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 |
|---|---|---|---|---|---|---|
LieModule.trace_toEnd_genWeightSpace | Mathlib/Algebra/Lie/Weights/Basic.lean | @[simp]
lemma trace_toEnd_genWeightSpace [IsDomain R] [IsPrincipalIdealRing R]
[Module.Free R M] [Module.Finite R M] (χ : L → R) (x : L) :
trace R _ (toEnd R L (genWeightSpace M χ) x) = finrank R (genWeightSpace M χ) • χ x | R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : Free R M
inst✝ : Module.Fini... | suffices _root_.IsNilpotent ((toEnd R L (genWeightSpace M χ) x) - χ x • LinearMap.id) by
replace this := (isNilpotent_trace_of_isNilpotent this).eq_zero
rwa [map_sub, map_smul, trace_id, sub_eq_zero, smul_eq_mul, mul_comm,
← nsmul_eq_mul] at this | R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : Free R M
inst✝ : Module.Fini... | 2105e30533b87f10 |
Module.reflection_mul_reflection_zpow_apply | Mathlib/LinearAlgebra/Reflection.lean | /-- A formula for $(r_1 r_2)^m z$, where $m$ is an integer and $z \in M$. -/
lemma reflection_mul_reflection_zpow_apply (m : ℤ) (z : M)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
a✝ :
∀ (n : ℕ),
((reflection hf * reflection hg) ^ ↑n) z =
z +
(Polynomial.eval t... | omega | no goals | ede6c0c006554b73 |
ite_eq_ite | Mathlib/.lake/packages/lean4/src/lean/Init/PropLemmas.lean | theorem ite_eq_ite (p : Prop) {h h' : Decidable p} (x y : α) :
(@ite _ p h x y = @ite _ p h' x y) ↔ True | α : Sort u_1
p : Prop
h h' : Decidable p
x y : α
⊢ (if p then x else y) = if p then x else y | congr | no goals | ff25be8a485bb34f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearI... | case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments un... | constructor | case right.right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignmen... | c90be518eb885cb1 |
hasFDerivAt_of_tendstoUniformlyOnFilter | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasFDerivAt g (g' x) x | case h
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G... | congr | case h.e_a
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[�... | d7a89a2808c33b6f |
ENNReal.toReal_eq_toReal_iff' | Mathlib/Data/ENNReal/Basic.lean | theorem toReal_eq_toReal_iff' {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) :
x.toReal = y.toReal ↔ x = y | x y : ℝ≥0∞
hx : x ≠ ⊤
hy : y ≠ ⊤
⊢ x.toReal = y.toReal ↔ x = y | simp only [ENNReal.toReal, NNReal.coe_inj, toNNReal_eq_toNNReal_iff' hx hy] | no goals | ea89e142acb8ae88 |
RootPairing.smul_coroot_eq_of_root_eq_smul | Mathlib/LinearAlgebra/RootSystem/Defs.lean | lemma smul_coroot_eq_of_root_eq_smul [Finite ι] [NoZeroSMulDivisors ℤ N] (i j : ι) (t : R)
(h : P.root j = t • P.root i) :
t • P.coroot j = P.coroot i | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
inst✝¹ : Finite ι
inst✝ : NoZeroSMulDivisors ℤ N
i j : ι
t : R
h : P.root j = t • P.root i
hij : t * P.pairing i j = 2
⊢ (P.root' i) (P.... | simp | no goals | b308f2233dbb1785 |
FreeGroup.Red.to_append_iff | Mathlib/GroupTheory/FreeGroup/Basic.lean | theorem to_append_iff : Red L (L₁ ++ L₂) ↔ ∃ L₃ L₄, L = L₃ ++ L₄ ∧ Red L₃ L₁ ∧ Red L₄ L₂ :=
Iff.intro
(by
generalize eq : L₁ ++ L₂ = L₁₂
intro h
induction' h with L' L₁₂ hLL' h ih generalizing L₁ L₂
· exact ⟨_, _, eq.symm, by rfl, by rfl⟩
· obtain @⟨s, e, a, b⟩ := h
rcases Li... | α : Type u
L L₁ L₂ L₁₂ : List (α × Bool)
eq : L₁ ++ L₂ = L₁₂
⊢ Red L L₁₂ → ∃ L₃ L₄, L = L₃ ++ L₄ ∧ Red L₃ L₁ ∧ Red L₄ L₂ | intro h | α : Type u
L L₁ L₂ L₁₂ : List (α × Bool)
eq : L₁ ++ L₂ = L₁₂
h : Red L L₁₂
⊢ ∃ L₃ L₄, L = L₃ ++ L₄ ∧ Red L₃ L₁ ∧ Red L₄ L₂ | 6fd81a62b1bdb428 |
List.getElem_of_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getElem_of_append {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i) :
l[i]'(eq ▸ h ▸ by simp_arith) = a := Option.some.inj <| by
rw [← getElem?_eq_getElem, eq, getElem?_append_right (h ▸ Nat.le_refl _), h]
simp
| α : Type u_1
l₁ : List α
a : α
l₂ : List α
i : Nat
l : List α
eq : l = l₁ ++ a :: l₂
h : l₁.length = i
⊢ (a :: l₂)[i - i]? = some a | simp | no goals | 5ad05d79585858af |
IsIntegrallyClosed.pow_dvd_pow_iff | Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean | theorem pow_dvd_pow_iff [IsDomain R] [IsIntegrallyClosed R]
{n : ℕ} (hn : n ≠ 0) {a b : R} : a ^ n ∣ b ^ n ↔ a ∣ b | case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : ¬a = 0
⊢ a ∣ b | let K := FractionRing R | case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : ¬a = 0
K : Type u_1 := FractionRing R
⊢ a ∣ b | fff7a7a39e81336f |
MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq | Mathlib/MeasureTheory/SetSemiring.lean | /-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`, there is a
finite set of sets in `C` whose union is `s \ ⋃₀ I`.
See `IsSetSemiring.disjointOfDiffUnion` for a definition that gives such a set. -/
lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
... | α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑... | rw [hJu_sUnion, hJu_sUnion] | α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : (↑J).PairwiseDisjoint id
h_eq : s \ ⋃₀ ↑... | 0bd1b80571fe1ad6 |
quasispectrum.mul_comm | Mathlib/Algebra/Algebra/Quasispectrum.lean | lemma quasispectrum.mul_comm {R A : Type*} [CommRing R] [NonUnitalRing A] [Module R A]
[IsScalarTower R A A] [SMulCommClass R A A] (a b : A) :
quasispectrum R (a * b) = quasispectrum R (b * a) | case h.e'_4
R : Type u_3
A : Type u_4
inst✝⁴ : CommRing R
inst✝³ : NonUnitalRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
⊢ quasispectrum R (a * b) ∩ {r | IsUnit r}ᶜ = quasispectrum R (b * a) ∩ {r | IsUnit r}ᶜ | rw [Set.inter_eq_right.mpr, Set.inter_eq_right.mpr] | case h.e'_4
R : Type u_3
A : Type u_4
inst✝⁴ : CommRing R
inst✝³ : NonUnitalRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
⊢ {r | IsUnit r}ᶜ ⊆ quasispectrum R (b * a)
case h.e'_4
R : Type u_3
A : Type u_4
inst✝⁴ : CommRing R
inst✝³ : NonUnitalRing A
inst✝² : Module R A
inst... | 8da711bcdac575e2 |
FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm | Mathlib/Analysis/Analytic/ChangeOrigin.lean | lemma changeOriginSeriesTerm_changeOriginIndexEquiv_symm (n t) :
let s := changeOriginIndexEquiv.symm ⟨n, t⟩
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ ↦ x) (fun _ ↦ y) =
p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
x y : E
n : ℕ
t : Finset (Fin n)
⊢ ∀ (m : ℕ) (hm : n = m),
(p n) (t.piecewise (fun x_1 => x) f... | rintro m rfl | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
x y : E
n : ℕ
t : Finset (Fin n)
⊢ (p n) (t.piecewise (fun x_1 => x) fun x => y) =
(p n) ((Fin... | 22d7f6c6c9378f52 |
IsArtinian.surjective_of_injective_endomorphism | Mathlib/RingTheory/Artinian/Module.lean | theorem surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Surjective f | R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : M →ₗ[R] M
s : Injective ⇑f
h : ¬Surjective ⇑f
