lean | theorem le_gronwallBound_of_liminf_deriv_right_le {f f' : ℝ → ℝ} {δ K ε : ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r)
(ha : f a ≤ δ) (bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε) :
∀ x ∈ Icc a b, f x ≤ gronwallBound δ K ε... | case hB
f f' : ℝ → ℝ
δ K ε a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r
ha : f a ≤ δ
bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε
x : ℝ
hx : x ∈ Icc a b
ε' : ℝ
hε' : ε' ∈ Ioi ε
⊢ ∀ (x : ℝ), HasDerivAt (fun x => gronwallBound δ K ε' (x -... | exact fun x => hasDerivAt_gronwallBound_shift δ K ε' x a | no goals | 7703692dd552179c |
Ideal.rank_eq | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | theorem Ideal.rank_eq {R S : Type*} [CommRing R] [StrongRankCondition R] [Ring S] [IsDomain S]
[Algebra R S] {n m : Type*} [Fintype n] [Fintype m] (b : Basis n R S) {I : Ideal S}
(hI : I ≠ ⊥) (c : Basis m R I) : Fintype.card m = Fintype.card n | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : StrongRankCondition R
inst✝⁴ : Ring S
inst✝³ : IsDomain S
inst✝² : Algebra R S
n : Type u_3
m : Type u_4
inst✝¹ : Fintype n
inst✝ : Fintype m
b : Basis n R S
I : Ideal S
hI : I ≠ ⊥
c : Basis m R ↥I
a : ↥I
ha : a ≠ 0
⊢ LinearIndependent R fun i => b i • a | have hb := b.linearIndependent | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : StrongRankCondition R
inst✝⁴ : Ring S
inst✝³ : IsDomain S
inst✝² : Algebra R S
n : Type u_3
m : Type u_4
inst✝¹ : Fintype n
inst✝ : Fintype m
b : Basis n R S
I : Ideal S
hI : I ≠ ⊥
c : Basis m R ↥I
a : ↥I
ha : a ≠ 0
hb : LinearIndependent R ⇑b
⊢ LinearIndependent R... | 1f1ec1d017dfcf9f |
Polynomial.reflect_mul_induction | Mathlib/Algebra/Polynomial/Reverse.lean | theorem reflect_mul_induction (cf cg : ℕ) :
∀ N O : ℕ,
∀ f g : R[X],
#f.support ≤ cf.succ →
#g.support ≤ cg.succ →
f.natDegree ≤ N →
g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g | case zero
R : Type u_1
inst✝ : Semiring R
cg : ℕ
⊢ ∀ (N O : ℕ) (f g : R[X]),
#f.support ≤ Nat.succ 0 →
#g.support ≤ cg.succ → f.natDegree ≤ N → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g | induction' cg with cg hcg | case zero.zero
R : Type u_1
inst✝ : Semiring R
⊢ ∀ (N O : ℕ) (f g : R[X]),
#f.support ≤ Nat.succ 0 →
#g.support ≤ Nat.succ 0 → f.natDegree ≤ N → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g
case zero.succ
R : Type u_1
inst✝ : Semiring R
cg : ℕ
hcg :
∀ (N O : ℕ) (f g : R[X]),
#f... | 0a2af005637b9810 |
Function.periodicOrbit_chain | Mathlib/Dynamics/PeriodicPts/Defs.lean | theorem periodicOrbit_chain (r : α → α → Prop) {f : α → α} {x : α} :
(periodicOrbit f x).Chain r ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | have hx' := minimalPeriod_pos_of_mem_periodicPts hx | case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | 13df9f8318a3fe41 |
padicNorm.add_eq_max_of_ne | Mathlib/NumberTheory/Padics/PadicNorm.lean | theorem add_eq_max_of_ne {q r : ℚ} (hne : padicNorm p q ≠ padicNorm p r) :
padicNorm p (q + r) = max (padicNorm p q) (padicNorm p r) | case a
p✝ p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hne : padicNorm p q ≠ padicNorm p r
hlt : padicNorm p r < padicNorm p q
this✝ : padicNorm p q ≤ padicNorm p (q + r) ⊔ padicNorm p r
hnge : padicNorm p r ≤ padicNorm p (q + r)
this : padicNorm p q ≤ padicNorm p (q + r)
⊢ padicNorm p q ⊔ padicNorm p r ≤ padicNorm p (q + r) | rwa [max_eq_left_of_lt hlt] | no goals | cd0b52cee2afa794 |
