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 : y ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (pf.unshift z n) fun x => y) (z + (f (x + y)) (x + y - x))
|
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 : E
hy : y ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (pf.unshift z (n + 1)) fun x => y) ((f (x + y)) (x + y - x))
|
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).Anisotropic
x : (i : ι) → Mᵢ i
hx : ∑ i : ι, (Q i) (x i) = 0
i j : ι
x✝ : j ∈ Finset.univ
⊢ 0 ≤ (Q j) (x j)
|
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_val := inv_val✝ } ∈ fromUnit.ker
v' i' : R
vi✝ : (algebraMap R K) v' * (algebraMap R K) i' = 1
vi : v' * i' = 1
iv : (algebraMap R K) i' * (algebraMap R K) v' = 1
hx :
(powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi✝, inv_val := iv } =
(Units.map ↑(algebraMap R K)) { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }
hv :
↑{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi✝, inv_val := iv } ^ n =
(algebraMap R K) ↑{ val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }
hi :
↑{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv, inv_val := vi✝ } ^ n =
(algebraMap R K) { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }.inv
⊢ { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ } ∈ (powMonoidHom n).range
|
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_val := inv_val✝ } ∈ fromUnit.ker
v' i' : R
vi✝ : (algebraMap R K) v' * (algebraMap R K) i' = 1
vi : v' * i' = 1
iv✝ : (algebraMap R K) i' * (algebraMap R K) v' = 1
iv : i' * v' = 1
hx :
(powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi✝, inv_val := iv✝ } =
(Units.map ↑(algebraMap R K)) { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }
hv :
↑{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi✝, inv_val := iv✝ } ^ n =
(algebraMap R K) ↑{ val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }
hi :
↑{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv✝, inv_val := vi✝ } ^ n =
(algebraMap R K) { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }.inv
⊢ { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ } ∈ (powMonoidHom n).range
|
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 (ReflTransGen r) e c
|
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 : ReflGen r b a
⊢ Join (ReflTransGen r) e c
|
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 i.hom)
case refine_2
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
x' : (Action.FintypeCat.ofMulAction G X).V.carrier
⊢ ∃ a, (ConcreteCategory.hom i.hom) a = x'
|
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 < ↑(Fintype.card Fq)
le_b : ⌈-log ε / log ↑(Fintype.card Fq)⌉₊ < b.natDegree
i₀ i₁ : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ
i_ne : i₀ ≠ i₁
deg_lt : (A i₁ % b - A i₀ % b).degree < ↑(b.natDegree - ⌈-log ε / log ↑(Fintype.card Fq)⌉₊)
h : ¬A i₁ % b = A i₀ % b
h' : A i₁ % b - A i₀ % b ≠ 0
⊢ ↑(A i₁ % b - A i₀ % b).natDegree = (A i₁ % b - A i₀ % b).degree
|
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 < ↑(Fintype.card Fq)
le_b : ⌈-log ε / log ↑(Fintype.card Fq)⌉₊ < b.natDegree
i₀ i₁ : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ
i_ne : i₀ ≠ i₁
deg_lt : (A i₁ % b - A i₀ % b).degree < ↑(b.natDegree - ⌈-log ε / log ↑(Fintype.card Fq)⌉₊)
