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
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|---|---|---|---|---|---|---|
Nat.getLast_digit_ne_zero
|
Mathlib/Data/Nat/Digits.lean
|
theorem getLast_digit_ne_zero (b : ℕ) {m : ℕ} (hm : m ≠ 0) :
(digits b m).getLast (digits_ne_nil_iff_ne_zero.mpr hm) ≠ 0
|
case succ.zero
m : ℕ
hm : m ≠ 0
⊢ ((0 + 1).digits m).getLast ⋯ ≠ 0
|
cases m
|
case succ.zero.zero
hm : 0 ≠ 0
⊢ ((0 + 1).digits 0).getLast ⋯ ≠ 0
case succ.zero.succ
n✝ : ℕ
hm : n✝ + 1 ≠ 0
⊢ ((0 + 1).digits (n✝ + 1)).getLast ⋯ ≠ 0
|
f80a47e31809cc02
|
Convex.helly_theorem_set
|
Mathlib/Analysis/Convex/Radon.lean
|
theorem helly_theorem_set {F : Finset (Set E)}
(h_card : finrank 𝕜 E + 1 ≤ #F)
(h_convex : ∀ X ∈ F, Convex 𝕜 X)
(h_inter : ∀ G : Finset (Set E), G ⊆ F → #G = finrank 𝕜 E + 1 → (⋂₀ G : Set E).Nonempty) :
(⋂₀ (F : Set (Set E))).Nonempty
|
case intro.intro.intro
𝕜 : Type u_2
E : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
F : Finset (Set E)
h_card : finrank 𝕜 E + 1 ≤ #F
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
I : Finset (Set E)
hI_ss : I ⊆ F
hI_card : #I ≤ finrank 𝕜 E + 1
J : Finset (Set E)
left✝ : I ⊆ J
hJ_ss : J ⊆ F
hJ_card : #J = finrank 𝕜 E + 1
this : ⋂₀ ↑J ⊆ ⋂₀ ↑I
⊢ (⋂₀ ↑J).Nonempty
|
exact h_inter J hJ_ss (by omega)
|
no goals
|
3b89e2c411c19e93
|
LieModule.exists_forall_mem_corootSpace_smul_add_eq_zero
|
Mathlib/Algebra/Lie/Weights/Chain.lean
|
/-- Given a (potential) root `α` relative to a Cartan subalgebra `H`, if we restrict to the ideal
`I = corootSpace α` of `H` (informally, `I = ⁅H(α), H(-α)⁆`), we may find an
integral linear combination between `α` and any weight `χ` of a representation.
This is Proposition 4.4 from [carter2005] and is a key step in the proof that the roots of a
semisimple Lie algebra form a root system. It shows that the restriction of `α` to `I` vanishes iff
the restriction of every root to `I` vanishes (which cannot happen in a semisimple Lie algebra). -/
lemma exists_forall_mem_corootSpace_smul_add_eq_zero
[IsDomain R] [IsPrincipalIdealRing R] [CharZero R] [NoZeroSMulDivisors R M] [IsNoetherian R M]
(hα : α ≠ 0) (hχ : genWeightSpace M χ ≠ ⊥) :
∃ a b : ℤ, 0 < b ∧ ∀ x ∈ corootSpace α, (a • α + b • χ) x = 0
|
R : Type u_1
L : Type u_2
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
M : Type u_3
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
inst✝⁶ : H.IsCartanSubalgebra
inst✝⁵ : IsNoetherian R L
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : CharZero R
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : IsNoetherian R M
hα : α ≠ 0
hχ : genWeightSpace M χ ≠ ⊥
p : ℤ
hp₀ : p < 0
q : ℤ
hq₀ : q > 0
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
a : ℤ := ∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ)) • i
b : ℕ := ∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ))
⊢ 0 < b
|
replace hχ : Nontrivial (genWeightSpace M χ) := by rwa [LieSubmodule.nontrivial_iff_ne_bot]
|
R : Type u_1
L : Type u_2
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
M : Type u_3
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
inst✝⁶ : H.IsCartanSubalgebra
inst✝⁵ : IsNoetherian R L
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : CharZero R
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : IsNoetherian R M
hα : α ≠ 0
p : ℤ
hp₀ : p < 0
q : ℤ
hq₀ : q > 0
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
a : ℤ := ∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ)) • i
b : ℕ := ∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ))
hχ : Nontrivial ↥(genWeightSpace M χ)
⊢ 0 < b
|
b168e9f902539ba8
|
LieAlgebra.isNilpotent_range_ad_iff
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem LieAlgebra.isNilpotent_range_ad_iff : IsNilpotent (ad R L).range ↔ IsNilpotent L
|
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
⊢ LieRing.IsNilpotent ↥(ad R L).range ↔ LieRing.IsNilpotent L
|
refine ⟨fun h => ?_, ?_⟩
|
case refine_1
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieRing.IsNilpotent ↥(ad R L).range
⊢ LieRing.IsNilpotent L
case refine_2
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
⊢ LieRing.IsNilpotent L → LieRing.IsNilpotent ↥(ad R L).range
|
b95fab998472ba6f
|
Monotone.isBoundedUnder_le_comp_iff
|
Mathlib/Order/LiminfLimsup.lean
|
theorem Monotone.isBoundedUnder_le_comp_iff [Nonempty β] [LinearOrder β] [Preorder γ] [NoMaxOrder γ]
{g : β → γ} {f : α → β} {l : Filter α} (hg : Monotone g) (hg' : Tendsto g atTop atTop) :
IsBoundedUnder (· ≤ ·) l (g ∘ f) ↔ IsBoundedUnder (· ≤ ·) l f
|
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Nonempty β
inst✝² : LinearOrder β
inst✝¹ : Preorder γ
inst✝ : NoMaxOrder γ
g : β → γ
f : α → β
l : Filter α
hg : Monotone g
hg' : Tendsto g atTop atTop
⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l (g ∘ f) → IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l f
|
rintro ⟨c, hc⟩
|
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Nonempty β
inst✝² : LinearOrder β
inst✝¹ : Preorder γ
inst✝ : NoMaxOrder γ
g : β → γ
f : α → β
l : Filter α
hg : Monotone g
hg' : Tendsto g atTop atTop
c : γ
hc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x1 x2 => x1 ≤ x2) x c
⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l f
|
27bb41283202a063
|
DFinsupp.lapply_comp_lsingle_of_ne
|
Mathlib/LinearAlgebra/DFinsupp.lean
|
theorem lapply_comp_lsingle_of_ne [DecidableEq ι] (i i' : ι) (h : i ≠ i') :
lapply i ∘ₗ lsingle i' = (0 : M i' →ₗ[R] M i)
|
case h
ι : Type u_1
R : Type u_2
M : ι → Type u_4
inst✝³ : Semiring R
inst✝² : (i : ι) → AddCommMonoid (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : DecidableEq ι
i i' : ι
h : i ≠ i'
x✝ : M i'
⊢ (lapply i ∘ₗ lsingle i') x✝ = 0 x✝
|
simp [h.symm]
|
no goals
|
b16b6e818e3e4df4
|
thickenedIndicatorAux_tendsto_indicator_closure
|
Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
|
theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞)))
|
case neg.intro.intro.intro
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEdist x E
N : ℕ
hN : ∀ b ≥ N, |δseq b| < ε
n : ℕ
n_large : n ≥ N
key : x ∉ thickening ε E
⊢ thickenedIndicatorAux (δseq n) E x ≤ 0
|
apply (thickenedIndicatorAux_mono (lt_of_abs_lt (hN n n_large)).le E x).trans
|
case neg.intro.intro.intro
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEdist x E
N : ℕ
hN : ∀ b ≥ N, |δseq b| < ε
n : ℕ
n_large : n ≥ N
key : x ∉ thickening ε E
⊢ thickenedIndicatorAux ε E x ≤ 0
|
72db20e07f758bc5
|
MeasureTheory.Lp_toLp_restrict_smul
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem Lp_toLp_restrict_smul (c : 𝕜) (f : Lp F p μ) (s : Set X) :
((Lp.memLp (c • f)).restrict s).toLp (⇑(c • f)) = c • ((Lp.memLp f).restrict s).toLp f
|
case h
X : Type u_1
F : Type u_4
mX : MeasurableSpace X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : ℝ≥0∞
μ : Measure X
c : 𝕜
f : ↥(Lp F p μ)
s : Set X
⊢ ∀ᵐ (x : X) ∂μ.restrict s,
↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x →
↑↑(c • f) x = (c • ↑↑f) x → ↑↑(MemLp.toLp ↑↑(c • f) ⋯) x = ↑↑(c • MemLp.toLp ↑↑f ⋯) x
|
refine (MemLp.coeFn_toLp ((Lp.memLp (c • f)).restrict s)).mp ?_
|
case h
X : Type u_1
F : Type u_4
mX : MeasurableSpace X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : ℝ≥0∞
μ : Measure X
c : 𝕜
f : ↥(Lp F p μ)
s : Set X
⊢ ∀ᵐ (x : X) ∂μ.restrict s,
↑↑(MemLp.toLp ↑↑(c • f) ⋯) x = ↑↑(c • f) x →
↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x →
↑↑(c • f) x = (c • ↑↑f) x → ↑↑(MemLp.toLp ↑↑(c • f) ⋯) x = ↑↑(c • MemLp.toLp ↑↑f ⋯) x
|
1d5bc8078fd879d3
|
sphere_prod
|
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
lemma sphere_prod (x : α × β) (r : ℝ) :
sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r
|
case inr.inr.h.mk.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
x' : α
y' : β
hr : 0 < dist y' x.2
⊢ dist x' x.1 ≤ dist y' x.2 ↔ dist x' x.1 ≤ dist y' x.2
|
rfl
|
no goals
|
949a8c74f6143514
|
NonUnitalAlgebra.commute_of_mem_adjoin_of_forall_mem_commute
|
Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
lemma commute_of_mem_adjoin_of_forall_mem_commute {a b : A} {s : Set A}
(hb : b ∈ adjoin R s) (h : ∀ b ∈ s, Commute a b) :
Commute a b
|
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalSemiring A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
s : Set A
hb : b ∈ adjoin R s
h : ∀ b ∈ s, Commute a b
⊢ a ∈ centralizer R s
