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 |
|---|---|---|---|---|---|---|
CategoryTheory.Functor.PreservesRightHomologyOf.mk' | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | /-- If a functor preserves a certain right homology data of a short complex `S`, then it
preserves the right homology of `S`. -/
lemma PreservesRightHomologyOf.mk' (h : S.RightHomologyData) [h.IsPreservedBy F] :
F.PreservesRightHomologyOf S where
isPreservedBy h' :=
{ f := ShortComplex.RightHomologyData.IsPre... | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{u_4, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
S : ShortComplex C
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
⊢ PreservesLimit (parallelPair h'.g' 0... | have := ShortComplex.RightHomologyData.IsPreservedBy.hg' h F | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{u_4, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
S : ShortComplex C
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
this : PreservesLimit (parallelPair h.... | 39fd2293ef3ce48f |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖... | exact this.le | no goals | e1f8d437480849f9 |
Filter.tendsto_atTop_finset_of_monotone | Mathlib/Order/Filter/AtTopBot/Finset.lean | theorem tendsto_atTop_finset_of_monotone [Preorder β] {f : β → Finset α} (h : Monotone f)
(h' : ∀ x : α, ∃ n, x ∈ f n) : Tendsto f atTop atTop | case intro
α : Type u_3
β : Type u_4
inst✝ : Preorder β
f : β → Finset α
h : Monotone f
h' : ∀ (x : α), ∃ n, x ∈ f n
a : α
b : β
hb : a ∈ f b
⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a} | exact (eventually_ge_atTop b).mono fun b' hb' => (Finset.singleton_subset_iff.2 hb).trans (h hb') | no goals | 7efa857265d50b55 |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : E.IsElliptic
inst✝¹ : E'.IsElliptic
p : ℕ
inst✝ : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝⁴ : NeZero 2
this✝³ : NeZero 4
this✝² : NeZero 6
this✝¹ : Invertible 2 := invertibleOfNonzero hchar2
this✝ : Invertible 3 := invertibleOf... | rwa [variableChange_comp] | no goals | 5599f8482dc6841e |
Finsupp.support_single_add | Mathlib/Data/Finsupp/Single.lean | theorem support_single_add {a : α} {b : M} {f : α →₀ M} (ha : a ∉ f.support) (hb : b ≠ 0) :
support (single a b + f) = cons a f.support ha | α : Type u_1
M : Type u_5
inst✝ : AddZeroClass M
a : α
b : M
f : α →₀ M
ha : a ∉ f.support
hb : b ≠ 0
⊢ (single a b + f).support = cons a f.support ha | have H := support_single_ne_zero a hb | α : Type u_1
M : Type u_5
inst✝ : AddZeroClass M
a : α
b : M
f : α →₀ M
ha : a ∉ f.support
hb : b ≠ 0
H : (single a b).support = {a}
⊢ (single a b + f).support = cons a f.support ha | e3a6e48c80b3900b |
IsLocalizedModule.mk'_eq_zero | Mathlib/Algebra/Module/LocalizedModule/Basic.lean | theorem mk'_eq_zero {m : M} (s : S) : mk' f m s = 0 ↔ f m = 0 | R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
M' : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
m : M
s : ↥S
⊢ mk' f m s = 0 ↔ f m = 0 | rw [mk'_eq_iff, smul_zero] | no goals | db62fb65c91114a2 |
PartENat.lt_def | Mathlib/Data/Nat/PartENat.lean | theorem lt_def (x y : PartENat) : x < y ↔ ∃ hx : x.Dom, ∀ hy : y.Dom, x.get hx < y.get hy | case h
x y : PartENat
h : ∀ (x_1 : x.Dom → y.Dom), ¬∀ (hy : x.Dom), y.get ⋯ ≤ x.get hy
hyx : y.Dom → x.Dom
H : ∀ (hy : y.Dom), x.get ⋯ ≤ y.get hy
hx : x.Dom
hy : y.Dom
⊢ x.get hx < y.get hy | specialize H hy | case h
x y : PartENat
h : ∀ (x_1 : x.Dom → y.Dom), ¬∀ (hy : x.Dom), y.get ⋯ ≤ x.get hy
hyx : y.Dom → x.Dom
hx : x.Dom
hy : y.Dom
H : x.get ⋯ ≤ y.get hy
⊢ x.get hx < y.get hy | d9afc030dbc26ee8 |
Finset.Nonempty.zero_smul | Mathlib/Algebra/GroupWithZero/Pointwise/Finset.lean | lemma Nonempty.zero_smul (ht : t.Nonempty) : (0 : Finset α) • t = 0 :=
t.zero_smul_subset.antisymm <| by simpa [mem_smul] using ht
| α : Type u_1
β : Type u_2
inst✝³ : DecidableEq β
inst✝² : Zero α
inst✝¹ : Zero β
inst✝ : SMulWithZero α β
t : Finset β
ht : t.Nonempty
⊢ 0 ⊆ 0 • t | simpa [mem_smul] using ht | no goals | 68cccc7d617ad1f6 |
Path.Homotopy.continuous_transReflReparamAux | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | theorem continuous_transReflReparamAux : Continuous transReflReparamAux | case refine_5
x : ↑I
hx : ↑x = 1 / 2
⊢ 2 * ↑x = 1 | simp [hx] | no goals | 92d41a2e994f3e1d |
LSeries_eventually_eq_zero_iff' | Mathlib/NumberTheory/LSeries/Injectivity.lean | /-- The `LSeries` of `f` is zero for large real arguments if and only if either `f n = 0`
for all `n ≠ 0` or the L-series converges nowhere. -/
lemma LSeries_eventually_eq_zero_iff' {f : ℕ → ℂ} :
(fun x : ℝ ↦ LSeries f x) =ᶠ[atTop] 0 ↔ (∀ n ≠ 0, f n = 0) ∨ abscissaOfAbsConv f = ⊤ | case neg.refine_1.ind
f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) ... | suffices Tendsto (fun x : ℝ ↦ n ^ (x : ℂ) * LSeries F x) atTop (nhds (F n)) by
replace this := this.congr' <| H' n
simp only [tendsto_const_nhds_iff] at this
exact this.symm | case neg.refine_1.ind
f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) ... | 7103bb783fd652c6 |
MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero | Mathlib/MeasureTheory/Function/Egorov.lean | theorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]
(hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g) (hsm : MeasurableSet s)
(hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
Tendsto (fun j => μ (s ∩ notConvergentSeq f g n ... | case inr
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Te... | rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter] | case inr
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Te... | bd38541566755b41 |
List.getElem!_cons_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getElem!_cons_succ [Inhabited α] {l : List α} : (a::l)[i+1]! = l[i]! | α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
⊢ (a :: l)[i + 1]! = l[i]! | by_cases h : i < l.length | case pos
α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
h : i < l.length
⊢ (a :: l)[i + 1]! = l[i]!
