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 |
|---|---|---|---|---|---|---|
Valuation.Integers.mem_of_integral | Mathlib/RingTheory/Valuation/Integral.lean | theorem mem_of_integral {x : R} (hx : IsIntegral O x) : x ∈ v.integer :=
let ⟨p, hpm, hpx⟩ := hx
le_of_not_lt fun hvx : 1 < v x => by
rw [hpm.as_sum, eval₂_add, eval₂_pow, eval₂_X, eval₂_finset_sum, add_eq_zero_iff_eq_neg] at hpx
replace hpx := congr_arg v hpx; refine ne_of_gt ?_ hpx
rw [v.map_neg, v.ma... | case inr
R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : R
hx : IsIntegral O x
p : O[X]
hpm : p.Monic
hvx : 1 < v x
hpx : v (x ^ p.natDegree) = v (-∑ i ∈ Finset.range p.natDegree, eval₂ (alg... | rw [hvpi, one_mul, one_mul] | case inr
R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : R
hx : IsIntegral O x
p : O[X]
hpm : p.Monic
hvx : 1 < v x
hpx : v (x ^ p.natDegree) = v (-∑ i ∈ Finset.range p.natDegree, eval₂ (alg... | 26df39b52d6316e9 |
MeasureTheory.mem_generateSetAlgebra_elim | Mathlib/MeasureTheory/SetAlgebra.lean | theorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :
∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Finite) ∧
(∀ᵉ (a ∈ A) (t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t | case compl.intro.intro.intro.intro.refine_2.h.mp
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∀ i ... | choose f hf using hx | case compl.intro.intro.intro.intro.refine_2.h.mp
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
f : (i : ... | 06f0f1876d6d12a0 |
intervalIntegral.integrableOn_Ioo_cpow_iff | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integrableOn_Ioo_cpow_iff {s : ℂ} {t : ℝ} (ht : 0 < t) :
IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioo (0 : ℝ) t) ↔ -1 < s.re | s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x => ↑x ^ s) (Set.Ioo 0 t) volume
⊢ -1 < s.re | have B : IntegrableOn (fun a ↦ a ^ s.re) (Ioo 0 t) := by
apply (integrableOn_congr_fun _ measurableSet_Ioo).1 h.norm
intro a ha
simp [Complex.norm_cpow_eq_rpow_re_of_pos ha.1] | s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x => ↑x ^ s) (Set.Ioo 0 t) volume
B : IntegrableOn (fun a => a ^ s.re) (Set.Ioo 0 t) volume
⊢ -1 < s.re | 24f3ae559755ae1d |
NumberField.hermiteTheorem.finite_of_discr_bdd_of_isComplex | Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean | theorem finite_of_discr_bdd_of_isComplex :
{K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
haveI : NumberField K := @NumberField.mk _ _ inferInstance K.prop
{w : InfinitePlace K | IsComplex w}.Nonempty ∧ |discr K| ≤ N }.Finite | case intro.intro
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈(sqrt (1 + B ^ 2) ⊔ 1) ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.N... | refine ⟨x, ⟨⟨minpoly ℤ (x : K), ⟨?_, fun i ↦ ?_⟩, ?_⟩, ?_⟩⟩ | case intro.intro.refine_1
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈(sqrt (1 + B ^ 2) ⊔ 1) ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsC... | 182a6ab055207691 |
ONote.cmp_compares | Mathlib/SetTheory/Ordinal/Notation.lean | theorem cmp_compares : ∀ (a b : ONote) [NF a] [NF b], (cmp a b).Compares a b
| 0, 0, _, _ => rfl
| oadd _ _ _, 0, _, _ => oadd_pos _ _ _
| 0, oadd _ _ _, _, _ => oadd_pos _ _ _
| o₁@(oadd e₁ n₁ a₁), o₂@(oadd e₂ n₂ a₂), h₁, h₂ => by -- TODO: golf
rw [cmp]
have IHe := @cmp_compares _ _ h₁.fst h₂.fst
s... | o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe :
match Ordering.eq, e₁, e₂ with
| Ordering.lt, a, b => a < b
| Ordering.eq, a, b => a = b
| Ordering.... | dsimp at IHe | o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe : e₁ = e₂
⊢ (Ordering.eq.then ((_root_.cmp ↑n₁ ↑n₂).then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a... | 52fdbaf9e931cd5c |
Nat.factors_eq | Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean | lemma factors_eq : ∀ n : ℕ, normalizedFactors n = n.primeFactorsList
| 0 => by simp
| n + 1 => by
rw [← Multiset.rel_eq, ← associated_eq_eq]
apply UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor _
· rw [Multiset.prod_coe, Nat.prod_primeFactorsList n.succ_ne_zero]
exact pr... | n : ℕ
⊢ Associated (normalizedFactors (n + 1)).prod (↑(n + 1).primeFactorsList).prod | rw [Multiset.prod_coe, Nat.prod_primeFactorsList n.succ_ne_zero] | n : ℕ
⊢ Associated (normalizedFactors (n + 1)).prod n.succ | f0db5791a238bfcf |
TopologicalSpace.isOpen_biUnion_countable | Mathlib/Topology/Bases.lean | theorem isOpen_biUnion_countable [SecondCountableTopology α] {ι : Type*} (I : Set ι) (s : ι → Set α)
(H : ∀ i ∈ I, IsOpen (s i)) : ∃ T ⊆ I, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i ∈ I, s i | α : Type u
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
ι : Type u_1
I : Set ι
s : ι → Set α
H : ∀ i ∈ I, IsOpen (s i)
⊢ ∃ s_1, (Subtype.val '' s_1).Countable ∧ ⋃ y ∈ s_1, s ↑y = ⋃ i ∈ I, s i | rcases isOpen_iUnion_countable (fun i : I ↦ s i) fun i ↦ H i i.2 with ⟨T, hTc, hU⟩ | case intro.intro
α : Type u
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
ι : Type u_1
I : Set ι
s : ι → Set α
H : ∀ i ∈ I, IsOpen (s i)
T : Set ↑I
hTc : T.Countable
hU : ⋃ i ∈ T, s ↑i = ⋃ i, s ↑i
⊢ ∃ s_1, (Subtype.val '' s_1).Countable ∧ ⋃ y ∈ s_1, s ↑y = ⋃ i ∈ I, s i | 6c906d0543581002 |
CategoryTheory.GrothendieckTopology.Plus.exists_of_sep | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
(hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
(X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
∃ t : ToType ((J.plusObj P).obj (op X)), Meq.mk S t = s | case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShap... | apply inj IV.Y | case h.h.a
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfSh... | adc5a5926d46b74e |
minpoly.degree_le_of_ne_zero | Mathlib/FieldTheory/Minpoly/Field.lean | theorem degree_le_of_ne_zero {p : A[X]} (pnz : p ≠ 0) (hp : Polynomial.aeval x p = 0) :
degree (minpoly A x) ≤ degree p :=
calc
degree (minpoly A x) ≤ degree (p * C (leadingCoeff p)⁻¹) :=
min A x (monic_mul_leadingCoeff_inv pnz) (by simp [hp])
_ = degree p := degree_mul_leadingCoeff_inv p pnz
| A : Type u_1
B : Type u_2
inst✝² : Field A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
pnz : p ≠ 0
hp : (Polynomial.aeval x) p = 0
⊢ (Polynomial.aeval x) (p * C p.leadingCoeff⁻¹) = 0 | simp [hp] | no goals | d52f2e4810a15947 |
jacobiSym.trichotomy | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | theorem trichotomy (a : ℤ) (b : ℕ) : J(a | b) = 0 ∨ J(a | b) = 1 ∨ J(a | b) = -1 :=
((@SignType.castHom ℤ _ _).toMonoidHom.mrange.copy {0, 1, -1} <| by
rw [Set.pair_comm]
exact (SignType.range_eq SignType.castHom).symm).list_prod_mem
(by
intro _ ha'
rcases List.mem_pmap.mp ha' with ⟨p, h... | a : ℤ
b : ℕ
⊢ {0, -1, 1} = ↑(MonoidHom.mrange ↑SignType.castHom) | exact (SignType.range_eq SignType.castHom).symm | no goals | 08838733967291cc |
FunctionField.InftyValuation.map_add_le_max' | Mathlib/NumberTheory/FunctionField.lean | theorem InftyValuation.map_add_le_max' (x y : RatFunc Fq) :