|
simpa only [Commute.symm_iff (a := a)] using h
|
no goals
|
ccb5feb11561b3f7
|
Topology.IsUpperSet.isSheaf_of_isRightKanExtension
|
Mathlib/Topology/Sheaves/Alexandrov.lean
|
theorem Topology.IsUpperSet.isSheaf_of_isRightKanExtension
(P : (Opens X)ᵒᵖ ⥤ C)
(η : Alexandrov.principals X ⋙ P ⟶ F)
[P.IsRightKanExtension η] :
Presheaf.IsSheaf (Opens.grothendieckTopology X) P
|
X : Type v
inst✝⁵ : TopologicalSpace X
inst✝⁴ : Preorder X
inst✝³ : Topology.IsUpperSet X
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
F : X ⥤ C
P : (Opens X)ᵒᵖ ⥤ C
η : principals X ⋙ P ⟶ F
inst✝ : P.IsRightKanExtension η
γ : principals X ⋙ principalsKanExtension F ⟶ F := (principals X).pointwiseRightKanExtensionCounit F
x✝¹ : (principalsKanExtension F).IsRightKanExtension γ := inferInstance
this : P ≅ principalsKanExtension F
x✝ : Preorder ↑(of X) := inferInstanceAs (Preorder X)
⊢ TopCat.Presheaf.IsSheaf (principalsKanExtension F)
|
have _ : Topology.IsUpperSet (TopCat.of X) := inferInstanceAs <| Topology.IsUpperSet X
|
X : Type v
inst✝⁵ : TopologicalSpace X
inst✝⁴ : Preorder X
inst✝³ : Topology.IsUpperSet X
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
F : X ⥤ C
P : (Opens X)ᵒᵖ ⥤ C
η : principals X ⋙ P ⟶ F
inst✝ : P.IsRightKanExtension η
γ : principals X ⋙ principalsKanExtension F ⟶ F := (principals X).pointwiseRightKanExtensionCounit F
x✝² : (principalsKanExtension F).IsRightKanExtension γ := inferInstance
this : P ≅ principalsKanExtension F
x✝¹ : Preorder ↑(of X) := inferInstanceAs (Preorder X)
x✝ : Topology.IsUpperSet ↑(of X)
⊢ TopCat.Presheaf.IsSheaf (principalsKanExtension F)
|
484b46eb3ee8177c
|
iteratedFDerivWithin_eventually_congr_set'
|
Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
|
theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
|
case succ
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 t : Set E
f : E → F
y : E
n : ℕ
ihn : ∀ {x : E}, s =ᶠ[𝓝[{y}ᶜ] x] t → iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
x : E
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ iteratedFDerivWithin 𝕜 (n + 1) f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 (n + 1) f t
|
refine (eventually_nhds_nhdsWithin.2 h).mono fun y hy => ?_
|
case succ
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 t : Set E
f : E → F
y✝ : E
n : ℕ
ihn : ∀ {x : E}, s =ᶠ[𝓝[{y✝}ᶜ] x] t → iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
x : E
h : s =ᶠ[𝓝[{y✝}ᶜ] x] t
y : E
hy : ∀ᶠ (x : E) in 𝓝[{y✝}ᶜ] y, s x = t x
⊢ iteratedFDerivWithin 𝕜 (n + 1) f s y = iteratedFDerivWithin 𝕜 (n + 1) f t y
|
148a306dc1a96a2a
|
List.map_rotate
|
Mathlib/Data/List/Rotate.lean
|
theorem map_rotate {β : Type*} (f : α → β) (l : List α) (n : ℕ) :
map f (l.rotate n) = (map f l).rotate n
|
case succ
α : Type u
β : Type u_1
f : α → β
n : ℕ
hn : ∀ (l : List α), map f (l.rotate n) = (map f l).rotate n
l : List α
⊢ map f (l.rotate (n + 1)) = (map f l).rotate (n + 1)
|
rcases l with - | ⟨hd, tl⟩
|
case succ.nil
α : Type u
β : Type u_1
f : α → β
n : ℕ
hn : ∀ (l : List α), map f (l.rotate n) = (map f l).rotate n
⊢ map f ([].rotate (n + 1)) = (map f []).rotate (n + 1)
case succ.cons
α : Type u
β : Type u_1
f : α → β
n : ℕ
hn : ∀ (l : List α), map f (l.rotate n) = (map f l).rotate n
hd : α
tl : List α
⊢ map f ((hd :: tl).rotate (n + 1)) = (map f (hd :: tl)).rotate (n + 1)
|
b4ad2f0f7827e283
|
recursion'
|
Mathlib/Data/Real/Pi/Irrational.lean
|
/--
Auxiliary for the proof that `π` is irrational.
While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction!).
Despite the usually inconvenient subtraction, this in fact allows deducing both of the above facts
with significantly fewer analysis computations.
In addition, note the `0 ^ n` on the right hand side - this is intentional, and again allows
combining the proof of the "usual" recursion formula and the base case `I 1 θ`.
-/
private lemma recursion' (n : ℕ) :
I (n + 1) θ * θ ^ 2 = - (2 * 2 * ((n + 1) * (0 ^ n * cos θ))) +
2 * (n + 1) * (2 * n + 1) * I n θ - 4 * (n + 1) * n * I (n - 1) θ
|
θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)
v₂ : ℝ → ℝ := fun x => cos (x * θ)
v₂' : ℝ → ℝ := fun x => -sin (x * θ) * θ
hfd : Continuous f
hu₁d : Continuous u₁'
hv₁d : Continuous v₁'
hu₂d : Continuous u₂'
hv₂d : Continuous v₂'
⊢ (∫ (x : ℝ) in -1 ..1, (1 - x ^ 2) ^ (n + 1) * cos (x * θ)) * θ ^ 2 =
-(2 * 2 * ((↑n + 1) * (0 ^ n * cos θ))) + 2 * (↑n + 1) * (2 * ↑n + 1) * I n θ - 4 * (↑n + 1) * ↑n * I (n - 1) θ
|
have hu₁_eval_one : u₁ 1 = 0 := by simp only [u₁, f]; simp
|
θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)
v₂ : ℝ → ℝ := fun x => cos (x * θ)
v₂' : ℝ → ℝ := fun x => -sin (x * θ) * θ
hfd : Continuous f
hu₁d : Continuous u₁'
hv₁d : Continuous v₁'
hu₂d : Continuous u₂'
hv₂d : Continuous v₂'
hu₁_eval_one : u₁ 1 = 0
⊢ (∫ (x : ℝ) in -1 ..1, (1 - x ^ 2) ^ (n + 1) * cos (x * θ)) * θ ^ 2 =
-(2 * 2 * ((↑n + 1) * (0 ^ n * cos θ))) + 2 * (↑n + 1) * (2 * ↑n + 1) * I n θ - 4 * (↑n + 1) * ↑n * I (n - 1) θ
|
834ac76d1ec60c71
|
lt_iff_transGen_covBy
|
Mathlib/Order/Interval/Finset/Basic.lean
|
/-- In a locally finite preorder, `<` is the transitive closure of `⋖`. -/
lemma lt_iff_transGen_covBy [Preorder α] [LocallyFiniteOrder α] {x y : α} :
x < y ↔ TransGen (· ⋖ ·) x y
|
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
x y : α
h : TransGen (fun x1 x2 => x1 ⋖ x2) x y
⊢ x < y
|
induction h with
| single hx => exact hx.1
| tail _ hb ih => exact ih.trans hb.1
|
no goals
|
2dfe5e14ad9034b4
|
LinearMap.toMatrix'_apply
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
theorem LinearMap.toMatrix'_apply (f : (n → R) →ₗ[R] m → R) (i j) :
LinearMap.toMatrix' f i j = f (fun j' ↦ if j' = j then 1 else 0) i
|
case pos
R : Type u_1
inst✝² : CommSemiring R
m : Type u_4
n : Type u_5
inst✝¹ : DecidableEq n
inst✝ : Fintype n
f : (n → R) →ₗ[R] m → R
i : m
j j' : n
h : j' = j
⊢ Pi.single j 1 j' = 1
|
rw [h, Pi.single_eq_same]
|
no goals
|
914bf9aa225ab3c0
|
HahnSeries.min_orderTop_le_orderTop_add
|
Mathlib/RingTheory/HahnSeries/Addition.lean
|
theorem min_orderTop_le_orderTop_add {Γ} [LinearOrder Γ] {x y : HahnSeries Γ R} :
min x.orderTop y.orderTop ≤ (x + y).orderTop
|
case neg
R : Type u_3
inst✝¹ : AddMonoid R
Γ : Type u_8
inst✝ : LinearOrder Γ
x y : HahnSeries Γ R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
⊢ x.orderTop ⊓ y.orderTop ≤ (x + y).orderTop
|
rw [orderTop_of_ne hx, orderTop_of_ne hy, orderTop_of_ne hxy, ← WithTop.coe_min,
WithTop.coe_le_coe]
|
case neg
R : Type u_3
inst✝¹ : AddMonoid R
Γ : Type u_8
inst✝ : LinearOrder Γ
x y : HahnSeries Γ R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
⊢ ⋯.min ⋯ ⊓ ⋯.min ⋯ ≤ ⋯.min ⋯
|
2ae7da1a5993815d
|
Odd.map
|
Mathlib/Algebra/Ring/Parity.lean
|
lemma Odd.map [FunLike F α β] [RingHomClass F α β] (f : F) : Odd a → Odd (f a)
|
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Semiring α
inst✝² : Semiring β
inst✝¹ : FunLike F α β
inst✝ : RingHomClass F α β
f : F
a : α
⊢ f (2 * a + 1) = 2 * f a + 1
|
simp [two_mul]
|
no goals
|
89d2d0c4c4fac8c0
|
MeasureTheory.tendsto_diracProba_iff_tendsto
|
Mathlib/MeasureTheory/Measure/DiracProba.lean
|
lemma tendsto_diracProba_iff_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X) :
Tendsto diracProba L (𝓝 (diracProba x)) ↔ Tendsto id L (𝓝 x)
|
case mpr
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
⊢ Tendsto id L (𝓝 x) → Tendsto diracProba L (𝓝 (diracProba x))
|
intro h
|
case mpr
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : Tendsto id L (𝓝 x)
⊢ Tendsto diracProba L (𝓝 (diracProba x))
|
a577f032dda19d3d
|
MeasureTheory.measure_mul_closure_one
|
Mathlib/MeasureTheory/Group/Measure.lean
|
@[to_additive (attr := simp)]
lemma measure_mul_closure_one (s : Set G) (μ : Measure G) :
μ (s * (closure {1} : Set G)) = μ s
|
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
⊢ μ (s * closure {1}) ≤ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t
|
simp only [le_iInf_iff]
|
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
⊢ ∀ (i : Set G), s ⊆ i → MeasurableSet i → μ (s * closure {1}) ≤ μ i
|
6ce7197066cf35a1
|
IsCauSeq.of_abv_le
|
Mathlib/Algebra/Order/CauSeq/BigOperators.lean
|
lemma of_abv_le (n : ℕ) (hm : ∀ m, n ≤ m → abv (f m) ≤ a m) :
IsCauSeq abs (fun n ↦ ∑ i ∈ range n, a i) → IsCauSeq abv fun n ↦ ∑ i ∈ range n, f i
|
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