case neg
α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
h : ¬i < l.length
⊢ (a :: l)[i + 1]! = l[i]! | 5553a5f71d6d6c08 |
prod_generateFrom_generateFrom_eq | Mathlib/Topology/Constructions.lean | theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}
(hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
@instTopologicalSpaceProd X Y (generateFrom s) (generateFrom t) =
generateFrom (image2 (· ×ˢ ·) s t) :=
let G := generateFrom (image2 (· ×ˢ ·) s t)
le_antisymm
(le... | X : Type u_5
Y : Type u_6
s : Set (Set X)
t : Set (Set Y)
hs : ⋃₀ s = univ
ht : ⋃₀ t = univ
G : TopologicalSpace (X × Y) := generateFrom (image2 (fun x1 x2 => x1 ×ˢ x2) s t)
v : Set Y
hv : v ∈ t
this : ⋃ u ∈ s, u ×ˢ v = Prod.snd ⁻¹' v
⊢ TopologicalSpace.IsOpen (⋃ u ∈ s, u ×ˢ v) | exact
isOpen_iUnion fun u =>
isOpen_iUnion fun hu => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩ | no goals | 6bf5cf316a327555 |
MeasureTheory.eLpNorm_sub_le_of_dist_bdd | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem eLpNorm_sub_le_of_dist_bdd (μ : Measure α)
{p : ℝ≥0∞} (hp' : p ≠ ∞) {s : Set α} (hs : MeasurableSet[m] s)
{f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c) :
eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal) | case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x => c) x‖
⊢ eLpNorm (s.indica... | rw [eLpNorm_indicator_const hs hp hp'] | case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x => c) x‖
⊢ ‖c‖ₑ * μ s ^ (1 /... | 5a09cd678840d75b |
Padic.exi_rat_seq_conv | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem exi_rat_seq_conv {ε : ℚ} (hε : 0 < ε) :
∃ N, ∀ i ≥ N, padicNormE (f i - (limSeq f i : ℚ_[p]) : ℚ_[p]) < ε | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ (↑i + 1) * ε | rw [right_distrib] | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ ↑i * ε + 1 * ε | a4713483bdf9656c |
MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
(haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ 1 ∈ V₀ | simp only [V₀, mem_iInter] | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ ∀ i ∈ t, 1 ∈ i.carrier | 25c220e55fac0398 |
intervalIntegral.continuousAt_parametric_primitive_of_dominated | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | theorem continuousAt_parametric_primitive_of_dominated [FirstCountableTopology X]
{F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ)
{a₀ b₀ : ℝ} {x₀ : X} (hF_meas : ∀ x, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
(h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ.restrict <| Ι a b, ‖F x t‖ ≤ bound t)
(bound_integrable : I... | E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.r... | have Ioo_nhds : Ioo a b ∈ 𝓝 b₀ := Ioo_mem_nhds hb₀.1 hb₀.2 | E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.r... | 700abcaa08c19cdb |
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod | Mathlib/LinearAlgebra/FreeModule/Int.lean | /-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
... | case h.refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM... | ext i | case h.refine_2.h
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map ... | 3d7a9cb7455a5c0f |
bind₁_rename_expand_wittPolynomial | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | theorem bind₁_rename_expand_wittPolynomial (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n + 1 →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
bind₁ (fun b => rename (fun i => (b, i)) (expand p (W_ ℤ n))) Φ =
... | case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ fu... | have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm | case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
(bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ (wittStructureRat... | 227cfd398809aa75 |
Set.le_einfsep_pair | Mathlib/Topology/MetricSpace/Infsep.lean | theorem le_einfsep_pair : edist x y ⊓ edist y x ≤ ({x, y} : Set α).einfsep | case inr.inl
α : Type u_1
inst✝ : EDist α
a b : α
hab : a ≠ b
⊢ edist b a ≤ edist a b ∨ edist a b ≤ edist a b | simp only [le_refl, true_or, or_true] | no goals | a64ce198c4883813 |
CategoryTheory.Adjunction.leftAdjointUniq_hom_counit | Mathlib/CategoryTheory/Adjunction/Unique.lean | theorem leftAdjointUniq_hom_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
whiskerLeft G (leftAdjointUniq adj1 adj2).hom ≫ adj2.counit = adj1.counit | case w.h
C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F F' : C ⥤ D
G : D ⥤ C
adj1 : F ⊣ G
adj2 : F' ⊣ G
x : D
⊢ F.map (adj2.unit.app (G.obj x)) ≫ adj1.counit.app (F'.obj (G.obj x)) ≫ adj2.counit.app x = adj1.counit.app x | rw [← adj1.counit_naturality, ← Category.assoc, ← F.map_comp] | case w.h
C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F F' : C ⥤ D
G : D ⥤ C
adj1 : F ⊣ G
adj2 : F' ⊣ G
x : D
⊢ F.map (adj2.unit.app (G.obj x) ≫ G.map (adj2.counit.app x)) ≫ adj1.counit.app x = adj1.counit.app x | 9669f69f339158ab |
finprod_mem_inter_mul_diff' | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_mem_inter_mul_diff' (t : Set α) (h : (s ∩ mulSupport f).Finite) :
((∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i) = ∏ᶠ i ∈ s, f i | case hs
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
h : (s ∩ mulSupport f).Finite
⊢ (s ∩ t ∩ mulSupport f).Finite
case ht
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
h : (s ∩ mulSupport f).Finite
⊢ (s \ t ∩ mulSupport f).Finite | exacts [h.subset fun x hx => ⟨hx.1.1, hx.2⟩, h.subset fun x hx => ⟨hx.1.1, hx.2⟩] | no goals | b5ec9483672c1fac |
PrimeSpectrum.isTopologicalBasis_basic_opens | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem isTopologicalBasis_basic_opens :
TopologicalSpace.IsTopologicalBasis
(Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) | R : Type u
inst✝ : CommSemiring R
⊢ TopologicalSpace.IsTopologicalBasis (Set.range fun r => ↑(basicOpen r)) | apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds | case h_open
R : Type u
inst✝ : CommSemiring R
⊢ ∀ u ∈ Set.range fun r => ↑(basicOpen r), IsOpen u
case h_nhds
R : Type u
inst✝ : CommSemiring R
⊢ ∀ (a : PrimeSpectrum R) (u : Set (PrimeSpectrum R)),
a ∈ u → IsOpen u → ∃ v ∈ Set.range fun r => ↑(basicOpen r), a ∈ v ∧ v ⊆ u | 38672fba2e8d5a02 |
MeasureTheory.Content.innerContent_iSup_nat | Mathlib/MeasureTheory/Measure/Content.lean | theorem innerContent_iSup_nat [R1Space G] (U : ℕ → Opens G) :
μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) | case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K... | convert le_trans (h3 t L) _ | case h.e'_3.h.e'_6
