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 |
|---|---|---|---|---|---|---|
HasFPowerSeriesOnBall.unshift | Mathlib/Analysis/Analytic/Constructions.lean | theorem HasFPowerSeriesOnBall.unshift (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (fun y ↦ z + f y (y - x)) (pf.unshift z) x r where
r_le | 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → E →L[𝕜] F
pf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
x : E
r : ℝ≥0∞
z : F
hf : HasFPowerSeriesOnBall f pf x r
y : E
hy :... | apply HasSum.zero_add | case h
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → E →L[𝕜] F
pf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
x : E
r : ℝ≥0∞
z : F
hf : HasFPowerSeriesOnBall f pf x r
y :... | 8771ecd745213522 |
QuadraticMap.posDef_pi_iff | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | theorem posDef_pi_iff {P} [Fintype ι] [OrderedAddCommMonoid P] [Module R P]
{Q : ∀ i, QuadraticMap R (Mᵢ i) P} : (pi Q).PosDef ↔ ∀ i, (Q i).PosDef | ι : Type u_1
R : Type u_2
Mᵢ : ι → Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝³ : (i : ι) → Module R (Mᵢ i)
P : Type u_10
inst✝² : Fintype ι
inst✝¹ : OrderedAddCommMonoid P
inst✝ : Module R P
Q : (i : ι) → QuadraticMap R (Mᵢ i) P
h : ∀ (i : ι), (∀ (x : Mᵢ i), 0 ≤ (Q i) x) ∧ (Q i).Anis... | exact (h j).1 _ | no goals | 77c7beb36c84b56a |
IsDedekindDomain.selmerGroup.fromUnit_ker | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | theorem fromUnit_ker [hn : Fact <| 0 < n] :
(@fromUnit R _ _ K _ _ _ n).ker = (powMonoidHom n : Rˣ →* Rˣ).range | case h.mk.mp.intro.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
n : ℕ
hn : Fact (0 < n)
val✝ inv✝ : R
val_inv✝ : val✝ * inv✝ = 1
inv_val✝ : inv✝ * val✝ = 1
hx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_... | rw [← map_mul, map_eq_one_iff _ <| FaithfulSMul.algebraMap_injective R K] at iv | case h.mk.mp.intro.mk.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
n : ℕ
hn : Fact (0 < n)
val✝ inv✝ : R
val_inv✝ : val✝ * inv✝ = 1
inv_val✝ : inv✝ * val✝ = 1
hx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_... | 252946ccdda732f7 |
Turing.Tape.move_left_right | Mathlib/Computability/Tape.lean | theorem Tape.move_left_right {Γ} [Inhabited Γ] (T : Tape Γ) :
(T.move Dir.left).move Dir.right = T | Γ : Type u_1
inst✝ : Inhabited Γ
T : Tape Γ
⊢ move Dir.right (move Dir.left T) = T | cases T | case mk
Γ : Type u_1
inst✝ : Inhabited Γ
head✝ : Γ
left✝ right✝ : ListBlank Γ
⊢ move Dir.right (move Dir.left { head := head✝, left := left✝, right := right✝ }) =
{ head := head✝, left := left✝, right := right✝ } | 798d0e8480a2164b |
List.findIdx?_isSome | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
(xs.findIdx? p).isSome = xs.any p | case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : (findIdx? p xs).isSome = xs.any p
⊢ (findIdx? p (x :: xs)).isSome = (x :: xs).any p | simp only [findIdx?_cons] | case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : (findIdx? p xs).isSome = xs.any p
⊢ (if p x = true then some 0 else Option.map (fun i => i + 1) (findIdx? p xs)).isSome = (x :: xs).any p | f5b414e8399ca944 |
Relation.church_rosser | Mathlib/Logic/Relation.lean | theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
(hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c | case tail.intro.intro
α : Type u_1
r : α → α → Prop
a b✝ c : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
this : ∃ a, ReflTransGen r e a ∧ ReflGen r b a
⊢ Join (Re... | rcases this with ⟨a, hea, hba⟩ | case tail.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
a✝¹ b✝ c : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a✝¹ c
d e : α
a✝ : ReflTransGen r a✝¹ d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
a : α
hea : ReflTransGen r e a
hba : Re... | b76c57a616feebb5 |
Cardinal.lift_lt_nat_iff | Mathlib/SetTheory/Cardinal/Basic.lean | theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n | a : Cardinal.{u}
n : ℕ
⊢ lift.{v, u} a < ↑n ↔ a < ↑n | rw [← lift_natCast.{v,u}, lift_lt] | no goals | 8732111f112099cd |
CategoryTheory.FintypeCat.Action.isConnected_of_transitive | Mathlib/CategoryTheory/Galois/Examples.lean | theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
[MulAction.IsPretransitive G X] [h : Nonempty X] :
IsConnected (Action.FintypeCat.ofMulAction G X) where
notInitial := not_initial_of_inhabited (Action.forget _ _) h.some
noTrivialComponent Y i hm hni | G : Type u
inst✝² : Group G
X : FintypeCat
inst✝¹ : MulAction G X.carrier
inst✝ : MulAction.IsPretransitive G X.carrier
h : Nonempty X.carrier
Y : Action FintypeCat G
i : Y ⟶ Action.FintypeCat.ofMulAction G X
hm : Mono i
hni : IsInitial Y → False
y : Y.V.carrier
⊢ IsIso i.hom | refine (ConcreteCategory.isIso_iff_bijective i.hom).mpr ⟨?_, fun x' ↦ ?_⟩ | case refine_1
G : Type u
inst✝² : Group G
X : FintypeCat
inst✝¹ : MulAction G X.carrier
inst✝ : MulAction.IsPretransitive G X.carrier
h : Nonempty X.carrier
Y : Action FintypeCat G
i : Y ⟶ Action.FintypeCat.ofMulAction G X
hm : Mono i
hni : IsInitial Y → False
y : Y.V.carrier
⊢ Function.Injective ⇑(ConcreteCategory.hom... | 598f0ed4a0644baa |
Polynomial.exists_approx_polynomial | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | theorem exists_approx_polynomial {b : Fq[X]} (hb : b ≠ 0) {ε : ℝ} (hε : 0 < ε)
(A : Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊).succ → Fq[X]) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ (cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε | case h.e'_3
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
b : Fq[X]
hb : b ≠ 0
ε : ℝ
hε : 0 < ε
A : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ → Fq[X]
hbε : 0 < cardPowDegree b • ε
one_lt_q : 1 < Fintype.card Fq
one_lt_q' : 1 < ↑(Fintype.card Fq)
q_pos : 0 < Fintype.card Fq
q_pos' : 0 < ↑(Fint... | rw [degree_eq_natDegree h'] | case h.e'_3
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
b : Fq[X]
hb : b ≠ 0
ε : ℝ
hε : 0 < ε
A : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ → Fq[X]
hbε : 0 < cardPowDegree b • ε
one_lt_q : 1 < Fintype.card Fq
one_lt_q' : 1 < ↑(Fintype.card Fq)
q_pos : 0 < Fintype.card Fq
q_pos' : 0 < ↑(Fint... | 0dfa674178256fe6 |
ENNReal.inv_rpow | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem inv_rpow (x : ℝ≥0∞) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ | case inr
x : ℝ≥0∞
y : ℝ
hy : y ≠ 0
⊢ x⁻¹ ^ y = (x ^ y)⁻¹ | replace hy := hy.lt_or_lt | case inr
x : ℝ≥0∞
y : ℝ
hy : y < 0 ∨ 0 < y
⊢ x⁻¹ ^ y = (x ^ y)⁻¹ | b19d8c607ccb3ad9 |
PolynomialModule.aeval_equivPolynomial | Mathlib/Algebra/Polynomial/Module/Basic.lean | @[simp]
lemma aeval_equivPolynomial {S : Type*} [CommRing S] [Algebra S R]
(f : PolynomialModule S S) (x : R) :
aeval x (equivPolynomial f) = eval x (map R (Algebra.linearMap S R) f) | case hadd
R : Type u_1
inst✝² : CommRing R
S : Type u_6
inst✝¹ : CommRing S
inst✝ : Algebra S R
f✝ : PolynomialModule S S
x : R
f g : PolynomialModule S S
e₁ : (aeval x) (equivPolynomial f) = (eval x) ((map R (Algebra.linearMap S R)) f)
e₂ : (aeval x) (equivPolynomial g) = (eval x) ((map R (Algebra.linearMap S R)) g)
⊢... | simp_rw [map_add, e₁, e₂] | no goals | 9bbec61022753076 |