⊢ ∃ i, ∀ j ≥ i, abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) i) < ε
|
exists max n i
|
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
⊢ ∀ j ≥ n ⊔ i, abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (n ⊔ i)) < ε
|
b6f4f6743be2f6bd
|
CategoryTheory.ShortComplex.ShortExact.extClass_hom
|
Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.lean
|
@[simp]
lemma extClass_hom [HasDerivedCategory.{w'} C] : hS.extClass.hom = hS.singleδ
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
S : ShortComplex C
hS : S.ShortExact
inst✝ : HasDerivedCategory C
⊢ (isoOfHom Q W (CochainComplex.mappingCone.descShortComplex (S.map (CochainComplex.singleFunctor C 0))) ⋯).inv ≫
Q.map (CochainComplex.mappingCone.triangle ((CochainComplex.singleFunctor C 0).map S.f)).mor₃ ≫
(Q.commShiftIso 1).hom.app ((CochainComplex.singleFunctor C 0).obj S.X₁) =
(𝟙 ((singleFunctors C).functor 0)).app S.X₃ ≫
inv
(Q.map
(CochainComplex.mappingCone.descShortComplex (S.map (HomologicalComplex.single C (ComplexShape.up ℤ) 0)))) ≫
Q.map (CochainComplex.mappingCone.triangle ((HomologicalComplex.single C (ComplexShape.up ℤ) 0).map S.f)).mor₃ ≫
(Q.commShiftIso 1).hom.app ((HomologicalComplex.single C (ComplexShape.up ℤ) 0).obj S.X₁) ≫
(shiftFunctor (DerivedCategory C) 1).map
((𝟙 (((CochainComplex.singleFunctors C).postcomp Q).functor 0)).app S.X₁)
|
erw [Category.id_comp, Functor.map_id, Category.comp_id]
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
S : ShortComplex C
hS : S.ShortExact
inst✝ : HasDerivedCategory C
⊢ (isoOfHom Q W (CochainComplex.mappingCone.descShortComplex (S.map (CochainComplex.singleFunctor C 0))) ⋯).inv ≫
Q.map (CochainComplex.mappingCone.triangle ((CochainComplex.singleFunctor C 0).map S.f)).mor₃ ≫
(Q.commShiftIso 1).hom.app ((CochainComplex.singleFunctor C 0).obj S.X₁) =
inv
(Q.map
(CochainComplex.mappingCone.descShortComplex (S.map (HomologicalComplex.single C (ComplexShape.up ℤ) 0)))) ≫
Q.map (CochainComplex.mappingCone.triangle ((HomologicalComplex.single C (ComplexShape.up ℤ) 0).map S.f)).mor₃ ≫
(Q.commShiftIso 1).hom.app ((HomologicalComplex.single C (ComplexShape.up ℤ) 0).obj S.X₁)
|
ecc4c8a588d5dbf9
|
Set.mk_preimage_tprod
|
Mathlib/Data/Prod/TProd.lean
|
theorem mk_preimage_tprod :
∀ (l : List ι) (t : ∀ i, Set (α i)), TProd.mk l ⁻¹' Set.tprod l t = { i | i ∈ l }.pi t
| [], t => by simp [Set.tprod]
| i :: l, t => by
ext f
have h : TProd.mk l f ∈ Set.tprod l t ↔ ∀ i : ι, i ∈ l → f i ∈ t i
|
case h
ι : Type u
α : ι → Type v
i : ι
l : List ι
t : (i : ι) → Set (α i)
f : (i : ι) → α i
⊢ f ∈ TProd.mk (i :: l) ⁻¹' Set.tprod (i :: l) t ↔ f ∈ {i_1 | i_1 ∈ i :: l}.pi t
|
have h : TProd.mk l f ∈ Set.tprod l t ↔ ∀ i : ι, i ∈ l → f i ∈ t i := by
change f ∈ TProd.mk l ⁻¹' Set.tprod l t ↔ f ∈ { x | x ∈ l }.pi t
rw [mk_preimage_tprod l t]
|
case h
ι : Type u
α : ι → Type v
i : ι
l : List ι
t : (i : ι) → Set (α i)
f : (i : ι) → α i
h : TProd.mk l f ∈ Set.tprod l t ↔ ∀ i ∈ l, f i ∈ t i
⊢ f ∈ TProd.mk (i :: l) ⁻¹' Set.tprod (i :: l) t ↔ f ∈ {i_1 | i_1 ∈ i :: l}.pi t
|
2802ceb5fdebf838
|
Filter.HasBasis.isVonNBounded_iff
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
theorem _root_.Filter.HasBasis.isVonNBounded_iff {q : ι → Prop} {s : ι → Set E} {A : Set E}
(h : (𝓝 (0 : E)).HasBasis q s) : IsVonNBounded 𝕜 A ↔ ∀ i, q i → Absorbs 𝕜 (s i) A
|
𝕜 : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
q : ι → Prop
s : ι → Set E
A : Set E
h : (𝓝 0).HasBasis q s
hA : ∀ (i : ι), q i → Absorbs 𝕜 (s i) A
V : Set E
hV : V ∈ 𝓝 0
⊢ Absorbs 𝕜 V A
|
rcases h.mem_iff.mp hV with ⟨i, hi, hV⟩
|
case intro.intro
𝕜 : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
q : ι → Prop
s : ι → Set E
A : Set E
h : (𝓝 0).HasBasis q s
hA : ∀ (i : ι), q i → Absorbs 𝕜 (s i) A
V : Set E
hV✝ : V ∈ 𝓝 0
i : ι
hi : q i
hV : s i ⊆ V
⊢ Absorbs 𝕜 V A
|
d51ba1066824d9b9
|
dist_left_midpoint_eq_dist_right_midpoint
|
Mathlib/Analysis/Normed/Affine/AddTorsor.lean
|
theorem dist_left_midpoint_eq_dist_right_midpoint (p₁ p₂ : P) :
dist p₁ (midpoint 𝕜 p₁ p₂) = dist p₂ (midpoint 𝕜 p₁ p₂)
|
V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₁ (midpoint 𝕜 p₁ p₂) = dist p₂ (midpoint 𝕜 p₁ p₂)
|
rw [dist_left_midpoint p₁ p₂, dist_right_midpoint p₁ p₂]
|
no goals
|
69da2c93168aa2f0
|
Nat.Partrec.Code.evaln_mono
|
Mathlib/Computability/PartrecCode.lean
|
theorem evaln_mono : ∀ {k₁ k₂ c n x}, k₁ ≤ k₂ → x ∈ evaln k₁ c n → x ∈ evaln k₂ c n
| 0, k₂, c, n, x, _, h => by simp [evaln] at h
| k + 1, k₂ + 1, c, n, x, hl, h => by
have hl' := Nat.le_of_succ_le_succ hl
have :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ → (x ∈ o₁ → x ∈ o₂) →
x ∈ do { guard (n ≤ k); o₁ } → x ∈ do { guard (n ≤ k₂); o₂ }
|
case zero
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
pure 0)
n =
some x
h : x ∈ pure 0
⊢ x ∈ pure 0
case succ
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
pure n.succ)
n =
some x
h : x ∈ pure n.succ
⊢ x ∈ pure n.succ
case left
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
pure (unpair n).1)
n =
some x
h : x ∈ pure (unpair n).1
⊢ x ∈ pure (unpair n).1
case right
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
pure (unpair n).2)
n =
some x
h : x ∈ pure (unpair n).2
⊢ x ∈ pure (unpair n).2
case pair
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
Nat.pair <$> evaln (k + 1) cf n <*> evaln (k + 1) cg n)
n =
some x
h : x ∈ Nat.pair <$> evaln (k + 1) cf n <*> evaln (k + 1) cg n
⊢ x ∈ Nat.pair <$> evaln (k₂ + 1) cf n <*> evaln (k₂ + 1) cg n
case comp
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
let x ← evaln (k + 1) cg n
evaln (k + 1) cf x)
n =
some x
h :
x ∈ do
let x ← evaln (k + 1) cg n
evaln (k + 1) cf x
⊢ x ∈ do
let x ← evaln (k₂ + 1) cg n
evaln (k₂ + 1) cf x
case prec
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
unpaired
(fun a n =>
Nat.casesOn n (evaln (k + 1) cf a) fun y => do
let i ← evaln k (cf.prec cg) (Nat.pair a y)
evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
n)
n =
some x
h :
x ∈
unpaired
(fun a n =>
Nat.casesOn n (evaln (k + 1) cf a) fun y => do
let i ← evaln k (cf.prec cg) (Nat.pair a y)
evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
n
⊢ x ∈
unpaired
(fun a n =>
Nat.casesOn n (evaln (k₂ + 1) cf a) fun y => do
let i ← evaln k₂ (cf.prec cg) (Nat.pair a y)
evaln (k₂ + 1) cg (Nat.pair a (Nat.pair y i)))
n
case rfind'
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
unpaired
(fun a m => do
let x ← evaln (k + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k cf.rfind' (Nat.pair a (m + 1)))
n)
n =
some x
h :
x ∈
unpaired
(fun a m => do
let x ← evaln (k + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k cf.rfind' (Nat.pair a (m + 1)))
n
⊢ x ∈
unpaired
(fun a m => do
let x ← evaln (k₂ + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k₂ cf.rfind' (Nat.pair a (m + 1)))
n
|
iterate 4 exact h
|
case pair
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
Nat.pair <$> evaln (k + 1) cf n <*> evaln (k + 1) cg n)
n =
some x
h : x ∈ Nat.pair <$> evaln (k + 1) cf n <*> evaln (k + 1) cg n
⊢ x ∈ Nat.pair <$> evaln (k₂ + 1) cf n <*> evaln (k₂ + 1) cg n
case comp
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
let x ← evaln (k + 1) cg n
evaln (k + 1) cf x)
n =
some x
h :
x ∈ do
let x ← evaln (k + 1) cg n
evaln (k + 1) cf x
⊢ x ∈ do
let x ← evaln (k₂ + 1) cg n
evaln (k₂ + 1) cf x
case prec
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
unpaired
(fun a n =>
Nat.casesOn n (evaln (k + 1) cf a) fun y => do
let i ← evaln k (cf.prec cg) (Nat.pair a y)
evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
n)
n =
some x
h :
x ∈
unpaired
(fun a n =>
Nat.casesOn n (evaln (k + 1) cf a) fun y => do
let i ← evaln k (cf.prec cg) (Nat.pair a y)
evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
n
⊢ x ∈
unpaired
(fun a n =>
Nat.casesOn n (evaln (k₂ + 1) cf a) fun y => do
let i ← evaln k₂ (cf.prec cg) (Nat.pair a y)
evaln (k₂ + 1) cg (Nat.pair a (Nat.pair y i)))
n
case rfind'
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
cf : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
n x : ℕ
h✝ :
(fun n => do
guard (n ≤ k)
unpaired
(fun a m => do
let x ← evaln (k + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k cf.rfind' (Nat.pair a (m + 1)))
n)
n =
some x
h :
x ∈
unpaired
(fun a m => do