h : ¬A i₁ % b = A i₀ % b
h' : A i₁ % b - A i₀ % b ≠ 0
⊢ ↑(A i₁ % b - A i₀ % b).natDegree = ↑(A i₁ % b - A i₀ % b).natDegree
|
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)
⊢ (aeval x) (equivPolynomial (f + g)) = (eval x) ((map R (Algebra.linearMap S R)) (f + 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) :
IsConstructible (f ⁻¹' 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 (f ⁻¹' t)
⊢ IsConstructible (f ⁻¹' (s ∪ t))
|
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 (f ⁻¹' t)
⊢ IsConstructible (f ⁻¹' s ∪ f ⁻¹' t)
|
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)
, Sheaf.IsConstant (coherentTopology Profinite)
((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj M)
, ∀ S : Profinite.{u}, Nonempty
(IsColimit <| (profiniteToCompHaus.op ⋙ M.val).mapCocone S.asLimitCone.op)
]
|
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).essImage M,
IsIso ((LocallyConstant.adjunction R).counit.app M),
Sheaf.IsConstant (coherentTopology Profinite)
((Condensed.ProfiniteCompHaus.equivalence (ModuleCat R)).inverse.obj M),
∀ (S : Profinite), Nonempty (IsColimit ((profiniteToCompHaus.op ⋙ M.val).mapCocone S.asLimitCone.op))].TFAE
|
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 (ModuleCat R)).counit.app M),
(Condensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
IsIso ((LocallyConstant.adjunction R).counit.app M),
Sheaf.IsConstant (coherentTopology Profinite)
((Condensed.ProfiniteCompHaus.equivalence (ModuleCat R)).inverse.obj M),
∀ (S : Profinite), Nonempty (IsColimit ((profiniteToCompHaus.op ⋙ M.val).mapCocone S.asLimitCone.op))].TFAE
|
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 Y.basicOpen s = X.basicOpen ((CommRingCat.Hom.hom (f.app ↑U)) s)
|
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 : C} (f : Y ⟶ Z), 0 ≫ f = 0
w : ∀ (X Y : C), Zero.zero = Zero.zero
this : zero = zero
⊢ HEq zero_comp✝¹ zero_comp✝
|
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`. -/
lemma schwartz_zippel_sup_sum :
∀ {n} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Fin n → Finset R),
#{x ∈ S ^^ n | eval x p = 0} / ∏ i, (#(S i) : ℚ≥0) ≤
p.support.sup fun s ↦ ∑ i, (s i / #(S i) : ℚ≥0)
| 0, p, hp, S => by
-- Because `p` is a polynomial over zero variables, it is constant.
rw [p.eq_C_of_isEmpty] at *
simp [C_ne_zero.mp hp]
-- Now, assume that the theorem holds for all polynomials in `n` variables.
| n + 1, p, hp, S => by
-- We can consider `p` to be a polynomial over multivariable polynomials in one fewer variables.
set p' : Polynomial (MvPolynomial (Fin n) R) := finSuccEquiv R n p with hp'
-- Since `p` is not identically zero, there is some `k` such that `pₖ` is not identically zero.
-- WLOG `k` is the largest such.
set k := p'.natDegree with hk
set pₖ := p'.leadingCoeff with hpₖ
have hp'₀ : p' ≠ 0 := EmbeddingLike.map_ne_zero_iff.2 hp
have hpₖ₀ : pₖ ≠ 0
|
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 (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ filter (fun xₜ => (eval xₜ) pₖ ≠ 0) (piFinset fun i => tail S i) ⊆ piFinset fun i => tail S i
|
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) section `σ`, `g₁` is injective with a
(set-theoretic) retraction `ρ`, and `π₁` is surjective, then `K₂ -δ→ C₁ -G→ C₂` is exact.