LinearIndependent.repr_eq_single | Mathlib/LinearAlgebra/LinearIndependent/Defs.lean | theorem LinearIndependent.repr_eq_single (i) (x : span R (range v)) (hx : ↑x = v i) :
hv.repr x = Finsupp.single i 1 | ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hv : LinearIndependent R v
i : ι
x : ↥(span R (range v))
hx : ↑x = v i
⊢ (Finsupp.linearCombination R v) (Finsupp.single i 1) = ↑x | simp [Finsupp.linearCombination_single, hx] | no goals | 20149aae7f7911e9 |
Nat.minSqFacAux_has_prop | Mathlib/Data/Nat/Squarefree.lean | theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
(ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) | case inr
n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
n' : ℕ
nd' : n' ∣ n
nk : ¬k ∣ n'
hn' : n' ≤ n
this : n'.sqrt - k < n.sqrt + 2 - k
m : ℕ
m2 : Prime m
d : m ∣ n'
ml : k < m
me : k.succ = m
⊢ False | rw [← me, e] at d | case inr
n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
n' : ℕ
nd' : n' ∣ n
nk : ¬k ∣ n'
hn' : n' ≤ n
this : n'.sqrt - k < n.sqrt + 2 - k
m : ℕ
m2 : Prime m
d : (2 * i + 3).succ ∣ n'
ml : k < m
me : k.succ = m
⊢ False | 6662ac2b7e23ff85 |
UniformFun.tendsto_iff_tendstoUniformly | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | theorem tendsto_iff_tendstoUniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (toFun ∘ F) (toFun f) p | α : Type u_1
β : Type u_2
ι : Type u_4
p : Filter ι
inst✝ : UniformSpace β
F : ι → α →ᵤ β
f : α →ᵤ β
⊢ (∀ i ∈ 𝓤 β, ∀ᶠ (x : ι) in p, F x ∈ {g | (f, g) ∈ UniformFun.gen α β i}) ↔
∀ u ∈ 𝓤 β, ∀ᶠ (n : ι) in p, ∀ (x : α), (toFun f x, (⇑toFun ∘ F) n x) ∈ u | simp only [mem_setOf, UniformFun.gen, Function.comp_def] | no goals | e2c1c54083b2cb04 |
Topology.IsConstructible.preimage | Mathlib/Topology/Constructible.lean | /-- If `f` is continuous and is such that preimages of retrocompact sets are retrocompact, then
preimages of constructible sets are constructible. -/
@[stacks 005I]
lemma IsConstructible.preimage {s : Set Y} (hfcont : Continuous f)
(hf : ∀ s, IsRetrocompact s → IsRetrocompact (f ⁻¹' s)) (hs : IsConstructible s) :
... | case union
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s✝ : Set Y
hfcont : Continuous f
hf : ∀ (s : Set Y), IsRetrocompact s → IsRetrocompact (f ⁻¹' s)
s : Set Y
hs : IsConstructible s
t : Set Y
ht : IsConstructible t
hs' : IsConstructible (f ⁻¹' s)
ht' : IsConstructible (... | rw [preimage_union] | case union
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s✝ : Set Y
hfcont : Continuous f
hf : ∀ (s : Set Y), IsRetrocompact s → IsRetrocompact (f ⁻¹' s)
s : Set Y
hs : IsConstructible s
t : Set Y
ht : IsConstructible t
hs' : IsConstructible (f ⁻¹' s)
ht' : IsConstructible (... | fc0dbcb56e879f3b |
min_eq_iff | Mathlib/Order/MinMax.lean | theorem min_eq_iff : min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a | α : Type u
inst✝ : LinearOrder α
a b c : α
h : a ⊓ b = c
h' : b ≤ a
⊢ b = c | simpa [h'] using h | no goals | 45de0eaec408f730 |
PiNat.mem_cylinder_iff_eq | Mathlib/Topology/MetricSpace/PiNat.lean | theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n | case mp.h₁
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder y n
i : ℕ
hi : i < n
⊢ z i = x i | rw [← hy i hi] | case mp.h₁
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder y n
i : ℕ
hi : i < n
⊢ z i = y i | 1650d6a7263b81d8 |
Set.mem_ite_empty_right | Mathlib/Data/Set/Basic.lean | theorem mem_ite_empty_right (p : Prop) [Decidable p] (t : Set α) (x : α) :
x ∈ ite p t ∅ ↔ p ∧ x ∈ t :=
(mem_dite_empty_right p (fun _ => t) x).trans (by simp)
| α : Type u
p : Prop
inst✝ : Decidable p
t : Set α
x : α
⊢ (∃ (_ : p), x ∈ t) ↔ p ∧ x ∈ t | simp | no goals | 9b2f6e8db686c2f7 |
CondensedMod.isDiscrete_tfae | Mathlib/Condensed/Discrete/Characterization.lean | theorem isDiscrete_tfae (M : CondensedMod.{u} R) :
TFAE
[ M.IsDiscrete
, IsIso ((Condensed.discreteUnderlyingAdj _).counit.app M)
, (Condensed.discrete _).essImage M
, (CondensedMod.LocallyConstant.functor R).essImage M
, IsIso ((CondensedMod.LocallyConstant.adjunction R).counit.app M)
, Sh... | R : Type (u + 1)
inst✝ : Ring R
M : CondensedMod R
tfae_1_iff_2 : Condensed.IsDiscrete M ↔ IsIso ((Condensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
⊢ [Condensed.IsDiscrete M, IsIso ((Condensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
(Condensed.discrete (ModuleCat R)).essImage M, (functor R... | tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩ | R : Type (u + 1)
inst✝ : Ring R
M : CondensedMod R
tfae_1_iff_2 : Condensed.IsDiscrete M ↔ IsIso ((Condensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : Condensed.IsDiscrete M ↔ (Condensed.discrete (ModuleCat R)).essImage M
⊢ [Condensed.IsDiscrete M, IsIso ((Condensed.discreteUnderlyingAdj (Module... | 6b931ea1420905ba |
map_prime_of_factor_orderIso | Mathlib/RingTheory/ChainOfDivisors.lean | theorem map_prime_of_factor_orderIso {m p : Associates M} {n : Associates N} (hn : n ≠ 0)
(hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) :
Prime (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : Associates N) | M : Type u_1
inst✝³ : CancelCommMonoidWithZero M
N : Type u_2
inst✝² : CancelCommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ Prime ↑(d ⟨p, ⋯⟩) | rw [← irreducible_iff_prime] | M : Type u_1
inst✝³ : CancelCommMonoidWithZero M
N : Type u_2
inst✝² : CancelCommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ Irreducible ↑(d ⟨p, ⋯⟩) | b9f5949373dcb4ed |
AlgebraicGeometry.Scheme.Hom.ker_apply | Mathlib/AlgebraicGeometry/IdealSheaf.lean | @[simp]
lemma Hom.ker_apply (f : X.Hom Y) [QuasiCompact f] (U : Y.affineOpens) :
f.ker.ideal U = RingHom.ker (f.app U).hom | X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
U✝ U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
this : IsLocalization.Away s ↑Γ(Y, Y.basicOpen s)
x : ↑Γ(Y, ↑U)
n : ℕ
hx :
IsLocalization.mk' (↑Γ(Y, Y.basicOpen s)) x ⟨(fun x => s ^ x) n, ⋯⟩ ∈
(fun U => RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) (Y.affineBasicOpen s)
⊢ f ⁻¹ᵁ... | simp | no goals | 443ed8a730592383 |
topologicalClosure_subgroupClosure_toSubmonoid | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | theorem topologicalClosure_subgroupClosure_toSubmonoid (s : Set G) :
(Subgroup.closure s).toSubmonoid.topologicalClosure =
(Submonoid.closure s).topologicalClosure | G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
s : Set G
⊢ (Subgroup.closure s).topologicalClosure ≤ (Submonoid.closure s).topologicalClosure | refine Submonoid.topologicalClosure_minimal _ ?_ isClosed_closure | G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
s : Set G
⊢ (Subgroup.closure s).toSubmonoid ≤ (Submonoid.closure s).topologicalClosure | fccf0f4d21f9338e |
CategoryTheory.Limits.HasZeroMorphisms.ext_aux | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | theorem ext_aux (I J : HasZeroMorphisms C)
(w : ∀ X Y : C, (I.zero X Y).zero = (J.zero X Y).zero) : I = J | case mk.mk.h.e_5
C : Type u
inst✝ : Category.{v, u} C
zero✝¹ : (X Y : C) → Zero (X ⟶ Y)
comp_zero✝¹ : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ 0 = 0
zero_comp✝¹ : ∀ (X : C) {Y Z : C} (f : Y ⟶ Z), 0 ≫ f = 0
zero✝ : (X Y : C) → Zero (X ⟶ Y)
comp_zero✝ : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ 0 = 0