let x ← evaln (k + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k cf.rfind' (Nat.pair a (m + 1)))
n
⊢ x ∈
unpaired
(fun a m => do
let x ← evaln (k₂ + 1) cf (Nat.pair a m)
if x = 0 then pure m else evaln k₂ cf.rfind' (Nat.pair a (m + 1)))
n
|
231a1123011d0046
|
Turing.PartrecToTM2.codeSupp'_supports
|
Mathlib/Computability/TMToPartrec.lean
|
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S
|
case comp.refine_1
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (f.comp g) k ⊆ S
H'✝ : trStmts₁ (trNormal (f.comp g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k) ⊆ S
⊢ codeSupp f k ⊆ S
|
simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
|
case comp.refine_1
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H'✝ : trStmts₁ (trNormal (f.comp g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ⊆ S ∧ trStmts₁ (trNormal f k) ⊆ S ∧ codeSupp' f k ⊆ S
H :
(trStmts₁ (trNormal (f.comp g) k) ⊆ S ∧
codeSupp' g (Cont'.comp f k) ⊆ S ∧ trStmts₁ (trNormal f k) ⊆ S ∧ codeSupp' f k ⊆ S) ∧
contSupp k ⊆ S
⊢ codeSupp' f k ⊆ S ∧ contSupp k ⊆ S
|
5fb2278c84599fe0
|
ENNReal.trichotomy₂
|
Mathlib/Data/ENNReal/Real.lean
|
theorem trichotomy₂ {p q : ℝ≥0∞} (hpq : p ≤ q) :
p = 0 ∧ q = 0 ∨
p = 0 ∧ q = ∞ ∨
p = 0 ∧ 0 < q.toReal ∨
p = ∞ ∧ q = ∞ ∨
0 < p.toReal ∧ q = ∞ ∨ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal
|
case inr.inr.h.h.h.h
p q : ℝ≥0∞
hpq : p ≤ q
hp : 0 < p
hq : q < ⊤
⊢ 0 < p.toReal ∧ q = ⊤ ∨ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal
|
right
|
case inr.inr.h.h.h.h.h
p q : ℝ≥0∞
hpq : p ≤ q
hp : 0 < p
hq : q < ⊤
⊢ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal
|
3be24298098eb436
|
HahnSeries.ofPowerSeries_X_pow
|
Mathlib/RingTheory/HahnSeries/PowerSeries.lean
|
theorem ofPowerSeries_X_pow {R} [Semiring R] (n : ℕ) :
ofPowerSeries Γ R (PowerSeries.X ^ n) = single (n : Γ) 1
|
Γ : Type u_1
inst✝¹ : StrictOrderedSemiring Γ
R : Type u_3
inst✝ : Semiring R
n : ℕ
⊢ (ofPowerSeries Γ R) (PowerSeries.X ^ n) = (single ↑n) 1
|
simp
|
no goals
|
727042d063ddfdf9
|
Finset.mulDysonETransform_idem
|
Mathlib/Combinatorics/Additive/ETransform.lean
|
theorem mulDysonETransform_idem :
mulDysonETransform e (mulDysonETransform e x) = mulDysonETransform e x
|
case snd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ x.2 ∩ e⁻¹ • x.1 ∩ e⁻¹ • (x.1 ∪ e • x.2) = x.2 ∩ e⁻¹ • x.1
|
rw [smul_finset_union, inv_smul_smul, union_comm, inter_eq_left]
|
case snd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ x.2 ∩ e⁻¹ • x.1 ⊆ x.2 ∪ e⁻¹ • x.1
|
8c8645060ea3b1ed
|
Nat.bisect_add_one_false
|
Mathlib/.lake/packages/batteries/Batteries/Data/Nat/Bisect.lean
|
theorem bisect_add_one_false {p : Nat → Bool} (h : start < stop) (hstart : p start = true)
(hstop : p stop = false) : p (bisect h hstart hstop + 1) = false
|
start stop : Nat
p : Nat → Bool
h : start < stop
hstart : p start = true
hstop : p stop = false
h' : ¬start < start.avg stop
heq : start.avg stop = start
⊢ p (start.avg stop + 1) = false
|
rw [← hstop, heq]
|
start stop : Nat
p : Nat → Bool
h : start < stop
hstart : p start = true
hstop : p stop = false
h' : ¬start < start.avg stop
heq : start.avg stop = start
⊢ p (start + 1) = p stop
|
ac8252c3fc1bc6fa
|
GenContFract.compExactValue_correctness_of_stream_eq_some_aux_comp
|
Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean
|
theorem compExactValue_correctness_of_stream_eq_some_aux_comp {a : K} (b c : K)
(fract_a_ne_zero : Int.fract a ≠ 0) :
((⌊a⌋ : K) * b + c) / Int.fract a + b = (b * a + c) / Int.fract a
|
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
a b c : K
fract_a_ne_zero : Int.fract a ≠ 0
⊢ ↑⌊a⌋ * b + c + b * Int.fract a = b * a + c
|
rw [Int.fract]
|
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
a b c : K
fract_a_ne_zero : Int.fract a ≠ 0
⊢ ↑⌊a⌋ * b + c + b * (a - ↑⌊a⌋) = b * a + c
|
4bc31c2d1c2c0d60
|
QuaternionGroup.quaternionGroup_one_isCyclic
|
Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
|
theorem quaternionGroup_one_isCyclic : IsCyclic (QuaternionGroup 1)
|
case hx
⊢ orderOf ?x = Nat.card (QuaternionGroup 1)
|
rw [Nat.card_eq_fintype_card, card, mul_one]
|
case hx
⊢ orderOf ?x = 4
|
eb8b7a5c1af5daba
|
CategoryTheory.HasExt.standard
|
Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
|
lemma HasExt.standard : HasExt.{max u v} C
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : (C : Type ?u.37386) → [inst : Category.{?u.37387, ?u.37386} C] → [inst_1 : Abelian C] → HasDerivedCategory C :=
HasDerivedCategory.standard
⊢ HasExt C
|
exact hasExt_of_hasDerivedCategory _
|
no goals
|
fa5cffabd0d06fd7
|
CategoryTheory.ShortComplex.HomologyData.exact_iff_i_p_zero
|
Mathlib/Algebra/Homology/ShortComplex/Exact.lean
|
lemma HomologyData.exact_iff_i_p_zero (h : S.HomologyData) :
S.Exact ↔ h.left.i ≫ h.right.p = 0
|
case mp
C : Type u_1
inst✝¹ : Category.{u_3, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
this : S.HasHomology
⊢ IsZero h.left.H → h.left.π ≫ h.iso.hom ≫ h.right.ι = 0
|
intro z
|
case mp
C : Type u_1
inst✝¹ : Category.{u_3, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
this : S.HasHomology
z : IsZero h.left.H
⊢ h.left.π ≫ h.iso.hom ≫ h.right.ι = 0
|
d036aacb3bf9ed60
|
Array.getElem_insertIdx_loop
|
Mathlib/.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean
|
theorem getElem_insertIdx_loop {as : Array α} {i : Nat} {j : Nat} {hj : j < as.size} {k : Nat} {h} :
(insertIdx.loop i as ⟨j, hj⟩)[k] =
if h₁ : k < i then
as[k]'(by simpa using h)
else
if h₂ : k = i then
if i ≤ j then as[j] else as[i]'(by simpa [h₂] using h)
else
if k ≤ j then as[k-1]'(by simp at h; omega) else as[k]'(by simpa using h)
|
α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
⊢ (insertIdx.loop i as ⟨j, hj⟩)[k] =
if h₁ : k < i then as[k]
else if h₂ : k = i then if i ≤ j then as[j] else as[i] else if k ≤ j then as[k - 1] else as[k]
|
split <;> rename_i h₁
|
case isTrue
α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
h₁ : k < i
⊢ (insertIdx.loop i as ⟨j, hj⟩)[k] = as[k]
case isFalse
α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
h₁ : ¬k < i
⊢ (insertIdx.loop i as ⟨j, hj⟩)[k] =
if h₂ : k = i then if i ≤ j then as[j] else as[i] else if k ≤ j then as[k - 1] else as[k]
|
7047eb0b3b596b09
|
ContinuousLinearMap.finset_sum_comp
|
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
theorem finset_sum_comp {ι : Type*} {s : Finset ι}
[ContinuousAdd M₃] (g : ι → M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂) : (∑ i ∈ s, g i).comp f = ∑ i ∈ s, (g i).comp f
|
R₁ : Type u_1
R₂ : Type u_2
R₃ : Type u_3
inst✝¹³ : Semiring R₁
inst✝¹² : Semiring R₂
inst✝¹¹ : Semiring R₃
σ₁₂ : R₁ →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R₁ →+* R₃
M₁ : Type u_4
inst✝¹⁰ : TopologicalSpace M₁
inst✝⁹ : AddCommMonoid M₁
M₂ : Type u_6
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : AddCommMonoid M₂
M₃ : Type u_7
inst✝⁶ : TopologicalSpace M₃
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Module R₁ M₁
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
ι : Type u_9
s : Finset ι
inst✝ : ContinuousAdd M₃
g : ι → M₂ →SL[σ₂₃] M₃
f : M₁ →SL[σ₁₂] M₂
⊢ (∑ i ∈ s, g i).comp f = ∑ i ∈ s, (g i).comp f
|
ext
|
case h
R₁ : Type u_1
R₂ : Type u_2
R₃ : Type u_3
inst✝¹³ : Semiring R₁
inst✝¹² : Semiring R₂
inst✝¹¹ : Semiring R₃
σ₁₂ : R₁ →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R₁ →+* R₃
M₁ : Type u_4
inst✝¹⁰ : TopologicalSpace M₁
inst✝⁹ : AddCommMonoid M₁
M₂ : Type u_6
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : AddCommMonoid M₂
M₃ : Type u_7
inst✝⁶ : TopologicalSpace M₃
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Module R₁ M₁
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
ι : Type u_9
s : Finset ι
inst✝ : ContinuousAdd M₃
g : ι → M₂ →SL[σ₂₃] M₃
f : M₁ →SL[σ₁₂] M₂
x✝ : M₁
⊢ ((∑ i ∈ s, g i).comp f) x✝ = (∑ i ∈ s, (g i).comp f) x✝
|
8c914d1a3ca1f3ba
|
List.getLast?_take
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean
|
theorem getLast?_take {l : List α} : (l.take n).getLast? = if n = 0 then none else l[n - 1]?.or l.getLast?
|
α : Type u_1
n : Nat
l : List α
⊢ (if min n l.length - 1 < n then l[min n l.length - 1]? else none) = if n = 0 then none else l[n - 1]?.or l.getLast?
|
split
|
case isTrue
α : Type u_1
n : Nat
l : List α
h✝ : min n l.length - 1 < n
⊢ l[min n l.length - 1]? = if n = 0 then none else l[n - 1]?.or l.getLast?