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i :... | 47868df373390eca |
MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner {u : E → F'}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
(hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ | case bc
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : Cont... | convert ENNReal.lintegral_mul_le_Lp_mul_Lq μ
(.symm <| .conjExponent <| show 1 < (p : ℝ) from hp) ?_ ?_ using 5 | case h.e'_4.h.e'_5.h.e'_5.h.e'_4.h
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSp... | 728bbe5fb5258103 |
Std.Tactic.BVDecide.BVExpr.bitblast.mkOverflowBit.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Carry.lean | theorem go_decl_eq {aig : AIG α} {cin} {lhs rhs : AIG.RefVec aig w} :
∀ (idx : Nat) (h1) (h2),
(go aig lhs rhs curr cin).aig.decls[idx]'h2 = aig.decls[idx]'h1 | case isTrue.h2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w curr : Nat
aig : AIG α
cin : aig.Ref
lhs rhs : aig.RefVec w
res : AIG.Entrypoint α
h✝ : curr < w
hgo :
go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
(curr + 1) (mkFullAdde... | apply AIG.LawfulOperator.lt_size_of_lt_aig_size (f := mkFullAdderCarry) | case isTrue.h2.h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w curr : Nat
aig : AIG α
cin : aig.Ref
lhs rhs : aig.RefVec w
res : AIG.Entrypoint α
h✝ : curr < w
hgo :
go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
(curr + 1) (mkFullAd... | 0e1562c4999c3b0c |
BooleanSubalgebra.sdiff_mem | Mathlib/Order/BooleanSubalgebra.lean | lemma sdiff_mem (ha : a ∈ L) (hb : b ∈ L) : a \ b ∈ L | α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ a \ b ∈ L | simpa [sdiff_eq] using L.infClosed ha (compl_mem hb) | no goals | 0a19dc16f6028850 |
ExistsContDiffBumpBase.y_smooth | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem y_smooth : ContDiffOn ℝ ∞ (uncurry y) (Ioo (0 : ℝ) 1 ×ˢ (univ : Set E)) | case refine_3
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
⊢ ContDiffOn ℝ ∞ (↿w) (Ioo 0 1 ×ˢ univ) | apply ContDiffOn.mul | case refine_3.hf
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
⊢ ContDiffOn ℝ ∞ (fun x => ((∫ (x : E), u x ∂μ) * |x.1| ^ finrank ℝ E)⁻¹) (Ioo 0 1 ×ˢ univ)
case refi... | 5d103a5b75b6c303 |
MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) | case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
one_lt_p : -1 < p - 1
g : ℝ → ℝ := fun t => t ^ (p - 1)
obs : ∀ (x : ℝ), intervalIntegral g 0 x volume = x ^ p / p
t : ℝ
t_pos : t ∈ Ioi 0
⊢ 0 ≤ g t | exact Real.rpow_nonneg (mem_Ioi.mp t_pos).le (p - 1) | no goals | 720d77f4c064a30f |
Module.End.exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow | Mathlib/LinearAlgebra/JordanChevalley.lean | theorem exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow {P : K[X]} {k : ℕ}
(sep : P.Separable) (nil : minpoly K f ∣ P ^ k) :
∃ᵉ (n ∈ adjoin K {f}) (s ∈ adjoin K {f}), IsNilpotent n ∧ IsSemisimple s ∧ f = n + s | K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k
ff : ↥(adjoin K {f}) := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
⊢ ∃ n ∈ adjoin K {f}, ∃ s ∈ adjoin K {f}, ... | obtain ⟨⟨s, mem⟩, ⟨⟨k, hk⟩, hss⟩, -⟩ := existsUnique_nilpotent_sub_and_aeval_eq_zero nil' sep' | case intro.mk.intro.intro.intro
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥(adjoin K {f}) := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End ... | d5adbca57ec910fe |
PNat.gcd_eq_left_iff_dvd | Mathlib/Data/PNat/Prime.lean | theorem gcd_eq_left_iff_dvd {m n : ℕ+} : m ∣ n ↔ m.gcd n = m | m n : ℕ+
⊢ (↑m).gcd ↑n = ↑m ↔ m.gcd n = m | rw [← coe_inj] | m n : ℕ+
⊢ (↑m).gcd ↑n = ↑m ↔ ↑(m.gcd n) = ↑m | 2ad2bac44da65d21 |
AnalyticOnNhd.is_constant_or_isOpen | Mathlib/Analysis/Complex/OpenMapping.lean | theorem AnalyticOnNhd.is_constant_or_isOpen (hg : AnalyticOnNhd ℂ g U) (hU : IsPreconnected U) :
(∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s) | case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
g : E → ℂ
hg : AnalyticOnNhd ℂ g U
hU : IsPreconnected U
h : ∀ z₀ ∈ U, ¬∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀
s : Set E
hs1 : s ⊆ U
hs2 : IsOpen s
⊢ ∀ x ∈ g '' s, g '' s ∈ 𝓝 x | rintro z ⟨w, hw1, rfl⟩ | case neg.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
g : E → ℂ
hg : AnalyticOnNhd ℂ g U
hU : IsPreconnected U
h : ∀ z₀ ∈ U, ¬∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀
s : Set E
hs1 : s ⊆ U
hs2 : IsOpen s
w : E
hw1 : w ∈ s
⊢ g '' s ∈ 𝓝 (g w) | bcea65785f6ffd2a |
intervalIntegral.continuousWithinAt_primitive | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | theorem continuousWithinAt_primitive (hb₀ : μ {b₀} = 0)
(h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)) :
ContinuousWithinAt (fun b => ∫ x in a..b, f x ∂μ) (Icc b₁ b₂) b₀ | case intro.h₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (a ⊓ b₁) (a ⊔ b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : b₁ ⊓ b₂ = b₁
x : ℝ
h₁ : b₁ ≤ x
h₂ : x ≤ b₂
⊢ x ∈ [[a ⊓ b₁, a ⊔ b₂]] | exact ⟨min_le_of_left_le <| (min_le_right _ _).trans h₁,
le_max_of_le_right <| h₂.trans <| le_max_right _ _⟩ | no goals | 9163a1d4de5d2966 |
Mathlib.Tactic.Bicategory.naturality_leftUnitor | Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean | theorem naturality_leftUnitor {p : a ⟶ b} {f : b ⟶ c} {pf : a ⟶ c} (η_f : p ≫ f ≅ pf) :
p ◁ (λ_ f) ≪≫ η_f = normalizeIsoComp (ρ_ p) η_f :=
Iso.ext (by simp)
| B : Type u
inst✝ : Bicategory B
a b c : B
p : a ⟶ b
f : b ⟶ c
pf : a ⟶ c
η_f : p ≫ f ≅ pf
⊢ (p ◁ λ_ f ≪≫ η_f).hom = (normalizeIsoComp (ρ_ p) η_f).hom | simp | no goals | 98745ba2aaaa51f8 |
eventually_homothety_mem_of_mem_interior | Mathlib/Analysis/Normed/Affine/AddTorsor.lean | theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s) :
∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s | case h
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
x : Q
s : Set Q
y : Q
hy : y ∈ interior s
h : y = x
⊢ 0 < 1 ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < 1} → (homothety x x_1) y ∈ s | simp [h.symm, interior_subset hy] | no goals | 1289875882d40d72 |
Real.continuousOn_tan_Ioo | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | theorem continuousOn_tan_Ioo : ContinuousOn tan (Ioo (-(π / 2)) (π / 2)) | x : ℝ
⊢ x ∈ Ioo (-(π / 2)) (π / 2) → x ∈ {x | cos x ≠ 0} | simp only [and_imp, mem_Ioo, mem_setOf_eq, Ne] | x : ℝ
⊢ -(π / 2) < x → x < π / 2 → ¬cos x = 0 | d91f87528b6a23d6 |
GromovHausdorff.hausdorffDist_optimal | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y | case intro.intro.intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHS... | have : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb :=