inftyValuationDef Fq (x + y) ≤ max (inftyValuationDef Fq x) (inftyValuationDef Fq y) | case pos
Fq : Type u_1
inst✝¹ : Field Fq
inst✝ : DecidableEq (RatFunc Fq)
x y : RatFunc Fq
hx : ¬x = 0
hy : y = 0
⊢ inftyValuationDef Fq x ≤ 0 ⊔ inftyValuationDef Fq x | rw [max_eq_right (WithZero.zero_le (inftyValuationDef Fq x))] | no goals | e10fb237486450a7 |
dNext_nat | Mathlib/Algebra/Homology/Homotopy.lean | theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i | V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ C.d i ((ComplexShape.down ℕ).next i) ≫ f ((ComplexShape.down ℕ).next i) i = C.d i (i - 1) ≫ f (i - 1) i | cases i | case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ C.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = C.d 0 (0 - 1) ≫ f (0 - 1) 0
case succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V... | 4a2ff132f37223e6 |
Homotopy.dNext_succ_chainComplex | Mathlib/Algebra/Homology/Homotopy.lean | theorem dNext_succ_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
dNext (i + 1) f = P.d _ _ ≫ f i (i + 1) | V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
i : ℕ
⊢ (dNext (i + 1)) f = P.d (i + 1) i ≫ f i (i + 1) | dsimp [dNext] | V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
i : ℕ
⊢ P.d (i + 1) ((ComplexShape.down ℕ).next (i + 1)) ≫ f ((ComplexShape.down ℕ).next (i + 1)) (i + 1) =
P.d (i + 1) i ≫ f i (i + 1) | b67a40e3b49092f8 |
CategoryTheory.IsFiltered.sup_objs_exists | Mathlib/CategoryTheory/Filtered/Basic.lean | theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
⊢ ∀ {X : C}, X ∈ ∅ → Nonempty (X ⟶ Classical.choice ⋯) | intro | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
X✝ : C
⊢ X✝ ∈ ∅ → Nonempty (X✝ ⟶ Classical.choice ⋯) | 6340905fdf09a34b |
IsUltrametricDist.exists_norm_multiset_prod_le | Mathlib/Analysis/Normed/Group/Ultra.lean | theorem exists_norm_multiset_prod_le (s : Multiset ι) [Nonempty ι] {f : ι → M} :
∃ i : ι, (s ≠ 0 → i ∈ s) ∧ ‖(s.map f).prod‖ ≤ ‖f i‖ | M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f M‖ ≤ ‖f a‖
⊢ a ::ₘ t ≠ 0 → a ∈ a ::ₘ t | simp | no goals | 76b69fa578fb2276 |
LipschitzWith.norm_compLp_sub_le | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem norm_compLp_sub_le (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : Lp E p μ) :
‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ c * ‖f - f'‖ | case h
α : Type u_1
E : Type u_4
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f f' : ↥(Lp E p μ)
⊢ ∀ᵐ (x : α) ∂μ, ‖↑↑(hg.compLp g0 f - hg.compLp g0 f') x‖ ≤ ↑c * ‖↑↑(f - f') x‖ | filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',
Lp.coeFn_sub (hg.compLp g0 f) (hg.compLp g0 f'), Lp.coeFn_sub f f'] with a ha1 ha2 ha3 ha4 | case h
α : Type u_1
E : Type u_4
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f f' : ↥(Lp E p μ)
a : α
ha1 : ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a
ha2 : ↑↑(hg.compLp g0 f') a = (g ∘ ↑↑f') a
ha3 : ... | ca81e6717d588801 |
Polynomial.isUnit_or_eq_zero_of_separable_expand | Mathlib/FieldTheory/Separable.lean | theorem isUnit_or_eq_zero_of_separable_expand {f : F[X]} (n : ℕ) (hp : 0 < p)
(hf : (expand F (p ^ n) f).Separable) : IsUnit f ∨ n = 0 | F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
n : ℕ
hp : 0 < p
hf : ((expand F (p ^ n)) f).Separable
hn : n ≠ 0
hf2 : derivative ((expand F (p ^ n)) f) = 0
⊢ IsUnit f | rw [separable_def, hf2, isCoprime_zero_right, isUnit_iff] at hf | F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
n : ℕ
hp : 0 < p
hf : ∃ r, IsUnit r ∧ C r = (expand F (p ^ n)) f
hn : n ≠ 0
hf2 : derivative ((expand F (p ^ n)) f) = 0
⊢ IsUnit f | 720e5d3b175c804d |
MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ upperCrossingTime a b f N n ω = N | refine le_antisymm upperCrossingTime_le (not_lt.1 ?_) | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ ¬upperCrossingTime a b f N n ω < N | 1f9b166b684779cd |
Subtype.preimage_val_subset_preimage_val_iff | Mathlib/Data/Set/Image.lean | lemma preimage_val_subset_preimage_val_iff (s t u : Set α) :
(Subtype.val ⁻¹' t : Set s) ⊆ Subtype.val ⁻¹' u ↔ s ∩ t ⊆ s ∩ u | case mp
α : Type u_1
s t u : Set α
⊢ val ⁻¹' t ⊆ val ⁻¹' u → s ∩ t ⊆ s ∩ u | rw [← image_preimage_coe, ← image_preimage_coe] | case mp
α : Type u_1
s t u : Set α
⊢ val ⁻¹' t ⊆ val ⁻¹' u → val '' (val ⁻¹' t) ⊆ val '' (val ⁻¹' u) | c026cf47ee0cc9f2 |
integral_le_sum_mul_Ico_of_antitone_monotone | Mathlib/Analysis/SumIntegralComparisons.lean | lemma integral_le_sum_mul_Ico_of_antitone_monotone
(hab : a ≤ b) (hf : AntitoneOn f (Icc a b)) (hg : MonotoneOn g (Icc (a - 1) (b - 1)))
(fpos : 0 ≤ f b) (gpos : 0 ≤ g (a - 1)) :
∫ x in a..b, f x * g (x - 1) ≤ ∑ i ∈ Finset.Ico a b, f i * g i | case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
hg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑b
gpos : 0 ≤ g (↑a - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ x - 1 ∈ Icc (↑a - 1) (↑b - 1) | simp only [mem_Icc, tsub_le_iff_right, sub_add_cancel] | case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
hg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑b
gpos : 0 ≤ g (↑a - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ ↑a ≤ x ∧ x ≤ ↑b | 53aa89cbac366b1c |
Real.cos_pi_div_four | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem cos_pi_div_four : cos (π / 4) = √2 / 2 | case e_x.e_a
⊢ 4 = 2 ^ 2 | norm_num | no goals | 8f4c0eb7a93e459f |
eq_wittStructureInt | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | theorem eq_wittStructureInt (Φ : MvPolynomial idx ℤ) (φ : ℕ → MvPolynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ φ (wittPolynomial p ℤ n) = bind₁ (fun i => rename (Prod.mk i) (W_ ℤ n)) Φ) :
φ = wittStructureInt p Φ | p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
φ : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ φ) (W_ ℤ n) = (bind₁ fun i => (rename (Prod.mk i)) (W_ ℤ n)) Φ
⊢ φ = wittStructureInt p Φ | funext k | case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
φ : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ φ) (W_ ℤ n) = (bind₁ fun i => (rename (Prod.mk i)) (W_ ℤ n)) Φ
k : ℕ
⊢ φ k = wittStructureInt p Φ k | 65e5a366a741e702 |
Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq | Mathlib/Algebra/Polynomial/Roots.lean | theorem prod_multiset_X_sub_C_of_monic_of_roots_card_eq (hp : p.Monic)
(hroots : Multiset.card p.roots = p.natDegree) : (p.roots.map fun a => X - C a).prod = p | case h.e'_2
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp : p.Monic
hroots : p.roots.card = p.natDegree
⊢ (Multiset.map (fun a => X - C a) p.roots).prod = C p.leadingCoeff * (Multiset.map (fun a => X - C a) p.roots).prod | rw [hp.leadingCoeff, C_1, one_mul] | no goals | 85f76366fad12fa0 |
Fintype.prod_eq_mul | Mathlib/Data/Fintype/BigOperators.lean | theorem prod_eq_mul {f : α → M} (a b : α) (h₁ : a ≠ b) (h₂ : ∀ x, x ≠ a ∧ x ≠ b → f x = 1) :
∏ x, f x = f a * f b | α : Type u_1
M : Type u_4
inst✝¹ : Fintype α
inst✝ : CommMonoid M
f : α → M
a b : α
h₁ : a ≠ b
h₂ : ∀ (x : α), x ≠ a ∧ x ≠ b → f x = 1