zero_comp✝ : ∀ (X : C) {Y Z :... | apply proof_irrel_heq | no goals | e7a423b46e9eb497 |
MvPolynomial.schwartz_zippel_sup_sum | Mathlib/Algebra/MvPolynomial/SchwartzZippel.lean | /-- The **Schwartz-Zippel lemma**
For a nonzero multivariable polynomial `p` over an integral domain, the probability that `p`
evaluates to zero at points drawn at random from a product of finite subsets `S i` of the integral
domain is bounded by the supremum of `∑ i, degᵢ s / #(S i)` ranging over monomials `s` of `p`... | case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (F... | exact filter_subset .. | no goals | 0aa93d268fe4b9d8 |
SnakeLemma.exact_δ_left | Mathlib/Algebra/Module/SnakeLemma.lean | /--
Suppose we have an exact commutative diagram
```
K₃
|
ι₃
↓
M₁ -f₁→ M₂ -f₂→ M₃
| | |
i₁ i₂ i₃
↓ ↓ ↓
N₁ -g₁→ N₂ -g₂→ N₃
| |
π₁ π₂
↓ ↓
C₁ -G-→ C₂
```
such that `f₂` is surjective with a (set-theoretic) s... | R : Type u_1
inst✝¹⁸ : CommRing R
M₁ : Type u_7
M₂ : Type u_8
M₃ : Type u_9
N₁ : Type u_4
N₂ : Type u_5
N₃ : Type u_10
inst✝¹⁷ : AddCommGroup M₁
inst✝¹⁶ : Module R M₁
inst✝¹⁵ : AddCommGroup M₂
inst✝¹⁴ : Module R M₂
inst✝¹³ : AddCommGroup M₃
inst✝¹² : Module R M₃
inst✝¹¹ : AddCommGroup N₁
inst✝¹⁰ : Module R N₁
inst✝⁹ : ... | haveI H₁ : ∀ x, f₂ (σ x) = x := congr_fun hσ | R : Type u_1
inst✝¹⁸ : CommRing R
M₁ : Type u_7
M₂ : Type u_8
M₃ : Type u_9
N₁ : Type u_4
N₂ : Type u_5
N₃ : Type u_10
inst✝¹⁷ : AddCommGroup M₁
inst✝¹⁶ : Module R M₁
inst✝¹⁵ : AddCommGroup M₂
inst✝¹⁴ : Module R M₂
inst✝¹³ : AddCommGroup M₃
inst✝¹² : Module R M₃
inst✝¹¹ : AddCommGroup N₁
inst✝¹⁰ : Module R N₁
inst✝⁹ : ... | 003c564d5000ab77 |
Real.sInf_smul_of_nonneg | Mathlib/Data/Real/Pointwise.lean | theorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s | case inr.inl
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sInf (0 • s) = 0 • sInf s | rw [zero_smul_set hs, zero_smul] | case inr.inl
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sInf 0 = 0 | e87faa52ebef3829 |
Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem ae_tendsto_measure_inter_div_of_measurableSet (μ : Measure β) [IsLocallyFiniteMeasure μ]
{s : Set β} (hs : MeasurableSet s) :
∀ᵐ x ∂μ,
Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0)
(𝓝 (s.indicator 1 x)) | β : Type u
inst✝⁵ : MetricSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : HasBesicovitchCovering β
μ : Measure β
inst✝ : IsLocallyFiniteMeasure μ
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) ... | filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet
(Besicovitch.vitaliFamily μ) hs] | case h
β : Type u
inst✝⁵ : MetricSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : HasBesicovitchCovering β
μ : Measure β
inst✝ : IsLocallyFiniteMeasure μ
s : Set β
hs : MeasurableSet s
⊢ ∀ (a : β),
Tendsto (fun a => μ (s ∩ a) / μ a) ((Besicovitch.vitaliFamily μ).f... | b2265b4585ab8741 |
Filter.map₂_sup_right | Mathlib/Order/Filter/NAry.lean | theorem map₂_sup_right : map₂ m f (g₁ ⊔ g₂) = map₂ m f g₁ ⊔ map₂ m f g₂ | α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g₁ g₂ : Filter β
⊢ map₂ m f (g₁ ⊔ g₂) = map₂ m f g₁ ⊔ map₂ m f g₂ | simp_rw [← map_prod_eq_map₂, prod_sup, map_sup] | no goals | aaa94fa941e0727d |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | case neg
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < as... | rcases Nat.lt_or_eq_of_le <| Nat.le_of_lt_succ k_property with k_lt_units_size | k_eq_units_size | case neg.inl
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val ... | 4e9908103f3f6d5f |
Polynomial.IsWeaklyEisensteinAt.map | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | theorem map (hf : f.IsWeaklyEisensteinAt 𝓟) {A : Type v} [CommRing A] (φ : R →+* A) :
(f.map φ).IsWeaklyEisensteinAt (𝓟.map φ) | R : Type u
inst✝¹ : CommSemiring R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
A : Type v
inst✝ : CommRing A
φ : R →+* A
n✝ : ℕ
hn : n✝ < (Polynomial.map φ f).natDegree
⊢ φ (f.coeff n✝) ∈ Ideal.map φ 𝓟 | exact mem_map_of_mem _ (hf.mem (lt_of_lt_of_le hn natDegree_map_le)) | no goals | aa9cfbf730e9ec74 |
MvQPF.liftR_map | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | theorem liftR_map {α β : TypeVec n} {F' : TypeVec n → Type u} [MvFunctor F'] [LawfulMvFunctor F']
(R : β ⊗ β ⟹ «repeat» n Prop) (x : F' α) (f g : α ⟹ β) (h : α ⟹ Subtype_ R)
(hh : subtypeVal _ ⊚ h = (f ⊗' g) ⊚ prod.diag) : LiftR' R (f <$$> x) (g <$$> x) | n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ «repeat» n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ (f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (g ⊚ TypeVec.id) <$$> x = g <$$> x | dsimp [LiftR'] | n : ℕ
α β : TypeVec.{u_1} n
F' : TypeVec.{u_1} n → Type u
inst✝¹ : MvFunctor F'
inst✝ : LawfulMvFunctor F'
R : β ⊗ β ⟹ «repeat» n Prop
x : F' α
f g : α ⟹ β
h : α ⟹ Subtype_ R
hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag
⊢ f <$$> x = f <$$> x ∧ g <$$> x = g <$$> x | 69a7a718513674ac |
RCLike.tendsto_add_mul_div_add_mul_atTop_nhds | Mathlib/Analysis/SpecificLimits/RCLike.lean | theorem RCLike.tendsto_add_mul_div_add_mul_atTop_nhds (a b c : 𝕜) {d : 𝕜} (hd : d ≠ 0) :
Tendsto (fun k : ℕ ↦ (a + c * k) / (b + d * k)) atTop (𝓝 (c / d)) | 𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ Tendsto (fun k => (↑k)⁻¹) atTop (𝓝 0) | exact RCLike.tendsto_inverse_atTop_nhds_zero_nat 𝕜 | no goals | 6d8178cde5b196f7 |
CochainComplex.mappingCone.inl_snd | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | @[simp]
lemma inl_snd :
(inl φ).comp (snd φ) (add_zero (-1)) = 0 | case h
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
p q : ℤ
hpq : p + -1 = q
⊢ ((inl φ).comp (snd φ) ⋯).v p q hpq = Cochain.v 0 p q hpq | simp [Cochain.comp_v _ _ (add_zero (-1)) p q q (by omega) (by omega)] | no goals | 44f1f38cc4972470 |
ZMod.unitsMap_comp | Mathlib/Data/ZMod/Units.lean | lemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :
(unitsMap hm).comp (unitsMap hd) = unitsMap (dvd_trans hm hd) | n m d : ℕ
hm : n ∣ m
hd : m ∣ d
⊢ Units.map ((↑(castHom hm (ZMod n))).comp ↑(castHom hd (ZMod m))) = Units.map ↑(castHom ⋯ (ZMod n)) | exact congr_arg Units.map <| congr_arg RingHom.toMonoidHom <| castHom_comp hm hd | no goals | da72c543c0a8baaf |
AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop | Mathlib/AlgebraicGeometry/AffineScheme.lean | lemma isoSpec_inv_appTop :
hU.isoSpec.inv.appTop = U.topIso.hom ≫ (Scheme.ΓSpecIso Γ(X, U)).inv | X : Scheme
U : X.Opens
hU : IsAffineOpen U
⊢ Scheme.Hom.app (Spec.map (X.presheaf.map (eqToHom ⋯).op)) (inv (↑U).toSpecΓ ⁻¹ᵁ ⊤) =
Scheme.Hom.appTop (Spec.map (X.presheaf.map (eqToHom ⋯).op)) | simp only [Opens.map_top] | no goals | 973fd42a0973af38 |
SimpleGraph.Iso.card_edgeFinset_eq | Mathlib/Combinatorics/SimpleGraph/Operations.lean | theorem card_edgeFinset_eq [Fintype G.edgeSet] [Fintype G'.edgeSet] :
#G.edgeFinset = #G'.edgeFinset | V : Type u_1
G : SimpleGraph V
W : Type u_2
G' : SimpleGraph W
f : G ≃g G'
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype ↑G'.edgeSet
⊢ #G.edgeFinset = #G'.edgeFinset | apply Finset.card_eq_of_equiv | case i
V : Type u_1
G : SimpleGraph V
W : Type u_2
G' : SimpleGraph W
f : G ≃g G'
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype ↑G'.edgeSet