Liouville.irrational | Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean | theorem irrational {x : ℝ} (h : Liouville x) : Irrational x | case intro.mk'.intro.intro.intro.intro
a : ℤ
b : ℕ
bN0 : b ≠ 0
cop : a.natAbs.Coprime b
h : Liouville (↑a / ↑b)
p q : ℤ
q1 : 1 < q
a0 : ↑a / ↑b ≠ ↑p / ↑q
a1 : |↑a / ↑b - ↑p / ↑q| < 1 / ↑q ^ (b + 1)
⊢ False | have qR0 : (0 : ℝ) < q := Int.cast_pos.mpr (zero_lt_one.trans q1) | case intro.mk'.intro.intro.intro.intro
a : ℤ
b : ℕ
bN0 : b ≠ 0
cop : a.natAbs.Coprime b
h : Liouville (↑a / ↑b)
p q : ℤ
q1 : 1 < q
a0 : ↑a / ↑b ≠ ↑p / ↑q
a1 : |↑a / ↑b - ↑p / ↑q| < 1 / ↑q ^ (b + 1)
qR0 : 0 < ↑q
⊢ False | 09ee53349f7b9322 |
List.filter_length_eq_length | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filter_length_eq_length {l} : (filter p l).length = l.length ↔ ∀ a ∈ l, p a | case nil
α✝ : Type u_1
p : α✝ → Bool
⊢ (filter p []).length = [].length ↔ ∀ (a : α✝), a ∈ [] → p a = true | simp | no goals | 4995a39048903be0 |
Real.exp_log_of_neg | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | theorem exp_log_of_neg (hx : x < 0) : exp (log x) = -x | x : ℝ
hx : x < 0
⊢ rexp (log x) = -x | rw [exp_log_eq_abs (ne_of_lt hx)] | x : ℝ
hx : x < 0
⊢ |x| = -x | 4618596bc513109b |
Vector.map_uncurry_zip_eq_zipWith | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Zip.lean | theorem map_uncurry_zip_eq_zipWith (f : α → β → γ) (l : Vector α n) (l' : Vector β n) :
map (Function.uncurry f) (l.zip l') = zipWith f l l' | α : Type u_1
β : Type u_2
γ : Type u_3
n : Nat
f : α → β → γ
l : Vector α n
l' : Vector β n
⊢ map (Function.uncurry f) (l.zip l') = zipWith f l l' | rcases l with ⟨l, rfl⟩ | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
l : Array α
l' : Vector β l.size
⊢ map (Function.uncurry f) ({ toArray := l, size_toArray := ⋯ }.zip l') =
zipWith f { toArray := l, size_toArray := ⋯ } l' | d4b1144fc4910464 |
RingHom.Flat.holdsForLocalizationAway | Mathlib/RingTheory/RingHom/Flat.lean | lemma holdsForLocalizationAway : HoldsForLocalizationAway Flat | case h.e'_5.h.e'_5.h
R S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : R
h : IsLocalization.Away r S
this : Module.Flat R S
r✝ : R
x✝ : S
⊢ (algebraMap R S) r✝ * x✝ = (algebraMap R S) r✝ * x✝ | rfl | no goals | b5305d6f669570d2 |
NormedSpace.eq_zero_iff_forall_dual_eq_zero | Mathlib/Analysis/Normed/Module/Dual.lean | theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
⟨fun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero 𝕜 h⟩
| 𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
hx : x = 0
⊢ ∀ (g : Dual 𝕜 E), g x = 0 | simp [hx] | no goals | 1f7df1e7292e7f8e |
Topology.IsScott.isOpen_iff_Iic_compl_or_univ | Mathlib/Topology/Order/ScottTopology.lean | lemma isOpen_iff_Iic_compl_or_univ [TopologicalSpace α] [Topology.IsScott α univ] (U : Set α) :
IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U | case mpr.inl
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
⊢ IsOpen univ | exact isOpen_univ | no goals | 76ab52ec9f6d0240 |
AddConstMapClass.rel_map_of_Icc | Mathlib/Algebra/AddConstMap/Basic.lean | theorem rel_map_of_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [AddGroup H]
[AddConstMapClass F G H a b] {f : F} {R : H → H → Prop} [IsTrans H R]
[hR : CovariantClass H H (fun x y ↦ y + x) R] (ha : 0 < a) {l : G}