hausdorffDist_optimal_le_HD _ _ (candidatesBOfCandidates_mem F Fgood) | case intro.intro.intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHS... | e1b74305bb180259 |
Metric.Sum.mem_uniformity_iff_glueDist | Mathlib/Topology/MetricSpace/Gluing.lean | theorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s | X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
⊢ s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ (a b : X ⊕ Y), glueDist Φ Ψ ε a b < δ → (a, b) ∈ s | simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage] | X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
⊢ ((∃ ε > 0, ∀ ⦃a b : X⦄, dist a b < ε → Prod.map Sum.inl Sum.inl (a, b) ∈ s) ∧
∃ ε > 0, ∀ ⦃a b : Y⦄, dist a b < ε → Prod.map Sum.inr Sum.inr (a, b) ∈ s) ↔
∃ δ > 0, ∀ ... | fe4b5204c103372a |
StieltjesFunction.outer_trim | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem outer_trim : f.outer.trim = f.outer | f : StieltjesFunction
s : Set ℝ
t : ℕ → Set ℝ
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ⊤
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i) | have hl :=
ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne' | f : StieltjesFunction
s : Set ℝ
t : ℕ → Set ℝ
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ⊤
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hl : f.length (t i) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i) | f9dd01cdb673361c |
Real.cos_pi_div_five | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem cos_pi_div_five : cos (π / 5) = (1 + √5) / 4 | case inr
c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ c = (1 + √5) / 4 | absurd (show 0 ≤ c from cos_nonneg_of_mem_Icc <| by constructor <;> linarith [pi_pos.le]) | case inr
c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ ¬0 ≤ c | 386d26d34e57890c |
Polynomial.cyclotomic_injective | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | theorem cyclotomic_injective [CharZero R] : Function.Injective fun n => cyclotomic n R | case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))
hmzero : NeZero m
⊢ n = m | rw [isRoot_cyclotomic_iff (R := ℂ)] at hroot | case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
⊢ n = m | 95f81674fb37b3a0 |
CategoryTheory.Comma.inv_right | Mathlib/CategoryTheory/Comma/Basic.lean | @[simp]
lemma inv_right [IsIso e] : (inv e).right = inv e.right | A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
e : X ⟶ Y
inst✝ : IsIso e
⊢ (inv e).right = inv e.right | apply IsIso.eq_inv_of_hom_inv_id | case hom_inv_id
A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
e : X ⟶ Y
inst✝ : IsIso e
⊢ e.right ≫ (inv e).right = 𝟙 X.right | c0e969a93d7f9594 |
Complex.continuousOn_norm_circleTransformBoundingFunction | Mathlib/MeasureTheory/Integral/CircleTransform.lean | theorem continuousOn_norm_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
ContinuousOn ((‖·‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) | case hg.hf
R r : ℝ
hr : r < R
z : ℂ
⊢ ContinuousOn (fun x => deriv (circleMap z R) x.2) (closedBall z r ×ˢ univ) | simp only [deriv_circleMap] | case hg.hf
R r : ℝ
hr : r < R
z : ℂ
⊢ ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r ×ˢ univ) | eed8430be825edc0 |
NumberField.mixedEmbedding.negAt_symm | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem negAt_symm :
(negAt s).symm = negAt s | case neg
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∉ s
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w | simp_rw [negAt_apply_isReal_and_not_mem _ hw, negAt, prod_symm,
ContinuousLinearEquiv.prod_apply, piCongrRight_symm_apply, if_neg hw, refl_symm, refl_apply] | no goals | b65992f9db9e340a |
Seminorm.ball_norm_mul_subset | Mathlib/Analysis/Seminorm.lean | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | case inl
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
⊢ p.ball 0 (‖0‖ * r) ⊆ 0 • p.ball 0 r | rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl] | case inl
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
⊢ ∅ ⊆ 0 • p.ball 0 r | b87a373bd1754990 |
cyclotomic_comp_X_add_one_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 | case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < ((cyclotomic p ℤ).comp (X + 1)).natDegree
⊢ (∑ x ∈ range p, if x = i then ↑(p.choose (x + 1)) else 0) ∈ Submodule.span ℤ {↑p} | rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
natDegree_cyclotomic, Nat.totient_prime hp.out] at hi | case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < p - 1
⊢ (∑ x ∈ range p, if x = i then ↑(p.choose (x + 1)) else 0) ∈ Submodule.span ℤ {↑p} | bff95b337ffb159a |
ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz | Mathlib/Analysis/Normed/Operator/Banach.lean | lemma bijective_iff_dense_range_and_antilipschitz (f : E →SL[σ] F) :
Bijective f ↔ (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c f | 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : Complete... | rwa [← LinearMap.range_eq_top, ← closed_range_of_antilipschitz hf] | no goals | 1255c3d2e1f157b7 |
OrderIso.strictConcaveOn_symm | Mathlib/Analysis/Convex/Function.lean | theorem OrderIso.strictConcaveOn_symm (f : α ≃o β) (hf : StrictConvexOn 𝕜 univ f) :
StrictConcaveOn 𝕜 univ f.symm | case intro.intro
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a ... | rw [← f.lt_iff_lt, OrderIso.apply_symm_apply] | case intro.intro
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a ... | 798e65840fb2b195 |
Set.inl_compl_union_inr_compl | Mathlib/Data/Set/Basic.lean | lemma inl_compl_union_inr_compl {α β : Type*} {s : Set α} {t : Set β} :
Sum.inl '' sᶜ ∪ Sum.inr '' tᶜ = (Sum.inl '' s ∪ Sum.inr '' t)ᶜ | α : Type u_1
β : Type u_2
s : Set α
t : Set β
⊢ Sum.inl '' sᶜ ∪ Sum.inr '' tᶜ = (Sum.inl '' s)ᶜ ∩ (Sum.inr '' t)ᶜ | aesop | no goals | 912b559fa88b7d10 |
IsLindelof.disjoint_nhdsSet_left | Mathlib/Topology/Compactness/Lindelof.lean | theorem IsLindelof.disjoint_nhdsSet_left {l : Filter X} [CountableInterFilter l]
(hs : IsLindelof s) :
Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l | X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
l : Filter X
inst✝ : CountableInterFilter l
hs : IsLindelof s
⊢ Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l | refine ⟨fun h x hx ↦ h.mono_left <| nhds_le_nhdsSet hx, fun H ↦ ?_⟩ | X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
l : Filter X
inst✝ : CountableInterFilter l
hs : IsLindelof s
H : ∀ x ∈ s, Disjoint (𝓝 x) l