⊢ ∏ x : α, f x = f a * f b | apply Finset.prod_eq_mul a b h₁ fun x _ hx => h₂ x hx <;>
exact fun hc => (hc (Finset.mem_univ _)).elim | no goals | 256c4b85f5010f99 |
not_disjoint_segment_convexHull_triple | Mathlib/Analysis/Convex/StoneSeparation.lean | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) | case intro.intro.intro.intro.intro.inr.intro.intro.intro.intro.intro.inr
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u x y : E
hu : u ∈ segment 𝕜 x p
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
... | obtain ⟨au, bu, hau, hbu, habu, rfl⟩ := hu | case intro.intro.intro.intro.intro.inr.intro.intro.intro.intro.intro.inr.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤... | b7449eac49163f66 |
Stream'.WSeq.join_map_ret | Mathlib/Data/Seq/WSeq.lean | theorem join_map_ret (s : WSeq α) : join (map ret s) ~ʷ s | α : Type u
s✝¹ s' s✝ : WSeq α
h : (map ret s✝).join = s'
c1 c2 : Computation (Option (α × WSeq α))
s : WSeq α
this :
∀ (s : WSeq α),
∃ s',
(map (fun a => cons a nil) s).join.destruct = (map (fun a => cons a nil) s').join.destruct ∧
s.destruct = s'.destruct
⊢ LiftRelAux (LiftRelO (fun x1 x2 => x1 = x... | induction' s using WSeq.recOn with a s s <;>
simp (config := { unfoldPartialApp := true }) [ret, ret_mem, this, Option.exists] | no goals | 2c42fba3f41af6bb |
NormedSpace.expSeries_radius_eq_top | Mathlib/Analysis/Normed/Algebra/Exponential.lean | theorem expSeries_radius_eq_top : (expSeries 𝕂 𝔸).radius = ∞ | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun n => ‖(↑(n + 1))⁻¹‖) atTop (𝓝 0) | apply norm_zero (E := 𝕂) ▸ Filter.Tendsto.norm | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun x => (↑(x + 1))⁻¹) atTop (𝓝 0) | 1dac2526a3967658 |
MonoidHom.compAddChar_injective_right | Mathlib/Algebra/Group/AddChar.lean | lemma _root_.MonoidHom.compAddChar_injective_right (f : M →* N) (hf : Injective f) :
Injective fun ψ : AddChar B M ↦ f.compAddChar ψ | B : Type u_2
M : Type u_3
N : Type u_4
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
f : M →* N
hf : Injective ⇑f
ψ χ : AddChar B M
h : ⇑((fun ψ => f.compAddChar ψ) ψ) = ⇑((fun ψ => f.compAddChar ψ) χ)
⊢ ⇑ψ = ⇑χ | exact hf.comp_left h | no goals | c90bdf231ebd1ff3 |
ProbabilityTheory.cond_iInter | Mathlib/Probability/Independence/Basic.lean | /-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
lemma cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i))
(hindep : iIndepFun (fun _ ↦ mα.prod mβ) (fun i ω ↦ (X i ω, Y i ω)) μ)
(hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i))
(hy : ∀ i ∉... | ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω... | congr | case e_a
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i... | 299952e2d8d690c9 |
zpow_mem | Mathlib/Algebra/Group/Subgroup/Defs.lean | theorem zpow_mem {x : M} (hx : x ∈ K) : ∀ n : ℤ, x ^ n ∈ K
| (n : ℕ) => by
rw [zpow_natCast]
exact pow_mem hx n
| -[n+1] => by
rw [zpow_negSucc]
exact inv_mem (pow_mem hx n.succ)
| M : Type u_3
S : Type u_4
inst✝¹ : DivInvMonoid M
inst✝ : SetLike S M
hSM : SubgroupClass S M
K : S
x : M
hx : x ∈ K
n : ℕ
⊢ (x ^ (n + 1))⁻¹ ∈ K | exact inv_mem (pow_mem hx n.succ) | no goals | 525620767b8239e3 |
DividedPowers.dpow_smul_right | Mathlib/RingTheory/DividedPowers/Basic.lean | theorem dpow_smul_right {n : ℕ} (ha : a ∈ I) :
hI.dpow n (a • b) = hI.dpow n a • b ^ n | A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
a b : A
hI : DividedPowers I
n : ℕ
ha : a ∈ I
⊢ hI.dpow n (a • b) = hI.dpow n a • b ^ n | rw [smul_eq_mul, hI.dpow_mul_right ha, smul_eq_mul] | no goals | f20ed7a0fe76a17c |
Ordinal.one_toPGame_moveLeft | Mathlib/SetTheory/Game/Ordinal.lean | theorem one_toPGame_moveLeft (x) : (toPGame 1).moveLeft x = toPGame 0 | x : (toPGame 1).LeftMoves
⊢ (toPGame 1).moveLeft x = toPGame 0 | simp | no goals | 377af6bbb74053bf |
AEMeasurable.exists_ae_eq_range_subset | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | theorem exists_ae_eq_range_subset (H : AEMeasurable f μ) {t : Set β} (ht : ∀ᵐ x ∂μ, f x ∈ t)
(h₀ : t.Nonempty) : ∃ g, Measurable g ∧ range g ⊆ t ∧ f =ᵐ[μ] g | case pos
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
H : AEMeasurable f μ
t : Set β
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
h₀ : t.Nonempty
s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ
g : α → β := s.piecewise (fun x => h₀.some) (mk f H)
x : α
hx : x ∈ s
⊢ g x ∈... | simpa [g, hx] using h₀.some_mem | no goals | 2f7513f9dc192863 |
IsMinOn.of_isLocalMinOn_of_convexOn | Mathlib/Analysis/Convex/Extrema.lean | theorem IsMinOn.of_isLocalMinOn_of_convexOn {f : E → β} {a : E} (a_in_s : a ∈ s)
(h_localmin : IsLocalMinOn f s a) (h_conv : ConvexOn ℝ s f) : IsMinOn f s a | E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : Conv... | have hg0 : g 0 = a := AffineMap.lineMap_apply_zero a x | E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : Conv... | b4da7187b7747bac |
MeasureTheory.Martingale.stoppedValue_ae_eq_condExp_of_le_const_of_countable_range | Mathlib/Probability/Martingale/OptionalSampling.lean | theorem stoppedValue_ae_eq_condExp_of_le_const_of_countable_range (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n) (h_countable_range : (Set.range τ).Countable)
[SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] :
stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace] | Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ... | exact fun i _ => stoppedValue_ae_eq_restrict_eq h _ hτ_le i | no goals | 428db120036cef51 |
RatFunc.liftMonoidWithZeroHom_injective | Mathlib/FieldTheory/RatFunc/Basic.lean | theorem liftMonoidWithZeroHom_injective [Nontrivial R] (φ : R[X] →*₀ G₀) (hφ : Function.Injective φ)
(hφ' : R[X]⁰ ≤ G₀⁰.comap φ := nonZeroDivisors_le_comap_nonZeroDivisors_of_injective _ hφ) :
Function.Injective (liftMonoidWithZeroHom φ hφ') | case ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_2
G₀ : Type u_1
R : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : CommRing R
inst✝ : Nontrivial R
φ : R[X] →*₀ G₀
hφ : Function.Injective ⇑φ
hφ' : optParam (R[X]⁰ ≤ Submonoid.comap φ G₀⁰) ⋯
a a' : R[X] × ↥R[X]⁰
h : φ a.1 / φ ↑a.2 = φ a'.1 / φ ↑a'.2
⊢ φ ↑a... | exact map_ne_zero_of_mem_nonZeroDivisors _ hφ (SetLike.coe_mem _) | no goals | 81fff59902e9b407 |
MeasureTheory.exists_pos_setLIntegral_lt_of_measure_lt | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem exists_pos_setLIntegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
ε₂ : ℝ≥0∞
hε₂0 : 0 < ε₂
hε₂ε : ε₂ < ε
ε₁ : ℝ≥0∞
hε₁0 : 0 < ε₁
hε₁₂ : ε₁ < ε₂
φ : α →ₛ ℝ≥0
left✝ : ∀ (x : α), ↑(φ x) ≤ f x
hφ : ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lint... | simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,
SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)] | no goals | df20228eb7834d7d |
ContinuousOn.isBigOWith_rev_principal | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem isBigOWith_rev_principal
(hf : ContinuousOn f s) (hs : IsCompact s) (hC : ∀ i ∈ s, f i ≠ 0) (c : F) :