(hf : ∀ x ∈ Icc l (l + a), ∀ y ∈ Icc l (l + a), x < y → R (f x) (f y)) :
((· < ·) ⇒ R... | case inl
F : Type u_1
G : Type u_2
H : Type u_3
inst✝⁵ : FunLike F G H
a : G
b : H
inst✝⁴ : LinearOrderedAddCommGroup G
inst✝³ : Archimedean G
inst✝² : AddGroup H
inst✝¹ : AddConstMapClass F G H a b
f : F
R : H → H → Prop
inst✝ : IsTrans H R
ha : 0 < a
l : G
hf : ∀ x ∈ Icc l (l + a), ∀ y ∈ Icc l (l + a), x < y → R (f x... | assumption | no goals | cb5d009734788901 |
finprod_mem_finset_product' | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_mem_finset_product' [DecidableEq α] [DecidableEq β] (s : Finset (α × β))
(f : α × β → M) :
(∏ᶠ (ab) (_ : ab ∈ s), f ab) =
∏ᶠ (a) (b) (_ : b ∈ (s.filter fun ab => Prod.fst ab = a).image Prod.snd), f (a, b) | α : Type u_1
β : Type u_2
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s : Finset (α × β)
f : α × β → M
this :
∀ (a : α),
∏ i ∈ Finset.image Prod.snd (Finset.filter (fun ab => ab.1 = a) s), f (a, i) =
(Finset.filter (fun x => x.1 = a) s).prod f
⊢ ∏ i ∈ s, f i = ∏ᶠ (a : α) ... | simp_rw [finprod_mem_finset_eq_prod, this] | α : Type u_1
β : Type u_2
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s : Finset (α × β)
f : α × β → M
this :
∀ (a : α),
∏ i ∈ Finset.image Prod.snd (Finset.filter (fun ab => ab.1 = a) s), f (a, i) =
(Finset.filter (fun x => x.1 = a) s).prod f
⊢ ∏ i ∈ s, f i = ∏ᶠ (a : α),... | 298b89fc063aa9c0 |
IsMin.not_isMax | Mathlib/Order/Directed.lean | theorem IsMin.not_isMax [IsDirected β (· ≥ ·)] {b : β} (hb : IsMin b) : ¬ IsMax b | β : Type v
inst✝² : PartialOrder β
inst✝¹ : Nontrivial β
inst✝ : IsDirected β fun x1 x2 => x1 ≥ x2
b : β
hb : IsMin b
⊢ ¬IsMax b | intro hb' | β : Type v
inst✝² : PartialOrder β
inst✝¹ : Nontrivial β
inst✝ : IsDirected β fun x1 x2 => x1 ≥ x2
b : β
hb : IsMin b
hb' : IsMax b
⊢ False | 7d03a959ac3d3292 |
List.map_eq_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem map_eq_cons_iff {f : α → β} {l : List α} :
map f l = b :: l₂ ↔ ∃ a l₁, l = a :: l₁ ∧ f a = b ∧ map f l₁ = l₂ | case mpr.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
a : α
l₁ : List α
⊢ f a = f a ∧ map f l₁ = map f l₁ | constructor <;> rfl | no goals | 94b4ac1f7a541098 |
AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | /-- If `P` holds for `f` over affine opens `U₂` of `Y` and `V₂` of `X` and `U₁` (resp. `V₁`) are
open affine neighborhoods of `x` (resp. `f.base x`), then `P` also holds for `f`
over some basic open of `U₁` (resp. `V₁`). -/
lemma exists_basicOpen_le_appLE_of_appLE_of_isAffine
(hPa : StableUnderCompositionWithLocali... | case intro.intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
hPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] => P
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
x : ↑↑X.toPresheafedS... | have hxa : x ∈ X.basicOpen (f.appLE U₂ V₂ e₂ r') := by
simpa [Scheme.Hom.appLE, ← Scheme.preimage_basicOpen] using And.intro hx₂ (hBrr' ▸ hBfx) | case intro.intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
hPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] => P
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
x : ↑↑X.toPresheafedS... | cdf0cd1399279970 |
ModuleCat.MonoidalCategory.leftUnitor_naturality | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | theorem leftUnitor_naturality {M N : ModuleCat R} (f : M ⟶ N) :