case isFalse
α : Type u_1
n : Nat
l : List α
h✝ : ¬min n l.length - 1 < n
⊢ none = if n = 0 then none else l[n - 1]?.or l.getLast?
|
05a9802408c6ca2f
|
Complex.exp_eq_exp_ℂ
|
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ
|
x : ℂ
⊢ cexp x = NormedSpace.exp ℂ x
|
rw [Complex.exp, exp_eq_tsum_div]
|
x : ℂ
⊢ (exp' x).lim = (fun x => ∑' (n : ℕ), x ^ n / ↑n !) x
|
61d2d71496263d1b
|
Basis.reindexRange_repr'
|
Mathlib/LinearAlgebra/Basis/Defs.lean
|
theorem reindexRange_repr' (x : M) {bi : M} {i : ι} (h : b i = bi) :
b.reindexRange.repr x ⟨bi, ⟨i, h⟩⟩ = b.repr x i
|
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
x bi : M
i : ι
h : b i = bi
⊢ (b.reindexRange.repr x) ⟨bi, ⋯⟩ = (b.repr x) i
|
nontriviality
|
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
x bi : M
i : ι
h : b i = bi
a✝ : Nontrivial R
⊢ (b.reindexRange.repr x) ⟨bi, ⋯⟩ = (b.repr x) i
|
d778555276f771cb
|
Array.flatten_eq_push_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem flatten_eq_push_iff {xs : Array (Array α)} {ys : Array α} {y : α} :
xs.flatten = ys.push y ↔
∃ (as : Array (Array α)) (bs : Array α) (cs : Array (Array α)),
xs = as.push (bs.push y) ++ cs ∧ (∀ l, l ∈ cs → l = #[]) ∧ ys = as.flatten ++ bs
|
case of.mk.mp.inr.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
y : α
as : List (List α)
bs : List α
c : α
cs : List α
ds : List (List α)
h : [y] = c :: cs ++ ds.flatten
⊢ ∃ as_1 bs_1 cs_1,
(List.map List.toArray (as ++ (bs ++ c :: cs) :: ds)).toArray = as_1.push (bs_1.push y) ++ cs_1 ∧
(∀ (l : Array α), l ∈ cs_1 → l = #[]) ∧ { toList := as.flatten ++ bs } = as_1.flatten ++ bs_1
|
rw [List.singleton_eq_append_iff] at h
|
case of.mk.mp.inr.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
y : α
as : List (List α)
bs : List α
c : α
cs : List α
ds : List (List α)
h : c :: cs = [] ∧ ds.flatten = [y] ∨ c :: cs = [y] ∧ ds.flatten = []
⊢ ∃ as_1 bs_1 cs_1,
(List.map List.toArray (as ++ (bs ++ c :: cs) :: ds)).toArray = as_1.push (bs_1.push y) ++ cs_1 ∧
(∀ (l : Array α), l ∈ cs_1 → l = #[]) ∧ { toList := as.flatten ++ bs } = as_1.flatten ++ bs_1
|
dbe592481896286a
|
LinearMap.map_le_map_iff
|
Mathlib/LinearAlgebra/Span/Basic.lean
|
theorem map_le_map_iff (f : F) {p p'} : map f p ≤ map f p' ↔ p ≤ p' ⊔ ker f
|
R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁸ : Semiring R
inst✝⁷ : Semiring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝² : RingHomSurjective τ₁₂
F : Type u_8
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
f : F
p p' : Submodule R M
⊢ map f p ≤ map f p' ↔ p ≤ p' ⊔ ker f
|
rw [map_le_iff_le_comap, Submodule.comap_map_eq]
|
no goals
|
4f08d29df089c851
|
MeasureTheory.setIntegral_abs_condExp_le
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
|
theorem setIntegral_abs_condExp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ
|
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
|
have : ∫ x in s, |(μ[f|m]) x| ∂μ = ∫ x, |(μ[s.indicator f|m]) x| ∂μ := by
rw [← integral_indicator (hnm _ hs)]
refine integral_congr_ae ?_
have : (fun x => |(μ[s.indicator f|m]) x|) =ᵐ[μ] fun x => |s.indicator (μ[f|m]) x| :=
(condExp_indicator hfint hs).fun_comp abs
refine EventuallyEq.trans (Eventually.of_forall fun x => ?_) this.symm
rw [← Real.norm_eq_abs, norm_indicator_eq_indicator_norm]
simp only [Real.norm_eq_abs]
|
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
this : ∫ (x : α) in s, |(μ[f|m]) x| ∂μ = ∫ (x : α), |(μ[s.indicator f|m]) x| ∂μ
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
|
f75c894a42ad0f54
|
le_nhdsAdjoint_iff'
|
Mathlib/Topology/Order.lean
|
theorem le_nhdsAdjoint_iff' {a : α} {f : Filter α} {t : TopologicalSpace α} :
t ≤ nhdsAdjoint a f ↔ @nhds α t a ≤ pure a ⊔ f ∧ ∀ b ≠ a, @nhds α t b = pure b
|
α : Type u
a : α
f : Filter α
t : TopologicalSpace α
⊢ t ≤ nhdsAdjoint a f ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b
|
simp_rw [le_iff_nhds, nhds_nhdsAdjoint, forall_update_iff, (pure_le_nhds _).le_iff_eq]
|
no goals
|
c347397418b97452
|
differentiableAt_iff_comp_add_const
|
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
lemma differentiableAt_iff_comp_add_const {a b : 𝕜} :
DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (x + b)) (a - b)
|
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a b : 𝕜
⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x => f (x + b)) (a - b)
|
simp [differentiableAt_comp_add_const]
|
no goals
|
76441202ae7eef99
|
Isometry.norm_map_of_map_one
|
Mathlib/Analysis/Normed/Group/Uniform.lean
|
theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
‖f x‖ = ‖x‖
|
E : Type u_2
F : Type u_3
inst✝¹ : SeminormedGroup E
inst✝ : SeminormedGroup F
f : E → F
hi : Isometry f
h₁ : f 1 = 1
x : E
⊢ ‖f x‖ = ‖x‖
|
rw [← dist_one_right, ← h₁, hi.dist_eq, dist_one_right]
|
no goals
|
4199a2679c9513dc
|
LieSubmodule.lie_baseChange
|
Mathlib/Algebra/Lie/BaseChange.lean
|
lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆
|
case refine_2.intro.intro.intro.intro.add
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ[R] ⁅x, n⁆ = m}
this : ⇑((TensorProduct.mk R A M) 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s
x : A ⊗[R] L
hx : x ∈ baseChange A I
⊢ ∀ (x y : A ⊗[R] L) (hx : x ∈ Submodule.span A ↑(Submodule.map ((TensorProduct.mk R A L) 1) ↑I))
(hy : y ∈ Submodule.span A ↑(Submodule.map ((TensorProduct.mk R A L) 1) ↑I)),
(fun x' x => ∀ m' ∈ baseChange A N, ⁅x', m'⁆ ∈ Submodule.span A s) x hx →
(fun x' x => ∀ m' ∈ baseChange A N, ⁅x', m'⁆ ∈ Submodule.span A s) y hy →
(fun x' x => ∀ m' ∈ baseChange A N, ⁅x', m'⁆ ∈ Submodule.span A s) (x + y) ⋯
|
intro x y _ _ hx hy m' hm'
|
case refine_2.intro.intro.intro.intro.add
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSubmodule R L M
s : Set (A ⊗[R] M) := {m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ[R] ⁅x, n⁆ = m}
this : ⇑((TensorProduct.mk R A M) 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s
x✝ : A ⊗[R] L
hx✝¹ : x✝ ∈ baseChange A I
x y : A ⊗[R] L
hx✝ : x ∈ Submodule.span A ↑(Submodule.map ((TensorProduct.mk R A L) 1) ↑I)
hy✝ : y ∈ Submodule.span A ↑(Submodule.map ((TensorProduct.mk R A L) 1) ↑I)
hx : ∀ m' ∈ baseChange A N, ⁅x, m'⁆ ∈ Submodule.span A s
hy : ∀ m' ∈ baseChange A N, ⁅y, m'⁆ ∈ Submodule.span A s
m' : A ⊗[R] M
hm' : m' ∈ baseChange A N
⊢ ⁅x + y, m'⁆ ∈ Submodule.span A s
|
c0e93f202362c69d
|
MeasureTheory.hasFDerivAt_convolution_right_with_param
|
Mathlib/Analysis/Convolution.lean
|
theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)) (q₀ : P × G)
(hq₀ : q₀.1 ∈ s) :
HasFDerivAt (fun q : P × G => (f ⋆[L, μ] g q.1) q.2)
((f ⋆[L.precompR (P × G), μ] fun x : G => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀
|
case neg
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A✝ : ∀ p ∈ s, Continuous (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
A : IsCompact ({q₀.1} ×ˢ k)
t : Set (P × G)
kt : {q₀.1} ×ˢ k ⊆ t
t_open : IsOpen t
ht : Bornology.IsBounded (g' '' t)
ε : ℝ
εpos : 0 < ε
hε : thickening ε ({q₀.1} ×ˢ k) ⊆ t
h'ε : ball q₀.1 ε ⊆ s
C : ℝ
Cpos : 0 < C
hC : g' '' t ⊆ closedBall 0 C
p : P
x : G
hp : ‖p - q₀.1‖ < ε
hps : p ∈ s
hx : x ∉ k
this : g' (p, x) = 0
⊢ ‖0‖ ≤ C
|
simpa only [norm_zero] using Cpos.le
|
no goals
|
681ec723c7fd0b70
|
MvPolynomial.support_sdiff_support_subset_support_add
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
theorem support_sdiff_support_subset_support_add [DecidableEq σ] (p q : MvPolynomial σ R) :
p.support \ q.support ⊆ (p + q).support
|
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p q : MvPolynomial σ R
⊢ p.support \ q.support ⊆ (p + q).support
|
intro m hm
|
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
p q : MvPolynomial σ R
m : σ →₀ ℕ
hm : m ∈ p.support \ q.support
⊢ m ∈ (p + q).support
|
fb17947fed01177e
|
nhds_nhdsAdjoint_same
|
Mathlib/Topology/Order.lean
|
theorem nhds_nhdsAdjoint_same (a : α) (f : Filter α) :
@nhds α (nhdsAdjoint a f) a = pure a ⊔ f
|
α : Type u
a : α
f : Filter α
⊢ 𝓝 a = pure a ⊔ f
|
let _ := nhdsAdjoint a f
|
α : Type u
a : α
f : Filter α
x✝ : TopologicalSpace α := nhdsAdjoint a f
⊢ 𝓝 a = pure a ⊔ f
|
fb86baaa4749a61f
|
Cardinal.nat_coe_dvd_iff
|
Mathlib/SetTheory/Cardinal/Divisibility.lean
|
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m
|
case intro
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
this : ↑m < ℵ₀
⊢ n ∣ m
|
rw [hk, mul_lt_aleph0_iff] at this
|
case intro