-/
lemma SnakeLemma.exact_δ_left (G : C₁ →ₗ[R] C₂) (hF : G.comp π₁ = π₂.comp g₁) (h : Surjective π₁) :
Exact (δ i₁ i₂ i₃ f₁ f₂ hf g₁ g₂ hg h₁ h₂ σ hσ ρ hρ ι₃ hι₃ π₁ hπ₁) G
|
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 : AddCommGroup N₂
inst✝⁸ : Module R N₂
inst✝⁷ : AddCommGroup N₃
inst✝⁶ : Module R N₃
i₁ : M₁ →ₗ[R] N₁
i₂ : M₂ →ₗ[R] N₂
i₃ : M₃ →ₗ[R] N₃
f₁ : M₁ →ₗ[R] M₂
f₂ : M₂ →ₗ[R] M₃
hf : Exact ⇑f₁ ⇑f₂
g₁ : N₁ →ₗ[R] N₂
g₂ : N₂ →ₗ[R] N₃
hg : Exact ⇑g₁ ⇑g₂
h₁ : g₁ ∘ₗ i₁ = i₂ ∘ₗ f₁
h₂ : g₂ ∘ₗ i₂ = i₃ ∘ₗ f₂
σ : M₃ → M₂
hσ : ⇑f₂ ∘ σ = id
ρ : N₂ → N₁
hρ : ρ ∘ ⇑g₁ = id
K₃ : Type u_6
C₁ : Type u_2
C₂ : Type u_3
inst✝⁵ : AddCommGroup K₃
inst✝⁴ : Module R K₃
inst✝³ : AddCommGroup C₁
inst✝² : Module R C₁
inst✝¹ : AddCommGroup C₂
inst✝ : Module R C₂
ι₃ : K₃ →ₗ[R] M₃
hι₃ : Exact ⇑ι₃ ⇑i₃
π₁ : N₁ →ₗ[R] C₁
hπ₁ : Exact ⇑i₁ ⇑π₁
π₂ : N₂ →ₗ[R] C₂
hπ₂ : Exact ⇑i₂ ⇑π₂
G : C₁ →ₗ[R] C₂
hF : G ∘ₗ π₁ = π₂ ∘ₗ g₁
h : Surjective ⇑π₁
⊢ Exact ⇑(δ i₁ i₂ i₃ f₁ f₂ hf g₁ g₂ hg h₁ h₂ σ hσ ρ hρ ι₃ hι₃ π₁ hπ₁) ⇑G
|
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 : AddCommGroup N₂
inst✝⁸ : Module R N₂
inst✝⁷ : AddCommGroup N₃
inst✝⁶ : Module R N₃
i₁ : M₁ →ₗ[R] N₁
i₂ : M₂ →ₗ[R] N₂
i₃ : M₃ →ₗ[R] N₃
f₁ : M₁ →ₗ[R] M₂
f₂ : M₂ →ₗ[R] M₃
hf : Exact ⇑f₁ ⇑f₂
g₁ : N₁ →ₗ[R] N₂
g₂ : N₂ →ₗ[R] N₃
hg : Exact ⇑g₁ ⇑g₂
h₁ : g₁ ∘ₗ i₁ = i₂ ∘ₗ f₁
h₂ : g₂ ∘ₗ i₂ = i₃ ∘ₗ f₂
σ : M₃ → M₂
hσ : ⇑f₂ ∘ σ = id
ρ : N₂ → N₁
hρ : ρ ∘ ⇑g₁ = id
K₃ : Type u_6
C₁ : Type u_2
C₂ : Type u_3
inst✝⁵ : AddCommGroup K₃
inst✝⁴ : Module R K₃
inst✝³ : AddCommGroup C₁
inst✝² : Module R C₁
inst✝¹ : AddCommGroup C₂
inst✝ : Module R C₂
ι₃ : K₃ →ₗ[R] M₃
hι₃ : Exact ⇑ι₃ ⇑i₃
π₁ : N₁ →ₗ[R] C₁
hπ₁ : Exact ⇑i₁ ⇑π₁
π₂ : N₂ →ₗ[R] C₂
hπ₂ : Exact ⇑i₂ ⇑π₂
G : C₁ →ₗ[R] C₂
hF : G ∘ₗ π₁ = π₂ ∘ₗ g₁
h : Surjective ⇑π₁
H₁ : ∀ (x : M₃), f₂ (σ x) = x
⊢ Exact ⇑(δ i₁ i₂ i₃ f₁ f₂ hf g₁ g₂ hg h₁ h₂ σ hσ ρ hρ ι₃ hι₃ π₁ hπ₁) ⇑G
|
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) (𝓝 (s.indicator 1 x))
|
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).filterAt a) (𝓝 (s.indicator 1 a)) →
Tendsto (fun r => μ (s ∩ closedBall a r) / μ (closedBall a r)) (𝓝[>] 0) (𝓝 (s.indicator 1 a))
|
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)) :
InsertUnitInvariant assignments0 assignments0_size units assignments assignments_size →
let update_res := insertUnit (units, assignments, foundContradiction) l
have update_res_size : update_res.snd.fst.size = n
|
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 < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
k :
Fin
(if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
else
(units.push (l.fst, l.snd), assignments.modify l.fst.val (addAssignment l.snd),
foundContradiction || assignments[l.fst.val]! != unassigned)).fst.size
k_ne_l : ¬k = mostRecentUnitIdx
k_ne_j : ¬k = ⟨↑j, j_lt_updatedUnits_size⟩
h : ¬↑k < units.size
k_property : ↑k < units.size + 1
⊢ False
|
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 < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
k :
Fin
(if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
else
(units.push (l.fst, l.snd), assignments.modify l.fst.val (addAssignment l.snd),
foundContradiction || assignments[l.fst.val]! != unassigned)).fst.size
k_ne_l : ¬k = mostRecentUnitIdx
k_ne_j : ¬k = ⟨↑j, j_lt_updatedUnits_size⟩