IsBigOWith (‖c‖ / sInf (Norm.norm '' (f '' s))) (𝓟 s) (fun _ => c) f | case intro.intro.intro.intro
α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn f s
hC : ∀ i ∈ s, f i ≠ 0
c : F
x✝ : α
hx✝ : x✝ ∈ s
hs : IsCompact ((fun a => ‖a‖) '' (f '' s))
h_sInf : IsGLB ((fun a => ‖a‖) '' (... | exact (norm_ne_zero_iff.mpr (hC x hx)).symm.lt_of_le (norm_nonneg _) | no goals | eb77d590216beaa7 |
LocalizedModule.subsingleton_iff_support_subset | Mathlib/RingTheory/Support.lean | lemma LocalizedModule.subsingleton_iff_support_subset {f : R} :
Subsingleton (LocalizedModule (.powers f) M) ↔
Module.support R M ⊆ PrimeSpectrum.zeroLocus {f} | case pos
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
H : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}
m : M
h : (Submodule.span R {m}).annihilator = ⊤
⊢ ∃ r ∈ Submonoid.powers f, r • m = 0 | rw [Submodule.annihilator_eq_top_iff, Submodule.span_singleton_eq_bot] at h | case pos
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
H : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}
m : M
h : m = 0
⊢ ∃ r ∈ Submonoid.powers f, r • m = 0 | 75188a38276e1985 |
ContDiffWithinAt.fderivWithin | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem ContDiffWithinAt.fderivWithin {f : E → F → G} {g : E → F} {t : Set F}
(hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (s ×ˢ t) (x₀, g x₀))
(hg : ContDiffWithinAt 𝕜 m g s x₀) (ht : UniqueDiffOn 𝕜 t) (hmn : m + 1 ≤ n) (hx₀ : x₀ ∈ s)
(hst : s ⊆ g ⁻¹' t) : ContDiffWithinAt 𝕜 m (fun x => fderivWithin... | 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
x₀ : E
m n : WithTop ℕ∞
f : E → F → G
g : E → F
t : Set F... | rw [← insert_eq_self.mpr hx₀] at hf | 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
x₀ : E
m n : WithTop ℕ∞
f : E → F → G
g : E → F
t : Set F... | 3ca83c31446aec8c |
SubmodulesRingBasis.toRing_subgroups_basis | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
RingSubgroupsBasis fun i => (B i).toAddSubgroup | ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB : SubmodulesRingBasis B
⊢ RingSubgroupsBasis fun i => (B i).toAddSubgroup | apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul | ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB : SubmodulesRingBasis B
⊢ ∀ (x : A) (i : ι), ∃ j, ↑(B j).toAddSubgroup ⊆ (fun y => x * y) ⁻¹' ↑(B i).toAddSubgroup | 78c087b5f4afed56 |
uniformEquicontinuous_iInf_rng | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem uniformEquicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} :
UniformEquicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, UniformEquicontinuous (uα := u k) F | ι : Type u_1
κ : Type u_2
α' : Type u_7
β : Type u_8
uβ : UniformSpace β
u : κ → UniformSpace α'
F : ι → β → α'
⊢ UniformEquicontinuous F ↔ ∀ (k : κ), UniformEquicontinuous F | simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)] | ι : Type u_1
κ : Type u_2
α' : Type u_7
β : Type u_8
uβ : UniformSpace β
u : κ → UniformSpace α'
F : ι → β → α'
⊢ UniformContinuous (⇑ofFun ∘ swap F) ↔ ∀ (k : κ), UniformContinuous (⇑ofFun ∘ swap F) | ae52e8c079f1a71a |
map_pow_le_pow' | Mathlib/Analysis/Normed/Ring/Seminorm.lean | theorem map_pow_le_pow' {F α : Type*} [Ring α] [FunLike F α ℝ] [RingSeminormClass F α ℝ] {f : F}
(hf1 : f 1 ≤ 1) (a : α) : ∀ n : ℕ, f (a ^ n) ≤ f a ^ n
| 0 => by simp only [pow_zero, hf1]
| n + 1 => by
simp only [pow_succ _ n]
exact le_trans (map_mul_le_mul f _ a)
(mul_le_mul_of_nonneg_right (map_... | F : Type u_2
α : Type u_3
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f : F
hf1 : f 1 ≤ 1
a : α
n : ℕ
⊢ f (a ^ n * a) ≤ f a ^ n * f a | exact le_trans (map_mul_le_mul f _ a)
(mul_le_mul_of_nonneg_right (map_pow_le_pow' hf1 _ n) (apply_nonneg f a)) | no goals | e3d0381ed09beebf |
Equiv.Perm.mclosure_swap_castSucc_succ | Mathlib/GroupTheory/Perm/Sign.lean | theorem mclosure_swap_castSucc_succ (n : ℕ) :
Submonoid.closure (Set.range fun i : Fin n ↦ swap i.castSucc i.succ) = ⊤ | case zero
n : ℕ
i : Fin (n + 1)
ne : i ≠ 0
lt : i < 0
⊢ swap i 0 ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ)) | cases lt | no goals | 6a1796e117466965 |
Cardinal.mk_finsupp_nat | Mathlib/SetTheory/Cardinal/Finsupp.lean | theorem mk_finsupp_nat (α : Type u) [Nonempty α] : #(α →₀ ℕ) = max #α ℵ₀ | α : Type u
inst✝ : Nonempty α
⊢ #(α →₀ ℕ) = #α ⊔ ℵ₀ | simp | no goals | 2eb3fb145d2b986a |
PhragmenLindelof.eq_zero_on_right_half_plane_of_superexponential_decay | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem eq_zero_on_right_half_plane_of_superexponential_decay (hd : DiffContOnCl ℂ f {z | 0 < z.re})
(hexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c))
(hre : SuperpolynomialDecay atTop expR fun x => ‖f x‖) (him : ∃ C, ∀ x : ℝ, ‖f (x * I)‖ ≤ C) :
EqOn f 0 {z... | case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E :... | suffices H : ∀ n : ℕ, ‖g n z‖ ≤ C by
contrapose! H
simp only [hg]
exact (((tendsto_pow_atTop_atTop_of_one_lt (Real.one_lt_exp_iff.2 hz)).atTop_mul
(norm_pos_iff.2 H) tendsto_const_nhds).eventually (eventually_gt_atTop C)).exists | case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E :... | 981cd793120919a3 |
Int.tdiv_sign | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem tdiv_sign : ∀ a b, a.tdiv (sign b) = a * sign b
| _, succ _ => by simp [sign, Int.mul_one]
| _, 0 => by simp [sign, Int.mul_zero]
| _, -[_+1] => by simp [sign, Int.mul_neg, Int.mul_one]
| x✝ : Int
a✝ : Nat
⊢ x✝.tdiv -[a✝+1].sign = x✝ * -[a✝+1].sign | simp [sign, Int.mul_neg, Int.mul_one] | no goals | 3e9d1e87a6399f78 |
Ordering.isGE_iff_eq_gt_or_eq_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Ord.lean | theorem isGE_iff_eq_gt_or_eq_eq {o : Ordering} : o.isGE ↔ o = .gt ∨ o = .eq | o : Ordering
⊢ o.isGE = true ↔ o = gt ∨ o = eq | cases o <;> simp | no goals | 824060222deb2510 |
PadicInt.isUnit_iff | Mathlib/NumberTheory/Padics/PadicIntegers.lean | theorem isUnit_iff {z : ℤ_[p]} : IsUnit z ↔ ‖z‖ = 1 :=
⟨fun h => by
rcases isUnit_iff_dvd_one.1 h with ⟨w, eq⟩
refine le_antisymm (norm_le_one _) ?_
have := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z)
rwa [mul_one, ← norm_mul, ← eq, norm_one] at this, fun h =>
⟨⟨z, z.inv, mul_inv h, ... | p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
⊢ ‖z‖ = 1 | rcases isUnit_iff_dvd_one.1 h with ⟨w, eq⟩ | case intro
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
⊢ ‖z‖ = 1 | 45ba828ff95aefca |
MeasureTheory.convolution_assoc' | Mathlib/Analysis/Convolution.lean | theorem convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G} (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
(hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt g k x L₄ μ)
(hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)) :
((f ⋆[L, ν] ... | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : Norme... | refine ((quasiMeasurePreserving_sub_left_of_right_invariant ν x₀).ae hgk).mono fun t ht => ?_ | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : Norme... | c9badc48511fd05a |