⊢ ∀ (n : ℕ), (fun x => LinearMap.range (f ^ x)) (n + 1) < (fun x => LinearMap.range (f ^ x)) n | intro n | R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : M →ₗ[R] M
s : Injective ⇑f
h : ¬Surjective ⇑f
n : ℕ
⊢ (fun x => LinearMap.range (f ^ x)) (n + 1) < (fun x => LinearMap.range (f ^ x)) n | b91f29bf146c34f4 |
ZMod.natCast_eq_iff | Mathlib/Data/ZMod/Basic.lean | theorem natCast_eq_iff (p : ℕ) (n : ℕ) (z : ZMod p) [NeZero p] :
↑n = z ↔ ∃ k, n = z.val + p * k | case mpr.intro
p : ℕ
z : ZMod p
inst✝ : NeZero p
k : ℕ
⊢ ↑(z.val + p * k) = z | rw [Nat.cast_add, natCast_zmod_val, Nat.cast_mul, natCast_self, zero_mul,
add_zero] | no goals | ab99dcba150b55bd |
ZMod.castHom_bijective | Mathlib/Data/ZMod/Basic.lean | theorem castHom_bijective [Fintype R] (h : Fintype.card R = n) :
Function.Bijective (ZMod.castHom (dvd_refl n) R) | n : ℕ
R : Type u_1
inst✝² : Ring R
inst✝¹ : CharP R n
inst✝ : Fintype R
h : Fintype.card R = n
this : NeZero n
⊢ Injective ⇑(castHom ⋯ R) | apply ZMod.castHom_injective | no goals | f5948c31783484d7 |
QuaternionAlgebra.Basis.ext | Mathlib/Algebra/QuaternionBasis.lean | theorem ext ⦃q₁ q₂ : Basis A c₁ c₂ c₃⦄ (hi : q₁.i = q₂.i)
(hj : q₁.j = q₂.j) : q₁ = q₂ | case mk
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q₂ : Basis A c₁ c₂ c₃
i✝ j✝ k✝ : A
i_mul_i✝ : i✝ * i✝ = c₁ • 1 + c₂ • i✝
j_mul_j✝ : j✝ * j✝ = c₃ • 1
i_mul_j✝ : i✝ * j✝ = k✝
j_mul_i✝ : j✝ * i✝ = c₂ • j✝ - k✝
hi :
{ i := i✝, j := j✝, k := k✝, i_mul_i := i_mul_i✝, j... | rename_i q₁_i_mul_j _ | case mk
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q₂ : Basis A c₁ c₂ c₃
i✝ j✝ k✝ : A
i_mul_i✝ : i✝ * i✝ = c₁ • 1 + c₂ • i✝
j_mul_j✝ : j✝ * j✝ = c₃ • 1
q₁_i_mul_j : i✝ * j✝ = k✝
j_mul_i✝ : j✝ * i✝ = c₂ • j✝ - k✝
hi :
{ i := i✝, j := j✝, k := k✝, i_mul_i := i_mul_i✝,... | 4fecd0862ec9d558 |
Set.image_const_sub_Iio | Mathlib/Algebra/Order/Group/Pointwise/Interval.lean | theorem image_const_sub_Iio : (fun x => a - x) '' Iio b = Ioi (a - b) | α : Type u_1
inst✝ : OrderedAddCommGroup α
a b : α
this : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)
⊢ (fun x => a - x) '' Iio b = Ioi (a - b) | dsimp [Function.comp_def] at this | α : Type u_1
inst✝ : OrderedAddCommGroup α
a b : α
this : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)
⊢ (fun x => a - x) '' Iio b = Ioi (a - b) | 5983d5134060a94c |
CategoryTheory.ShortComplex.isIso_homologyMap_of_epi_of_isIso_of_mono' | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | lemma isIso_homologyMap_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology]
(h₁ : Epi φ.τ₁) (h₂ : IsIso φ.τ₂) (h₃ : Mono φ.τ₃) :
IsIso (homologyMap φ) | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h₁ : Epi φ.τ₁
h₂ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
⊢ IsIso (homologyMap φ) | dsimp only [homologyMap] | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h₁ : Epi φ.τ₁
h₂ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
⊢ IsIso (homologyMap' φ S₁.homologyData S₂.homologyData) | ba7976bb8d6998a3 |
Set.Countable.isPathConnected_compl_of_one_lt_rank | Mathlib/Analysis/NormedSpace/Connected.lean | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ | case intro.inr
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
⊢ J... | have Ia : c - x = a := by
simp only [c, x]
module | case intro.inr
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia ... | a2405a5e761a2c4d |
DirectSum.decompose_lhom_ext | Mathlib/Algebra/DirectSum/Decomposition.lean | theorem decompose_lhom_ext {N} [AddCommMonoid N] [Module R N] ⦃f g : M →ₗ[R] N⦄
(h : ∀ i, f ∘ₗ (ℳ i).subtype = g ∘ₗ (ℳ i).subtype) : f = g :=
LinearMap.ext <| (decomposeLinearEquiv ℳ).symm.surjective.forall.mpr <|
suffices f ∘ₗ (decomposeLinearEquiv ℳ).symm
= (g ∘ₗ (decomposeLinearEquiv ℳ).symm : (... | ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁶ : DecidableEq ι
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
ℳ : ι → Submodule R M
inst✝² : Decomposition ℳ
N : Type u_5
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
f g : M →ₗ[R] N
h : ∀ (i : ι), f ∘ₗ (ℳ i).subtype = g ∘ₗ (ℳ i).subtype
i : ι
⊢ (f ∘ₗ ↑(de... | simp_rw [LinearMap.comp_assoc, decomposeLinearEquiv_symm_comp_lof ℳ i, h] | no goals | df4cf824c2973bb8 |
Set.Nonempty.eq_univ | Mathlib/Data/Set/Basic.lean | theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ | case intro
α : Type u
s : Set α
inst✝ : Subsingleton α
x : α
hx : x ∈ s
⊢ s = univ | exact eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x] | no goals | 5ca5e8752663b705 |
Nat.divisors_subset_properDivisors | Mathlib/NumberTheory/Divisors.lean | theorem divisors_subset_properDivisors {m : ℕ} (hzero : n ≠ 0) (h : m ∣ n) (hdiff : m ≠ n) :
divisors m ⊆ properDivisors n | n m : ℕ
hzero : n ≠ 0
h : m ∣ n
hdiff : m ≠ n
⊢ ∀ ⦃x : ℕ⦄, x ∈ m.divisors → x ∈ n.properDivisors | intro x hx | n m : ℕ
hzero : n ≠ 0
h : m ∣ n
hdiff : m ≠ n
x : ℕ
hx : x ∈ m.divisors
⊢ x ∈ n.properDivisors | b0a1218cac480e84 |
AlgebraicGeometry.StructureSheaf.comap_comp | Mathlib/AlgebraicGeometry/StructureSheaf.lean | theorem comap_comp (f : R →+* S) (g : S →+* P) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (W : Opens (PrimeSpectrum.Top P))
(hUV : ∀ p ∈ V, PrimeSpectrum.comap f p ∈ U) (hVW : ∀ p ∈ W, PrimeSpectrum.comap g p ∈ V) :
(comap (g.comp f) U W fun p hpW => hUV (PrimeSpectrum.comap g p) (h... | R : Type u
inst✝² : CommRing R
S : Type u
inst✝¹ : CommRing S
P : Type u
inst✝ : CommRing P
f : R →+* S
g : S →+* P
U : Opens ↑(PrimeSpectrum.Top R)
V : Opens ↑(PrimeSpectrum.Top S)
W : Opens ↑(PrimeSpectrum.Top P)
hUV : ∀ p ∈ V, (PrimeSpectrum.comap f) p ∈ U
hVW : ∀ p ∈ W, (PrimeSpectrum.comap g) p ∈ V
s : ↑((structur... | rw [Localization.localRingHom_comp _ (PrimeSpectrum.comap g p.1).asIdeal] <;>
rfl | no goals | a319f413ced4abd9 |
PerfectPairing.exists_basis_basis_of_span_eq_top_of_mem_algebraMap | Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean | /-- If a perfect pairing over a field `L` takes values in a subfield `K` along two `K`-subspaces
whose `L` span is full, then these subspaces induce a `K`-structure in the sense of
[*Algebra I*, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989]. -/
lemma exists_basis_basis_of_span_eq_top_of_mem_algebraMap
(M'... | case intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Subm... | have hv' : LinearIndependent K v' := by
replace hv₃ := hv₃.restrict_scalars (R := K) <| by
simp_rw [← Algebra.algebraMap_eq_smul_one]
exact FaithfulSMul.algebraMap_injective K L
rw [show ((↑) : v → M) = M'.subtype ∘ v' by ext; simp [v']] at hv₃
exact hv₃.of_comp | case intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Subm... | 28e714660261030d |
MeasureTheory.isStoppingTime_hitting_isStoppingTime | Mathlib/Probability/Process/HittingTime.lean | theorem isStoppingTime_hitting_isStoppingTime [ConditionallyCompleteLinearOrder ι]