tensorHom (𝟙 (ModuleCat.of R R)) f ≫ (leftUnitor N).hom = (leftUnitor M).hom ≫ f | case hf
R : Type u
inst✝ : CommRing R
M N : ModuleCat R
f : M ⟶ N
⊢ Hom.hom (tensorHom (𝟙 (of R R)) f ≫ (leftUnitor N).hom) = Hom.hom ((leftUnitor M).hom ≫ f) | apply TensorProduct.ext | case hf.H
R : Type u
inst✝ : CommRing R
M N : ModuleCat R
f : M ⟶ N
⊢ (mk R ↑(of R R) ↑M).compr₂ (Hom.hom (tensorHom (𝟙 (of R R)) f ≫ (leftUnitor N).hom)) =
(mk R ↑(of R R) ↑M).compr₂ (Hom.hom ((leftUnitor M).hom ≫ f)) | 046daff1428f99f6 |
List.ofFn_get | Mathlib/Data/List/OfFn.lean | theorem ofFn_get : ∀ l : List α, (ofFn (get l)) = l
| [] => by rw [ofFn_zero]
| a :: l => by
rw [ofFn_succ]
congr
exact ofFn_get l
| α : Type u
a : α
l : List α
⊢ ofFn (a :: l).get = a :: l | rw [ofFn_succ] | α : Type u
a : α
l : List α
⊢ ((a :: l).get 0 :: ofFn fun i => (a :: l).get i.succ) = a :: l | d40ad03eedddb1a3 |
ZMod.unitsMap_surjective | Mathlib/Data/ZMod/Units.lean | theorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :
Function.Surjective (unitsMap h) | case pos
n m : ℕ
hm : NeZero m
h : n ∣ m
x : ℕ
hx : x.Coprime n
ps : Finset ℕ := Finset.filter (fun p => ¬p ∣ x) m.primeFactors
p : ℕ
pp : Nat.Prime p
hp : p ∣ x + ps.prod id * n
hpn : p ∣ m
hpx : p ∣ x
⊢ False | have h := Nat.dvd_sub' hp hpx | case pos
n m : ℕ
hm : NeZero m
h✝ : n ∣ m
x : ℕ
hx : x.Coprime n
ps : Finset ℕ := Finset.filter (fun p => ¬p ∣ x) m.primeFactors
p : ℕ
pp : Nat.Prime p
hp : p ∣ x + ps.prod id * n
hpn : p ∣ m
hpx : p ∣ x
h : p ∣ x + ps.prod id * n - x
⊢ False | 1e42d6deae752fa9 |
CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite | Mathlib/CategoryTheory/Sites/DenseSubsite/SheafEquiv.lean | lemma isIso_ranCounit_app_of_isDenseSubsite (Y : Sheaf J A) (U X) :
IsIso ((yoneda.map ((G.op.ranCounit.app Y.val).app (op U))).app (op X)) | case right
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U ... | simp only [c, id_obj, yoneda_map_app, this] | case right
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U ... | b6d9495444a0fb1a |
CategoryTheory.Limits.colimit.post_post | Mathlib/CategoryTheory/Limits/HasLimits.lean | theorem colimit.post_post {E : Type u''} [Category.{v''} E] (H : D ⥤ E)
-- H G (colimit F) ⟶ H (colimit (F ⋙ G)) ⟶ colimit ((F ⋙ G) ⋙ H) equals
-- H G (colimit F) ⟶ colimit (F ⋙ (G ⋙ H))
[h : HasColimit ((F ⋙ G) ⋙ H)] : haveI : HasColimit (F ⋙ G ⋙ H) := h
colimit.post (F ⋙ G) H ≫ H.map (colimit.post F G... | case w
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
C : Type u
inst✝⁴ : Category.{v, u} C
F : J ⥤ C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasColimit F
G : C ⥤ D
inst✝¹ : HasColimit (F ⋙ G)
E : Type u''
inst✝ : Category.{v'', u''} E
H : D ⥤ E
h : HasColimit ((F ⋙ G) ⋙ H)
j : J
⊢ H.map (G.map (ι F j)) = ι ((F ⋙ G... | haveI : HasColimit (F ⋙ G ⋙ H) := h | case w
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
C : Type u
inst✝⁴ : Category.{v, u} C
F : J ⥤ C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasColimit F
G : C ⥤ D
inst✝¹ : HasColimit (F ⋙ G)
E : Type u''
inst✝ : Category.{v'', u''} E
H : D ⥤ E
h : HasColimit ((F ⋙ G) ⋙ H)
j : J
this : HasColimit (F ⋙ G ⋙ H)
⊢ H.m... | 73eed82337eb03f3 |
Complex.betaIntegral_recurrence | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v | u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ 0 < (v + 1).re | rw [add_re, one_re] | u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x => ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ 0 < v.re + 1 | 16621814b8d90795 |