⊢ { x // x ∈ G.edgeFinset } ≃ { x // x ∈ G'.edgeFinset } | 85bae14cfc5d0190 |
CFC.negPart_mul_posPart | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean | @[simp]
lemma negPart_mul_posPart (a : A) : a⁻ * a⁺ = 0 | case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ x⁻ * x⁺ = 0 x | simp only [_root_.posPart_def, _root_.negPart_def] | case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ (-x ⊔ 0) * (x ⊔ 0) = 0 ... | ef2285912fb9ffa3 |
Submodule.mem_span_finite_of_mem_span | Mathlib/LinearAlgebra/Span/Defs.lean | theorem mem_span_finite_of_mem_span {S : Set M} {x : M} (hx : x ∈ span R S) :
∃ T : Finset M, ↑T ⊆ S ∧ x ∈ span R (T : Set M) | case refine_3.intro.intro.intro.intro
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set M
x✝ : M
hx : x✝ ∈ span R S
x y : M
X : Finset M
hX : ↑X ⊆ S
hxX : x ∈ span R ↑X
Y : Finset M
hY : ↑Y ⊆ S
hyY : y ∈ span R ↑Y
⊢ ∃ T, ↑T ⊆ S ∧ x + y ∈ span R ↑T | refine ⟨X ∪ Y, ?_, ?_⟩ | case refine_3.intro.intro.intro.intro.refine_1
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set M
x✝ : M
hx : x✝ ∈ span R S
x y : M
X : Finset M
hX : ↑X ⊆ S
hxX : x ∈ span R ↑X
Y : Finset M
hY : ↑Y ⊆ S
hyY : y ∈ span R ↑Y
⊢ ↑(X ∪ Y) ⊆ S
case refine_3.intro.intro.intro.i... | 56bd86754d77402c |
covariant_le_of_covariant_lt | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | theorem covariant_le_of_covariant_lt [PartialOrder N] :
Covariant M N μ (· < ·) → Covariant M N μ (· ≤ ·) | M : Type u_1
N : Type u_2
μ : M → N → N
inst✝ : PartialOrder N
⊢ (Covariant M N μ fun x1 x2 => x1 < x2) → Covariant M N μ fun x1 x2 => x1 ≤ x2 | intro h a b c bc | M : Type u_1
N : Type u_2
μ : M → N → N
inst✝ : PartialOrder N
h : Covariant M N μ fun x1 x2 => x1 < x2
a : M
b c : N
bc : b ≤ c
⊢ μ a b ≤ μ a c | d2d038fc353a5ff4 |
Matrix.Pivot.isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow
(hM : M (inr unit) (inr unit) ≠ 0) :
IsTwoBlockDiagonal ((listTransvecCol M).prod * M * (listTransvecRow M).prod) | case right.a
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Unit
j : Fin r
⊢ ((listTransvecCol M).prod * M * (listTransvecRow M).prod).toBlocks₂₁ i j = 0 i j | have : i = unit := by simp only [eq_iff_true_of_subsingleton] | case right.a
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Unit
j : Fin r
this : i = ()
⊢ ((listTransvecCol M).prod * M * (listTransvecRow M).prod).toBlocks₂₁ i j = 0 i j | 82335502dab8b550 |
cfc_comp | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean | lemma cfc_comp (g f : R → R) (a : A) (ha : p a | R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
g f :... | have := hg.comp hf <| (spectrum R a).mapsTo_image f | R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
g f :... | c2e8482f645781c9 |
Set.mulIndicator_iInter_apply | Mathlib/Algebra/Order/Group/Indicator.lean | @[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x | ι : Sort u_1
α : Type u_2
M : Type u_3
inst✝² : CompleteLattice M
inst✝¹ : One M
inst✝ : Nonempty ι
h1 : ⊥ = 1
s : ι → Set α
f : α → M
x : α
⊢ (⋂ i, s i).mulIndicator f x = ⨅ i, (s i).mulIndicator f x | by_cases hx : x ∈ ⋂ i, s i | case pos
ι : Sort u_1
α : Type u_2
M : Type u_3
inst✝² : CompleteLattice M
inst✝¹ : One M
inst✝ : Nonempty ι
h1 : ⊥ = 1
s : ι → Set α
f : α → M
x : α
hx : x ∈ ⋂ i, s i
⊢ (⋂ i, s i).mulIndicator f x = ⨅ i, (s i).mulIndicator f x
case neg
ι : Sort u_1
α : Type u_2
M : Type u_3
inst✝² : CompleteLattice M
inst✝¹ : One M
i... | 1f86a6e9fd525ad5 |
Finset.supIndep_product_iff | Mathlib/Order/SupIndep.lean | theorem supIndep_product_iff {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} :
(s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i'))
∧ t.SupIndep fun i' => s.sup fun i => f (i, i') | case refine_1
α : Type u_1
ι : Type u_3
ι' : Type u_4
inst✝¹ : DistribLattice α
inst✝ : OrderBot α
s : Finset ι
t : Finset ι'
f : ι × ι' → α
h : (↑(s.product t)).PairwiseDisjoint f
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
hij : i ≠ j
i' : ι'
hi' : i' ∈ t
j' : ι'
hj' : j' ∈ t
⊢ Disjoint (f (i, i')) (f (j, j')) | exact h (mk_mem_product hi hi') (mk_mem_product hj hj') (ne_of_apply_ne Prod.fst hij) | no goals | 0ebb2b0ffa7364cc |
thickening_thickening | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem thickening_thickening (hε : 0 < ε) (hδ : 0 < δ) (s : Set E) :
thickening ε (thickening δ s) = thickening (ε + δ) s :=
(thickening_thickening_subset _ _ _).antisymm fun x => by
simp_rw [mem_thickening_iff]
rintro ⟨z, hz, hxz⟩
rw [add_comm] at hxz
obtain ⟨y, hxy, hyz⟩ := exists_dist_lt_lt hε... | case intro.intro
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 < ε
hδ : 0 < δ
s : Set E
x z : E
hz : z ∈ s
hxz : dist x z < δ + ε
⊢ ∃ z, (∃ z_1 ∈ s, dist z z_1 < δ) ∧ dist x z < ε | obtain ⟨y, hxy, hyz⟩ := exists_dist_lt_lt hε hδ hxz | case intro.intro.intro.intro
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 < ε
hδ : 0 < δ
s : Set E
x z : E
hz : z ∈ s
hxz : dist x z < δ + ε
y : E
hxy : dist x y < ε
hyz : dist y z < δ
⊢ ∃ z, (∃ z_1 ∈ s, dist z z_1 < δ) ∧ dist x z < ε | 9e14a4b557febbe2 |
SimplexCategory.len_le_of_mono | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem len_le_of_mono {x y : SimplexCategory} {f : x ⟶ y} : Mono f → x.len ≤ y.len | x y : SimplexCategory
f : x ⟶ y
hyp_f_mono : Mono f
⊢ x.len ≤ y.len | have f_inj : Function.Injective f.toOrderHom.toFun := mono_iff_injective.1 hyp_f_mono | x y : SimplexCategory
f : x ⟶ y
hyp_f_mono : Mono f
f_inj : Function.Injective (Hom.toOrderHom f).toFun
⊢ x.len ≤ y.len | 7453314deb953b9d |
Finsupp.range_linearCombination | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem range_linearCombination : LinearMap.range (linearCombination R v) = span R (range v) | case h.mpr.a
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
x : M
⊢ Set.range v ⊆ ↑(LinearMap.range (linearCombination R v)) | intro x hx | case h.mpr.a
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
x✝ x : M
hx : x ∈ Set.range v
⊢ x ∈ ↑(LinearMap.range (linearCombination R v)) | 28c3b4ab3746ce9c |
Topology.IsScott.scottContinuous_iff_continuous | Mathlib/Topology/Order/ScottTopology.lean | @[simp] lemma scottContinuous_iff_continuous {D : Set (Set α)} [Topology.IsScott α D]
(hD : ∀ a b : α, a ≤ b → {a, b} ∈ D) : ScottContinuousOn D f ↔ Continuous f | case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
hf : Continuous f
t : Set α
h₀ : t ∈ D
d₁ : t.Nonempty
d₂ : DirectedOn (fun x... | by_contra h | case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
hf : Continuous f
t : Set α
h₀ : t ∈ D
d₁ : t.Nonempty
d₂ : DirectedOn (fun x... | ca207f5b40605d41 |
iteratedFDerivWithin_neg_apply | Mathlib/Analysis/Calculus/ContDiff/Operations.lean | theorem iteratedFDerivWithin_neg_apply {f : E → F} (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x | case zero.H
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
x✝ : Fin 0 → E
⊢ (iteratedFDerivWithin 𝕜 0 (-f) s x) x✝ = (-it... | simp | no goals | bb20b634e45c45b1 |
List.dropInfix?_eq_some_iff | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem dropInfix?_eq_some_iff [BEq α] {l i p s : List α} :
dropInfix? l i = some (p, s) ↔