[WellFoundedLT ι] [Countable ι] [TopologicalSpace ι] [OrderTopology ι]
[FirstCountableTopology ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace β]
[BorelSpace β] {f : Filtration ι m} {u : ι → Ω → β} {τ : Ω →... | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁹ : ConditionallyCompleteLinearOrder ι
inst✝⁸ : WellFoundedLT ι
inst✝⁷ : Countable ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : OrderTopology ι
inst✝⁴ : FirstCountableTopology ι
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSp... | have h₂ : ⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n} = ∅ := by
ext x
simp only [gt_iff_lt, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop,
Set.mem_empty_iff_false, iff_false, not_exists, not_and, not_le]
rintro m hm rfl
exact lt_of_lt_of_le hm (le_hitting (hτbdd _) _) | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁹ : ConditionallyCompleteLinearOrder ι
inst✝⁸ : WellFoundedLT ι
inst✝⁷ : Countable ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : OrderTopology ι
inst✝⁴ : FirstCountableTopology ι
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSp... | 2ade5d53007f19b3 |
Orientation.norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle (x - y) x) = ‖x‖ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
⊢ ‖y‖ / (o.oangle (x - y) x).tan = ‖x‖ | rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ‖y‖ / ((-o).oangle x (x - y)).tan = ‖x‖ | cd182e180412192d |
Matrix.circulant_mul | Mathlib/LinearAlgebra/Matrix/Circulant.lean | theorem circulant_mul [Semiring α] [Fintype n] [AddGroup n] (v w : n → α) :
circulant v * circulant w = circulant (circulant v *ᵥ w) | case a
α : Type u_1
n : Type u_3
inst✝² : Semiring α
inst✝¹ : Fintype n
inst✝ : AddGroup n
v w : n → α
i j : n
⊢ ∀ (x : n), v (i - x) * w (x - j) = v (i - j - (Equiv.subRight j) x) * w ((Equiv.subRight j) x) | intro x | case a
α : Type u_1
n : Type u_3
inst✝² : Semiring α
inst✝¹ : Fintype n
inst✝ : AddGroup n
v w : n → α
i j x : n
⊢ v (i - x) * w (x - j) = v (i - j - (Equiv.subRight j) x) * w ((Equiv.subRight j) x) | 9247ee4e7f05243a |
AccPt.nhds_inter | Mathlib/Topology/Perfect.lean | theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) | α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
x : α
U : Set α
h_acc : AccPt x (𝓟 C)
hU : U ∈ 𝓝 x
⊢ 𝓝[≠] x ≤ 𝓟 U | rw [le_principal_iff] | α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
x : α
U : Set α
h_acc : AccPt x (𝓟 C)
hU : U ∈ 𝓝 x
⊢ U ∈ 𝓝[≠] x | e1c3c6891dd944be |
Fin.predAbove_left_monotone | Mathlib/Order/Fin/Basic.lean | lemma predAbove_left_monotone (i : Fin (n + 1)) : Monotone fun p ↦ predAbove p i := fun a b H ↦ by
dsimp [predAbove]
split_ifs with ha hb hb
· rfl
· exact pred_le _
· have : b < a := castSucc_lt_castSucc_iff.mpr (hb.trans_le (le_of_not_gt ha))
exact absurd H this.not_le
· rfl
| case pos
n : ℕ
i : Fin (n + 1)
a b : Fin n
H : a ≤ b
ha : a.castSucc < i
hb : b.castSucc < i
⊢ i.pred ⋯ ≤ i.pred ⋯ | rfl | no goals | 74aeb5d3b481472b |
CategoryTheory.DifferentialObject.eqToHom_f | Mathlib/CategoryTheory/DifferentialObject.lean | theorem eqToHom_f {X Y : DifferentialObject S C} (h : X = Y) :
Hom.f (eqToHom h) = eqToHom (congr_arg _ h) | S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X : DifferentialObject S C
⊢ (eqToHom ⋯).f = eqToHom ⋯ | rw [eqToHom_refl, eqToHom_refl] | S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X : DifferentialObject S C
⊢ (𝟙 X).f = 𝟙 X.obj | ffcc110b6830f2f9 |
EReal.continuousAt_mul_top_top | Mathlib/Topology/Instances/EReal/Lemmas.lean | private lemma continuousAt_mul_top_top :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, ⊤) | case h.refine_2.refine_2.refine_2
x : ℝ
⊢ (⊤, ⊤).2 ∈ Ioi 1 | rw [Set.mem_Ioi, ← EReal.coe_one] | case h.refine_2.refine_2.refine_2
x : ℝ
⊢ ↑1 < (⊤, ⊤).2 | 14ad4956648676bd |
Complex.integral_rpow_mul_exp_neg_mul_rpow | Mathlib/MeasureTheory/Integral/Gamma.lean | theorem Complex.integral_rpow_mul_exp_neg_mul_rpow {p q b : ℝ} (hp : 1 ≤ p) (hq : - 2 < q)
(hb : 0 < b) :
∫ x : ℂ, ‖x‖ ^ q * rexp (- b * ‖x‖ ^ p) = (2 * π / p) *
b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p) | p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-b * ‖x‖ ^ p) = 2 * π / p * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p) | calc
_ = ∫ x in Ioi (0 : ℝ) ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (- b * |x.1| ^ p)) := by
rw [← Complex.integral_comp_polarCoord_symm, polarCoord_target]
simp_rw [Complex.norm_polarCoord_symm, smul_eq_mul]
_ = (∫ x in Ioi (0 : ℝ), x * |x| ^ q * rexp (- b * |x| ^ p)) * ∫ _ in Ioo (-π) π, 1 := by
rw [← ... | no goals | c4a815d7c64d67fa |
Equiv.Perm.cycleOf_zpow_apply_self | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycleOf_zpow_apply_self (f : Perm α) [DecidableRel f.SameCycle] (x : α) :
∀ n : ℤ, (cycleOf f x ^ n) x = (f ^ n) x | case negSucc
α : Type u_2
f : Perm α
inst✝ : DecidableRel f.SameCycle
x : α
z : ℕ
⊢ (f.cycleOf x ^ Int.negSucc z) x = (f ^ Int.negSucc z) x | rw [zpow_negSucc, ← inv_pow, cycleOf_inv, zpow_negSucc, ← inv_pow, cycleOf_pow_apply_self] | no goals | 47a5061368da539b |
Polynomial.splits_prod_iff | Mathlib/Algebra/Polynomial/Splits.lean | theorem splits_prod_iff {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, s j ≠ 0) → ((∏ x ∈ t, s x).Splits i ↔ ∀ j ∈ t, (s j).Splits i) | K : Type v
L : Type w
inst✝¹ : Field K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t : Finset ι
⊢ (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j)) | refine
Finset.induction_on t (fun _ =>
⟨fun _ _ h => by simp only [Finset.not_mem_empty] at h, fun _ => splits_one i⟩)
fun a t hat ih ht => ?_ | K : Type v
L : Type w
inst✝¹ : Field K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t✝ : Finset ι
a : ι
t : Finset ι
hat : a ∉ t
ih : (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))
ht : ∀ j ∈ insert a t, s j ≠ 0
⊢ Splits i (∏ x ∈ insert a t, s x) ↔ ∀ j ∈ insert a t, Splits i (s j) | 72d77f00dc774bee |
affineIndependent_iff_indicator_eq_of_affineCombination_eq | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem affineIndependent_iff_indicator_eq_of_affineCombination_eq (p : ι → P) :
AffineIndependent k p ↔
∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
∑ i ∈ s1, w1 i = 1 →
∑ i ∈ s2, w2 i = 1 →
s1.affineCombination k p w1 = s2.affineCombination k p w2 →
Set.indicator (↑s1) w... | case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
s1 s2 : Finset ι
w1 w2 : ι → k
hw1 : ∑ i ∈ s1, w1 i = 1
hw2 : ∑ i ∈ s2, w2 i = 1
heq : (affineCombination k s1 p) w1 = (affineCombination k... | ext i | case mp.h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
s1 s2 : Finset ι
w1 w2 : ι → k
hw1 : ∑ i ∈ s1, w1 i = 1
hw2 : ∑ i ∈ s2, w2 i = 1
heq : (affineCombination k s1 p) w1 = (affineCombination... | 4f9d6693c0403c52 |