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
this : ↑n = 0 ∨ k = 0 ∨ ↑n < ℵ₀ ∧ k < ℵ₀
⊢ n ∣ m
|
76e842e0713287f3
|
CategoryTheory.Triangulated.Subcategory.isoClosure_W
|
Mathlib/CategoryTheory/Triangulated/Subcategory.lean
|
lemma isoClosure_W : S.isoClosure.W = S.W
|
case h.mp.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
S : Subcategory C
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f g h ∈ distinguishedTriangles
Z' : C
hZ' : S.P Z'
e : Z ≅ Z'
⊢ Triangle.mk f (g ≫ e.hom) (e.inv ≫ h) ≅ Triangle.mk f g h
|
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) e.symm
|
no goals
|
860294c19a783fa0
|
Finset.mem_map_equiv
|
Mathlib/Data/Finset/Image.lean
|
theorem mem_map_equiv {f : α ≃ β} {b : β} : b ∈ s.map f.toEmbedding ↔ f.symm b ∈ s
|
α : Type u_1
β : Type u_2
s : Finset α
f : α ≃ β
b : β
⊢ b ∈ map f.toEmbedding s ↔ f.symm b ∈ s
|
rw [mem_map]
|
α : Type u_1
β : Type u_2
s : Finset α
f : α ≃ β
b : β
⊢ (∃ a ∈ s, f.toEmbedding a = b) ↔ f.symm b ∈ s
|
40c90e1b99d7e288
|
ContinuousLinearMap.projKerOfRightInverse_apply_idem
|
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
theorem projKerOfRightInverse_apply_idem [IsTopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
(f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : LinearMap.ker f₁) :
f₁.projKerOfRightInverse f₂ h x = x
|
case a
R : Type u_1
inst✝⁹ : Ring R
R₂ : Type u_2
inst✝⁸ : Ring R₂
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
M₂ : Type u_5
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M
inst✝² : Module R₂ M₂
σ₁₂ : R →+* R₂
σ₂₁ : R₂ →+* R
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : IsTopologicalAddGroup M
f₁ : M →SL[σ₁₂] M₂
f₂ : M₂ →SL[σ₂₁] M
h : Function.RightInverse ⇑f₂ ⇑f₁
x : ↥(LinearMap.ker f₁)
⊢ ↑((f₁.projKerOfRightInverse f₂ h) ↑x) = ↑x
|
simp
|
no goals
|
8e79a5a2228c5cb7
|
Nat.floorRoot_zero_right
|
Mathlib/Data/Nat/Factorization/Root.lean
|
@[simp] lemma floorRoot_zero_right (n : ℕ) : floorRoot n 0 = 0
|
n : ℕ
⊢ n.floorRoot 0 = 0
|
simp [floorRoot]
|
no goals
|
d167cce764104f8a
|
discrim_le_zero
|
Mathlib/Algebra/QuadraticDiscriminant.lean
|
theorem discrim_le_zero (h : ∀ x : K, 0 ≤ a * (x * x) + b * x + c) : discrim a b c ≤ 0
|
case inr.inl.inr
K : Type u_1
inst✝ : LinearOrderedField K
b c : K
h : ∀ (x : K), 0 ≤ 0 * (x * x) + b * x + c
hb : b ≠ 0
this : 0 ≤ 0 * ((-c - 1) / b * ((-c - 1) / b)) + b * ((-c - 1) / b) + c
⊢ b * b - 4 * 0 * c ≤ 0
|
rw [mul_div_cancel₀ _ hb] at this
|
case inr.inl.inr
K : Type u_1
inst✝ : LinearOrderedField K
b c : K
h : ∀ (x : K), 0 ≤ 0 * (x * x) + b * x + c
hb : b ≠ 0
this : 0 ≤ 0 * ((-c - 1) / b * ((-c - 1) / b)) + (-c - 1) + c
⊢ b * b - 4 * 0 * c ≤ 0
|
1756922daf2602f9
|
AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
|
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
|
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
(hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q
|
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
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)
|
rw [mul_comm]
|
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
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (i * m)
|
e11b14794a0cd9b6
|
IsLocalMin.fderiv_eq_zero
|
Mathlib/Analysis/Calculus/LocalExtr/Basic.lean
|
theorem IsLocalMin.fderiv_eq_zero (h : IsLocalMin f a) : fderiv ℝ f a = 0
|
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
h : IsLocalMin f a
⊢ fderiv ℝ f a = 0
|
classical
exact if hf : DifferentiableAt ℝ f a then h.hasFDerivAt_eq_zero hf.hasFDerivAt
else fderiv_zero_of_not_differentiableAt hf
|
no goals
|
8ae5d0ec02b23fbb
|
jacobiTheta_S_smul
|
Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean
|
theorem jacobiTheta_S_smul (τ : ℍ) :
jacobiTheta ↑(ModularGroup.S • τ) = (-I * τ) ^ (1 / 2 : ℂ) * jacobiTheta τ
|
τ : ℍ
⊢ jacobiTheta ↑(ModularGroup.S • τ) = (-I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ
|
have h0 : (τ : ℂ) ≠ 0 := ne_of_apply_ne im (zero_im.symm ▸ ne_of_gt τ.2)
|
τ : ℍ
h0 : ↑τ ≠ 0
⊢ jacobiTheta ↑(ModularGroup.S • τ) = (-I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ
|
0586d0787d847c58
|
LieModule.iSup_ucs_eq_genWeightSpace_zero
|
Mathlib/Algebra/Lie/Weights/Basic.lean
|
/-- See also `LieModule.iInf_lowerCentralSeries_eq_posFittingComp`. -/
lemma iSup_ucs_eq_genWeightSpace_zero [IsNoetherian R M] :
⨆ k, (⊥ : LieSubmodule R L M).ucs k = genWeightSpace M (0 : L → R)
|
case intro
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : LieRing.IsNilpotent L
inst✝ : IsNoetherian R M
k : ℕ
hk : genWeightSpace M 0 ≤ LieSubmodule.ucs k ⊥
⊢ LieSubmodule.ucs k ⊥ ≤ ⨆ k, LieSubmodule.ucs k ⊥
|
exact le_iSup (fun k ↦ (⊥ : LieSubmodule R L M).ucs k) k
|
no goals
|
953c26500f119b1f
|
Nat.image_div_divisors_eq_divisors
|
Mathlib/NumberTheory/Divisors.lean
|
theorem image_div_divisors_eq_divisors (n : ℕ) :
image (fun x : ℕ => n / x) n.divisors = n.divisors
|
case neg.h.mp.intro.intro
n : ℕ
hn : ¬n = 0
a x : ℕ
hx1 : x ∣ n ∧ n ≠ 0
hx2 : n / x = a
⊢ a ∣ n ∧ n ≠ 0
|
refine ⟨?_, hn⟩
|
case neg.h.mp.intro.intro
n : ℕ
hn : ¬n = 0
a x : ℕ
hx1 : x ∣ n ∧ n ≠ 0
hx2 : n / x = a
⊢ a ∣ n
|
fa9501818042115c
|
Set.chainHeight_le_chainHeight_TFAE
|
Mathlib/Order/Height.lean
|
theorem chainHeight_le_chainHeight_TFAE (s : Set α) (t : Set β) :
TFAE [s.chainHeight ≤ t.chainHeight, ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l = length l',
∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l ≤ length l']
|
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s : Set α
t : Set β
⊢ s.chainHeight ≤ t.chainHeight ↔ ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, l.length ≤ l'.length
|
convert ← chainHeight_add_le_chainHeight_add s t 0 0 <;> apply add_zero
|
no goals
|
c0f9e0f54d2e27df
|
MulChar.isQuadratic_iff_sq_eq_one
|
Mathlib/NumberTheory/MulChar/Basic.lean
|
/-- A multiplicative character `χ` into an integral domain is quadratic
if and only if `χ^2 = 1`. -/
lemma isQuadratic_iff_sq_eq_one {M R : Type*} [CommMonoid M] [CommRing R] [NoZeroDivisors R]
[Nontrivial R] {χ : MulChar M R} :
IsQuadratic χ ↔ χ ^ 2 = 1
|
case refine_1.inl
M : Type u_4
R : Type u_5
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
χ : MulChar M R
h : χ.IsQuadratic
x : Mˣ
H : χ ↑x = 0
⊢ χ ↑x ^ 2 = 1
|
exact (not_isUnit_zero <| H ▸ IsUnit.map χ <| x.isUnit).elim
|
no goals
|
6577a1512c271721
|
CategoryTheory.mapPair_equifibered
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
NatTrans.Equifibered α
|
case mk.right.mk.up.up
C : Type u
inst✝ : Category.{v, u} C
F F' : Discrete WalkingPair ⥤ C
α : F ⟶ F'
⊢ IsPullback (𝟙 (F.obj { as := WalkingPair.right })) (α.app { as := WalkingPair.right })
(α.app { as := WalkingPair.right }) (𝟙 (F'.obj { as := WalkingPair.right }))
|
exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩
|
no goals
|
75bbeab18b3afdbf
|
FractionalIdeal.div_spanSingleton
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J
|
case pos
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : d = 0
⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J
|
simp only [hd, spanSingleton_zero, div_zero, zero_mul]
|
no goals
|
3a4f3cf63940a3cf
|
CategoryTheory.Limits.cokernelZeroIsoTarget_hom
|
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
theorem cokernelZeroIsoTarget_hom :
cokernelZeroIsoTarget.hom = cokernel.desc (0 : X ⟶ Y) (𝟙 Y) (by simp)
|
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
X Y : C
⊢ coequalizer.π 0 0 ≫ cokernelZeroIsoTarget.hom = coequalizer.π 0 0 ≫ cokernel.desc 0 (𝟙 Y) ⋯
|
simp [cokernelZeroIsoTarget]
|
no goals
|
10fc23a19e773db4
|
ConcaveOn.slope_anti_adjacent
|
Mathlib/Analysis/Convex/Slope.lean
|
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
|
have := neg_le_neg (ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz)
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : -(((-f) z - (-f) y) / (z - y)) ≤ -(((-f) y - (-f) x) / (y - x))
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
|
266390e102ca535a
|
EuclideanGeometry.reflection_mem_of_le_of_mem
|
Mathlib/Geometry/Euclidean/Basic.lean
|
theorem reflection_mem_of_le_of_mem {s₁ s₂ : AffineSubspace ℝ P} [Nonempty s₁]
[HasOrthogonalProjection s₁.direction] (hle : s₁ ≤ s₂) {p : P} (hp : p ∈ s₂) :
reflection s₁ p ∈ s₂
|
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥s₁
inst✝ : HasOrthogonalProjection s₁.direction
hle : s₁ ≤ s₂
p : P
hp : p ∈ s₂