h : ¬↑k < units.size
k_property : ↑k < units.size + 1
k_lt_units_size : ↑k < units.size
⊢ False
case neg.inr
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 < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
k :
Fin
(if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
else
(units.push (l.fst, l.snd), assignments.modify l.fst.val (addAssignment l.snd),
foundContradiction || assignments[l.fst.val]! != unassigned)).fst.size
k_ne_l : ¬k = mostRecentUnitIdx
k_ne_j : ¬k = ⟨↑j, j_lt_updatedUnits_size⟩
h : ¬↑k < units.size
k_property : ↑k < units.size + 1
k_eq_units_size : ↑k = units.size
⊢ False
|
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 x
|
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.intro.refine_2
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 ∈ span R ↑(X ∪ Y)
|
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 : R → R
a : A
ha : autoParam (p a) _auto✝
hg : autoParam (ContinuousOn g (f '' spectrum R a)) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
⊢ cfc (g ∘ f) a = cfc g (cfc f a)
|
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 : R → R
a : A
ha : autoParam (p a) _auto✝
hg : autoParam (ContinuousOn g (f '' spectrum R a)) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
this : ContinuousOn (g ∘ f) (spectrum R a)
⊢ cfc (g ∘ f) a = cfc g (cfc f a)
|
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
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
|
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 hδ hxz
exact ⟨y, ⟨_, hz, hyz⟩, hxy⟩
|
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 x1 x2 => x1 ≤ x2) t
a : α
d₃ : IsLUB t a
b : β
hb : b ∈ upperBounds (f '' t)
⊢ f a ≤ b
|
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 x1 x2 => x1 ≤ x2) t
a : α
d₃ : IsLUB t a
b : β
hb : b ∈ upperBounds (f '' t)
h : ¬f a ≤ b
⊢ False
|
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✝ = (-iteratedFDerivWithin 𝕜 0 f s x) x✝
|
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), l = p' ++ i' ++ s' → (i' == i) = true → p'.length ≥ p.length
|
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✝ : IsTopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
|
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✝ : IsTopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
|
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 ^ (-s - 1)
|
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 refine_3.H
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
y✝ : { j // j ≠ i } → ↑I
⊢ H ((0, x✝).1, (Cube.insertAt i) ((0, x✝).2, y✝)) = p ((Cube.insertAt i) (x✝, y✝))
case refine_5
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)
⊢ ∀ (t x_1 : ↑I),
x_1 ∈ {0, 1} →
{
toFun := fun x_2 =>
{ toFun := fun t => ⟨(homotopyTo i H) t, ⋯⟩, continuous_toFun := ⋯, map_zero_left := ⋯,
map_one_left := ?refine_4 }.toFun
(t, x_2),
continuous_toFun := ⋯ }
x_1 =
(toLoop i p).toContinuousMap x_1
|
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
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
y✝ : { j // j ≠ i } → ↑I
⊢ H ((0, x✝).1, (Cube.insertAt i) ((0, x✝).2, y✝)) = p ((Cube.insertAt i) (x✝, y✝))
case refine_5
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)
⊢ ∀ (t x_1 : ↑I),
x_1 ∈ {0, 1} →
{
toFun := fun x_2 =>