EMetric.hausdorffEdist_triangle | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | theorem hausdorffEdist_triangle : hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u | α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u | rw [hausdorffEdist_def] | α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ (⨆ x ∈ s, infEdist x u) ⊔ ⨆ y ∈ u, infEdist y s ≤ hausdorffEdist s t + hausdorffEdist t u | 91d661feb399f899 |
solvableByRad.induction | Mathlib/FieldTheory/AbelRuffini.lean | theorem induction (P : solvableByRad F E → Prop)
(base : ∀ α : F, P (algebraMap F (solvableByRad F E) α))
(add : ∀ α β : solvableByRad F E, P α → P β → P (α + β))
(neg : ∀ α : solvableByRad F E, P α → P (-α))
(mul : ∀ α β : solvableByRad F E, P α → P β → P (α * β))
(inv : ∀ α : solvableByRad F E, P ... | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | rw [← hα₀] | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | 856cf0182c98d000 |
Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ((-o).oangle y (x + y)).cos * ‖x + y‖ = ‖y‖ | rw [add_comm] | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ((-o).oangle y (y + x)).cos * ‖y + x‖ = ‖y‖ | ff44f08a8b90fea3 |
orbit_fixingSubgroup_compl_subset | Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean | theorem orbit_fixingSubgroup_compl_subset {s : Set α} {a : α} (a_in_s : a ∈ s) :
MulAction.orbit (fixingSubgroup M sᶜ) a ⊆ s | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
a : α
a_in_s : a ∈ s
⊢ orbit (↥(fixingSubgroup M sᶜ)) a ⊆ s | intro b b_in_orbit | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
a : α
a_in_s : a ∈ s
b : α
b_in_orbit : b ∈ orbit (↥(fixingSubgroup M sᶜ)) a
⊢ b ∈ s | 5bc157162cc1d042 |
Cardinal.ofENat_mul_aleph0 | Mathlib/SetTheory/Cardinal/ENat.lean | @[simp] lemma ofENat_mul_aleph0 {m : ℕ∞} (hm : m ≠ 0) : ↑m * ℵ₀ = ℵ₀ | case top
hm : ⊤ ≠ 0
⊢ ↑⊤ * ℵ₀ = ℵ₀ | exact aleph0_mul_aleph0 | no goals | 148366b6dd4a869f |
Subgroup.conj_smul_le_of_le | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | theorem conj_smul_le_of_le {P H : Subgroup G} (hP : P ≤ H) (h : H) :
MulAut.conj (h : G) • P ≤ H | G : Type u_2
inst✝ : Group G
P H : Subgroup G
hP : P ≤ H
h : ↥H
⊢ MulAut.conj ↑h • P ≤ H | rintro - ⟨g, hg, rfl⟩ | case intro.intro
G : Type u_2
inst✝ : Group G
P H : Subgroup G
hP : P ≤ H
h : ↥H
g : G
hg : g ∈ ↑P
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj ↑h)) g ∈ H | 53dad7a591b125b9 |
ProbabilityTheory.IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel κ] [IsFiniteKernel ν] (a : α) (q : ℚ) {A : Set β} (hA : MeasurableSet A) :
∫ t in A, ⨅ r : Ioi q, f (a, t) r ∂(ν a) = (κ a (A ×ˢ Iic (q : ℝ))).toReal | case refine_2.refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
⊢ (fun c => f (a, c) q) ≤ᶠ[ae (ν a)] fun t =... | filter_upwards [hf.mono a] with c h_mono using le_ciInf (fun r ↦ h_mono (le_of_lt r.prop)) | no goals | 5552b387e425f2b5 |
AlgebraicGeometry.Scheme.Opens.nonempty_iff | Mathlib/AlgebraicGeometry/Restrict.lean | @[simp]
lemma nonempty_iff : Nonempty U.toScheme ↔ (U : Set X).Nonempty | X : Scheme
U : X.Opens
⊢ Nonempty ↑↑(↑U).toPresheafedSpace ↔ (↑U).Nonempty | simp only [toScheme_carrier, SetLike.coe_sort_coe, nonempty_subtype] | X : Scheme
U : X.Opens
⊢ (∃ x, x ∈ U) ↔ (↑U).Nonempty | e60fa5e3d9b34049 |
Asymptotics.IsEquivalent.smul | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | theorem IsEquivalent.smul {α E 𝕜 : Type*} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{a b : α → 𝕜} {u v : α → E} {l : Filter α} (hab : a ~[l] b) (huv : u ~[l] v) :
(fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x | case intro.intro.intro.intro
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ ... | specialize hφ (c / 2 / C) (div_pos (div_pos hc zero_lt_two) hC) | case intro.intro.intro.intro
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ ... | 1fc300a902bb7f39 |
Nat.pepin_primality | Mathlib/NumberTheory/Fermat.lean | /-- `Fₙ = 2^(2^n)+1` is prime if `3^(2^(2^n-1)) = -1 mod Fₙ` (**Pépin's test**). -/
lemma pepin_primality (n : ℕ) (h : 3 ^ (2 ^ (2 ^ n - 1)) = (-1 : ZMod (fermatNumber n))) :
(fermatNumber n).Prime | case ha
n : ℕ
h : 3 ^ 2 ^ (2 ^ n - 1) = -1
this : Fact (2 < n.fermatNumber)
key : 2 ^ n = 2 ^ n - 1 + 1
⊢ 3 ^ (2 ^ 2 ^ n + 1 - 1) = 1 | rw [Nat.add_sub_cancel, key, pow_succ, pow_mul, ← pow_succ, ← key, h, neg_one_sq] | no goals | 663d69b05282f8ff |
ONote.NFBelow.lt | Mathlib/SetTheory/Ordinal/Notation.lean | theorem NFBelow.lt {e n a b} (h : NFBelow (ONote.oadd e n a) b) : repr e < b | e : ONote
n : ℕ+
a : ONote
b : Ordinal.{0}
h : (e.oadd n a).NFBelow b
⊢ e.repr < b | obtain - | ⟨h₁, h₂, h₃⟩ := h | case oadd'
e : ONote
n : ℕ+
a : ONote
b eb✝ : Ordinal.{0}
h₁ : e.NFBelow eb✝
h₂ : a.NFBelow e.repr
h₃ : e.repr < b
⊢ e.repr < b | e4c846269cddce61 |
IsMax.withBot | Mathlib/Order/WithBot.lean | theorem _root_.IsMax.withBot (h : IsMax a) : IsMax (a : WithBot α) :=
fun x ↦ by cases x <;> simp; simpa using @h _
| α : Type u_1
a : α
inst✝ : LE α
h : IsMax a
x : WithBot α
⊢ ↑a ≤ x → x ≤ ↑a | cases x <;> simp | case coe
α : Type u_1
a : α
inst✝ : LE α
h : IsMax a
a✝ : α
⊢ a ≤ a✝ → a✝ ≤ a | dae9b9feff5d1505 |
NonUnitalSubsemiring.unitization_range | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | theorem unitization_range :
(unitization s).range = subalgebraOfSubsemiring (.closure s) | R : Type u_1
S : Type u_2
inst✝¹ : Semiring R
inst✝ : SetLike S R
hSR : NonUnitalSubsemiringClass S R
s : S
⊢ (unitization s).range = subalgebraOfSubsemiring (Subsemiring.closure ↑s) | have := AddSubmonoidClass.nsmulMemClass (S := S) | R : Type u_1
S : Type u_2
inst✝¹ : Semiring R
inst✝ : SetLike S R
hSR : NonUnitalSubsemiringClass S R
s : S
this : SMulMemClass S ℕ R
⊢ (unitization s).range = subalgebraOfSubsemiring (Subsemiring.closure ↑s) | bef695ba967db83c |
Polynomial.exists_root_of_degree_eq_one | Mathlib/Algebra/Polynomial/FieldDivision.lean | theorem exists_root_of_degree_eq_one (h : degree p = 1) : ∃ x, IsRoot p x :=
⟨-((p.coeff 1)⁻¹ * p.coeff 0), by
rw [← mem_roots (by simp [← zero_le_degree_iff, h])]
simp [roots_degree_eq_one h]⟩
| R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 1
⊢ -((p.coeff 1)⁻¹ * p.coeff 0) ∈ p.roots | simp [roots_degree_eq_one h] | no goals | 48d63a9d139e135c |
Relation.reflGen_eq_self | Mathlib/Logic/Relation.lean | lemma reflGen_eq_self (hr : Reflexive r) : ReflGen r = r | α : Type u_1
r : α → α → Prop
hr : Reflexive r
⊢ ReflGen r = r | ext x y | case h.h.a
α : Type u_1
r : α → α → Prop
hr : Reflexive r
x y : α
⊢ ReflGen r x y ↔ r x y | 1d961a1ffd12f4cb |
Real.cosh_add | Mathlib/Data/Complex/Trigonometric.lean | theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y | x y : ℝ
⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh y | rw [← ofReal_inj] | x y : ℝ
⊢ ↑(cosh (x + y)) = ↑(cosh x * cosh y + sinh x * sinh y) | 3284e2e8c4a39b73 |
Equiv.Perm.ofSubtype_swap_eq | Mathlib/GroupTheory/Perm/Support.lean | theorem ofSubtype_swap_eq {p : α → Prop} [DecidablePred p] (x y : Subtype p) :