ChartedSpace.discreteTopology | Mathlib/Geometry/Manifold/ChartedSpace.lean | theorem ChartedSpace.discreteTopology [DiscreteTopology H] : DiscreteTopology M | H : Type u
M : Type u_2
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : DiscreteTopology H
x : M
this : IsOpen ((chartAt H x).source ∩ ↑(chartAt H x) ⁻¹' {↑(chartAt H x) x})
⊢ {x} ⊆ (chartAt H x).source ∩ ↑(chartAt H x) ⁻¹' {↑(chartAt H x) x} | simp | no goals | 1439e53cafa424b2 |
Odd.neg_pow | Mathlib/Algebra/Ring/Parity.lean | lemma Odd.neg_pow : Odd n → ∀ a : α, (-a) ^ n = -a ^ n | case intro
α : Type u_2
inst✝¹ : Monoid α
inst✝ : HasDistribNeg α
c : ℕ
a : α
⊢ (-a) ^ (2 * c + 1) = -a ^ (2 * c + 1) | simp_rw [pow_add, pow_mul, neg_sq, pow_one, mul_neg] | no goals | a8648371186ac69e |
RingSeminorm.seminorm_one_eq_one_iff_ne_zero | Mathlib/Analysis/Normed/Ring/Seminorm.lean | theorem seminorm_one_eq_one_iff_ne_zero (hp : p 1 ≤ 1) : p 1 = 1 ↔ p ≠ 0 | case inl
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : p 1 = 0
⊢ p 1 = 1 | exfalso | case inl
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : p 1 = 0
⊢ False | 2e0d471a9b8c810e |
Batteries.HashMap.Imp.erase_WF | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem erase_WF [BEq α] [Hashable α] {m : Imp α β} {k}
(h : m.buckets.WF) : (erase m k).buckets.WF | α : Type u_1
β : Type u_2
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
k : α
h : m.buckets.WF
⊢ (m.erase k).buckets.WF | dsimp [erase, cond] | α : Type u_1
β : Type u_2
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
k : α
h : m.buckets.WF
⊢ (match AssocList.contains k m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat] with
| true =>
{ size := m.size - 1,
buckets :=
m.buckets.update (mkIdx ⋯ (hash k).toUSize).val
... | 5aad4902f05c9a8d |
EuclideanGeometry.Sphere.dist_div_cos_oangle_center_div_two_eq_radius | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem dist_div_cos_oangle_center_div_two_eq_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) :
dist p₁ p₂ / Real.Angle.cos (∡ p₂ p₁ s.center) / 2 = s.radius | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h : p₁ ≠ p₂
⊢ 2⁻¹ * dist p₁ p₂ = dist p₁ (midpoint ℝ p₁ p₂) | simp | no goals | 11090c85f50f31fd |
CFC.posPart_negPart_unique | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean | /-- The positive and negative parts of a selfadjoint element `a` are unique. That is, if
`a = b - c` is the difference of nonnegative elements whose product is zero, then these are
precisely `a⁺` and `a⁻`. -/
lemma posPart_negPart_unique {a b c : A} (habc : a = b - c) (hbc : b * c = 0)
(hb : 0 ≤ b | case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrum... | have mul₁ (f g : C(s, ℝ)₀) :
(cfcₙHomSuperset hb' hbs f) * (cfcₙHomSuperset hc' hcs g) = 0 := by
refine f.nonUnitalStarAlgHom_apply_mul_eq_zero s0 _ _ ?id ?star_id
(cfcₙHomSuperset_continuous hb' hbs)
case' star_id => rw [star_trivial]
all_goals
refine g.mul_nonUnitalStarAlgHom_apply_eq_zero s0 _ _ ?_... | case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrum... | 702687b54a399521 |
Int.ceil_one | Mathlib/Algebra/Order/Floor.lean | theorem ceil_one : ⌈(1 : α)⌉ = 1 | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
⊢ ⌈1⌉ = 1 | rw [← cast_one, ceil_intCast] | no goals | 3eac4ccef2b719d9 |
RootPairing.isOrthogonal_comm | Mathlib/LinearAlgebra/RootSystem/Defs.lean | lemma isOrthogonal_comm (h : IsOrthogonal P i j) : Commute (P.reflection i) (P.reflection j) | case h
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j : ι
v : M
h : P.pairing i j = 0 ∧ P.pairing j i = 0
⊢ v - (P.toPerfectPairing v) (P.coroot i) • P.root i - (P.toPerfectPairing... | abel | no goals | 12ff8d55500e6e55 |
Nat.bitCasesOn_bit | Mathlib/Data/Nat/BinaryRec.lean | theorem bitCasesOn_bit (h : ∀ b n, motive (bit b n)) (b : Bool) (n : Nat) :
bitCasesOn (bit b n) h = h b n | motive : Nat → Sort u
h : (b : Bool) → (n : Nat) → motive (bit b n)
b : Bool
n : Nat
e : motive (bit ((bit b n).testBit 0) (bit b n >>> 1)) = motive (bit b n)
⊢ e ▸ h ((bit b n).testBit 0) (bit b n >>> 1) = h b n | revert e | motive : Nat → Sort u
h : (b : Bool) → (n : Nat) → motive (bit b n)
b : Bool
n : Nat
⊢ ∀ (e : motive (bit ((bit b n).testBit 0) (bit b n >>> 1)) = motive (bit b n)),
e ▸ h ((bit b n).testBit 0) (bit b n >>> 1) = h b n | 06d170367d1d27ec |
List.firstM_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem firstM_toArray [Alternative m] (as : List α) (f : α → m β) :
as.toArray.firstM f = as.firstM f | case succ.isFalse
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Alternative m
as : List α
f : α → m β
i : Nat
ih : i ≤ as.length → firstM.go f as.toArray (as.length - i) = firstM f (drop (as.length - i) as)
h : as.length ≤ as.length - (i + 1)
this : as.length = 0
⊢ i + 1 ≤ as.length → failure = firstM f (dr... | simp_all | no goals | 8ff826aeb0a69b7f |
Polynomial.isIntegral_isLocalization_polynomial_quotient | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Idea... | case intro.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
ins... | rw [RingHom.comp_apply] | no goals | 4f255f198e9b3f92 |
NonUnitalStarSubalgebra.iSupLift_inclusion | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | theorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :
iSupLift K dir f hf T hT (inclusion h x) = f i x | R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : ... | apply Set.iUnionLift_inclusion | case h
R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
i... | 99a36d2072712ece |
Setoid.IsPartition.ncard_eq_finsum | Mathlib/Data/Setoid/Partition/Card.lean | theorem Setoid.IsPartition.ncard_eq_finsum {α : Type*} {P : Set (Set α)}
(hP : Setoid.IsPartition P) (s : Set α) (hs : s.Finite | α : Type u_1
P : Set (Set α)
hP : IsPartition P
s : Set α
hs : autoParam s.Finite _auto✝
hst : ∀ (t : Set α), (s ∩ t).Finite
hst' : ∀ (t : Set α), Nat.card ↑(s ∩ t) = ⋯.toFinset.card
f : ↑(Function.support fun t => (s ∩ ↑t).ncard) → ↑s :=
fun x =>
match x with
| ⟨t, ht⟩ => ⟨Exists.choose ⋯, ⋯⟩
⊢ (Function.sup... | have hf (t : Function.support (fun (t : P) ↦ (s ∩ (t : Set α)).ncard)) :
↑↑t ∈ P ∧ (f t : α) ∈ (t : Set α) :=
⟨(↑t : P).prop, (Set.nonempty_of_ncard_ne_zero t.prop).choose_spec.2⟩ | α : Type u_1
P : Set (Set α)
hP : IsPartition P
s : Set α
hs : autoParam s.Finite _auto✝
hst : ∀ (t : Set α), (s ∩ t).Finite
hst' : ∀ (t : Set α), Nat.card ↑(s ∩ t) = ⋯.toFinset.card
f : ↑(Function.support fun t => (s ∩ ↑t).ncard) → ↑s :=
fun x =>
match x with
| ⟨t, ht⟩ => ⟨Exists.choose ⋯, ⋯⟩
hf : ∀ (t : ↑(F... | 04849ad22be2c673 |
MeasureTheory.Measure.ext_of_generateFrom_of_cover | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem ext_of_generateFrom_of_cover {S T : Set (Set α)} (h_gen : ‹_› = generateFrom S)
(hc : T.Countable) (h_inter : IsPiSystem S) (hU : ⋃₀ T = univ) (htop : ∀ t ∈ T, μ t ≠ ∞)
(ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)) (T_eq : ∀ t ∈ T, μ t = ν t) : μ = ν | case h.compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ⊤