ProbabilityTheory.isRatCondKernelCDFAux_preCDF | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | lemma isRatCondKernelCDFAux_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
IsRatCondKernelCDFAux (fun p r ↦ (preCDF ρ r p.2).toReal)
(Kernel.const Unit ρ) (Kernel.const Unit ρ.fst) where
measurable := measurable_preCDF'.comp measurable_snd
mono' a r r' hrr' | α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : ℕ → ℚ
x✝ : Antitone s
hs_tendsto : Tendsto s atTop atBot
⊢ Tendsto (fun m => ((ρ.IicSnd ↑(s m)) univ).toReal) atTop (𝓝 0) | have h := ρ.tendsto_IicSnd_atBot MeasurableSet.univ | α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : ℕ → ℚ
x✝ : Antitone s
hs_tendsto : Tendsto s atTop atBot
h : Tendsto (fun r => (ρ.IicSnd ↑r) univ) atBot (𝓝 0)
⊢ Tendsto (fun m => ((ρ.IicSnd ↑(s m)) univ).toReal) atTop (𝓝 0) | 09d99cd916c68ac2 |
tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux
(hs : MeasurableSet s) (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂... | α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (... | filter_upwards [this] with i hi | case h
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ... | d95d2d6a6855f169 |
MeasureTheory.condExp_add | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_add (hf : Integrable f μ) (hg : Integrable g μ) (m : MeasurableSpace α) :
μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] | α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
⊢ μ[f + g|m] =ᶠ[ae μ] μ[f|m] + μ[g|m] | by_cases hm : m ≤ m₀ | case pos
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : Integrable f μ
hg : Integrable g μ
m : MeasurableSpace α
hm : m ≤ m₀
⊢ μ[f + g|m] =ᶠ[ae μ] μ[f|m] + μ[g|m]
case neg
α : Type u_1
E : Type u_3
m₀ : Meas... | 6f21a27293cfd3a1 |
DirectSum.induction_on | Mathlib/Algebra/DirectSum/Basic.lean | theorem induction_on {C : (⨁ i, β i) → Prop} (x : ⨁ i, β i) (H_zero : C 0)
(H_basic : ∀ (i : ι) (x : β i), C (of β i x))
(H_plus : ∀ x y, C x → C y → C (x + y)) : C x | ι : Type v
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : DecidableEq ι
C : (⨁ (i : ι), β i) → Prop
x : ⨁ (i : ι), β i
H_zero : C 0
H_basic : ∀ (i : ι) (x : β i), C ((of β i) x)
H_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)
i : ι
b : β i
f : Π₀ (i : ι), β i
h1 : f i = 0
h2 : b ≠ 0
ih : C f
⊢ C... | solve_by_elim | no goals | 6f34496e3b658879 |
BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_isClosedEmbedding | Mathlib/Topology/TietzeExtension.lean | theorem exists_extension_forall_exists_le_ge_of_isClosedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
{e : X → Y} (he : IsClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f | case refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : ... | rw [Set.disjoint_left] | case refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : ... | c0f84dcf09f7662d |
Mathlib.Meta.NormNum.IsInt.to_isNat | Mathlib/Tactic/NormNum/Result.lean | theorem IsInt.to_isNat {α} [Ring α] : ∀ {a : α} {n}, IsInt a (.ofNat n) → IsNat a n
| _, _, ⟨rfl⟩ => ⟨by simp⟩
| α : Type u_1
inst✝ : Ring α
a✝ : ℕ
⊢ ↑(Int.ofNat a✝) = ↑a✝ | simp | no goals | 3a86c54d19b31452 |
Matroid.IsCircuit.strong_multi_elimination_set | Mathlib/Data/Matroid/Circuit.lean | /-- A version of `Circuit.strong_multi_elimination` where the collection of circuits is