ofSubtype (Equiv.swap x y) = Equiv.swap ↑x ↑y :=
Equiv.ext fun z => by
by_cases hz : p z
· rw [swap_apply_def, ofSubtype_apply_of_mem _ hz]
split_ifs with hzx hzy
· simp_rw [hzx, Subtype.coe_eta, swap_apply_left]
... | case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
p : α → Prop
inst✝ : DecidablePred p
x y : Subtype p
z : α
hz : ¬p z
h : z = ↑x
⊢ p ↑x | exact Subtype.prop x | no goals | 7c967a4cc58d1491 |
Lean.Order.csup_conj | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Basic.lean | theorem csup_conj (c P : α → Prop) (hchain : chain c) (h : ∀ x, c x → ∃ y, c y ∧ x ⊑ y ∧ P y) :
csup c = csup (fun x => c x ∧ P x) | case a.intro.intro.intro
α : Sort u
inst✝ : CCPO α
c P : α → Prop
hchain : chain c
h : ∀ (x : α), c x → ∃ y, c y ∧ x ⊑ y ∧ P y
x : α
hcx : c x
y : α
hcy : c y
hxy : x ⊑ y
hPy : P y
⊢ y ⊑ csup fun x => c x ∧ P x | clear x hcx hxy | case a.intro.intro.intro
α : Sort u
inst✝ : CCPO α
c P : α → Prop
hchain : chain c
h : ∀ (x : α), c x → ∃ y, c y ∧ x ⊑ y ∧ P y
y : α
hcy : c y
hPy : P y
⊢ y ⊑ csup fun x => c x ∧ P x | 3024f009497d9407 |
Prod.comul_comp_inl | Mathlib/RingTheory/Coalgebra/Basic.lean | theorem comul_comp_inl :
comul ∘ₗ inl R A B = TensorProduct.map (.inl R A B) (.inl R A B) ∘ₗ comul | R : Type u
A : Type v
B : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : Module R A
inst✝² : Module R B
inst✝¹ : Coalgebra R A
inst✝ : Coalgebra R B
⊢ comul ∘ₗ inl R A B = TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul | ext | case h
R : Type u
A : Type v
B : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : Module R A
inst✝² : Module R B
inst✝¹ : Coalgebra R A
inst✝ : Coalgebra R B
x✝ : A
⊢ (comul ∘ₗ inl R A B) x✝ = (TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul) x✝ | 0d8c212aec19a778 |
convexJoin_singleton_left | Mathlib/Analysis/Convex/Join.lean | theorem convexJoin_singleton_left (t : Set E) (x : E) :
convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y | 𝕜 : Type u_2
E : Type u_3
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
t : Set E
x : E
⊢ convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y | simp [convexJoin] | no goals | 772f52dc249e0189 |
Finset.sum_mul_self_eq_zero_iff | Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | theorem sum_mul_self_eq_zero_iff [LinearOrderedSemiring R] [ExistsAddOfLE R] (s : Finset ι)
(f : ι → R) : ∑ i ∈ s, f i * f i = 0 ↔ ∀ i ∈ s, f i = 0 | ι : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedSemiring R
inst✝ : ExistsAddOfLE R
s : Finset ι
f : ι → R
⊢ (∀ i ∈ s, f i * f i = 0) ↔ ∀ i ∈ s, f i = 0 | simp | no goals | 0e4c80553bdf342d |
GromovHausdorff.ghDist_le_nonemptyCompacts_dist | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem ghDist_le_nonemptyCompacts_dist (p q : NonemptyCompacts X) :
dist p.toGHSpace q.toGHSpace ≤ dist p q | X : Type u
inst✝ : MetricSpace X
p q : NonemptyCompacts X
ha : Isometry Subtype.val
hb : Isometry Subtype.val
A : dist p q = hausdorffDist ↑p ↑q
I : ↑p = range Subtype.val
J : ↑q = range Subtype.val
⊢ dist p.toGHSpace q.toGHSpace ≤ dist p q | rw [A, I, J] | X : Type u
inst✝ : MetricSpace X
p q : NonemptyCompacts X
ha : Isometry Subtype.val
hb : Isometry Subtype.val
A : dist p q = hausdorffDist ↑p ↑q
I : ↑p = range Subtype.val
J : ↑q = range Subtype.val
⊢ dist p.toGHSpace q.toGHSpace ≤ hausdorffDist (range Subtype.val) (range Subtype.val) | 08300c0b3db15994 |
CategoryTheory.Abelian.Ext.mk₀_zero | Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean | @[simp]
lemma mk₀_zero : mk₀ (0 : X ⟶ Y) = 0 | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ mk₀ 0 = 0 | ext | case h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ (mk₀ 0).hom = hom 0 | 76b7e767784f943a |
Class.sUnion_apply | Mathlib/SetTheory/ZFC/Basic.lean | theorem sUnion_apply {x : Class} {y : ZFSet} : (⋃₀ x) y ↔ ∃ z : ZFSet, x z ∧ y ∈ z | x : Class.{u_1}
y : ZFSet.{u_1}
⊢ (⋃₀ x) y ↔ ∃ z, x z ∧ y ∈ z | constructor | case mp
x : Class.{u_1}
y : ZFSet.{u_1}
⊢ (⋃₀ x) y → ∃ z, x z ∧ y ∈ z
case mpr
x : Class.{u_1}
y : ZFSet.{u_1}
⊢ (∃ z, x z ∧ y ∈ z) → (⋃₀ x) y | de10961c81e49695 |
coeSubmodule_differentIdeal | Mathlib/RingTheory/DedekindDomain/Different.lean | lemma coeSubmodule_differentIdeal [NoZeroSMulDivisors A B] :
coeSubmodule L (differentIdeal A B) = 1 / Submodule.traceDual A K 1 | A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
ins... | apply IsLocalization.ringHom_ext A⁰ | case h
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomai... | de2215dd565ee658 |
cfcₙ_integral | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean | /-- The non-unital continuous functional calculus commutes with integration. -/
lemma cfcₙ_integral [TopologicalSpace X] [OpensMeasurableSpace X] (f : X → 𝕜 → 𝕜)
(bound : X → ℝ) (a : A) [SecondCountableTopologyEither X C(quasispectrum 𝕜 a, 𝕜)₀]
(hf₁ : ∀ x, ContinuousOn (f x) (quasispectrum 𝕜 a))
(hf₂ :... | X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : NormedSpace 𝕜 A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : NonUnita... | rw [ContinuousMap.norm_le _ (norm_nonneg (bound x))] | X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : NormedSpace 𝕜 A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : NonUnita... | c8ceb0f73ee7838a |
MvPowerSeries.order_monomial | Mathlib/RingTheory/MvPowerSeries/Order.lean | theorem order_monomial {d : σ →₀ ℕ} {a : R} [Decidable (a = 0)] :
order (monomial R d a) = if a = 0 then (⊤ : ℕ∞) else ↑(degree d) | σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
⊢ ((monomial R d) a).order = if a = 0 then ⊤ else ↑((weight fun x => 1) d) | exact weightedOrder_monomial _ | no goals | e18d3565ea1400f2 |
String.join_eq | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem join_eq (ss : List String) : join ss = ⟨(ss.map data).flatten⟩ := go ss [] where
go : ∀ (ss : List String) cs, ss.foldl (· ++ ·) (mk cs) = ⟨cs ++ (ss.map data).flatten⟩
| [], _ => by simp
| ⟨s⟩::ss, _ => (go ss _).trans (by simp)
| ss✝ : List String
s : List Char
ss : List String
x✝ : List Char
⊢ { data := x✝ ++ s ++ (List.map data ss).flatten } = { data := x✝ ++ (List.map data ({ data := s } :: ss)).flatten } | simp | no goals | 2cc551e94da7df0b |
LieIdeal.map_bracket_le | Mathlib/Algebra/Lie/IdealOperations.lean | theorem map_bracket_le {I₁ I₂ : LieIdeal R L} : map f ⁅I₁, I₂⁆ ≤ ⁅map f I₁, map f I₂⁆ | case intro.mk.intro.mk
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I₁ I₂ : LieIdeal R L
x y₁ : L
hy₁ : y₁ ∈ I₁
y₂ : L
hy₂ : y₂ ∈ I₂
hx : ⁅↑⟨y₁, hy₁⟩, ↑⟨y₂, hy₂⟩⁆ = x
fy₁ : ↥(map f I₁) := ⟨f y₁, ⋯⟩
⊢ ⁅↑⟨y₁, h... | let fy₂ : ↥(map f I₂) := ⟨f y₂, mem_map hy₂⟩ | case intro.mk.intro.mk
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I₁ I₂ : LieIdeal R L
x y₁ : L
hy₁ : y₁ ∈ I₁
y₂ : L
hy₂ : y₂ ∈ I₂
hx : ⁅↑⟨y₁, hy₁⟩, ↑⟨y₂, hy₂⟩⁆ = x