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u... | rwa [← measure_inter_add_diff t hu, ← measure_inter_add_diff t hu, ← ihu,
ENNReal.add_right_inj] at this | case h.compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ⊤
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u... | a49685f1b0631501 |
BitVec.sub_le_sub_iff_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem sub_le_sub_iff_le {x y z : BitVec w} (hxz : z ≤ x) (hyz : z ≤ y) :
(x - z ≤ y - z) ↔ x ≤ y | w : Nat
x y z : BitVec w
hxz : z.toNat ≤ x.toNat
hyz : z.toNat ≤ y.toNat
⊢ x.toNat - z.toNat ≤ y.toNat - z.toNat ↔ x.toNat ≤ y.toNat | omega | no goals | f7f41f2a9a6b7554 |
Submodule.exists_fg_le_subset_range_rTensor_subtype | Mathlib/RingTheory/Finiteness/TensorProduct.lean | theorem exists_fg_le_subset_range_rTensor_subtype (s : Set (N ⊗[R] M)) (hs : s.Finite) :
∃ (J : Submodule R N) (_ : J.FG), s ⊆ LinearMap.range (rTensor M J.subtype) | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
s : Set (N ⊗[R] M)
hs : Finite ↑s
J : N ⊗[R] M → Submodule R N
fg : ∀ (x : N ⊗[R] M), (J x).FG
y : (x : N ⊗[R] M) → ↥(J x) ⊗[R] M
eq : ∀ (x : N ⊗[R] M), x = (rTensor M ... | rfl | no goals | 80ac97d07d695483 |
Types.Pushout.condition | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | lemma condition : f ≫ inl f g = g ≫ inr f g | S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
⊢ f ≫ inl f g = g ≫ inr f g | ext x | case h
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
x : S
⊢ (f ≫ inl f g) x = (g ≫ inr f g) x | df68459bb8711e54 |
Array.flatten_ne_empty_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem flatten_ne_empty_iff {xs : Array (Array α)} : xs.flatten ≠ #[] ↔ ∃ x, x ∈ xs ∧ x ≠ #[] | α : Type u_1
xs : Array (Array α)
⊢ xs.flatten ≠ #[] ↔ ∃ x, x ∈ xs ∧ x ≠ #[] | simp | no goals | 042c87bbcdd070fd |
iteratedDerivWithin_comp_const_smul | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | theorem iteratedDerivWithin_comp_const_smul (hf : ContDiffOn 𝕜 n f s) (c : 𝕜)
(hs : Set.MapsTo (c * ·) s s) :
iteratedDerivWithin n (fun x => f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x) | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x => c * x) s s
n : ℕ
ih :
∀ {x : 𝕜},
x ∈ s →
ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x => f (c * x)) s x... | apply DifferentiableWithinAt.comp | case hg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x => c * x) s s
n : ℕ
ih :
∀ {x : 𝕜},
x ∈ s →
ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x => f (c *... | 8405dfec972dab7c |
AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map | Mathlib/AlgebraicGeometry/AffineScheme.lean | lemma appLE_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (hU : IsAffineOpen U)
{V : X.Opens} (hV : IsAffineOpen V) (e) (r : Γ(Y, U)) :
letI := hU.isLocalization_basicOpen r
letI := hV.isLocalization_basicOpen (f.appLE U V e r)
f.appLE (Y.basicOpen r) (X.basicOpen (f.appLE U V e r)) (by simp ... | case hf
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
hU : IsAffineOpen U
V : X.Opens
hV : IsAffineOpen V
e : V ≤ f ⁻¹ᵁ U
r : ↑Γ(Y, U)
this✝ : IsLocalization.Away r ↑Γ(Y, Y.basicOpen r) := isLocalization_basicOpen hU r
this : IsLocalization.Away ((ConcreteCategory.hom (Scheme.Hom.appLE f U V e)) r)
↑Γ(X, X.basicOpen ((ConcreteC... | apply IsLocalization.ringHom_ext (.powers r) | case hf.h
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
hU : IsAffineOpen U
V : X.Opens
hV : IsAffineOpen V
e : V ≤ f ⁻¹ᵁ U
r : ↑Γ(Y, U)
this✝ : IsLocalization.Away r ↑Γ(Y, Y.basicOpen r) := isLocalization_basicOpen hU r
this : IsLocalization.Away ((ConcreteCategory.hom (Scheme.Hom.appLE f U V e)) r)
↑Γ(X, X.basicOpen ((Concret... | 31b8d115dc427c5d |
Lean.Omega.IntList.mul_distrib_left | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean | theorem mul_distrib_left (xs ys zs : IntList) : (xs + ys) * zs = xs * zs + ys * zs | case nil
ys zs : IntList
⊢ ([] + ys) * zs = [] * zs + ys * zs | cases ys with
| nil => simp
| cons _ _ =>
cases zs with
| nil => simp
| cons _ _ => simp_all [Int.add_mul] | no goals | ad435ae640ae9032 |
LaurentPolynomial.support_C_mul_T_of_ne_zero | Mathlib/Algebra/Polynomial/Laurent.lean | theorem support_C_mul_T_of_ne_zero {a : R} (a0 : a ≠ 0) (n : ℤ) :
Finsupp.support (C a * T n) = {n} | R : Type u_1
inst✝ : Semiring R
a : R
a0 : a ≠ 0
n : ℤ
⊢ (C a * T n).support = {n} | rw [← single_eq_C_mul_T] | R : Type u_1
inst✝ : Semiring R
a : R
a0 : a ≠ 0
n : ℤ
⊢ (Finsupp.single n a).support = {n} | ffcdd5a5ffdeb9e1 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem sat_of_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n)
(p : PosFin n → Bool) (pf : p ⊨ f) :
(insertRupUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false ... | simp only [addAssignment, ← b_eq_false, addNegAssignment, ite_false, reduceCtorEq] at h2 | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false ... | 6b3be4f74b21096f |
geometric_hahn_banach_open | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
(disj : Disjoint s t) : ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b | E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝ : 0 ∈ ... | exact disj.zero_not_mem_sub_set (vadd_mem_vadd_set_iff.1 hx₀) | no goals | 2f61733fca4a9fa5 |
CFC.monotoneOn_one_sub_one_add_inv | Mathlib/Analysis/CStarAlgebra/ApproximateUnit.lean | lemma CFC.monotoneOn_one_sub_one_add_inv :
MonotoneOn (cfcₙ (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹)) (Set.Ici (0 : A)) | case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : autoParam (0 ≤ c) _auto✝
⊢ ∀ x ∈ σ ℝ≥0 c, 1 + x ≠ 0 | intro _ _ | case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : autoParam (0 ≤ c) _auto✝
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 c
⊢ 1 + x✝ ≠ 0 | 66ae445c802b501c |
smul_orthogonalProjection_singleton | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem smul_orthogonalProjection_singleton {v : E} (w : E) :
((‖v‖ ^ 2 : ℝ) : 𝕜) • (orthogonalProjection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) (↑‖v‖ ^ 2 • w)) = ⟪v, w⟫_𝕜 • v | apply eq_orthogonalProjection_of_mem_of_inner_eq_zero | case hvm
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ⟪v, w⟫_𝕜 • v ∈ Submodule.span 𝕜 {v}
case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ∀ w_1 ∈ Submodule.span 𝕜 {v}... | eea63064d21a7d29 |
ProbabilityTheory.Kernel.tendsto_eLpNorm_one_densityProcess_limitProcess | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma tendsto_eLpNorm_one_densityProcess_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
- (countableFiltration γ).limitProcess (fun n x ↦ densityProcess κ ν n a x s) (ν a))
1 (ν a)) atTop (... | case refine_2.refine_2.refine_1.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ‖κ.... | rw [Real.nnnorm_of_nonneg (densityProcess_nonneg _ _ _ _ _ _)] | case refine_2.refine_2.refine_1.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
ε : ℝ
x✝ : 0 < ε
n : ℕ
x : γ