-- `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 = p' ++ i' ++ s' → i' == i → p'.length ≥ p.length) | α : Type u_1
inst✝ : BEq α
l i p s : List α
⊢ dropInfix?.go i l [] = some (p, s) ↔
(∃ i', l = p ++ i' ++ s ∧ (i' == i) = true) ∧
∀ (p' i' s' : List α), l = p' ++ i' ++ s' → (i' == i) = true → p'.length ≥ p.length | rw [dropInfix?_go_eq_some_iff] | α : Type u_1
inst✝ : BEq α
l i p s : List α
⊢ (∃ p',
p = [].reverse ++ p' ∧
(∃ i', l = p' ++ i' ++ s ∧ (i' == i) = true) ∧
∀ (p'' i'' s'' : List α), l = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length) ↔
(∃ i', l = p ++ i' ++ s ∧ (i' == i) = true) ∧
∀ (p' i' s' : List α)... | 670e84ee38da8bdb |
List.map_inj | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem map_inj : map f = map g ↔ f = g | case mpr
α✝¹ : Type u_1
α✝ : Type u_2
f g : α✝¹ → α✝
h : f = g
⊢ map f = map g | subst h | case mpr
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → α✝
⊢ map f = map f | 707d036b1fa4a9fd |
UniformConvergenceCLM.topologicalSpace_mono | Mathlib/Topology/Algebra/Module/StrongTopology.lean | theorem topologicalSpace_mono [TopologicalSpace F] [IsTopologicalAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂ | 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGrou... | haveI : UniformAddGroup F := uniformAddGroup_of_addCommGroup | 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGrou... | 798e9de145e94e05 |
Liouville.exists_pos_real_of_irrational_root | Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean | theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
(fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ,
(1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|α - a / (b + 1)| * A) | α : ℝ
ha : Irrational α
f : ℤ[X]
f0 : f ≠ 0
fR : ℝ[X] := map (algebraMap ℤ ℝ) f
fa : eval α fR = 0
⊢ ∃ A, 0 < A ∧ ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|α - ↑a / (↑b + 1)| * A) | obtain fR0 : fR ≠ 0 := fun fR0 =>
(map_injective (algebraMap ℤ ℝ) fun _ _ A => Int.cast_inj.mp A).ne f0
(fR0.trans (Polynomial.map_zero _).symm) | α : ℝ
ha : Irrational α
f : ℤ[X]
f0 : f ≠ 0
fR : ℝ[X] := map (algebraMap ℤ ℝ) f
fa : eval α fR = 0
fR0 : fR ≠ 0
⊢ ∃ A, 0 < A ∧ ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|α - ↑a / (↑b + 1)| * A) | 438cd78b42993df9 |
BitVec.ushiftRightRec_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem ushiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
ushiftRightRec x y n = x >>> (y.setWidth (n + 1)).setWidth w₂ | w₁ w₂ n : Nat
ih : ∀ (x : BitVec w₁) (y : BitVec w₂), x.ushiftRightRec y n = x >>> setWidth w₂ (setWidth (n + 1) y)
x : BitVec w₁
y : BitVec w₂
⊢ (x.ushiftRightRec y n >>> if y.getLsbD (n + 1) = true then twoPow w₂ (n + 1) else 0#w₂) =
x >>> setWidth w₂ (setWidth (n + 1 + 1) y) | rw [ih] | w₁ w₂ n : Nat
ih : ∀ (x : BitVec w₁) (y : BitVec w₂), x.ushiftRightRec y n = x >>> setWidth w₂ (setWidth (n + 1) y)
x : BitVec w₁
y : BitVec w₂
⊢ (x >>> setWidth w₂ (setWidth (n + 1) y) >>> if y.getLsbD (n + 1) = true then twoPow w₂ (n + 1) else 0#w₂) =
x >>> setWidth w₂ (setWidth (n + 1 + 1) y) | c716927de5030eb6 |
lemma₁ | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem lemma₁ (hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l))
{s : ℝ} (hs : 1 < s) :
IntegrableOn (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * (t : ℂ) ^ (-(s : ℂ) - 1)) (Set.Ici 1) | case refine_1
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
s : ℝ
hs : 1 < s
h₁ : LocallyIntegrableOn (fun t => (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume
h₂ : (fun t => ∑ k ∈ Icc 1 ⌊t⌋₊, f k) =O[atTop] fun t => t ^ 1
⊢ (fun t => ↑t ^ (-↑s - 1)) =O[atTop] fun t => t ... | exact (norm_ofReal_cpow_eventually_eq_atTop _).isBigO.of_norm_left | no goals | 44984ad91259849f |
GenLoop.homotopicTo | Mathlib/Topology/Homotopy/HomotopyGroup.lean | theorem homotopicTo (i : N) {p q : Ω^ N X x} :
Homotopic p q → (toLoop i p).Homotopic (toLoop i q) | case refine_4
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i : N
p q : ↑(Ω^ N X x)
H : (↑p).HomotopyRel (↑q) (Cube.boundary N)
⊢ ∀ (x_1 : ↑I),
{ toFun := fun t => ⟨(homotopyTo i H) t, ⋯⟩, continuous_toFun := ⋯ }.toFun (1, x_1) =
(toLoop i q).toContinuousMap x_1
case refin... | intro | case refine_4
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i : N
p q : ↑(Ω^ N X x)
H : (↑p).HomotopyRel (↑q) (Cube.boundary N)
x✝ : ↑I
⊢ { toFun := fun t => ⟨(homotopyTo i H) t, ⋯⟩, continuous_toFun := ⋯ }.toFun (1, x✝) = (toLoop i q).toContinuousMap x✝
case refine_3.H
N : Type u_1... | 5948809eedd27d88 |
List.filter_eq_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filter_eq_cons_iff {l} {a} {as} :
filter p l = a :: as ↔
∃ l₁ l₂, l = l₁ ++ a :: l₂ ∧ (∀ x, x ∈ l₁ → ¬p x) ∧ p a ∧ filter p l₂ = as | case mp.cons
α✝ : Type u_1
p : α✝ → Bool
a : α✝
as : List α✝
x : α✝
l : List α✝
ih :
filter p l = a :: as → ∃ l₁ l₂, l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝), x ∈ l₁ → ¬p x = true) ∧ p a = true ∧ filter p l₂ = as
h : (if p x = true then x :: filter p l else filter p l) = a :: as
⊢ ∃ l₁ l₂, x :: l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝)... | split at h <;> rename_i w | case mp.cons.isTrue
α✝ : Type u_1
p : α✝ → Bool
a : α✝
as : List α✝
x : α✝
l : List α✝
ih :
filter p l = a :: as → ∃ l₁ l₂, l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝), x ∈ l₁ → ¬p x = true) ∧ p a = true ∧ filter p l₂ = as
w : p x = true
h : x :: filter p l = a :: as
⊢ ∃ l₁ l₂, x :: l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝), x ∈ l₁ → ¬p x... | 40c82d08e6c727c5 |
LieAlgebra.IsSemisimple.isSimple_of_isAtom | Mathlib/Algebra/Lie/Semisimple/Basic.lean | lemma isSimple_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : IsSimple R I where
non_abelian := IsSemisimple.non_abelian_of_isAtom I hI
eq_bot_or_eq_top | case a.right
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
J' : LieIdeal R L :=
let __spread.0 := Submodule.map ↑I.incl ↑J;
{ toSubmodule := __spread.0, lie_mem := ⋯ }
hJ : J' = I
⊢ ⊤ ≤ J | rintro ⟨x, hx⟩ - | case a.right.mk
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
J' : LieIdeal R L :=
let __spread.0 := Submodule.map ↑I.incl ↑J;
{ toSubmodule := __spread.0, lie_mem := ⋯ }
hJ : J' = I
x : L
hx : x ∈ I... | 94384633946b96a7 |
tendsto_prod_nat_add | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | theorem tendsto_prod_nat_add [T2Space G] (f : ℕ → G) :
Tendsto (fun i ↦ ∏' k, f (k + i)) atTop (𝓝 1) | case neg
G : Type u_2
inst✝³ : CommGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalGroup G
inst✝ : T2Space G
f : ℕ → G
hf : ¬Multipliable f
n : ℕ
⊢ ¬Multipliable fun k => f (k + n) | rwa [multipliable_nat_add_iff n] | no goals | 33f63157f8fa4b6a |
ConvexOn.locallyLipschitzOn_iff_continuousOn | Mathlib/Analysis/Convex/Continuous.lean | lemma ConvexOn.locallyLipschitzOn_iff_continuousOn (hC : IsOpen C) (hf : ConvexOn ℝ C f) :
LocallyLipschitzOn C f ↔ ContinuousOn f C | case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
hC : IsOpen ∅
hf : ConvexOn ℝ ∅ f
⊢ LocallyLipschitzOn ∅ f ↔ ContinuousOn f ∅ | simp | no goals | 5e4fbafa12420a3c |
Finpartition.card_filter_equitabilise_small | Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | theorem card_filter_equitabilise_small (hm : m ≠ 0) :