fy₁ : ↥(map f I₁) := ⟨f y₁, ⋯⟩
fy₂ : ↥(ma... | f4b2fffd77926150 |
Rat.AbsoluteValue.is_prime_of_minimal_nat_zero_lt_and_lt_one | Mathlib/NumberTheory/Ostrowski.lean | /-- The minimal positive integer with absolute value smaller than 1 is a prime number. -/
lemma is_prime_of_minimal_nat_zero_lt_and_lt_one : p.Prime | f : AbsoluteValue ℚ ℝ
a b : ℕ
hp0 : 0 < f ↑(a * b)
hp1 : f ↑(a * b) < 1
hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → a * b ≤ m
ha₁ : a ≠ 1
hb₁ : b ≠ 1
ha₀ : a ≠ 0
hb₀ : b ≠ 0
hap : a < a * b
⊢ 1 < a | omega | no goals | 462fd42846234179 |
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 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
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsi... | simp only [List.get, l_eq_i] | no goals | 02ca1706b397b254 |
fourierIntegral_gaussian_pi' | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem _root_.fourierIntegral_gaussian_pi' (hb : 0 < b.re) (c : ℂ) :
(𝓕 fun x : ℝ => cexp (-π * b * x ^ 2 + 2 * π * c * x)) = fun t : ℝ =>
1 / b ^ (1 / 2 : ℂ) * cexp (-π / b * (t + I * c) ^ 2) | b : ℂ
hb : 0 < b.re
c : ℂ
this : b ≠ 0
⊢ (-↑π * b).re < 0 | simpa only [neg_mul, neg_re, re_ofReal_mul, neg_lt_zero] using mul_pos pi_pos hb | no goals | b9a2243045a942ca |
SimpleGraph.isIndepSet_induce | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem isIndepSet_induce {F : Set α} {s : Set F} :
((⊤ : Subgraph G).induce F).coe.IsIndepSet s ↔ G.IsIndepSet (Subtype.val '' s) | α : Type u_1
G : SimpleGraph α
F : Set α
s : Set ↑F
⊢ (⊤.induce F).coe.IsIndepSet s ↔ G.IsIndepSet (Subtype.val '' s) | simp [Set.Pairwise] | no goals | 058c7cd9290187ac |
HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2)
=O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p univ x r
hr : r' < r
⊢ (fun y => ... | simpa using hf.isBigO_image_sub_image_sub_deriv_principal hr | no goals | cc6a0632f4d3aa1e |
Multiset.isDershowitzMannaLT_singleton_insert | Mathlib/Data/Multiset/DershowitzManna.lean | private lemma isDershowitzMannaLT_singleton_insert (h : OneStep N (a ::ₘ M)) :
∃ M', N = a ::ₘ M' ∧ OneStep M' M ∨ N = M + M' ∧ ∀ x ∈ M', x < a | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝ : Preorder α
M : Multiset α
a : α
X Y : Multiset α
h0 : a ::ₘ M = X + {a}
h2 : ∀ y ∈ Y, y < a
⊢ X + Y = M + Y | simpa [add_comm _ {a}, singleton_add, eq_comm] using h0 | no goals | fe1657f5397146bc |
Nat.nth_of_forall_not | Mathlib/Data/Nat/Nth.lean | theorem nth_of_forall_not {n : ℕ} (hp : ∀ n' ≥ n, ¬p n') : nth p n = 0 | p : ℕ → Prop
n : ℕ
hp : ∀ n' ≥ n, ¬p n'
this : setOf p ⊆ ↑(range n)
⊢ ⋯.toFinset ⊆ range n | exact Set.Finite.toFinset_subset.mpr this | no goals | 6bf1c9d4ae0922f6 |
SeminormFamily.filter_eq_iInf | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem filter_eq_iInf (p : SeminormFamily 𝕜 E ι) :
p.moduleFilterBasis.toFilterBasis.filter = ⨅ i, (𝓝 0).comap (p i) | case refine_1.refine_1
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
i : ι
ε : ℝ
hε : 0 < ε
⊢ ({i}.sup p).ball 0 ε ∈ AddGroupFilterBasis.toFilterBasis | exact p.basisSets_mem {i} hε | no goals | 043221e62cb70beb |
ZMod.ker_intCastRingHom | Mathlib/RingTheory/ZMod.lean | theorem ZMod.ker_intCastRingHom (n : ℕ) :
RingHom.ker (Int.castRingHom (ZMod n)) = Ideal.span ({(n : ℤ)} : Set ℤ) | n : ℕ
⊢ RingHom.ker (Int.castRingHom (ZMod n)) = Ideal.span {↑n} | ext | case h
n : ℕ
x✝ : ℤ
⊢ x✝ ∈ RingHom.ker (Int.castRingHom (ZMod n)) ↔ x✝ ∈ Ideal.span {↑n} | b698aed1a1a94ada |
List.mapIdx_singleton | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean | theorem mapIdx_singleton {a : α} : mapIdx f [a] = [f 0 a] | α : Type u_1
α✝ : Type u_2
f : Nat → α → α✝
a : α
⊢ mapIdx f [a] = [f 0 a] | simp | no goals | c34353a5e52a0924 |
ExteriorAlgebra.ιMulti_span | Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean | /-- The union of the images of the maps `ExteriorAlgebra.ιMulti R n` for `n` running through
all natural numbers spans the exterior algebra. -/
lemma ιMulti_span :
Submodule.span R (Set.range fun x : Σ n, (Fin n → M) => ιMulti R x.1 x.2) = ⊤ | case h_homogeneous
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i✝ : ℕ
hm✝ : ↥(⋀[R]^i✝ M)
m : ExteriorAlgebra R M
hm : m ∈ ⋀[R]^i✝ M
⊢ ↑⟨m, hm⟩ ∈ Submodule.span R (Set.range fun x => (ιMulti R x.fst) x.snd) | apply Set.mem_of_mem_of_subset hm | case h_homogeneous
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i✝ : ℕ
hm✝ : ↥(⋀[R]^i✝ M)
m : ExteriorAlgebra R M
hm : m ∈ ⋀[R]^i✝ M
⊢ ↑(⋀[R]^i✝ M) ⊆ ↑(Submodule.span R (Set.range fun x => (ιMulti R x.fst) x.snd)) | 6122bc0f0c7cb07d |
Multiset.rel_refl_of_refl_on | Mathlib/Data/Multiset/ZeroCons.lean | theorem rel_refl_of_refl_on {m : Multiset α} {r : α → α → Prop} : (∀ x ∈ m, r x x) → Rel r m m | α : Type u_1
m : Multiset α
r : α → α → Prop
⊢ (∀ (x : α), x ∈ m → r x x) → Rel r m m | refine m.induction_on ?_ ?_ | case refine_1
α : Type u_1
m : Multiset α
r : α → α → Prop
⊢ (∀ (x : α), x ∈ 0 → r x x) → Rel r 0 0
case refine_2
α : Type u_1
m : Multiset α
r : α → α → Prop
⊢ ∀ (a : α) (s : Multiset α),
((∀ (x : α), x ∈ s → r x x) → Rel r s s) → (∀ (x : α), x ∈ a ::ₘ s → r x x) → Rel r (a ::ₘ s) (a ::ₘ s) | ab00a6fe45ebb271 |
Order.krullDim_le_one_iff | Mathlib/Order/KrullDimension.lean | lemma krullDim_le_one_iff : krullDim α ≤ 1 ↔ ∀ x : α, IsMin x ∨ IsMax x | case mpr.intro.intro.intro.intro
α : Type u_1
inst✝ : Preorder α
x y : α
hxy : y < x
z : α
hzx : x < z
⊢ ∃ i, 1 < ↑i.length | exact ⟨⟨2, ![y, x, z], fun i ↦ by fin_cases i <;> simpa⟩, by simp⟩ | no goals | 191756a8f12b45b2 |
BitVec.getMsbD_setWidth' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getMsbD_setWidth' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
getMsbD (setWidth' ge x) i = (decide (m - n ≤ i) && getMsbD x (i + n - m)) | case neg
m n : Nat
ge : m ≥ n
x : BitVec n
i : Nat
h₁ : ¬decide (i < m) = true
h₂ : ¬decide (m - n ≤ i) = true
h₃ : ¬decide (i + n - m < n) = true
h₄ : ¬n - 1 - (i + n - m) = m - 1 - i
⊢ (decide (i < m) && x.getLsbD (m - 1 - i)) =
(decide (m - n ≤ i) && (decide (i + n - m < n) && x.getLsbD (n - 1 - (i + n - m)))) | simp only [h₁, h₂, h₃, h₄] | case neg
m n : Nat
ge : m ≥ n
x : BitVec n
i : Nat
h₁ : ¬decide (i < m) = true
h₂ : ¬decide (m - n ≤ i) = true
h₃ : ¬decide (i + n - m < n) = true
h₄ : ¬n - 1 - (i + n - m) = m - 1 - i
⊢ (false && x.getLsbD (m - 1 - i)) = (false && (false && x.getLsbD (n - 1 - (i + n - m)))) | 03a8635c3d52111b |
UniformSpace.metrizable_uniformity | Mathlib/Topology/Metrizable/Uniformity.lean | theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]
[IsCountablyGenerated (𝓤 X)] : ∃ I : PseudoMetricSpace X, I.toUniformSpace = ‹_› | X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → Set (X × X)
hU_symm : ∀ (n : ℕ), SymmetricRel (U n)
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y ... | refine PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x₁ x₂ x₃ x₄ => ?_ | X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → Set (X × X)