⊢ Disjoint (𝓝ˢ s) l | f96889b611855166 |
ZMod.cast_sub_one | Mathlib/Data/ZMod/Basic.lean | theorem cast_sub_one {R : Type*} [Ring R] {n : ℕ} (k : ZMod n) :
(cast (k - 1 : ZMod n) : R) = (if k = 0 then (n : R) else cast k) - 1 | case neg.zero
R : Type u_1
inst✝ : Ring R
k : ZMod 0
hk : ¬k = 0
⊢ (k - 1).cast = k.cast - 1 | dsimp [ZMod, ZMod.cast] | case neg.zero
R : Type u_1
inst✝ : Ring R
k : ZMod 0
hk : ¬k = 0
⊢ ↑(k - 1) = ↑k - 1 | 4fdb3038bd174435 |
Set.Iic_union_Icc' | Mathlib/Order/Interval/Set/Basic.lean | theorem Iic_union_Icc' (h₁ : c ≤ b) : Iic b ∪ Icc c d = Iic (max b d) | case h
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c ≤ b
x : α
⊢ x ∈ Iic b ∪ Icc c d ↔ x ∈ Iic (b ⊔ d) | simp_rw [mem_union, mem_Iic, mem_Icc, le_max_iff] | case h
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c ≤ b
x : α
⊢ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ x ≤ b ∨ x ≤ d | cb2a065dca8b2fee |
UV.shadow_compression_subset_compression_shadow | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem shadow_compression_subset_compression_shadow (u v : Finset α)
(huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
∂ (𝓒 u v 𝒜) ⊆ 𝓒 u v (∂ 𝒜) | case intro.intro.refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜... | rwa [← insert_sdiff_of_not_mem _ ‹x ∉ u›, ← insert_union] | no goals | 9666ab7e87eab58f |
Commute.orderOf_mul_eq_right_of_forall_prime_mul_dvd | Mathlib/GroupTheory/OrderOfElement.lean | theorem orderOf_mul_eq_right_of_forall_prime_mul_dvd (h : Commute x y) (hy : IsOfFinOrder y)
(hdvd : ∀ p : ℕ, p.Prime → p ∣ orderOf x → p * orderOf x ∣ orderOf y) :
orderOf (x * y) = orderOf y | case hd
G : Type u_1
inst✝ : Monoid G
x y : G
h : Commute x y
hy : IsOfFinOrder y
hdvd : ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf x → p * orderOf x ∣ orderOf y
hoy : 0 < orderOf y
hxy : orderOf x ∣ orderOf y
⊢ ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf y → ¬orderOf (x * y) ∣ orderOf y / p | refine fun p hp hpy hd => hp.ne_one ?_ | case hd
G : Type u_1
inst✝ : Monoid G
x y : G
h : Commute x y
hy : IsOfFinOrder y
hdvd : ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf x → p * orderOf x ∣ orderOf y
hoy : 0 < orderOf y
hxy : orderOf x ∣ orderOf y
p : ℕ
hp : Nat.Prime p
hpy : p ∣ orderOf y
hd : orderOf (x * y) ∣ orderOf y / p
⊢ p = 1 | ad488aedf0f1c9e4 |
Int.mul_le_mul_of_nonneg_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem mul_le_mul_of_nonneg_left {a b c : Int}
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
if hba : b ≤ a then by
rw [Int.le_antisymm hba h₁]; apply Int.le_refl
else if hc0 : c ≤ 0 then by
simp [Int.le_antisymm hc0 h₂, Int.zero_mul]
else by
exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left
... | a b c : Int
h₁ : a ≤ b
h₂ : 0 ≤ c
hba : ¬b ≤ a
hc0 : c ≤ 0
⊢ c * a ≤ c * b | simp [Int.le_antisymm hc0 h₂, Int.zero_mul] | no goals | ebac8b4eee743453 |
clusterPt_iff_lift'_closure' | Mathlib/Topology/Basic.lean | theorem clusterPt_iff_lift'_closure' {F : Filter X} :
ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot | case mpr
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
⊢ (pure x ⊓ F.lift' closure).NeBot → pure x ≤ F.lift' closure | intro h U hU | case mpr
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
h : (pure x ⊓ F.lift' closure).NeBot
U : Set X
hU : U ∈ F.lift' closure
⊢ U ∈ pure x | 70879bd9b2cfc6e6 |
Set.sInter_prod_sInter | Mathlib/Data/Set/Lattice.lean | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
T : Set (Set β)
s₁ : Set α
h₁ : s₁ ∈ S
s₂ : Set β
h₂ : s₂ ∈ T
x : α × β
hx : ∀ i ∈ S ×ˢ T, x ∈ i.1 ×ˢ i.2
⊢ x ∈ ⋂₀ S ×ˢ ⋂₀ T | exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩ | no goals | 5614e7d5edc46357 |
Ordinal.sup_mul_nat | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem sup_mul_nat (o : Ordinal) : (sup fun n : ℕ => o * n) = o * ω | case inr
o : Ordinal.{u_1}
ho : 0 < o
⊢ (sup fun n => o * ↑n) = o * ω | exact (mul_isNormal ho).apply_omega0 | no goals | 4ffb26912b5f26d1 |
MulAction.IsBlockSystem.of_normal | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem IsBlockSystem.of_normal {N : Subgroup G} [N.Normal] :
IsBlockSystem G (Set.range fun a : X => orbit N a) | case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
⊢ ∀ ⦃B : Set X⦄, (B ∈ range fun a => orbit (↥N) a) → IsBlock G B | intro b | case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
b : Set X
⊢ (b ∈ range fun a => orbit (↥N) a) → IsBlock G b | 94f82df37ff2f4b9 |
MeasureTheory.volume_sum_rpow_le | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | theorem MeasureTheory.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ card ι *
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
ENNReal.ofReal r ^ ... | have nm_neg := fun x : ι → ℝ => norm_neg (E := PiLp (.ofReal p) (fun _ : ι => ℝ)) x | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-x‖ = ‖x‖
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 ... | b43945fead980666 |
ProbabilityTheory.Kernel.tendsto_integral_density_of_monotone | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma tendsto_integral_density_of_monotone (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 (κ a univ).toReal) | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), Measurab... | simp_rw [integral_density hκν a (hseq_meas _)] | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), Measurab... | e97057e55029f4e1 |
SimpleGraph.chromaticNumber_sum | Mathlib/Combinatorics/SimpleGraph/Sum.lean | theorem chromaticNumber_sum :
(G ⊕g H).chromaticNumber = max G.chromaticNumber H.chromaticNumber | α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
d : ℕ∞
hG : G.chromaticNumber ≤ d
hH : H.chromaticNumber ≤ d
⊢ (G ⊕g H).chromaticNumber ≤ d | cases d with
| top => simp
| coe n =>
let cG : G.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hG).some
let cH : H.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hH).some
exact chromaticNumber_le_iff_colorable.mpr (Nonempty.intro (cG.sum cH)) | no goals | 14daea5911720f3b |
RootPairing.range_polarization_domRestrict_le_span_coroot | Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean | theorem range_polarization_domRestrict_le_span_coroot :
LinearMap.range (P.Polarization.domRestrict P.rootSpan) ≤ P.corootSpan | case intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
y : N
x : ↥P.rootSpan
hx : (P.Polarization.domRestrict P.rootSpan) x = y
⊢ ∃ c, ∑ i : ι, c i • P.coroot i =... | use fun i => (P.toPerfectPairing x) (P.coroot i) | case h
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