{ toFun := fun t => ⟨(homotopyTo i H) t, ⋯⟩, continuous_toFun := ⋯, map_zero_left := ⋯,
map_one_left := ⋯ }.toFun
(t, x_2),
continuous_toFun := ⋯ }
x_1 =
(toLoop i p).toContinuousMap x_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✝), x ∈ l₁ → ¬p x = true) ∧ p a = true ∧ filter p l₂ = as
|
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 = true) ∧ p a = true ∧ filter p l₂ = as
case mp.cons.isFalse
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 : filter p l = a :: as
⊢ ∃ l₁ l₂, x :: l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝), x ∈ l₁ → ¬p x = true) ∧ p a = true ∧ filter p l₂ = as
|
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
⊢ ⟨x, hx⟩ ∈ J
|
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 u => #u = m + 1
i : Finset α
h : x i
⊢ ⊥ i
|
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 u => #u = m + 1
i : Finset α
h : x i
⊢ m.succ = m
|
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 : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
f g : P.Hom P'
x : ↥P.ker
⊢ ↑(P'.ker.cotangentEquivIdeal (mk ⟨f.toAlgHom ↑x, ⋯⟩ - mk ⟨g.toAlgHom ↑x, ⋯⟩).val) =
↑(P'.ker.cotangentEquivIdeal (P'.ker.toCotangent ((f.subToKer g) ↑x)))
|
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 : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
f g : P.Hom P'
x : ↥P.ker
⊢ (Ideal.Quotient.mk (P'.ker ^ 2)) (f.toAlgHom ↑x) - (Ideal.Quotient.mk (P'.ker ^ 2)) (g.toAlgHom ↑x) =
(Ideal.Quotient.mk (P'.ker ^ 2)) (f.toRingHom ↑x) - (Ideal.Quotient.mk (P'.ker ^ 2)) (g.toRingHom ↑x)
|
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 : Commute (f₁ - μ₁ • 1) (f₂ - μ₂ • 1)
x✝ : ℕ × ℕ
i j : ℕ
hij : (i, j) ∈ Finset.antidiagonal (l₁ + l₂)
⊢ ((l₁ + l₂).choose (i, j).1 • ((f₁ - μ₁ • 1) ^ (i, j).1 * (f₂ - μ₂ • 1) ^ (i, j).2)) m = 0
|
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 : Commute (f₁ - μ₁ • 1) (f₂ - μ₂ • 1)
x✝ : ℕ × ℕ
i j : ℕ
hij : (i, j) ∈ Finset.antidiagonal (l₁ + l₂)
⊢ ((f₁ - μ₁ • 1) ^ i * (f₂ - μ₂ • 1) ^ j) m = 0
|
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, ⋯⟩ ∈ normalClosure {⟨c * g * c⁻¹, hg'⟩}
|
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 → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →L[𝕜] E
A : (m : ℕ) → ContinuousMultilinearMap 𝕜 (fun i => E) F → ContinuousMultilinearMap 𝕜 (fun i => G) F :=
fun m h => h.compContinuousLinearMap fun x => g
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ ((p (g x) 0) fun x => 0) = (p (g x) 0) 0
|
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_insertIdx] at h; split at h <;> omega)
|
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, 2-213717653/363295959742218, 2-71906926/488934711121807,
2-29337101/797916288104986, 2-45326311/4931175952730065, 2-7506877/3266776448781479,
2-5854787/10191338039232571, 2-4538642/31601378399861717, 2-276149/7691013341581098,
2-350197/39013283396653714, 2-442757/197299283738495963, 2-632505/1127415566199968707,
2-1157/8249230030392285, 2-205461/5859619883403334178, 2-33721/3846807755987625852,
2-11654/5317837263222296743, 2-8162/14897610345776687857, 2-731/5337002285107943372,
2-1320/38549072592845336201, 2-707/82588467645883795866, 2-53/24764858756615791675,
2-237/442963888703240952920, 2-128/956951523274512100791, 2-32/956951523274512100783,