hU_symm : ∀ (n : ℕ), SymmetricRel (U n)
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y ... | 6b4b13aa9c62ee27 |
Ordinal.card_opow_le | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem card_opow_le (a b : Ordinal) : (a ^ b).card ≤ max ℵ₀ (max a.card b.card) | case inl.intro.inl.intro
n m : ℕ
⊢ (↑n ^ ↑m).card ≤ ℵ₀ ⊔ ((↑n).card ⊔ (↑m).card) | rw [← natCast_opow, card_nat] | case inl.intro.inl.intro
n m : ℕ
⊢ ↑(n ^ m) ≤ ℵ₀ ⊔ ((↑n).card ⊔ (↑m).card) | 128126bfa7317aa2 |
MeasureTheory.mul_upcrossingsBefore_le | Mathlib/Probability/Martingale/Upcrossing.lean | theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k : ℕ
⊢ ∑ n ∈ Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppe... | rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossi... | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k : ℕ
⊢ f (upperCrossingTime a b f N (k + 1) ω) ω - f (lowerCrossingTime a b f N k ω) ω =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N ... | fc87e1e3d79bf5b5 |
LieSubmodule.comap_normalizer | Mathlib/Algebra/Lie/Normalizer.lean | theorem comap_normalizer (f : M' →ₗ⁅R,L⁆ M) : N.normalizer.comap f = (N.comap f).normalizer | R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
N : LieS... | ext | case h
R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
N... | ec68f89d7c6a8532 |
MvQPF.has_good_supp_iff | Mathlib/Data/QPF/Multivariate/Basic.lean | theorem has_good_supp_iff {α : TypeVec n} (x : F α) :
(∀ p, LiftP p x ↔ ∀ (i), ∀ u ∈ supp x i, p i u) ↔
∃ a f, abs ⟨a, f⟩ = x ∧ ∀ i a' f', abs ⟨a', f'⟩ = x → f i '' univ ⊆ f' i '' univ | case mp.intro.intro.intro
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
x : F α
h : ∀ (p : (i : Fin2 n) → α i → Prop), LiftP p x ↔ ∀ (i : Fin2 n), ∀ u ∈ supp x i, p i u
a : (P F).A
f : (P F).B a ⟹ α
xeq : x = abs ⟨a, f⟩
h' : ∀ (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)
⊢ ∀ (i : Fin2 n) (a' : (P... | intro a' f' h'' | case mp.intro.intro.intro
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
x : F α
h : ∀ (p : (i : Fin2 n) → α i → Prop), LiftP p x ↔ ∀ (i : Fin2 n), ∀ u ∈ supp x i, p i u
a : (P F).A
f : (P F).B a ⟹ α
xeq : x = abs ⟨a, f⟩
h' : ∀ (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)
a' : Fin2 n
f' : (P F).A
... | c8763328353a1fcf |
Turing.TM2.step_supports | Mathlib/Computability/TuringMachine.lean | theorem step_supports (M : Λ → Stmt Γ Λ σ) {S : Finset Λ} (ss : Supports M S) :
∀ {c c' : Cfg Γ Λ σ}, c' ∈ step M c → c.l ∈ Finset.insertNone S → c'.l ∈ Finset.insertNone S
| ⟨some l₁, v, T⟩, c', h₁, h₂ => by
replace h₂ := ss.2 _ (Finset.some_mem_insertNone.1 h₂)
simp only [step, Option.mem_def, Option.so... | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : Inhabited Λ
inst✝ : DecidableEq K
M : Λ → Stmt Γ Λ σ
S : Finset Λ
ss : Supports M S
l₁ : Λ
v : σ
T : (k : K) → List (Γ k)
c' : Cfg Γ Λ σ
h₂ : SupportsStmt S (M l₁)
h₁ : stepAux (M l₁) v T = c'
⊢ c'.l ∈ Finset.insertNone S | subst c' | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : Inhabited Λ
inst✝ : DecidableEq K
M : Λ → Stmt Γ Λ σ
S : Finset Λ
ss : Supports M S
l₁ : Λ
v : σ
T : (k : K) → List (Γ k)
h₂ : SupportsStmt S (M l₁)
⊢ (stepAux (M l₁) v T).l ∈ Finset.insertNone S | 4820c6206a25e806 |
Language.reverse_mem_reverse | Mathlib/Computability/Language.lean | lemma reverse_mem_reverse : a.reverse ∈ l.reverse ↔ a ∈ l | α : Type u_1
l : Language α
a : List α
⊢ a.reverse ∈ l.reverse ↔ a ∈ l | rw [mem_reverse, List.reverse_reverse] | no goals | 59408426c58d4c28 |
Finsupp.iSup_lsingle_range | Mathlib/LinearAlgebra/Finsupp/Span.lean | theorem iSup_lsingle_range : ⨆ a, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) = ⊤ | α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : α →₀ M
x✝ : f ∈ ⊤
⊢ f.sum single ∈ ⨆ a, LinearMap.range (lsingle a) | exact sum_mem fun a _ => Submodule.mem_iSup_of_mem a ⟨_, rfl⟩ | no goals | c55a20c551a2d199 |
Ordnode.insertWith.valid_aux | Mathlib/Data/Ordmap/Ordset.lean | theorem insertWith.valid_aux [IsTotal α (· ≤ ·)] [DecidableRel (α := α) (· ≤ ·)] (f : α → α) (x : α)
(hf : ∀ y, x ≤ y ∧ y ≤ x → x ≤ f y ∧ f y ≤ x) :
∀ {t o₁ o₂},
Valid' o₁ t o₂ →
Bounded nil o₁ x →
Bounded nil x o₂ →
Valid' o₁ (insertWith f x t) o₂ ∧ Raised (size t) (size (in... | α : Type u_1
inst✝² : Preorder α
inst✝¹ : IsTotal α fun x1 x2 => x1 ≤ x2
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
f : α → α
x : α
hf : ∀ (y : α), x ≤ y ∧ y ≤ x → x ≤ f y ∧ f y ≤ x
sz : ℕ
l : Ordnode α
y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
h : Valid' o₁ (node sz l y r) o₂
bl : nil.Bounded o₁ ↑x
br : nil.Bou... | split_ifs with h_1 h_2 <;> dsimp only | case pos
α : Type u_1
inst✝² : Preorder α
inst✝¹ : IsTotal α fun x1 x2 => x1 ≤ x2
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
f : α → α
x : α
hf : ∀ (y : α), x ≤ y ∧ y ≤ x → x ≤ f y ∧ f y ≤ x
sz : ℕ
l : Ordnode α
y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
h : Valid' o₁ (node sz l y r) o₂
bl : nil.Bounded o₁ ↑x
br ... | 6bc0954662ae2391 |
EuclideanGeometry.Sphere.secondInter_secondInter | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | theorem Sphere.secondInter_secondInter (s : Sphere P) (p : P) (v : V) :
s.secondInter (s.secondInter p v) v = p | case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
hv : v = 0
⊢ s.secondInter (s.secondInter p v) v = p | simp [hv] | no goals | ee5a6023431a9a7c |
GenContFract.abs_sub_convs_le | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem abs_sub_convs_le (not_terminatedAt_n : ¬(of v).TerminatedAt n) :
|v - (of v).convs n| ≤ 1 / ((of v).dens n * ((of v).dens <| n + 1)) | case intro.intro.intro.intro.intro.intro.right
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := of v
nextConts : Pair K := g.contsAux (n + 2)
conts : Pair K := g.contsAux (n + 1)
conts_eq : conts = g.contsAux (n + 1)
pred_conts ... | suffices (ifp_succ_n.b : K) * conts.b ≤ ifp_n.fr⁻¹ * conts.b by rwa [← ifp_succ_n_b_eq_gp_b] | case intro.intro.intro.intro.intro.intro.right
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := of v
nextConts : Pair K := g.contsAux (n + 2)
conts : Pair K := g.contsAux (n + 1)
conts_eq : conts = g.contsAux (n + 1)
pred_conts ... | a2279d23d2560b5e |
Hindman.FP.finset_prod | Mathlib/Combinatorics/Hindman.lean | theorem FP.finset_prod {M} [CommMonoid M] (a : Stream' M) (s : Finset ℕ) (hs : s.Nonempty) :
(s.prod fun i => a.get i) ∈ FP a | case H
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
⊢ (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a) | rcases (s.erase (s.min' hs)).eq_empty_or_nonempty with h | h | case H.inl
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
h : s.erase (s.min' hs) = ∅
⊢ (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)
case H.inr... | 2807cb40e776fcc5 |