a `Set (Set α)` and the distinguished elements are a `Set α`, rather than both being indexed. -/
lemma IsCircuit.strong_multi_elimination_set (hC₀ : M.IsCircuit C₀) (X : Set α) (S : Set (Set α))
(z : α) (hCS : ∀ C ∈ S, M.IsCircu... | case refine_3
α : Type u_1
M : Matroid α
C₀ : Set α
hC₀ : M.IsCircuit C₀
X : Set α
S : Set (Set α)
z : α
hCS : ∀ C ∈ S, M.IsCircuit C
hXC₀ : X ⊆ C₀
hzC₀ : z ∈ C₀
hz : ∀ C ∈ S, z ∉ C
C : α → Set α
hC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}
⊢ ∀ a ∈ X, {a} ⊆ C a | exact fun e heX ↦ by simp [← hC.2 e heX] | no goals | 0e3ab0ae3a8bacff |
DirectSum.induction_on | Mathlib/Algebra/DirectSum/Basic.lean | theorem induction_on {C : (⨁ i, β i) → Prop} (x : ⨁ i, β i) (H_zero : C 0)
(H_basic : ∀ (i : ι) (x : β i), C (of β i x))
(H_plus : ∀ x y, C x → C y → C (x + y)) : C x | ι : Type v
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : DecidableEq ι
C : (⨁ (i : ι), β i) → Prop
x : ⨁ (i : ι), β i
H_zero : C 0
H_basic : ∀ (i : ι) (x : β i), C ((of β i) x)
H_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)
⊢ ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → C f → C... | intro i b f h1 h2 ih | ι : Type v
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
inst✝ : DecidableEq ι
C : (⨁ (i : ι), β i) → Prop
x : ⨁ (i : ι), β i
H_zero : C 0
H_basic : ∀ (i : ι) (x : β i), C ((of β i) x)
H_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)
i : ι
b : β i
f : Π₀ (i : ι), β i
h1 : f i = 0
h2 : b ≠ 0
ih : C f
⊢ C... | 6f34496e3b658879 |
MvPolynomial.aux_IH | Mathlib/RingTheory/Jacobson/Ring.lean | private lemma aux_IH {R : Type u} {S : Type v} {T : Type w}
[CommRing R] [CommRing S] [CommRing T] [IsJacobsonRing S] [Algebra R S] [Algebra R T]
(IH : ∀ (Q : Ideal S), (IsMaximal Q) → RingHom.IsIntegral (algebraMap R (S ⧸ Q)))
(v : S[X] ≃ₐ[R] T) (P : Ideal T) (hP : P.IsMaximal) :
RingHom.IsIntegral (algebraMap... | R : Type u
S : Type v
T : Type w
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : IsJacobsonRing S
inst✝¹ : Algebra R S
inst✝ : Algebra R T
IH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral
v : S[X] ≃ₐ[R] T
P : Ideal T
hP : P.IsMaximal
Q : Ideal S[X] := comap (↑v).toRingHom P
⊢ (... | have hw : Ideal.map v Q = P := map_comap_of_surjective v v.surjective P | R : Type u
S : Type v
T : Type w
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : IsJacobsonRing S
inst✝¹ : Algebra R S
inst✝ : Algebra R T
IH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral
v : S[X] ≃ₐ[R] T
P : Ideal T
hP : P.IsMaximal
Q : Ideal S[X] := comap (↑v).toRingHom P
hw ... | c0d8bd3b40af6504 |
ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_one | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | theorem exists_sq_eq_prime_iff_of_mod_four_eq_one (hp1 : p % 4 = 1) (hq1 : q ≠ 2) :
IsSquare (q : ZMod p) ↔ IsSquare (p : ZMod q) | case inr
p q : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact (Nat.Prime q)
hp1 : p % 4 = 1
hq1 : q ≠ 2
h : p ≠ q
⊢ IsSquare ↑q ↔ IsSquare ↑p | rw [← eq_one_iff' p (prime_ne_zero p q h), ← eq_one_iff' q (prime_ne_zero q p h.symm),
quadratic_reciprocity_one_mod_four hp1 hq1] | no goals | 4d6efc4ac8247a24 |
HasStrictFDerivAt.map_nhds_eq_of_surj | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
map f (𝓝 a) = 𝓝 (f a) | case intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : E → F