2-27/3229711391051478340136]
|
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 : MeasurableSet t
a✝ : μ t < ⊤
h_int_restrict : Integrable (t.indicator (μ.restrict s[f|m])) (μ.restrict s)
⊢ Integrable (s.indicator (t.indicator (μ.restrict s[f|m]))) μ
|
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 : MeasurableSet t
a✝ : μ t < ⊤
h_int_restrict : Integrable (t.indicator (μ.restrict s[f|m])) (μ.restrict s)
⊢ Integrable (t.indicator (μ.restrict s[f|m])) (μ.restrict s)
|
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 : ClearInsertInductionMotive f f_assignments_size units idx.1 assignments) :
ClearInsertInductionMotive f f_assignments_size units (idx.1 + 1) (clearUnit assignments units[idx])
|
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]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ units[j2] = (⟨↑i, ⋯⟩, false)
|
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]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ units[↑j2] = (⟨↑i, ⋯⟩, false)
|
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)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν =
∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν)
|
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)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) =
∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν
|
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, closedBall x (r x)) ∧ (∑' x : t, μ (closedBall x (r x))) ≤ μ s + ε
|
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 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u ≤ μ s + ε / 2
this : ∀ x ∈ s, ∃ R > 0, ball x R ⊆ u
⊢ ∃ t r,
t.Countable ∧
t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧ s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧ ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε
|
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 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
⊢ ∃ t r,
t.Countable ∧
t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧ s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧ ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε
|
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 : Set E
h : c = 0 → s.Nonempty
x : E
hc : c ≠ 0
⊢ egauge 𝕜 s (c • x) ≤ ‖c‖ₑ * egauge 𝕜 s x
|
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, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, hwb, wb⟩
refine
⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
⟨sb * ra + sa * rb,
add_mem (add_comm jb ja ▸ mul_mem_graded sb_mem ra_mem : sb * ra ∈ 𝒜 (ja + jb))
(mul_mem_graded sa_mem rb_mem)⟩,
⟨sa * sb, mul_mem_graded sa_mem sb_mem⟩, fun y ↦
y.1.asHomogeneousIdeal.toIdeal.primeCompl.mul_mem (hwa ⟨y.1, y.2.1⟩) (hwb ⟨y.1, y.2.2⟩), ?_⟩
rintro ⟨y, hy⟩
simp only [Subtype.forall, Opens.apply_mk] at wa wb
simp [wa y hy.1, wb y hy.2, ext_iff_val, add_mk, add_comm (sa * rb)]
|
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 : (isLocallyFraction 𝒜).pred a
hb : (isLocallyFraction 𝒜).pred b
x : ↥(unop U)
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ unop U
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
hwa : ∀ (x : ↥Va), ↑⟨sa, sa_mem⟩ ∉ (↑x).asHomogeneousIdeal
wa :
∀ (x : ↥Va),
(fun x => a (ia x)) x =