⊢ (reflection s₁) p ∈ s₂
|
rw [reflection_apply]
|
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥s₁
inst✝ : HasOrthogonalProjection s₁.direction
hle : s₁ ≤ s₂
p : P
hp : p ∈ s₂
⊢ (↑((orthogonalProjection s₁) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₁) p) ∈ s₂
|
d7a829dab8264775
|
IsCyclotomicExtension.adjoin_primitive_root_eq_top
|
Mathlib/NumberTheory/Cyclotomic/Basic.lean
|
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤
|
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A ((cyclotomic (↑n) A).rootSet B) = ⊤
|
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
|
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1} = ⊤
|
ad8b2bbb3401eddb
|
Polynomial.addSubmonoid_closure_setOf_eq_monomial
|
Mathlib/Algebra/Polynomial/Basic.lean
|
theorem addSubmonoid_closure_setOf_eq_monomial :
AddSubmonoid.closure { p : R[X] | ∃ n a, p = monomial n a } = ⊤
|
case h
R : Type u
inst✝ : Semiring R
⊢ ⊤ ≤ AddSubmonoid.closure {p | ∃ n a, p = (monomial n) a}
|
rw [← AddSubmonoid.map_equiv_top (toFinsuppIso R).symm.toAddEquiv, ←
Finsupp.add_closure_setOf_eq_single, AddMonoidHom.map_mclosure]
|
case h
R : Type u
inst✝ : Semiring R
⊢ AddSubmonoid.closure (⇑(toFinsuppIso R).symm.toAddEquiv '' {f | ∃ a b, f = Finsupp.single a b}) ≤
AddSubmonoid.closure {p | ∃ n a, p = (monomial n) a}
|
4cb938b2b4ef705e
|
AlgebraicGeometry.iSup_opensRange_sigmaι
|
Mathlib/AlgebraicGeometry/Limits.lean
|
lemma iSup_opensRange_sigmaι : ⨆ i, (Sigma.ι f i).opensRange = ⊤ :=
eq_top_iff.mpr fun x ↦ by simpa using exists_sigmaι_eq f x
|
ι : Type u
f : ι → Scheme
x : ↑↑(∐ f).toPresheafedSpace
⊢ x ∈ ↑⊤ → x ∈ ↑(⨆ i, Scheme.Hom.opensRange (Sigma.ι f i))
|
simpa using exists_sigmaι_eq f x
|
no goals
|
10b96ceb72ebd425
|
CategoryTheory.FreeBicategory.normalizeAux_congr
|
Mathlib/CategoryTheory/Bicategory/Coherence.lean
|
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g
|
case mk.h.left_unitor_inv
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p (𝟙 a✝ ≫ f✝)
|
funext
|
case mk.h.left_unitor_inv.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ f✝ = normalizeAux x✝ (𝟙 a✝ ≫ f✝)
|
023558b97807da1b
|
Convex.convexJoin
|
Mathlib/Analysis/Convex/Join.lean
|
theorem Convex.convexJoin (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) :
Convex 𝕜 (convexJoin 𝕜 s t)
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
x₁ : E
hx₁ : x₁ ∈ s
y₁ : E
hy₁ : y₁ ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
x₂ : E
hx₂ : x₂ ∈ s
y₂ : E
hy₂ : y₂ ∈ t
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
p q : 𝕜
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ ∃ i, ∃ (_ : i ∈ s), ∃ i_1, ∃ (_ : i_1 ∈ t), p • (a₁ • x₁ + b₁ • y₁) + q • (a₂ • x₂ + b₂ • y₂) ∈ [i-[𝕜]i_1]
|
rcases hs.exists_mem_add_smul_eq hx₁ hx₂ (mul_nonneg hp ha₁) (mul_nonneg hq ha₂) with ⟨x, hxs, hx⟩
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
x₁ : E
hx₁ : x₁ ∈ s
y₁ : E
hy₁ : y₁ ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
x₂ : E
hx₂ : x₂ ∈ s
y₂ : E
hy₂ : y₂ ∈ t
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
p q : 𝕜
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
x : E
hxs : x ∈ s
hx : (p * a₁ + q * a₂) • x = (p * a₁) • x₁ + (q * a₂) • x₂
⊢ ∃ i, ∃ (_ : i ∈ s), ∃ i_1, ∃ (_ : i_1 ∈ t), p • (a₁ • x₁ + b₁ • y₁) + q • (a₂ • x₂ + b₂ • y₂) ∈ [i-[𝕜]i_1]
|
8773104f02cc3ff3
|
integrable_rpow_mul_exp_neg_mul_sq
|
Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
|
theorem integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
Integrable fun x : ℝ => x ^ s * exp (-b * x ^ 2)
|
case hf
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ AEStronglyMeasurable (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (volume.restrict (Ioi 0))
|
apply Measurable.aestronglyMeasurable
|
case hf.hf
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ Measurable fun x => (-x) ^ s * rexp (-b * x ^ 2)
|
7d42832261d7a82c
|
MulAction.orbitRel_subgroupOf
|
Mathlib/GroupTheory/GroupAction/Basic.lean
|
@[to_additive]
lemma orbitRel_subgroupOf (H K : Subgroup G) :
orbitRel (H.subgroupOf K) α = orbitRel (H ⊓ K : Subgroup G) α
|
case a.refine_2.intro.mk
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H K : Subgroup G
b✝ : α
gv : G
gp : gv ∈ Subgroup.map K.subtype (H.subgroupOf K)
⊢ (fun m => m • b✝) ⟨gv, gp⟩ ∈ orbit (↥(H.subgroupOf K)) b✝
|
simp only [Submonoid.mk_smul]
|
case a.refine_2.intro.mk
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H K : Subgroup G
b✝ : α
gv : G
gp : gv ∈ Subgroup.map K.subtype (H.subgroupOf K)
⊢ ⟨gv, gp⟩ • b✝ ∈ orbit (↥(H.subgroupOf K)) b✝
|
8d6ed0da5cf28147
|
Seminorm.closedBall_zero_eq_preimage_closedBall
|
Mathlib/Analysis/Seminorm.lean
|
theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r
|
𝕜 : Type u_3
E : Type u_7
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
⊢ p.closedBall 0 r = ⇑p ⁻¹' Metric.closedBall 0 r
|
rw [closedBall_zero_eq, preimage_metric_closedBall]
|
no goals
|
f0da54906ed73db9
|
LieAlgebra.radical_eq_top_of_isSolvable
|
Mathlib/Algebra/Lie/Solvable.lean
|
@[simp] lemma radical_eq_top_of_isSolvable [IsSolvable L] :
radical R L = ⊤
|
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSolvable L
h : IsSolvable ↥⊤
⊢ ⊤ ≤ radical R L
|
exact le_sSup h
|
no goals
|
75f4078fc152f6b5
|
MvPowerSeries.lexOrder_mul
|
Mathlib/RingTheory/MvPowerSeries/LexOrder.lean
|
theorem lexOrder_mul [NoZeroDivisors R] (φ ψ : MvPowerSeries σ R) :
lexOrder (φ * ψ) = lexOrder φ + lexOrder ψ
|
case pos
σ : Type u_1
R : Type u_2
inst✝³ : Semiring R
inst✝² : LinearOrder σ
inst✝¹ : WellFoundedGT σ
inst✝ : NoZeroDivisors R
φ ψ : MvPowerSeries σ R
hφ : ¬φ = 0
hψ : ψ = 0
⊢ (φ * ψ).lexOrder = φ.lexOrder + ψ.lexOrder
|
simp only [hψ, mul_zero, lexOrder_zero, add_top]
|
no goals
|
445a75654a0141af
|
List.setIfInBounds_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean
|
theorem setIfInBounds_toArray (l : List α) (i : Nat) (a : α) :
l.toArray.setIfInBounds i a = (l.set i a).toArray
|
α : Type u_1
l : List α
i : Nat
a : α
⊢ l.toArray.setIfInBounds i a = (l.set i a).toArray
|
apply ext'
|
case h
α : Type u_1
l : List α
i : Nat
a : α
⊢ (l.toArray.setIfInBounds i a).toList = (l.set i a).toArray.toList
|
7a113e107f9a51d9
|
Polynomial.IsSeparableContraction.degree_eq
|
Mathlib/RingTheory/Polynomial/SeparableDegree.lean
|
theorem IsSeparableContraction.degree_eq [hF : ExpChar F q] (g : F[X])
(hg : IsSeparableContraction q f g) : g.natDegree = hf.degree
|
case prime.intro.intro.intro.intro
F : Type u_1
inst✝ : Field F
q : ℕ
f : F[X]
hf : HasSeparableContraction q f
g : F[X]
hprime✝ : Nat.Prime q
hchar✝ : CharP F q
hg : g.Separable
m : ℕ
hm : (expand F (q ^ m)) g = f
g' : F[X] := Classical.choose hf
hg' : (Classical.choose hf).Separable
m' : ℕ
hm' : (expand F (q ^ m')) (Classical.choose hf) = f
this : Fact (Nat.Prime q)
⊢ (expand F (q ^ m)) g = (expand F (q ^ m')) g'
|
rw [hm, hm']
|
no goals
|
e8a084ffc5e2adb4
|
FreeAlgebra.ι_ne_algebraMap
|
Mathlib/Algebra/FreeAlgebra.lean
|
theorem ι_ne_algebraMap [Nontrivial R] (x : X) (r : R) : ι R x ≠ algebraMap R _ r := fun h ↦ by
let f0 : FreeAlgebra R X →ₐ[R] R := lift R 0
let f1 : FreeAlgebra R X →ₐ[R] R := lift R 1
have hf0 : f0 (ι R x) = 0 := lift_ι_apply _ _
have hf1 : f1 (ι R x) = 1 := lift_ι_apply _ _
rw [h, f0.commutes, Algebra.id.map_eq_self] at hf0
rw [h, f1.commutes, Algebra.id.map_eq_self] at hf1
exact zero_ne_one (hf0.symm.trans hf1)
|
R : Type u_1
inst✝¹ : CommSemiring R
X : Type u_2
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : f0 (ι R x) = 0
⊢ False
|
have hf1 : f1 (ι R x) = 1 := lift_ι_apply _ _
|
R : Type u_1
inst✝¹ : CommSemiring R
X : Type u_2
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : f0 (ι R x) = 0
hf1 : f1 (ι R x) = 1
⊢ False
|
dcbb22fde95f412f
|
Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto
|
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
theorem circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto {c : ℂ}
{R : ℝ} (h0 : 0 < R) {f : ℂ → E} {y : E} {s : Set ℂ} (hs : s.Countable)
(hc : ContinuousOn f (closedBall c R \ {c}))
(hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z) (hy : Tendsto f (𝓝[{c}ᶜ] c) (𝓝 y)) :
(∮ z in C(c, R), (z - c)⁻¹ • f z) = (2 * π * I : ℂ) • y
|
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
⊢ 2 * π * r * ε = ε * (r * (2 * π))
|
ac_rfl
|
no goals
|
161c13e075bc65e7
|
CategoryTheory.Limits.WidePushout.eq_desc_of_comp_eq