#{u ∈ (P.equitabilise h).parts | #u = m} = a | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h✝ : a * m + b * (m + 1) = #s
hm : m ≠ 0
hunion :
(equitabilise h✝).parts =
filter (fun u => #u = m) (equitabilise h✝).parts ∪ filter (fun u => #u = m + 1) (equitabilise h✝).parts
x : Finset α → Prop
ha : x ≤ fun u => #u = m
hb : x ≤ fun... | apply succ_ne_self m _ | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h✝ : a * m + b * (m + 1) = #s
hm : m ≠ 0
hunion :
(equitabilise h✝).parts =
filter (fun u => #u = m) (equitabilise h✝).parts ∪ filter (fun u => #u = m + 1) (equitabilise h✝).parts
x : Finset α → Prop
ha : x ≤ fun u => #u = m
hb : x ≤ fun... | 6764f0dc61dac4e1 |
SimpleGraph.IsAlternating.spanningCoe | Mathlib/Combinatorics/SimpleGraph/Matching.lean | lemma IsAlternating.spanningCoe (halt : G.IsAlternating G') (H : Subgraph G) :
H.spanningCoe.IsAlternating G' | V : Type u_1
G G' : SimpleGraph V
halt : G.IsAlternating G'
H : G.Subgraph
⊢ H.spanningCoe.IsAlternating G' | intro v w w' hww' hvw hvv' | V : Type u_1
G G' : SimpleGraph V
halt : G.IsAlternating G'
H : G.Subgraph
v w w' : V
hww' : w ≠ w'
hvw : H.spanningCoe.Adj v w
hvv' : H.spanningCoe.Adj v w'
⊢ G'.Adj v w ↔ ¬G'.Adj v w' | ed3a8bfe083f2513 |
Algebra.Extension.Cotangent.map_sub_map | Mathlib/RingTheory/Kaehler/CotangentComplex.lean | lemma Cotangent.map_sub_map (f g : Hom P P') :
map f - map g = (f.sub g) ∘ₗ P.cotangentComplex | case h.e.intro.a.a
R : Type u
S : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
P : Extension R S
R' : Type u'
S' : Type v'
inst✝⁷ : CommRing R'
inst✝⁶ : CommRing S'
inst✝⁵ : Algebra R' S'
P' : Extension R' S'
inst✝⁴ : Algebra R R'
inst✝³ : Algebra S S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTowe... | simp only [val_sub, val_mk, map_sub, AddSubgroupClass.coe_sub, Ideal.cotangentEquivIdeal_apply,
Ideal.toCotangent_to_quotient_square, Submodule.mkQ_apply, Ideal.Quotient.mk_eq_mk,
Hom.subToKer_apply_coe] | case h.e.intro.a.a
R : Type u
S : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
P : Extension R S
R' : Type u'
S' : Type v'
inst✝⁷ : CommRing R'
inst✝⁶ : CommRing S'
inst✝⁵ : Algebra R' S'
P' : Extension R' S'
inst✝⁴ : Algebra R R'
inst✝³ : Algebra S S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTowe... | 0b659ae456a50538 |
Real.sqrt_eq_zero | Mathlib/Data/Real/Sqrt.lean | theorem sqrt_eq_zero (h : 0 ≤ x) : √x = 0 ↔ x = 0 | x : ℝ
h : 0 ≤ x
⊢ √x = 0 ↔ x = 0 | simpa using sqrt_inj h le_rfl | no goals | e4ec7c482c0284fd |
MeasureTheory.measurableSet_generateFrom_singleton_iff | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ | case neg
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
⊢ (fun x => x ∈ s) ⁻¹' x = ∅ ∨ (fun x => x ∈ s) ⁻¹' x = s ∨ (fun x => x ∈ s) ⁻¹' x = sᶜ ∨ (fun x => x ∈ s) ⁻¹' x = univ | have hx : x = {True} := by
ext p
refine ⟨fun hp ↦ mem_singleton_iff.2 ?_, fun hp ↦ hp ▸ hT⟩
by_contra hpneg
rw [eq_iff_iff, iff_true, ← false_iff] at hpneg
exact hF (by convert hp) | case neg
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
hx : x = {True}
⊢ (fun x => x ∈ s) ⁻¹' x = ∅ ∨ (fun x => x ∈ s) ⁻¹' x = s ∨ (fun x => x ∈ s) ⁻¹' x = sᶜ ∨ (fun x => x ∈ s) ⁻¹' x = univ | 39127bdfcef76ebf |
div_le_one_of_neg | Mathlib/Algebra/Order/Field/Basic.lean | theorem div_le_one_of_neg (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a | α : Type u_2
inst✝ : LinearOrderedField α
a b : α
hb : b < 0
⊢ a / b ≤ 1 ↔ b ≤ a | rw [div_le_iff_of_neg hb, one_mul] | no goals | 30b01843a216b560 |
Module.End.genEigenspace_inf_le_add | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma genEigenspace_inf_le_add
(f₁ f₂ : End R M) (μ₁ μ₂ : R) (k₁ k₂ : ℕ∞) (h : Commute f₁ f₂) :
(f₁.genEigenspace μ₁ k₁) ⊓ (f₂.genEigenspace μ₂ k₂) ≤
(f₁ + f₂).genEigenspace (μ₁ + μ₂) (k₁ + k₂) | case h.right
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f₁ f₂ : End R M
μ₁ μ₂ : R
k₁ k₂ : ℕ∞
m : M
l₁ : ℕ
hlk₁ : ↑l₁ ≤ k₁
hl₁ : ((f₁ - μ₁ • 1) ^ l₁) m = 0
l₂ : ℕ
hlk₂ : ↑l₂ ≤ k₂
hl₂ : ((f₂ - μ₂ • 1) ^ l₂) m = 0
this : f₁ + f₂ - (μ₁ + μ₂) • 1 = f₁ - μ₁ • 1 + (f₂ - μ₂ • 1)
h : Co... | suffices (((f₁ - μ₁ • 1) ^ i) * ((f₂ - μ₂ • 1) ^ j)) m = 0 by
rw [LinearMap.smul_apply, this, smul_zero] | case h.right
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f₁ f₂ : End R M
μ₁ μ₂ : R
k₁ k₂ : ℕ∞
m : M
l₁ : ℕ
hlk₁ : ↑l₁ ≤ k₁
hl₁ : ((f₁ - μ₁ • 1) ^ l₁) m = 0
l₂ : ℕ
hlk₂ : ↑l₂ ≤ k₂
hl₂ : ((f₂ - μ₂ • 1) ^ l₂) m = 0
this : f₁ + f₂ - (μ₁ + μ₂) • 1 = f₁ - μ₁ • 1 + (f₂ - μ₂ • 1)
h : Co... | 4aaf1ee24d6a3721 |
MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
(f_meas : Measurable f) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
(hf : ∀ᵐ x ∂μ.restrict s, f x < ∞) :
∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤
⊢ ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ | rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable _ f_meas hf] | no goals | 7b09b8562c7f6ee7 |
Mathlib.Tactic.Qify.intCast_ne | Mathlib/Tactic/Qify.lean | @[qify_simps] lemma intCast_ne (a b : ℤ) : a ≠ b ↔ (a : ℚ) ≠ (b : ℚ) | a b : ℤ
⊢ a ≠ b ↔ ↑a ≠ ↑b | simp only [ne_eq, Int.cast_inj] | no goals | 31cd91cd96368c69 |
Module.End.exists_eigenvalue | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
∃ c : K, f.HasEigenvalue c | K : Type u_1
V : Type u_2
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : IsAlgClosed K
inst✝¹ : FiniteDimensional K V
inst✝ : Nontrivial V
f : End K V
⊢ ∃ c, c ∈ spectrum K f | exact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f | no goals | ffbb664e5460280f |
IsConj.normalClosure_eq_top_of | Mathlib/Algebra/Group/Subgroup/Basic.lean | theorem normalClosure_eq_top_of {N : Subgroup G} [hn : N.Normal] {g g' : G} {hg : g ∈ N}
{hg' : g' ∈ N} (hc : IsConj g g') (ht : normalClosure ({⟨g, hg⟩} : Set N) = ⊤) :
normalClosure ({⟨g', hg'⟩} : Set N) = ⊤ | case intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
hn : N.Normal
g : G
hg : g ∈ N
ht : normalClosure {⟨g, hg⟩} = ⊤
c : G
hg' : c * g * c⁻¹ ∈ N
hc : IsConj g (c * g * c⁻¹)
h : ∀ (x : ↥N), (MulAut.conj c) ↑x ∈ N
hs : Surjective ⇑(((MulEquiv.toMonoidHom (MulAut.conj c)).restrict N).codRestrict N h)
⊢ ⟨c * g * c⁻¹, ⋯⟩ ... | exact subset_normalClosure (Set.mem_singleton _) | no goals | 8c82072c9d0e73b6 |
HasFTaylorSeriesUpToOn.compContinuousLinearMap | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap
(hf : HasFTaylorSeriesUpToOn n f p s) (g : G →L[𝕜] E) :
HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g)
(g ⁻¹' s) | case zero_eq
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : WithTop ℕ∞
p : E → FormalMulti... | rfl | no goals | 827bfe05b18c9500 |
List.getElem_insertIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/InsertIdx.lean | theorem getElem_insertIdx {l : List α} {x : α} {n k : Nat} (h : k < (insertIdx n x l).length) :
(insertIdx n x l)[k] =
if h₁ : k < n then
l[k]'(by simp [length_insertIdx] at h; split at h <;> omega)
else
if h₂ : k = n then
x
else
l[k-1]'(by simp [length_insert... | case isFalse.isFalse
α : Type u
l : List α
x : α
n k : Nat
h : k < (insertIdx n x l).length
h₁ : ¬k < n
h₂ : ¬k = n
⊢ (insertIdx n x l)[k] = l[k - 1] | rw [getElem_insertIdx_of_ge (by omega)] | no goals | 2303cbe5977d4288 |