⊢ ⟨κ.... | 32d63f22ec24ac11 |
ContDiffOn.differentiableOn_iteratedFDerivWithin | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
(hmn : m < n) (hs : UniqueDiffOn 𝕜 s) :
DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s | 𝕜 : Type u
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
n : WithTop ℕ∞
m : ℕ
h : ContDiffOn 𝕜 n f s
hmn : ↑m < n
hs : UniqueDiffOn 𝕜 s
⊢ DifferentiableOn 𝕜 (iteratedFDeri... | intro x hx | 𝕜 : Type u
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
n : WithTop ℕ∞
m : ℕ
h : ContDiffOn 𝕜 n f s
hmn : ↑m < n
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
⊢ DifferentiableWit... | a37aa9c26950a9e9 |
differentiableWithinAt_localInvariantProp | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | theorem differentiableWithinAt_localInvariantProp :
(contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
(DifferentiableWithinAtProp I I') :=
{ is_local | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | symm | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | 3d1405d37db22f10 |
List.dropWhile_replicate_eq_filter_not | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean | theorem dropWhile_replicate_eq_filter_not (p : α → Bool) :
(replicate n a).dropWhile p = (replicate n a).filter (fun a => !p a) | case succ
α : Type u_1
a : α
p : α → Bool
n : Nat
ih : dropWhile p (replicate n a) = filter (fun a => !p a) (replicate n a)
⊢ dropWhile p (replicate (n + 1) a) = filter (fun a => !p a) (replicate (n + 1) a) | simp only [replicate_succ, dropWhile_cons] | case succ
α : Type u_1
a : α
p : α → Bool
n : Nat
ih : dropWhile p (replicate n a) = filter (fun a => !p a) (replicate n a)
⊢ (if p a = true then dropWhile p (replicate n a) else a :: replicate n a) = filter (fun a => !p a) (a :: replicate n a) | a7b440356903b59a |
Orientation.inner_eq_norm_mul_norm_mul_cos_oangle | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :
⟪x, y⟫ = ‖x‖ * ‖y‖ * Real.Angle.cos (o.oangle x y) | case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
⊢ (o.kahler x) y ≠ 0 | exact o.kahler_ne_zero hx hy | no goals | f883f52cc9e285db |
List.mapIdxMGo_eq_mapIdxMAuxSpec | Mathlib/Data/List/Indexes.lean | theorem mapIdxMGo_eq_mapIdxMAuxSpec
[LawfulMonad m] {β} (f : ℕ → α → m β) (arr : Array β) (as : List α) :
mapIdxM.go f as arr = (arr.toList ++ ·) <$> mapIdxMAuxSpec f arr.size as | case e_a
α : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
β : Type u
f : ℕ → α → m β
len : ℕ
ih :
∀ (arr : Array β) (as : List α),
as.length = len → mapIdxM.go f as arr = (fun x => arr.toList ++ x) <$> mapIdxMAuxSpec f arr.size as
arr : Array β
as : List α
head : α
tail : List α
h : tail.leng... | conv => { lhs; intro x; rw [ih _ _ h]; } | case e_a
α : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
β : Type u
f : ℕ → α → m β
len : ℕ
ih :
∀ (arr : Array β) (as : List α),
as.length = len → mapIdxM.go f as arr = (fun x => arr.toList ++ x) <$> mapIdxMAuxSpec f arr.size as
arr : Array β
as : List α
head : α
tail : List α
h : tail.leng... | 484508954c433f55 |
RieszExtension.step | Mathlib/Analysis/Convex/Cone/Extension.lean | theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x | case intro.intro.intro.intro.refine_2.mk
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : ConvexCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ ... | rcases mem_sup.1 hz with ⟨x, hx, y', hy', rfl⟩ | case intro.intro.intro.intro.refine_2.mk.intro.intro.intro.intro
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : ConvexCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ... | 4bac7c59ac26110a |
Finset.prod_update_of_not_mem | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem prod_update_of_not_mem [DecidableEq α] {s : Finset α} {i : α} (h : i ∉ s) (f : α → β)
(b : β) : ∏ x ∈ s, Function.update f i b x = ∏ x ∈ s, f x | α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
i : α
h : i ∉ s
f : α → β
b : β
⊢ ∀ x ∈ s, update f i b x = f x | intros j hj | α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
i : α
h : i ∉ s
f : α → β
b : β
j : α
hj : j ∈ s
⊢ update f i b j = f j | 09131c9f87281507 |
not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
{f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (hne : a ≠ b) (hc : c ∈ [[a, b]])
(h_deriv : ∀ᶠ x in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x)
(h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop)
(hg : deriv ... | case inl
E : Type u_1
F : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : der... | rw [← Iic_diff_right] | case inl
E : Type u_1
F : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : der... | d700313f9978d7e7 |
charP_zero_or_prime_power | Mathlib/Algebra/CharP/LocalRing.lean | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [IsLocalRing R] (q : ℕ)
[char_R_q : CharP R q] : q = 0 ∨ IsPrimePow q | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
q : ℕ
char_R_q : CharP R q
q_pos : ¬q = 0
K : Type u_1 := IsLocalRing.ResidueField R
RM_char : CharP K (ringChar K)
r : ℕ := ringChar K
n : ℕ := q.factorization r
r_prime : Nat.Prime r
a : ℕ := q / r ^ n
q_eq_a_mul_rn : q = a * r ^ n
r_ne_dvd_a : ¬r ∣ q / r ^ q.fac... | rw [← IsLocalRing.mem_maximalIdeal] at g | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
q : ℕ
char_R_q : CharP R q
q_pos : ¬q = 0
K : Type u_1 := IsLocalRing.ResidueField R
RM_char : CharP K (ringChar K)
r : ℕ := ringChar K
n : ℕ := q.factorization r
r_prime : Nat.Prime r
a : ℕ := q / r ^ n
q_eq_a_mul_rn : q = a * r ^ n
r_ne_dvd_a : ¬r ∣ q / r ^ q.fac... | 8d79fa0321d945b6 |
VitaliFamily.ae_tendsto_lintegral_div' | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_tendsto_lintegral_div' {f : α → ℝ≥0∞} (hf : Measurable f) (h'f : (∫⁻ y, f y ∂μ) ≠ ∞) :
∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x)) | case h
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → ℝ≥0∞
hf : Measurable f
h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤
ρ : Measure α := μ.withDensity f
this : IsFiniteMeasure ρ
x : α
h... | apply hx.congr' _ | α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → ℝ≥0∞
hf : Measurable f
h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤
ρ : Measure α := μ.withDensity f
this : IsFiniteMeasure ρ
x : α
hx : Ten... | fd6be72e8d74c88b |
gosper_trick | Mathlib/Combinatorics/Enumerative/Catalan.lean | theorem gosper_trick {n i : ℕ} (h : i ≤ n) :
gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =
Nat.centralBinom i / (i + 1) * Nat.centralBinom (n - i) / (n - i + 1) | n i : ℕ
h : i ≤ n
l₁ : ↑i + 1 ≠ 0
l₂ : ↑n - ↑i + 1 ≠ 0
h₁ : ↑(i + 1).centralBinom = (↑i + 1) * ↑(i + 1).centralBinom / (↑i + 1)
h₂ : ↑(n - i + 1).centralBinom = (↑n - ↑i + 1) * ↑(n - i + 1).centralBinom / (↑n - ↑i + 1)
h₃ : (↑i + 1) * ↑(i + 1).centralBinom = 2 * (2 * ↑i + 1) * ↑i.centralBinom
h₄ : (↑n - ↑i + 1) * ↑(n -... | ring | no goals | cb2d12f1c6b672a9 |
Int.add_ediv_of_dvd_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem add_ediv_of_dvd_right {a b c : Int} (H : c ∣ b) : (a + b) / c = a / c + b / c :=