HomogeneousLocalization.mk { deg := ja, num := ⟨ra, ra_mem⟩, den := ⟨sa, sa_mem⟩, den_mem := ⋯ }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ unop U
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
hwb : ∀ (x : ↥Vb), ↑⟨sb, sb_mem⟩ ∉ (↑x).asHomogeneousIdeal
wb :
∀ (x : ↥Vb),
(fun x => b (ib x)) x =
HomogeneousLocalization.mk { deg := jb, num := ⟨rb, rb_mem⟩, den := ⟨sb, sb_mem⟩, den_mem := ⋯ }
⊢ ∀ (x : ↥(Va ⊓ Vb)),
(fun x => (a + b) ((Va.infLELeft Vb ≫ ia) x)) x =
HomogeneousLocalization.mk { deg := ja + jb, num := ⟨sb * ra + sa * rb, ⋯⟩, den := ⟨sa * sb, ⋯⟩, den_mem := ⋯ }
|
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 : (isLocallyFraction 𝒜).pred a
hb : (isLocallyFraction 𝒜).pred b
x : ↥(unop U)
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ unop U
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
hwa : ∀ (x : ↥Va), ↑⟨sa, sa_mem⟩ ∉ (↑x).asHomogeneousIdeal
wa :
∀ (x : ↥Va),
(fun x => a (ia x)) x =
HomogeneousLocalization.mk { deg := ja, num := ⟨ra, ra_mem⟩, den := ⟨sa, sa_mem⟩, den_mem := ⋯ }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ unop U
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
hwb : ∀ (x : ↥Vb), ↑⟨sb, sb_mem⟩ ∉ (↑x).asHomogeneousIdeal
wb :
∀ (x : ↥Vb),
(fun x => b (ib x)) x =
HomogeneousLocalization.mk { deg := jb, num := ⟨rb, rb_mem⟩, den := ⟨sb, sb_mem⟩, den_mem := ⋯ }
y : ↑(ProjectiveSpectrum.top 𝒜)
hy : y ∈ Va ⊓ Vb
⊢ (fun x => (a + b) ((Va.infLELeft Vb ≫ ia) x)) ⟨y, hy⟩ =
HomogeneousLocalization.mk { deg := ja + jb, num := ⟨sb * ra + sa * rb, ⋯⟩, den := ⟨sa * sb, ⋯⟩, den_mem := ⋯ }
|
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 * (∏ n ∈ m.toFinset, (m.count n)!)))
else 0
|
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)) fun j c ↦ c • b j ⊗ₜ[R] b k
|
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 fun i y => (b.repr y).sum fun j z => (b.repr (b i * x)) k • z • b j ⊗ₜ[R] b k
h₂ : ∀ (x : S), ∑ a ∈ (b.repr x).support, (b.repr x) a • b a = x
⊢ ∑ x ∈ f.support, ∑ x_1 ∈ (a x).support, ((a x) x_1 • f x) ⊗ₜ[R] b x_1 =
∑ x_1 ∈ f.support,
∑ x_2 ∈ f.support.biUnion fun i => (b.repr (b i * x)).support, ((b.repr (b x_1 * x)) x_2 • f x_1) ⊗ₜ[R] b x_2
|
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 fun i y => (b.repr y).sum fun j z => (b.repr (b i * x)) k • z • b j ⊗ₜ[R] b k
h₂ : ∀ (x : S), ∑ a ∈ (b.repr x).support, (b.repr x) a • b a = x
⊢ ∀ x_1 ∈ f.support,
∑ x ∈ (a x_1).support, ((a x_1) x • f x_1) ⊗ₜ[R] b x =
∑ x_2 ∈ f.support.biUnion fun i => (b.repr (b i * x)).support, ((b.repr (b x_1 * x)) x_2 • f x_1) ⊗ₜ[R] b x_2
|
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
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
(p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
⊢ (fun x_1 => (p.leftInv i x x_1.length) (p.applyComposition x_1 (⇑↑i.symm ∘ fun j => (p 1) fun x => v j))) = fun c =>
(p.leftInv i x c.length) (p.applyComposition c v)
|
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 ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
(p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
c : Composition (n + 2)
⊢ (p.leftInv i x c.length) (p.applyComposition c (⇑↑i.symm ∘ fun j => (p 1) fun x => v j)) =
(p.leftInv i x c.length) (p.applyComposition c v)
|
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
|
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