List.append_sublist_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem append_sublist_iff {l₁ l₂ : List α} :
l₁ ++ l₂ <+ r ↔ ∃ r₁ r₂, r = r₁ ++ r₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂ | case nil.mp
α : Type u_1
l₂ r : List α
w : [] ++ l₂ <+ r
⊢ ∃ r₁ r₂, r = r₁ ++ r₂ ∧ [] <+ r₁ ∧ l₂ <+ r₂ | refine ⟨[], r, by simp_all⟩ | no goals | 0739d14b8e5e6a1b |
Cardinal.power_nat_le_max | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem power_nat_le_max {c : Cardinal.{u}} {n : ℕ} : c ^ (n : Cardinal.{u}) ≤ max c ℵ₀ | case inl
c : Cardinal.{u}
n : ℕ
hc : ℵ₀ ≤ c
⊢ c ^ ↑n ≤ c ⊔ ℵ₀ | exact le_max_of_le_left (power_nat_le hc) | no goals | b97a0b119d137349 |
Real.pi_gt_d20 | Mathlib/Data/Real/Pi/Bounds.lean | theorem pi_gt_d20 : 3.14159265358979323846 < π | ⊢ 3.14159265358979323846 < π | pi_lower_bound [
671574048197/474874563549, 58134718954/31462283181, 3090459598621/1575502640777,
2-7143849599/741790664068, 8431536490061/4220852446654, 2-2725579171/4524814682468,
2-2494895647/16566776788806, 2-608997841/16175484287402, 2-942567063/100141194694075,
2-341084060/144951150987041, 2-213717653/363... | no goals | 6462fe7379859663 |
MeasureTheory.condExp_restrict_ae_eq_restrict | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | theorem condExp_restrict_ae_eq_restrict (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs_m : MeasurableSet[m] s) (hf_int : Integrable f μ) :
(μ.restrict s)[f|m] =ᵐ[μ.restrict s] μ[f|m] | α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : Measu... | rw [integrable_indicator_iff (hm _ hs_m), IntegrableOn] | α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : Measu... | 1b22860bba6f556e |
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.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 assignments units[idx]).siz... | simp only [Fin.getElem_fin] | case 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 assignments units[idx]).siz... | c90be518eb885cb1 |
FermatLastTheoremForThreeGen.lambda_sq_dvd_c | Mathlib/NumberTheory/FLT/Three.lean | /-- Given `S' : Solution'`, we have that `λ ^ 2` divides `S'.c`. -/
lemma lambda_sq_dvd_c : λ ^ 2 ∣ S'.c | case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hm : FiniteMultiplicity λ S'.c
this : 2 ≤ multiplicity λ S'.c
x : 𝓞 K
hx : S'.c = λ ^ multiplicity λ S'.c * x
⊢ λ ^ 2 ∣ S'.c | refine ⟨λ ^ (multiplicity (hζ.toInteger - 1) S'.c - 2) * x, ?_⟩ | case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hm : FiniteMultiplicity λ S'.c
this : 2 ≤ multiplicity λ S'.c
x : 𝓞 K
hx : S'.c = λ ^ multiplicity λ S'.c * x
⊢ S'.c = λ ^ 2 * (λ ^ (multiplicity λ S'.c - 2) * x) | c4ebb8ae9468f9d9 |
Batteries.RBNode.min?_eq_toList_head? | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem min?_eq_toList_head? {t : RBNode α} : t.min? = t.toList.head? | case node
α : Type u_1
c✝ : RBColor
l : RBNode α
v✝ : α
r✝ : RBNode α
ih : l.min? = l.toList.head?
r_ih✝ : r✝.min? = r✝.toList.head?
⊢ (node c✝ l v✝ r✝).min? = (node c✝ l v✝ r✝).toList.head? | cases l <;> simp [RBNode.min?, ih] | no goals | a0db33adc70d6f16 |
NatOrdinal.toOrdinal_cast_nat | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem toOrdinal_cast_nat (n : ℕ) : toOrdinal n = n | n : ℕ
⊢ toOrdinal ↑n = ↑n | induction' n with n hn | case zero
⊢ toOrdinal ↑0 = ↑0
case succ
n : ℕ
hn : toOrdinal ↑n = ↑n
⊢ toOrdinal ↑(n + 1) = ↑(n + 1) | f1ab3737731117b1 |
intervalIntegral.integral_const' | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | theorem integral_const' [CompleteSpace E] (c : E) :
∫ _ in a..b, c ∂μ = ((μ <| Ioc a b).toReal - (μ <| Ioc b a).toReal) • c | E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝ : CompleteSpace E
c : E
⊢ ∫ (x : ℝ) in a..b, c ∂μ = ((μ (Ioc a b)).toReal - (μ (Ioc b a)).toReal) • c | simp only [intervalIntegral, setIntegral_const, sub_smul] | no goals | 305e107ae5363c8d |
NoetherNormalization.sum_r_mul_neq | Mathlib/RingTheory/NoetherNormalization.lean | private lemma sum_r_mul_neq (vlt : ∀ i, v i < up) (wlt : ∀ i, w i < up) (neq : v ≠ w) :
∑ x : Fin (n + 1), r x * v x ≠ ∑ x : Fin (n + 1), r x * w x | k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
v w : Fin (n + 1) →₀ ℕ
vlt : ∀ (i : Fin (n + 1)), v i < up
wlt : ∀ (i : Fin (n + 1)), w i < up
neq : v ≠ w
h : ∑ x : Fin (n + 1), r x * v x = ∑ x : Fin (n + 1), r x * w x
⊢ ofDigits up (ofFn ⇑v) = ofDigits up (ofFn ⇑w) | simpa only [ofDigits_eq_sum_mapIdx, mapIdx_eq_ofFn, get_ofFn, length_ofFn,
Fin.coe_cast, mul_comm, sum_ofFn] using h | no goals | 954bc6c11dddb9ea |
MeasureTheory.lintegral_lintegral_mul_inv | Mathlib/MeasureTheory/Group/Prod.lean | theorem lintegral_lintegral_mul_inv [IsMulLeftInvariant ν] (f : G → G → ℝ≥0∞)
(hf : AEMeasurable (uncurry f) (μ.prod ν)) :
(∫⁻ x, ∫⁻ y, f (y * x) x⁻¹ ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ | G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹... | symm | G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹... | 5030442f8dec594f |
Besicovitch.exists_closedBall_covering_tsum_measure_le | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
[Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
(s ⊆ ⋃ x ∈ t, closedBa... | case intro.intro.intro
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo... | choose! R hR using this | case intro.intro.intro
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo... | 181682c2ed74f98b |
egauge_smul_right | Mathlib/Analysis/Convex/EGauge.lean | lemma egauge_smul_right (h : c = 0 → s.Nonempty) (x : E) :
egauge 𝕜 s (c • x) = ‖c‖ₑ * egauge 𝕜 s x | 𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s : Set E
h : c = 0 → s.Nonempty
x : E
⊢ egauge 𝕜 s (c • x) ≤ ‖c‖ₑ * egauge 𝕜 s x | rcases eq_or_ne c 0 with rfl | hc | case inl
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
h : 0 = 0 → s.Nonempty
⊢ egauge 𝕜 s (0 • x) ≤ ‖0‖ₑ * egauge 𝕜 s x
case inr
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s : Se... | 72e8d0e4301ff41d |
IsDiscreteValuationRing.eq_unit_mul_pow_irreducible | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsDiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
⊢ ∃ n u, x = ↑u * ϖ ^ n | obtain ⟨n, hn⟩ := associated_pow_irreducible hx hirr | case intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsDiscreteValuationRing R
x : R
hx : x ≠ 0
ϖ : R
hirr : Irreducible ϖ
n : ℕ
hn : Associated x (ϖ ^ n)
⊢ ∃ n u, x = ↑u * ϖ ^ n | f3a096b201723180 |
AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.add_mem' | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean | theorem add_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
(ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
(isLocallyFraction 𝒜).pred (a + b) := fun x => by
rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, hwa, wa⟩
rcases hb x with ⟨Vb... | case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.intro.mk.intro.mk.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥(unop U)) → at ↑x
ha : (isLoc... | rintro ⟨y, hy⟩ | case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.intro.mk.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥(unop U)) → at ↑x
ha : (is... | 02bfb46c59768ab9 |