y : N
x : ↥P.rootSpan
hx : (P.Polarization.domRestrict P.rootSpan) x = y
⊢ ∑ i : ι, (fun i => (P.toPerfectPairi... | f62e36c9fced14b2 |
WittVector.coeff_add_of_disjoint | Mathlib/RingTheory/WittVector/InitTail.lean | theorem coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n | case h
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n => y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n => if P n then x.coeff n else y.coeff n
n : ℕ
⊢ (select P z).coeff n = x.coeff n | rw [select, coeff_mk, coeff_mk] | case h
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n => y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n => if P n then x.coeff n else y.coeff n
n : ℕ
⊢ (if P n then if P n then x.coeff n else y.coeff n else 0) =... | e3aa7fcb498bb341 |
precise_refinement_set | Mathlib/Topology/Compactness/Paracompact.lean | theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X)
(uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i | ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
inst✝ : ParacompactSpace X
s : Set X
hs : IsClosed s
u : ι → Set X
uo : ∀ (i : ι), IsOpen (u i)
us : s ⊆ ⋃ i, u i
⊢ ⋃ i, Option.elim' sᶜ u i = univ | apply Subset.antisymm (subset_univ _) | ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
inst✝ : ParacompactSpace X
s : Set X
hs : IsClosed s
u : ι → Set X
uo : ∀ (i : ι), IsOpen (u i)
us : s ⊆ ⋃ i, u i
⊢ univ ⊆ ⋃ i, Option.elim' sᶜ u i | ecc14a28630beea6 |
RingHom.FormallyUnramified.propertyIsLocal | Mathlib/RingTheory/RingHom/Unramified.lean | lemma propertyIsLocal :
PropertyIsLocal FormallyUnramified | case localizationAwayPreserves
R S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u_3
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
H✝ : f.FormallyUnramified
algInst✝² : Alge... | exact Algebra.FormallyUnramified.of_comp R R' S' | no goals | f955845a3f1095b7 |
Stream'.Seq.cons_injective2 | Mathlib/Data/Seq/Seq.lean | theorem cons_injective2 : Function.Injective2 (cons : α → Seq α → Seq α) := fun x y s t h =>
⟨by rw [← Option.some_inj, ← get?_cons_zero, h, get?_cons_zero],
Seq.ext fun n => by simp_rw [← get?_cons_succ x s n, h, get?_cons_succ]⟩
| α : Type u
x y : α
s t : Seq α
h : cons x s = cons y t
n : ℕ
⊢ s.get? n = t.get? n | simp_rw [← get?_cons_succ x s n, h, get?_cons_succ] | no goals | c62240eacb0d23c4 |
Polynomial.Monic.comp | Mathlib/Algebra/Polynomial/Monic.lean | lemma comp (hp : p.Monic) (hq : q.Monic) (h : q.natDegree ≠ 0) : (p.comp q).Monic | R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p.Monic
hq : q.Monic
h : q.natDegree ≠ 0
a✝ : Nontrivial R
this : (p.comp q).natDegree = p.natDegree * q.natDegree
⊢ (p.comp q).Monic | rw [Monic.def, Polynomial.leadingCoeff, this, coeff_comp_degree_mul_degree h, hp.leadingCoeff,
hq.leadingCoeff, one_pow, mul_one] | no goals | 6796f7c453197a8d |
IsVisible.of_convexHull_of_pos | Mathlib/Analysis/Convex/Visible.lean | /-- If a point `x` sees a convex combination of points of a set `s` through `convexHull ℝ s ∌ x`,
then it sees all terms of that combination.
Note that the converse does not hold. -/
lemma IsVisible.of_convexHull_of_pos {ι : Type*} {t : Finset ι} {a : ι → V} {w : ι → 𝕜}
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i ∈ t, w ... | case h.e'_10
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ ... | rw [← one_smul 𝕜 (a i), ← hwi, eq_comm] | case h.e'_10
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ ... | 7c7847924d1006dd |
MvPowerSeries.coeff_mul_of_add_lexOrder | Mathlib/RingTheory/MvPowerSeries/LexOrder.lean | theorem coeff_mul_of_add_lexOrder {φ ψ : MvPowerSeries σ R}
{p q : σ →₀ ℕ} (hp : lexOrder φ = toLex p) (hq : lexOrder ψ = toLex q) :
coeff R (p + q) (φ * ψ) = coeff R p φ * coeff R q ψ | σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
⊢ ∑ p ∈ Finset.antidiagonal (p + q), (coeff R p.1) φ * (coeff R p.2) ψ = (coeff R p) φ * (coeff R q) ψ | apply Finset.sum_eq_single (⟨p, q⟩ : (σ →₀ ℕ) × (σ →₀ ℕ)) | case h₀
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
⊢ ∀ b ∈ Finset.antidiagonal (p + q), b ≠ (p, q) → (coeff R b.1) φ * (coeff R b.2) ψ = 0
case h₁
σ : Type u_1
R : Type u_2
i... | 0c23704cc23b543a |
isPrimePow_iff_factorization_eq_single | Mathlib/Data/Nat/Factorization/PrimePow.lean | theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k | n : ℕ
⊢ IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k | rw [isPrimePow_nat_iff] | n : ℕ
⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k | f949a688297a79db |
Algebra.FormallyUnramified.iff_exists_tensorProduct | Mathlib/RingTheory/Unramified/Finite.lean | theorem iff_exists_tensorProduct [EssFiniteType R S] :
FormallyUnramified R S ↔ ∃ t : S ⊗[R] S,
(∀ s, ((1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) * t = 0) ∧ TensorProduct.lmul' R t = 1 | R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1 | have : ∀ t : S ⊗[R] S, TensorProduct.lmul' R t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S := by
intro t
simp only [KaehlerDifferential.ideal, RingHom.mem_ker, map_sub, map_one,
sub_eq_zero, @eq_comm S 1] | R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
∃ t, (∀ (s : S... | 242718edd9112c3c |
Set.Countable.image | Mathlib/Data/Set/Countable.lean | theorem Countable.image {s : Set α} (hs : s.Countable) (f : α → β) : (f '' s).Countable | α : Type u
β : Type v
s : Set α
hs : s.Countable
f : α → β
this : Countable ↑s
⊢ (range fun x => f ↑x).Countable | apply countable_range | no goals | f0d955afdd57dc60 |
IsMulFreimanHom.prod | Mathlib/Combinatorics/Additive/FreimanHom.lean | @[to_additive]
lemma IsMulFreimanHom.prod (h₁ : IsMulFreimanHom n A₁ B₁ f₁) (h₂ : IsMulFreimanHom n A₂ B₂ f₂) :
IsMulFreimanHom n (A₁ ×ˢ A₂) (B₁ ×ˢ B₂) (Prod.map f₁ f₂) where
mapsTo := h₁.mapsTo.prodMap h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h | α₁ : Type u_5
α₂ : Type u_6
β₁ : Type u_7
β₂ : Type u_8
inst✝³ : CommMonoid α₁
inst✝² : CommMonoid α₂
inst✝¹ : CommMonoid β₁
inst✝ : CommMonoid β₂
A₁ : Set α₁
A₂ : Set α₂
B₁ : Set β₁
B₂ : Set β₂
f₁ : α₁ → β₁
f₂ : α₂ → β₂
n : ℕ
h₁ : IsMulFreimanHom n A₁ B₁ f₁
h₂ : IsMulFreimanHom n A₂ B₂ f₂
s t : Multiset (α₁ × α₂)
hs :... | simpa [@forall_swap α₁] using htA.2 | no goals | 33dda5fafc4888d5 |
WeierstrassCurve.variableChange_id | Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean | lemma variableChange_id : W.variableChange VariableChange.id = W | R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ { a₁ := 1 * (W.a₁ + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),