StrictMonoOn.exists_slope_lt_deriv_aux | Mathlib/Analysis/Convex/Deriv.lean | theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy hf A | case intro.intro.intro
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
a : ℝ
ha : deriv f a = (f y - f x) / (y - x)
hxa : x < a
hay : a < y
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | be3b59c8e97fd897 |
OrdinalApprox.lfpApprox_mem_fixedPoints_of_eq | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | /-- If the sequence of ordinal-indexed approximations takes a value twice,
then it actually stabilised at that value. -/
lemma lfpApprox_mem_fixedPoints_of_eq {a b c : Ordinal}
(h_init : x ≤ f x) (h_ab : a < b) (h_ac : a ≤ c) (h_fab : lfpApprox f x a = lfpApprox f x b) :
lfpApprox f x c ∈ fixedPoints f | α : Type u
inst✝ : CompleteLattice α
f : α →o α
x : α
a b c : Ordinal.{u}
h_init : x ≤ f x
h_ab : a < b
h_ac : a ≤ c
h_fab : lfpApprox f x a = lfpApprox f x b
⊢ lfpApprox f x (a + 1) = lfpApprox f x a | exact Monotone.eq_of_le_of_le (lfpApprox_monotone f x)
h_fab (SuccOrder.le_succ a) (SuccOrder.succ_le_of_lt h_ab) | no goals | fcaa28d818378714 |
UniformSpace.Completion.continuous_hatInv | Mathlib/Topology/Algebra/UniformField.lean | theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
ContinuousAt hatInv x | K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
⊢ ∀ᶠ (x : hat K) in 𝓝 x, ∃ c, Tendsto (fun x => ↑x⁻¹) (Filter.comap coe' (𝓝 x)) (𝓝 c) | apply mem_of_superset (compl_singleton_mem_nhds h) | K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
⊢ {0}ᶜ ⊆ {x | (fun x => ∃ c, Tendsto (fun x => ↑x⁻¹) (Filter.comap coe' (𝓝 x)) (𝓝 c)) x} | 2b79bfc09ae36473 |
FormalMultilinearSeries.changeOrigin_eval_of_finite | Mathlib/Analysis/Analytic/CPolynomialDef.lean | theorem changeOrigin_eval_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
(hn : ∀ (m : ℕ), n ≤ m → p m = 0) (x y : E) :
(p.changeOrigin x).sum y = p.sum (x + y) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
hn : ∀ (m : ℕ), n✝ ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=... | rw [← Pi.add_def, (p n).map_add_univ (fun _ ↦ x) fun _ ↦ y] | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
hn : ∀ (m : ℕ), n✝ ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=... | 8c604310c51ca63d |
ModuleCat.ExtendScalars.hom_ext | Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean | @[ext]
lemma hom_ext {M : ModuleCat R} {N : ModuleCat S}
{α β : (extendScalars f).obj M ⟶ N}
(h : ∀ (m : M), α ((1 : S) ⊗ₜ m) = β ((1 : S) ⊗ₜ m)) : α = β | case a.hf.H
R : Type u₁
S : Type u₂
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
M : ModuleCat R
N : ModuleCat S
α β : (extendScalars f).obj M ⟶ N
h : ∀ (m : ↑M), (ConcreteCategory.hom α) (1 ⊗ₜ[R] m) = (ConcreteCategory.hom β) (1 ⊗ₜ[R] m)
this : Algebra R S := f.toAlgebra
s : S
m : ↑M
⊢ (ConcreteCategory.hom α) (... | have : s ⊗ₜ[R] (m : M) = s • (1 : S) ⊗ₜ[R] m := by
rw [ExtendScalars.smul_tmul, mul_one] | case a.hf.H
R : Type u₁
S : Type u₂
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
M : ModuleCat R
N : ModuleCat S
α β : (extendScalars f).obj M ⟶ N
h : ∀ (m : ↑M), (ConcreteCategory.hom α) (1 ⊗ₜ[R] m) = (ConcreteCategory.hom β) (1 ⊗ₜ[R] m)
this✝ : Algebra R S := f.toAlgebra
s : S
m : ↑M
this : s ⊗ₜ[R] m = s • 1 ⊗ₜ... | 89a1ddede4e60741 |
eVariationOn.comp_eq_of_antitoneOn | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) | α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : AntitoneOn φ t
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t | cases isEmpty_or_nonempty β | case inl
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : AntitoneOn φ t
h✝ : IsEmpty β
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t
case inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetric... | 1c2ffe34f4a2d4cd |
AlgebraicGeometry.Scheme.affineBasisCover_is_basis | Mathlib/AlgebraicGeometry/Cover/Open.lean | theorem affineBasisCover_is_basis (X : Scheme.{u}) :
TopologicalSpace.IsTopologicalBasis
{x : Set X |
∃ a : X.affineBasisCover.J, x = Set.range (X.affineBasisCover.map a).base} | case h_nhds.intro.intro.intro.intro.intro.intro.intro.refine_2
X : Scheme
a : ↑↑X.toPresheafedSpace
U : Set ↑↑X.toPresheafedSpace
haU : a ∈ U
hU : IsOpen U
x : ↑↑(X.affineCover.obj (X.affineCover.f a)).toPresheafedSpace
e : (ConcreteCategory.hom (X.affineCover.map (X.affineCover.f a)).base) x = a
U' : Set ↑↑(X.affineCo... | rw [Set.image_subset_iff] | case h_nhds.intro.intro.intro.intro.intro.intro.intro.refine_2
X : Scheme
a : ↑↑X.toPresheafedSpace
U : Set ↑↑X.toPresheafedSpace
haU : a ∈ U
hU : IsOpen U
x : ↑↑(X.affineCover.obj (X.affineCover.f a)).toPresheafedSpace
e : (ConcreteCategory.hom (X.affineCover.map (X.affineCover.f a)).base) x = a
U' : Set ↑↑(X.affineCo... | f074a127ed56c597 |
top_le_span_of_aux | Mathlib/LinearAlgebra/Basis/Exact.lean | private lemma top_le_span_of_aux (v : κ ⊕ σ → M)
(hg : Function.Surjective g) (hslzero : ∀ i, s (v (.inl i)) = 0)
(hli : LinearIndependent R (s ∘ v ∘ .inr)) (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :
⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ .inl) | case intro
R✝ : Type u_1
M✝ : Type u_2
K✝ : Type u_3
P✝ : Type u_4
inst✝¹³ : Ring R✝
inst✝¹² : AddCommGroup M✝
inst✝¹¹ : AddCommGroup K✝
inst✝¹⁰ : AddCommGroup P✝
inst✝⁹ : Module R✝ M✝
inst✝⁸ : Module R✝ K✝
inst✝⁷ : Module R✝ P✝
g✝ : M✝ →ₗ[R✝] P✝
s✝ : M✝ →ₗ[R✝] K✝
κ✝ : Type u_6
σ✝ : Type u_7
R : Type u_1
M : Type u_2
K... | replace hli := (linearIndependent_iff'.mp hli) c.support.toRight (c ∘ .inr) h | case intro
R✝ : Type u_1
M✝ : Type u_2
K✝ : Type u_3
P✝ : Type u_4
inst✝¹³ : Ring R✝
inst✝¹² : AddCommGroup M✝
inst✝¹¹ : AddCommGroup K✝
inst✝¹⁰ : AddCommGroup P✝
inst✝⁹ : Module R✝ M✝
inst✝⁸ : Module R✝ K✝
inst✝⁷ : Module R✝ P✝
g✝ : M✝ →ₗ[R✝] P✝
s✝ : M✝ →ₗ[R✝] K✝
κ✝ : Type u_6
σ✝ : Type u_7
R : Type u_1
M : Type u_2
K... | d124f5729aeaf148 |
CategoryTheory.isSheaf_coherent | Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean | lemma isSheaf_coherent (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (coherentTopology C) P ↔
(∀ (B : C) (α : Type) [Finite α] (X : α → C) (π : (a : α) → (X a ⟶ B)),
EffectiveEpiFamily X π → (Presieve.ofArrows X π).IsSheafFor P) | case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Precoherent C
P : Cᵒᵖ ⥤ Type w
⊢ (∀ (B : C) (α : Type) [inst : Finite α] (X : α → C) (π : (a : α) → X a ⟶ B),
EffectiveEpiFamily X π → Presieve.IsSheafFor P (Presieve.ofArrows X π)) →
Presieve.IsSheaf (coherentTopology C) P | intro h | case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Precoherent C
P : Cᵒᵖ ⥤ Type w
h :
∀ (B : C) (α : Type) [inst : Finite α] (X : α → C) (π : (a : α) → X a ⟶ B),
EffectiveEpiFamily X π → Presieve.IsSheafFor P (Presieve.ofArrows X π)
⊢ Presieve.IsSheaf (coherentTopology C) P | 5dd76cf3a7d8a0a6 |
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