IsClosed.ae_eq_univ_iff_eq | Mathlib/MeasureTheory/Measure/OpenPos.lean | theorem _root_.IsClosed.ae_eq_univ_iff_eq (hF : IsClosed F) :
F =ᵐ[μ] univ ↔ F = univ | X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed F
⊢ F =ᶠ[ae μ] univ ↔ F = univ | refine ⟨fun h ↦ ?_, fun h ↦ by rw [h]⟩ | X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed F
h : F =ᶠ[ae μ] univ
⊢ F = univ | bd0f56f21b466be3 |
continuousAt_gaussian_integral | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | theorem continuousAt_gaussian_integral (b : ℂ) (hb : 0 < re b) :
ContinuousAt (fun c : ℂ => ∫ x : ℝ, cexp (-c * (x : ℂ) ^ 2)) b | case h
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_meas : ∀ (c : ℂ), AEStronglyMeasurable (f c) volume
f_cts : ∀ (x : ℝ), ContinuousAt (fun c => f c x) b
c : ℂ
hc : c ∈ re ⁻¹' Ioi d
x : ℝ
⊢ rexp (-c.re * x ^ 2) ≤ rexp (-d * x ^ 2) | gcongr | case h.h.h.a
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_meas : ∀ (c : ℂ), AEStronglyMeasurable (f c) volume
f_cts : ∀ (x : ℝ), ContinuousAt (fun c => f c x) b
c : ℂ
hc : c ∈ re ⁻¹' Ioi d
x : ℝ
⊢ d ≤ c.re | 7ef48b8cfcbec0f8 |
AlternatingGroup.card_of_cycleType | Mathlib/GroupTheory/SpecificGroups/Alternating/Centralizer.lean | theorem card_of_cycleType (m : Multiset ℕ) :
(Finset.univ.filter fun g : alternatingGroup α => (g : Equiv.Perm α).cycleType = m).card =
if (m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) ∧ Even (m.sum + Multiset.card m) then
(Fintype.card α)! /
((Fintype.card α - m.sum)! *
(m.prod * (∏... | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
m : Multiset ℕ
hm : ¬((m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) ∧ Even (m.sum + m.card))
hm' : Even (m.sum + m.card)
⊢ #(filter (fun g => g.cycleType = m) univ) = 0 | rw [Equiv.Perm.card_of_cycleType, if_neg] | case pos.hnc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
m : Multiset ℕ
hm : ¬((m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) ∧ Even (m.sum + m.card))
hm' : Even (m.sum + m.card)
⊢ ¬(m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) | a0b3827213bc0d24 |
BoxIntegral.Prepartition.IsPartition.exists_splitMany_le | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | theorem IsPartition.exists_splitMany_le {I : Box ι} {π : Prepartition I} (h : IsPartition π) :
∃ s, splitMany I s ≤ π | case hp
ι : Type u_1
inst✝ : Finite ι
I : Box ι
π : Prepartition I
h : π.IsPartition
s : Finset (ι × ℝ)
hs : π ⊓ splitMany I s = (splitMany I s).filter fun J => ↑J ⊆ ↑I
⊢ ∀ J ∈ splitMany I s, ↑J ⊆ ↑I | exact fun J hJ => le_of_mem _ hJ | no goals | cbb3b080275f2e9a |
PiNat.mem_cylinder_iff_eq | Mathlib/Topology/MetricSpace/PiNat.lean | theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n | case mp
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
⊢ cylinder y n = cylinder x n | apply Subset.antisymm | case mp.h₁
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
⊢ cylinder y n ⊆ cylinder x n
case mp.h₂
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
⊢ cylinder x n ⊆ cylinder y n | 1650d6a7263b81d8 |
Nat.ppred_eq_some | Mathlib/Data/Nat/PSub.lean | theorem ppred_eq_some {m : ℕ} : ∀ {n}, ppred n = some m ↔ succ m = n
| 0 => by constructor <;> intro h <;> contradiction
| n + 1 => by constructor <;> intro h <;> injection h <;> subst m <;> rfl
| m : ℕ
⊢ ppred 0 = some m ↔ m.succ = 0 | constructor <;> intro h <;> contradiction | no goals | b7a06cee7a813398 |
Algebra.FormallyUnramified.finite_of_free_aux | Mathlib/RingTheory/Unramified/Finite.lean | lemma finite_of_free_aux (I) [DecidableEq I] (b : Basis I R S)
(f : I →₀ S) (x : S) (a : I → I →₀ R) (ha : a = fun i ↦ b.repr (b i * x)) :
(1 ⊗ₜ[R] x * Finsupp.sum f fun i y ↦ y ⊗ₜ[R] b i) =
Finset.sum (f.support.biUnion fun i ↦ (a i).support) fun k ↦
Finsupp.sum (b.repr (f.sum fun i y ↦ a i k • y)) f... | R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
I : Type u_2
inst✝ : DecidableEq I
b : Basis I R S
f : I →₀ S
x : S
a : I → I →₀ R := fun i => b.repr (b i * x)
h₁ :
∀ (k : I),
((f.sum fun i y => (b.repr (b i * x)) k • b.repr y).sum fun j z => z • b j ⊗ₜ[R] b k) =
f.sum... | apply Finset.sum_congr rfl | R : Type u_3
S : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
I : Type u_2
inst✝ : DecidableEq I
b : Basis I R S
f : I →₀ S
x : S
a : I → I →₀ R := fun i => b.repr (b i * x)
h₁ :
∀ (k : I),
((f.sum fun i y => (b.repr (b i * x)) k • b.repr y).sum fun j z => z • b j ⊗ₜ[R] b k) =
f.sum... | 15cd50f5f7e23b0f |
Matrix.fromCols_inj | Mathlib/Data/Matrix/ColumnRowPartitioned.lean | lemma fromCols_inj : Function.Injective2 (@fromCols R m n₁ n₂) | R : Type u_1
m : Type u_2
n₁ : Type u_6
n₂ : Type u_7
x1 x2 : Matrix m n₁ R
y1 y2 : Matrix m n₂ R
⊢ x1.fromCols y1 = x2.fromCols y2 → x1 = x2 ∧ y1 = y2 | simp only [funext_iff, ← Matrix.ext_iff] | R : Type u_1
m : Type u_2
n₁ : Type u_6
n₂ : Type u_7
x1 x2 : Matrix m n₁ R
y1 y2 : Matrix m n₂ R
⊢ (∀ (i : m) (j : n₁ ⊕ n₂), x1.fromCols y1 i j = x2.fromCols y2 i j) →
(∀ (i : m) (j : n₁), x1 i j = x2 i j) ∧ ∀ (i : m) (j : n₂), y1 i j = y2 i j | 092b292eeeabf04f |
FormalMultilinearSeries.leftInv_comp | Mathlib/Analysis/Analytic/Inverse.lean | theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
(h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
(leftInv p i x).comp p = id 𝕜 E x | case convert_2.e_f
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
... | ext c | case convert_2.e_f.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑... | 66065fa083a1e448 |
Set.mul_eq_one_iff | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1 | case refine_1
α : Type u_2
inst✝ : DivisionMonoid α
s t : Set α
h : s * t = 1
hst : (s * t).Nonempty
⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1 | obtain ⟨a, ha⟩ := hst.of_image2_left | case refine_1.intro
α : Type u_2
inst✝ : DivisionMonoid α
s t : Set α
h : s * t = 1
hst : (s * t).Nonempty
a : α
ha : a ∈ s
⊢ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1 | e4590bbb0a7c7ea2 |
MonCat.Colimits.cocone_naturality_components | Mathlib/Algebra/Category/MonCat/Colimits.lean | theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x | J : Type v
inst✝ : Category.{u, v} J
F : J ⥤ MonCat
j j' : J
f : j ⟶ j'
x : ↑(F.obj j)
⊢ (ConcreteCategory.hom (coconeMorphism F j')) ((ConcreteCategory.hom (F.map f)) x) =
(ConcreteCategory.hom (F.map f ≫ coconeMorphism F j')) x | rfl | no goals | 159167e7fd0243a3 |
LieSubalgebra.lieSpan_le | Mathlib/Algebra/Lie/Subalgebra.lean | theorem lieSpan_le {K} : lieSpan R L s ≤ K ↔ s ⊆ K | case mpr
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
s : Set L
K : LieSubalgebra R L
hs : s ⊆ ↑K
m : L
hm : ∀ (K : LieSubalgebra R L), s ⊆ ↑K → m ∈ K
⊢ m ∈ K | exact hm _ hs | no goals | 69fce09aa84e6f88 |
SemiNormedGrp.explicitCokernelDesc_comp_eq_zero | Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean | theorem explicitCokernelDesc_comp_eq_zero {X Y Z W : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
{h : Z ⟶ W} (cond : f ≫ g = 0) (cond2 : g ≫ h = 0) : explicitCokernelDesc cond ≫ h = 0 | X Y Z W : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ W
cond : f ≫ g = 0
cond2 : g ≫ h = 0
⊢ g ≫ h = explicitCokernelπ f ≫ 0 | simp [cond2] | no goals | 20ae50867bd37233 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.