a₂ :=
1 ^ 2 *
(W.a₂ - { u := 1, r := 0, s := 0, t := 0 }.s * W.a₁ + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -
{ u := 1, r := 0, s := 0, t := 0 }.s ^ 2),
a₃ := ... | ext <;> (dsimp only; ring1) | no goals | be30df7f818a1ddd |
AlgebraicTopology.DoldKan.HigherFacesVanish.induction | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | theorem induction {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ) :
HigherFacesVanish (q + 1) (φ ≫ (𝟙 _ + Hσ q).f (n + 1)) | case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a +... | rw [X.δ_comp_σ_of_gt', j.pred_succ] | case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a +... | 69ef2d77296be0d2 |
minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow | Mathlib/FieldTheory/SeparableDegree.lean | theorem natSepDegree_eq_one_iff_eq_X_sub_C_pow : (minpoly F x).natSepDegree = 1 ↔
∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n | case refine_1.intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpol... | have hx := congr_arg (Polynomial.aeval x) h.symm | case refine_1.intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (Polynomial.aeval x) (X ^ q ^ n - C y) = ... | 731ae2c59cbb171a |
Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card | Mathlib/LinearAlgebra/Dimension/Finite.lean | theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
{t : Finset M} (h : finrank R M + 1 < t.card) :
∃ f : M → R, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 | R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Finset M := Finset.map shift (t.erase x₀)
h' : finrank R M... | simpa only [f, Embedding.coeFn_mk, sub_add_cancel, this.2, true_and, if_neg this.1] | no goals | 9ccb45a24efab47f |
Submonoid.LocalizationWithZeroMap.leftCancelMulZero_of_le_isLeftRegular | Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean | theorem leftCancelMulZero_of_le_isLeftRegular
(f : LocalizationWithZeroMap S N) [IsLeftCancelMulZero M]
(h : ∀ ⦃x⦄, x ∈ S → IsLeftRegular x) : IsLeftCancelMulZero N | M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
f : S.LocalizationWithZeroMap N
inst✝ : IsLeftCancelMulZero M
h : ∀ ⦃x : M⦄, x ∈ S → IsLeftRegular x
fl : S.LocalizationMap N := f.toLocalizationMap
g : M →* N := f.toMap
a z w : N
ha : a ≠ 0
hazw : a * z = a * w
b : M... | rw [← mul_assoc, ← mul_assoc _ w, mul_comm _ w, mul_assoc w, mul_assoc,
← mul_assoc w, ← mul_assoc w, mul_comm w] | no goals | 395e8d90995bfe86 |
Ring.DirectLimit.lift_unique | Mathlib/Algebra/Colimit/Ring.lean | theorem lift_unique (F : DirectLimit G f →+* P) (x) :
F x = lift G f P (fun i ↦ F.comp <| of G f i) (fun i j hij x ↦ by simp) x | ι : Type u_1
inst✝² : Preorder ι
G : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝ : CommRing P
F : DirectLimit G f →+* P
x : FreeCommRing ((i : ι) × G i)
⊢ ∀ (x y : FreeCommRing ((i : ι) × G i)),
F
((Ideal.Quotient.mk
(Ideal.span
... | simp+contextual | no goals | 051b292ba4e666cf |
Mon_.mul_braiding | Mathlib/CategoryTheory/Monoidal/Mon_.lean | theorem mul_braiding {X Y : Mon_ C} :
(X ⊗ Y).mul ≫ (β_ X.X Y.X).hom = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (Y ⊗ X).mul | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : SymmetricCategory C
X Y : Mon_ C
⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫
((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫
Y.X ◁ (α_ Y... | slice_lhs 1 2 =>
rw [← tensorHom_def] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : SymmetricCategory C
X Y : Mon_ C
⊢ (((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫
Y.X ◁ (α_ Y.X X.X X.X).hom) ≫
(α_ Y.X Y... | 6be9ea682fb53a86 |
pairwise_coprime_iff_coprime_prod | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem pairwise_coprime_iff_coprime_prod [DecidableEq I] :
Pairwise (IsCoprime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsCoprime (s i) (∏ j ∈ t \ {i}, s j) | case refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsCoprime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t ∧ ¬j = i
⊢ IsCoprime (s i) (s j) | obtain ⟨hj, ji⟩ := hj | case refine_1.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsCoprime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t
ji : ¬j = i
⊢ IsCoprime (s i) (s j) | d6dd92ae2640356e |
WeierstrassCurve.Jacobian.Y_ne_negY_of_Y_ne | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma Y_ne_negY_of_Y_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
(hQ : W'.Equation Q) (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3) :
P y ≠ W'.negY P | R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3
⊢ P y ≠ W'.negY P | have hy' : P y * Q z ^ 3 - W'.negY Q * P z ^ 3 = 0 :=
(mul_eq_zero.mp <| Y_sub_Y_mul_Y_sub_negY hP hQ hx).resolve_left <| sub_ne_zero_of_ne hy | R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3
hy' : P y * Q z ^ 3 - W'.negY Q * P z ^ 3 = 0
⊢ P y ≠ W'.negY P | 1dce46f4b895ab5b |
linearIndependent_iff_card_eq_finrank_span | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K | case mp
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
ι : Type u_2
inst✝ : Fintype ι
b : ι → V
h : LinearIndependent K b
⊢ Fintype.card ι = Set.finrank K (range b) | exact (finrank_span_eq_card h).symm | no goals | 911125b85a6b1f1b |
RCLike.nnnorm_natCast | Mathlib/Analysis/RCLike/Basic.lean | @[simp, rclike_simps, norm_cast] lemma nnnorm_natCast (n : ℕ) : ‖(n : K)‖₊ = n | K : Type u_1
inst✝ : RCLike K
n : ℕ
⊢ ‖↑n‖₊ = ↑n | simp [nnnorm] | no goals | 20b378915ca882c2 |
PFun.mem_fix_iff | Mathlib/Data/PFun.lean | theorem mem_fix_iff {f : α →. β ⊕ α} {a : α} {b : β} :
b ∈ f.fix a ↔ Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ f.fix a' :=
⟨fun h => by
let ⟨h₁, h₂⟩ := Part.mem_assert_iff.1 h
rw [WellFounded.fixFEq] at h₂
simp only [Part.mem_assert_iff] at h₂
obtain ⟨h₂, h₃⟩ := h₂
split at h₃
next e ... | case inl.intro.refine_2
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
⊢ b ∈
match e : (f a).get h₁ with
| Sum.inl b => Part.some b
| Sum.inr a' =>
(fun y p =>
WellFounded.fixF
(fun a IH =>
Part.assert (f a).Dom fun... | split | case inl.intro.refine_2.h_1
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
b✝ : β
heq✝ : (f a).get h₁ = Sum.inl b✝
⊢ b ∈ Part.some b✝
case inl.intro.refine_2.h_2
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
a'✝ : α
... | b0fcdf86c1a0e2fe |
List.mem_permutationsAux_of_perm | Mathlib/Data/List/Permutation.lean | theorem mem_permutationsAux_of_perm :
∀ {ts is l : List α},
l ~ is ++ ts → (∃ (is' : _) (_ : is' ~ is), l = is' ++ ts) ∨ l ∈ permutationsAux ts is | case inr