f' : E →L[𝕜] F
a : E
hf : HasStrictFDerivAt f f' a
h : ... | apply hs.map_nhds_eq f'symm s_nhds (Or.inr (NNReal.half_lt_self _)) | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : E → F
f' : E →L[𝕜] F
a : E
hf : HasStrictFDerivAt f f' a
h : LinearMap.range f... | a31ff4a0787a8f56 |
ciSup_partialSups_eq' | Mathlib/Order/PartialSups.lean | theorem ciSup_partialSups_eq' [ConditionallyCompleteLinearOrder α] (f : ι → α) :
⨆ i, partialSups f i = ⨆ i, f i | case pos
α : Type u_1
ι : Type u_2
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
h : BddAbove (Set.range f)
⊢ ⨆ i, (partialSups f) i = ⨆ i, f i | exact ciSup_partialSups_eq h | no goals | f2ac7a59015a456c |
SimpleGraph.Walk.bypass_eq_self_of_length_le | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma bypass_eq_self_of_length_le {u v : V} (p : G.Walk u v) (h : p.length ≤ p.bypass.length) :
p.bypass = p | case pos
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.length ≤ p.bypass.length → p.bypass = p
hb : u✝ ∈ p.bypass.support
h : p.length + 1 ≤ (p.bypass.dropUntil u✝ ⋯).length
⊢ False | apply Nat.not_succ_le_self p.length | case pos
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.length ≤ p.bypass.length → p.bypass = p
hb : u✝ ∈ p.bypass.support
h : p.length + 1 ≤ (p.bypass.dropUntil u✝ ⋯).length
⊢ p.length.succ ≤ p.length | 922bb886a39525ab |
Real.sign_mul_nonneg | Mathlib/Data/Real/Sign.lean | theorem sign_mul_nonneg (r : ℝ) : 0 ≤ sign r * r | case inr.inr
r : ℝ
hp : 0 < r
⊢ 0 ≤ r | exact hp.le | no goals | 63f4055a97fdab0e |
MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff | Mathlib/GroupTheory/GroupAction/Defs.lean | @[to_additive (attr := norm_cast, simp)]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff {H : Subgroup G} {x : orbitRel.Quotient G α}
{a b : x.orbit} : (a : α) ∈ MulAction.orbit H (b : α) ↔ a ∈ MulAction.orbit H b | G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
⊢ ↑a ∈ MulAction.orbit ↥H ↑b ↔ a ∈ MulAction.orbit (↥H) b | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩ | case refine_1
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
h : ↑a ∈ MulAction.orbit ↥H ↑b
⊢ a ∈ MulAction.orbit (↥H) b
case refine_2
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
h : a ∈... | ac31f088983cbb17 |
IsCyclotomicExtension.Rat.Three.eq_one_or_neg_one_of_unit_of_congruent | Mathlib/NumberTheory/Cyclotomic/Three.lean | theorem eq_one_or_neg_one_of_unit_of_congruent
[NumberField K] [IsCyclotomicExtension {3} ℚ K] (hcong : ∃ n : ℤ, λ ^ 2 ∣ (u - n : 𝓞 K)) :
u = 1 ∨ u = -1 | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hcong : ∃ n, λ ^ 2 ∣ ↑u - ↑n
⊢ ∃ n, 3 ∣ ↑u - ↑n | obtain ⟨n, x, hx⟩ := hcong | case intro.intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
n : ℤ
x : 𝓞 K
hx : ↑u - ↑n = λ ^ 2 * x
⊢ ∃ n, 3 ∣ ↑u - ↑n | 220139a45fa38b2d |
Matrix.det_ne_zero_of_sum_row_pos | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | /-- A matrix whose nondiagonal entries are negative with the sum of the entries of each
row positive has nonzero determinant. -/
lemma det_ne_zero_of_sum_row_pos [DecidableEq n] {S : Type*} [LinearOrderedCommRing S]
{A : Matrix n n S} (h1 : Pairwise fun i j => A i j < 0) (h2 : ∀ i, 0 < ∑ j, A i j) :
A.det ≠ 0 | case refine_1
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h2 : ∀ (i : n), 0 < ∑ j : n, A i j