if h : c = 0 then by simp [h] else by
let ⟨k, hk⟩ := H
rw [hk, Int.mul_comm c k, Int.add_mul_ediv_right _ _ h,
← Int.zero_add (k * c), Int.add_mul_ediv_right _ _ h, Int.zero_ediv, Int.zero_add]
| a b c : Int
H : c ∣ b
h : ¬c = 0
⊢ (a + b) / c = a / c + b / c | let ⟨k, hk⟩ := H | a b c : Int
H : c ∣ b
h : ¬c = 0
k : Int
hk : b = c * k
⊢ (a + b) / c = a / c + b / c | c40378507a747bb4 |
CategoryTheory.Functor.IsStronglyCartesian.map_self | Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean | @[simp]
lemma map_self : map p f φ (id_comp f).symm φ = 𝟙 a | case map
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
a✝ b✝ : 𝒳
φ : a✝ ⟶ b✝
R S : 𝒮
a b : 𝒳
inst✝ : p.IsStronglyCartesian (p.map φ) φ
⊢ map p (p.map φ) φ ⋯ φ = 𝟙 a✝ | symm | case map
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
a✝ b✝ : 𝒳
φ : a✝ ⟶ b✝
R S : 𝒮
a b : 𝒳
inst✝ : p.IsStronglyCartesian (p.map φ) φ
⊢ 𝟙 a✝ = map p (p.map φ) φ ⋯ φ | a80104e74779d5bc |
ProbabilityTheory.Kernel.eLpNorm_density_le | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma eLpNorm_density_le (hκν : fst κ ≤ ν) (a : α) (s : Set β) :
eLpNorm (fun x ↦ density κ ν a x s) 1 (ν a) ≤ ν a univ | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
s : Set β
⊢ eLpNorm (fun x => κ.density ν a x s) 1 (ν a) ≤ (ν a) univ | refine (eLpNorm_le_of_ae_bound (C := 1) (ae_of_all _ (fun t ↦ ?_))).trans ?_ | case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
a : α
s : Set β
t : γ
⊢ ‖κ.density ν a t s‖ ≤ 1
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : Measurable... | f687fb3af814580a |
Multipliable.map_iff_of_leftInverse | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | theorem Multipliable.map_iff_of_leftInverse [CommMonoid γ] [TopologicalSpace γ] {G G'}
[FunLike G α γ] [MonoidHomClass G α γ] [FunLike G' γ α] [MonoidHomClass G' γ α]
(g : G) (g' : G') (hg : Continuous g) (hg' : Continuous g') (hinv : Function.LeftInverse g' g) :
Multipliable (g ∘ f) ↔ Multipliable f :=
⟨... | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : CommMonoid α
inst✝⁶ : TopologicalSpace α
f : β → α
inst✝⁵ : CommMonoid γ
inst✝⁴ : TopologicalSpace γ
G : Type u_4
G' : Type u_5
inst✝³ : FunLike G α γ
inst✝² : MonoidHomClass G α γ
inst✝¹ : FunLike G' γ α
inst✝ : MonoidHomClass G' γ α
g : G
g' : G'
hg : Continuous ⇑g
hg' ... | rwa [← Function.comp_assoc, hinv.id] at this | no goals | 9e48d18fba79b9d6 |
Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le | Mathlib/Analysis/Calculus/MeanValue.lean | theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
(hs : Convex ℝ s) : LipschitzOnWith C f s | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
f' : E → E →L[𝕜] G
C : ℝ≥0
hf : ∀ x ∈ s, HasFDerivWithi... | intro x x_in y y_in | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
f' : E → E →L[𝕜] G
C : ℝ≥0
hf : ∀ x ∈ s, HasFDerivWithi... | a25a156b0fda3ea4 |
norm_iteratedFDerivWithin_comp_le | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem norm_iteratedFDerivWithin_comp_le {g : F → G} {f : E → F} {n : ℕ} {s : Set E} {t : Set F}
{x : E} {N : WithTop ℕ∞} (hg : ContDiffOn 𝕜 N g t) (hf : ContDiffOn 𝕜 N f s) (hn : n ≤ N)
(ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : MapsTo f s t) (hx : x ∈ s) {C : ℝ}
{D : ℝ} (hC : ∀ i, i ≤ n →... | 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
g : F → G
f : E → F
n : ℕ
s : Set E
t : Set F
x : E
N : WithTop ℕ∞
... | exact norm_iteratedFDerivWithin_comp_le_aux hgu hfu htu hs hstu hx hC hD | no goals | 03cc79be722d004a |
linearIndependent_le_infinite_basis | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι | R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
b : Basis ι R M
inst✝ : Infinite ι
κ : Type w
v : κ → M
i : LinearIndependent R v
h : ¬#κ ≤ #ι
⊢ False | rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h | R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
b : Basis ι R M
inst✝ : Infinite ι
κ : Type w
v : κ → M
i : LinearIndependent R v
h : #(Finset ι) < #κ
⊢ False | 41ace6beb6e412dd |
mul_inv_cancel_right | Mathlib/Algebra/Group/Defs.lean | theorem mul_inv_cancel_right (a b : G) : a * b * b⁻¹ = a | G : Type u_1
inst✝ : Group G
a b : G
⊢ a * b * b⁻¹ = a | rw [mul_assoc, mul_inv_cancel, mul_one] | no goals | 92a41dc7c587339a |
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
⊢ cylinder y n ⊆ cylinder x n | intro z hz 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 = x i | 1650d6a7263b81d8 |
Disjoint.exists_thickenings | Mathlib/Topology/MetricSpace/Thickening.lean | theorem _root_.Disjoint.exists_thickenings (hst : Disjoint s t) (hs : IsCompact s)
(ht : IsClosed t) :
∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) | case intro.intro.intro
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
hst : Disjoint s t
hs : IsCompact s
ht : IsClosed t
r : ℝ≥0
hr : 0 < r
h : ∀ x ∈ s, ∀ y ∈ t, ↑r < edist x y
z : α
hzs : ∃ z_1 ∈ s, edist z z_1 < ENNReal.ofReal (↑r / 2)
hzt : ∃ z_1 ∈ t, edist z z_1 < ENNReal.ofReal (↑r / 2)
⊢ z ∈ ⊥ | rw [← NNReal.coe_two, ← NNReal.coe_div, ENNReal.ofReal_coe_nnreal] at hzs hzt | case intro.intro.intro
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
hst : Disjoint s t
hs : IsCompact s
ht : IsClosed t
r : ℝ≥0
hr : 0 < r
h : ∀ x ∈ s, ∀ y ∈ t, ↑r < edist x y
z : α
hzs : ∃ z_1 ∈ s, edist z z_1 < ↑(r / 2)
hzt : ∃ z_1 ∈ t, edist z z_1 < ↑(r / 2)
⊢ z ∈ ⊥ | 1ab9118e3fded447 |
MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | /-- Auxiliary lemma for `rnDeriv_withDensity_left`. -/
lemma rnDeriv_withDensity_withDensity_rnDeriv_left (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
(hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν | α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤
this : SigmaFinite ((μ.singularPart ν).withDensity f)
⊢ ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᶠ[ae ν]
((ν.withDensity (μ.rnDeriv ν)).withDensity f + (μ.s... | have : SigmaFinite ((ν.withDensity (μ.rnDeriv ν)).withDensity f) :=
SigmaFinite.withDensity_of_ne_top (ae_mono (Measure.withDensity_rnDeriv_le _ _) hf_ne_top) | α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤
this✝ : SigmaFinite ((μ.singularPart ν).withDensity f)
this : SigmaFinite ((ν.withDensity (μ.rnDeriv ν)).withDensity f)
⊢ ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ... | 10ba1e7b1fd99023 |
Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C | Mathlib/FieldTheory/Separable.lean | theorem separable_C_mul_X_pow_add_C_mul_X_add_C
{n : ℕ} (a b c : R) (hn : (n : R) = 0) (hb : IsUnit b) :
(C a * X ^ n + C b * X + C c).Separable | R : Type u
inst✝ : CommRing R
n : ℕ
a b c : R
hn : ↑n = 0
hb : IsUnit b
f : R[X] := C a * X ^ n + C b * X + C c
⊢ f.Separable | obtain ⟨e, hb⟩ := hb.exists_left_inv | case intro
R : Type u
inst✝ : CommRing R
n : ℕ
a b c : R
hn : ↑n = 0
hb✝ : IsUnit b
f : R[X] := C a * X ^ n + C b * X + C c
e : R
hb : e * b = 1
⊢ f.Separable | 28c40967b2dee714 |
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