α : Type u_1
t : α
ts is : List α
IH1 :
∀ (l : List α), l ~ t :: is ++ ts → (∃ is', ∃ (_ : is' ~ t :: is), l = is' ++ ts) ∨ l ∈ ts.permutationsAux (t :: is)
IH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is', ∃ (_ : is' ~ []), l = is' ++ is) ∨ l ∈ is.permutationsAux []
l : List α
p : l ~ is ++ t :: ts
m : l ∈ ts.pe... | exact Or.inr (Or.inl m) | no goals | c51a06076c0dbbc1 |
Pell.xz_sub | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem xz_sub {m n} (h : n ≤ m) :
xz a1 (m - n) = xz a1 m * xz a1 n - d a1 * yz a1 m * yz a1 n | a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n - ↑(Pell.d a1) * yz a1 m * yz a1 n | rw [sub_eq_add_neg, ← mul_neg] | a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n + ↑(Pell.d a1) * yz a1 m * -yz a1 n | 8c1df321155b9108 |
OmegaCompletePartialOrder.ωSup_eq_of_isLUB | Mathlib/Order/OmegaCompletePartialOrder.lean | lemma ωSup_eq_of_isLUB {c : Chain α} {a : α} (h : IsLUB (Set.range c) a) : a = ωSup c | case right
α : Type u_2
inst✝ : OmegaCompletePartialOrder α
c : Chain α
a : α
h : (∀ (a_1 : ℕ), c a_1 ≤ a) ∧ ∀ ⦃a_1 : α⦄, (∀ (a : ℕ), c a ≤ a_1) → a ≤ a_1
⊢ ∀ (i : ℕ), c i ≤ a | apply h.1 | no goals | 05205937dddfa3ff |
Finset.offDiag_card | Mathlib/Data/Finset/Prod.lean | theorem offDiag_card : (offDiag s).card = s.card * s.card - s.card :=
suffices (diag s).card + (offDiag s).card = s.card * s.card by rw [s.diag_card] at this; omega
by rw [← card_product, diag, offDiag]
conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun a => a.1 = a.2)]
| α : Type u_1
inst✝ : DecidableEq α
s : Finset α
⊢ #(filter (fun a => a.1 = a.2) (s ×ˢ s)) + #(filter (fun a => a.1 ≠ a.2) (s ×ˢ s)) = #(s ×ˢ s) | conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun a => a.1 = a.2)] | no goals | 953ae524eef68703 |
WithCStarModule.prod_norm_le_norm_add | Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean | lemma prod_norm_le_norm_add (x : C⋆ᵐᵒᵈ (E × F)) : ‖x‖ ≤ ‖x.1‖ + ‖x.2‖ | A : Type u_1
inst✝⁹ : NonUnitalCStarAlgebra A
inst✝⁸ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : Module ℂ E
inst✝⁵ : SMul Aᵐᵒᵖ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : Module ℂ F
inst✝² : SMul Aᵐᵒᵖ F
inst✝¹ : CStarModule A E
inst✝ : CStarModule A F
x : C⋆ᵐᵒᵈ (E × F)
⊢ ‖⟪x.1, x.1⟫_A... | simp [norm_sq_eq] | no goals | 530d4c0602d88bbf |
Real.rpow_sub_intCast | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | lemma rpow_sub_intCast {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n | x : ℝ
hx : x ≠ 0
y : ℝ
n : ℕ
⊢ x ^ (y - ↑n) = x ^ y / x ^ n | simpa using rpow_add_intCast hx y (-n) | no goals | d0fad410096f2fd8 |
Equiv.symm_trans_swap_trans | Mathlib/Logic/Equiv/Basic.lean | theorem symm_trans_swap_trans [DecidableEq β] (a b : α) (e : α ≃ β) :
(e.symm.trans (swap a b)).trans e = swap (e a) (e b) :=
Equiv.ext fun x => by
have : ∀ a, e.symm x = a ↔ x = e a := fun a => by
rw [@eq_comm _ (e.symm x)]
constructor <;> intros <;> simp_all
simp only [trans_apply, swap_appl... | α : Sort u_1
β : Sort u_4
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
a b : α
e : α ≃ β
x : β
⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x | have : ∀ a, e.symm x = a ↔ x = e a := fun a => by
rw [@eq_comm _ (e.symm x)]
constructor <;> intros <;> simp_all | α : Sort u_1
β : Sort u_4
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
a b : α
e : α ≃ β
x : β
this : ∀ (a : α), e.symm x = a ↔ x = e a
⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x | 48f02afc5d1f6bd8 |
AlternatingMap.domCoprod.summand_eq_zero_of_smul_invariant | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁)
(b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : ιa ⊕ ιb → Mᵢ}
{i j : ιa ⊕ ιb} (hv : v i = v j) (hij : i ≠ j) :
swap i j • σ = σ → domCoprod.summand a b σ v = 0 | case h.e'_2.h.e'_10
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ :... | exact AlternatingMap.map_eq_zero_of_eq _ _ hv fun hij' => hij (hij' ▸ rfl) | no goals | 3328e6875d9d0122 |
Metric.subsingleton_closedBall | Mathlib/Topology/MetricSpace/Defs.lean | theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton | case inr
γ : Type w
inst✝ : MetricSpace γ
x : γ
hr : 0 ≤ 0
⊢ {x}.Subsingleton | exact subsingleton_singleton | no goals | c66d4373aa3b40df |
Vector.mem_set | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem mem_set (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
a ∈ v.set i a hi | α : Type u_1
n : Nat
v : Vector α n
i : Nat
hi : i < n
a : α
⊢ ∃ i_1 h, (v.set i a hi)[i_1] = a | exact ⟨i, (by simpa using hi), by simp⟩ | no goals | 78ca1d27ff802219 |
FractionalIdeal.mem_zero_iff | Mathlib/RingTheory/FractionalIdeal/Basic.lean | theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero], fun hx => ⟨0, rfl, by simp [hx]⟩⟩
| R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
x : P
x✝ : x ∈ 0
x' : R
x'_mem_zero : x' ∈ ↑0
x'_eq_x : (Algebra.linearMap R P) x' = x
x'_eq_zero : x' = 0
⊢ x = 0 | simp [x'_eq_x.symm, x'_eq_zero] | no goals | 0682cefbc56573ad |
List.dropInfix?_go_eq_some_iff | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l ... | case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
a' : List α
h : (a :: (a' ++ s)).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), a :: (a' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ [].length
h₂ :... | rw [← cons_append] at h | case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
a' : List α
h : (a :: a' ++ s).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), a :: (a' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ [].length
h₂ : (... | 2a30116d5cc9db54 |
OrthogonalFamily.range_linearIsometry | Mathlib/Analysis/InnerProductSpace/l2Space.lean | theorem range_linearIsometry [∀ i, CompleteSpace (G i)] :
LinearMap.range hV.linearIsometry.toLinearMap =
(⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure | ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ ... | refine le_antisymm ?_ ?_ | case refine_1
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 ... | e67b0f768537d5e1 |
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite | Mathlib/MeasureTheory/Measure/SeparableMeasure.lean | theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
(h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ ... | simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul,
Nat.cast_add, Nat.cast_one] | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ ... | 92d6fe7992cc346b |
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