⊢ Pairwise fun i j => Aᵀ i j < 0 | simp_rw [Matrix.transpose_apply] | case refine_1
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h2 : ∀ (i : n), 0 < ∑ j : n, A i j
⊢ Pairwise fun i j => A j i < 0 | 69225adc294561fd |
Basis.reindexFinsetRange_apply | Mathlib/LinearAlgebra/Basis/Defs.lean | theorem reindexFinsetRange_apply (x : Finset.univ.image b) : b.reindexFinsetRange x = x | case mk
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
inst✝¹ : Fintype ι
inst✝ : DecidableEq M
bi : M
hbi : bi ∈ Finset.image (⇑b) Finset.univ
⊢ b.reindexFinsetRange ⟨bi, hbi⟩ = ↑⟨bi, hbi⟩ | rcases Finset.mem_image.mp hbi with ⟨i, -, rfl⟩ | case mk.intro.intro
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
inst✝¹ : Fintype ι
inst✝ : DecidableEq M
i : ι
hbi : b i ∈ Finset.image (⇑b) Finset.univ
⊢ b.reindexFinsetRange ⟨b i, hbi⟩ = ↑⟨b i, hbi⟩ | 4ee8ce0da4fcb480 |
IntermediateField.rank_bot' | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem rank_bot' : Module.rank (⊥ : IntermediateField F E) E = Module.rank F E | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
⊢ Module.rank (↥⊥) E = Module.rank F E | rw [← rank_mul_rank F (⊥ : IntermediateField F E) E, IntermediateField.rank_bot, one_mul] | no goals | e380e6977ffb2182 |
ProbabilityTheory.measurableSet_kernel_integrable | Mathlib/Probability/Kernel/MeasurableIntegral.lean | theorem ProbabilityTheory.measurableSet_kernel_integrable ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) :
MeasurableSet {x | Integrable (f x) (κ x)} | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : IsSFiniteKernel κ
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | HasFiniteIntegral (f x) (κ x)} | exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.enorm) measurable_const | no goals | 4b4febcad75b7007 |
uniq_inv_of_isField | Mathlib/Algebra/Field/IsField.lean | theorem uniq_inv_of_isField (R : Type u) [Ring R] (hf : IsField R) :
∀ x : R, x ≠ 0 → ∃! y : R, x * y = 1 | case hex
R : Type u
inst✝ : Ring R
hf : IsField R
x : R
hx : x ≠ 0
⊢ ∃ x_1, x * x_1 = 1 | exact hf.mul_inv_cancel hx | no goals | 3e56e3416b5a7e33 |
cardinal_eq_of_mem_nhds_zero | Mathlib/Topology/Algebra/Module/Cardinality.lean | /-- In a topological vector space over a nontrivially normed field, any neighborhood of zero has
the same cardinality as the whole space.
See also `cardinal_eq_of_mem_nhds`. -/
lemma cardinal_eq_of_mem_nhds_zero
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSp... | case intro
E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
A : ∀ (x : E), ∀ᶠ (n : ℕ) in atTop, x ∈ c ^ n • s
B : ∀ (n : ℕ), #↑(c ^ ... | apply (Cardinal.mk_of_countable_eventually_mem A B).symm | no goals | 892055d13be3c7d1 |
MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top | Mathlib/MeasureTheory/Measure/Regular.lean | theorem restrict_of_measure_ne_top [BorelSpace α] [WeaklyRegular μ] {A : Set α}
(h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) | α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
inst✝ : μ.WeaklyRegular
A : Set α
h'A : μ A ≠ ⊤
⊢ (μ.restrict A).WeaklyRegular | haveI : Fact (μ A < ∞) := ⟨h'A.lt_top⟩ | α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
inst✝ : μ.WeaklyRegular
A : Set α
h'A : μ A ≠ ⊤
this : Fact (μ A < ⊤)
⊢ (μ.restrict A).WeaklyRegular | a0ec9e412b15eaf5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.