|
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
|
theorem eq_desc_of_comp_eq (g : widePushout _ _ arrows ⟶ X) :
(∀ j : J, ι arrows j ≫ g = fs j) → head arrows ≫ g = f → g = desc f fs w
|
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
⊢ (∀ (j : J), ι arrows j ≫ g = fs j) → head arrows ≫ g = f → g = desc f fs w
|
intro h1 h2
|
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ g = desc f fs w
|
9b335e052985e4ad
|
MeasureTheory.Measure.isEverywherePos_everywherePosSubset
|
Mathlib/MeasureTheory/Measure/EverywherePos.lean
|
/-- In a space with an inner regular measure, the everywhere positive subset of a measurable set
is itself everywhere positive. This is not obvious as `μ.everywherePosSubset s` is defined as
the points whose neighborhoods intersect `s` along positive measure subsets, but this does not
say they also intersect `μ.everywherePosSubset s` along positive measure subsets. -/
lemma isEverywherePos_everywherePosSubset
[OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
μ.IsEverywherePos (μ.everywherePosSubset s)
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
⊢ μ.IsEverywherePos (μ.everywherePosSubset s)
|
intro x hx n hn
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
⊢ 0 < μ n
|
b60d9263f112e009
|
Tropical.add_pow
|
Mathlib/Algebra/Tropical/Basic.lean
|
theorem add_pow [LinearOrder R] [AddMonoid R] [AddLeftMono R] [AddRightMono R]
(x y : Tropical R) (n : ℕ) :
(x + y) ^ n = x ^ n + y ^ n
|
R : Type u
inst✝³ : LinearOrder R
inst✝² : AddMonoid R
inst✝¹ : AddLeftMono R
inst✝ : AddRightMono R
x y : Tropical R
n : ℕ
⊢ (x + y) ^ n = x ^ n + y ^ n
|
rcases le_total x y with h | h
|
case inl
R : Type u
inst✝³ : LinearOrder R
inst✝² : AddMonoid R
inst✝¹ : AddLeftMono R
inst✝ : AddRightMono R
x y : Tropical R
n : ℕ
h : x ≤ y
⊢ (x + y) ^ n = x ^ n + y ^ n
case inr
R : Type u
inst✝³ : LinearOrder R
inst✝² : AddMonoid R
inst✝¹ : AddLeftMono R
inst✝ : AddRightMono R
x y : Tropical R
n : ℕ
h : y ≤ x
⊢ (x + y) ^ n = x ^ n + y ^ n
|
231576695e18102f
|
LinearEquiv.neg_apply
|
Mathlib/Algebra/Module/Equiv/Basic.lean
|
theorem neg_apply (x : M) : neg R x = -x
|
R : Type u_1
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : M
⊢ (neg R) x = -x
|
simp
|
no goals
|
2c4cd459210b0404
|
Polynomial.degreeLE_eq_span_X_pow
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem degreeLE_eq_span_X_pow [DecidableEq R] {n : ℕ} :
degreeLE R n = Submodule.span R ↑((Finset.range (n + 1)).image fun n => (X : R[X]) ^ n)
|
case a
R : Type u
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℕ
⊢ ↑(range (n + 1)) ⊆ (fun n => X ^ n) ⁻¹' ↑(degreeLE R ↑n)
|
intro k hk
|
case a
R : Type u
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n k : ℕ
hk : k ∈ ↑(range (n + 1))
⊢ k ∈ (fun n => X ^ n) ⁻¹' ↑(degreeLE R ↑n)
|
093264c8e0a4af45
|
QuadraticMap.associated_linMulLin
|
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
theorem associated_linMulLin [Invertible (2 : R)] (f g : M →ₗ[R] R) :
associated (R := R) (N := R) (linMulLin f g) =
⅟ (2 : R) • ((mul R R).compl₁₂ f g + (mul R R).compl₁₂ g f)
|
R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Invertible 2
f g : M →ₗ[R] R
⊢ associated (linMulLin f g) = ⅟2 • ((mul R R).compl₁₂ f g + (mul R R).compl₁₂ g f)
|
ext
|
case h.h
R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Invertible 2
f g : M →ₗ[R] R
x✝¹ x✝ : M
⊢ ((associated (linMulLin f g)) x✝¹) x✝ = ((⅟2 • ((mul R R).compl₁₂ f g + (mul R R).compl₁₂ g f)) x✝¹) x✝
|
380e1e77dd3b3c1a
|
AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean
|
theorem inter_nonempty_of_nonempty_of_sup_direction_eq_top {s₁ s₂ : AffineSubspace k P}
(h1 : (s₁ : Set P).Nonempty) (h2 : (s₂ : Set P).Nonempty)
(hd : s₁.direction ⊔ s₂.direction = ⊤) : ((s₁ : Set P) ∩ s₂).Nonempty
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
s₁ s₂ : AffineSubspace k P
h1 : (↑s₁).Nonempty
h2 : (↑s₂).Nonempty
hd : s₁.direction ⊔ s₂.direction = ⊤
h : ¬(↑s₁ ∩ ↑s₂).Nonempty
⊢ False
|
rw [Set.not_nonempty_iff_eq_empty] at h
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
s₁ s₂ : AffineSubspace k P
h1 : (↑s₁).Nonempty
h2 : (↑s₂).Nonempty
hd : s₁.direction ⊔ s₂.direction = ⊤
h : ↑s₁ ∩ ↑s₂ = ∅
⊢ False
|
29b6a92f85b1cedf
|
AlgebraicGeometry.RingedSpace.isUnit_res_of_isUnit_germ
|
Mathlib/Geometry/RingedSpace/Basic.lean
|
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : X) (hx : x ∈ U)
(h : IsUnit (X.presheaf.germ U x hx f)) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x ∈ V), IsUnit (X.presheaf.map i.op f)
|
case intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↑↑X.toPresheafedSpace
hx : x ∈ U
h : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)
g' : ↑(X.presheaf.stalk x)
heq : (ConcreteCategory.hom (X.presheaf.germ U x hx)) f * g' = 1
⊢ ∃ V i, ∃ (_ : x ∈ V), IsUnit ((ConcreteCategory.hom (X.presheaf.map i.op)) f)
|
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x g'
|
case intro.intro.intro.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↑↑X.toPresheafedSpace
hx : x ∈ U
h : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : x ∈ V
g : ToType (X.presheaf.obj (op V))
heq : (ConcreteCategory.hom (X.presheaf.germ U x hx)) f * (ConcreteCategory.hom (X.presheaf.germ V x hxV)) g = 1
⊢ ∃ V i, ∃ (_ : x ∈ V), IsUnit ((ConcreteCategory.hom (X.presheaf.map i.op)) f)
|
c4e0030823594ac8
|
BitVec.zero_or
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem zero_or {x : BitVec w} : 0#w ||| x = x
|
w : Nat
x : BitVec w
⊢ 0#w ||| x = x
|
ext i
|
case pred
w : Nat
x : BitVec w
i : Nat
a✝ : i < w
⊢ (0#w ||| x).getLsbD i = x.getLsbD i
|
179340e5607fd2c1
|
Submodule.span_smul_of_span_eq_top
|
Mathlib/Algebra/Algebra/Tower.lean
|
theorem span_smul_of_span_eq_top {s : Set S} (hs : span R s = ⊤) (t : Set A) :
span R (s • t) = (span S t).restrictScalars R :=
le_antisymm
(span_le.2 fun _x ⟨p, _hps, _q, hqt, hpqx⟩ ↦ hpqx ▸ (span S t).smul_mem p (subset_span hqt))
fun _ hp ↦ closure_induction (hx := hp) (zero_mem _) (fun _ _ _ _ ↦ add_mem) fun s0 y hy ↦ by
refine span_induction (fun x hx ↦ subset_span <| by exact ⟨x, hx, y, hy, rfl⟩) ?_ ?_ ?_
(hs ▸ mem_top : s0 ∈ span R s)
· rw [zero_smul]; apply zero_mem
· intro _ _ _ _; rw [add_smul]; apply add_mem
· intro r s0 _ hy; rw [IsScalarTower.smul_assoc]; exact smul_mem _ r hy
|
case refine_1
R : Type u
S : Type v
A : Type w
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R S
inst✝² : Module S A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
s : Set S
hs : span R s = ⊤
t : Set A
x✝ : A
hp : x✝ ∈ restrictScalars R (span S t)
s0 : S
y : A
hy : y ∈ t
⊢ 0 ∈ span R (s • t)
|
apply zero_mem
|
no goals
|
9949108e14d83982
|
MvPolynomial.exists_rename_eq_of_vars_subset_range
|
Mathlib/Algebra/MvPolynomial/Variables.lean
|
theorem exists_rename_eq_of_vars_subset_range (p : MvPolynomial σ R) (f : τ → σ) (hfi : Injective f)
(hf : ↑p.vars ⊆ Set.range f) : ∃ q : MvPolynomial τ R, rename f q = p :=
⟨aeval (fun i : σ => Option.elim' 0 X <| partialInv f i) p,
by
show (rename f).toRingHom.comp _ p = RingHom.id _ p
refine hom_congr_vars ?_ ?_ ?_
· ext1
simp [algebraMap_eq]
· intro i hip _
rcases hf hip with ⟨i, rfl⟩
simp [partialInv_left hfi]
· rfl⟩
|
case refine_3
R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
f : τ → σ
hfi : Injective f
hf : ↑p.vars ⊆ range f
⊢ p = p
|
rfl
|
no goals
|
8ffc8e66df6fb3bc
|
Batteries.BinomialHeap.Imp.Heap.realSize_tail?
|
Mathlib/.lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
|
theorem Heap.realSize_tail? {s : Heap α} : s.tail? le = some s' →
s.realSize = s'.realSize + 1
|
α : Type u_1
le : α → α → Bool
s' s : Heap α
eq : Option.map (fun x => x.snd) (deleteMin le s) = some s'
a : α
tl : Heap α
eq₂ : deleteMin le s = some (a, tl)
⊢ s.realSize = ((fun x => x.snd) (a, tl)).realSize + 1
|
exact realSize_deleteMin eq₂
|
no goals
|
28fc4ef177708b16
|
Matrix.aeval_self_charpoly
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean
|
theorem aeval_self_charpoly (M : Matrix n n R) : aeval M M.charpoly = 0
|
R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (aeval M) M.charpoly = 0
|
have h : M.charpoly • (1 : Matrix n n R[X]) = adjugate (charmatrix M) * charmatrix M :=
(adjugate_mul _).symm
|
R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
h : M.charpoly • 1 = M.charmatrix.adjugate * M.charmatrix
⊢ (aeval M) M.charpoly = 0
|
650c4547af86a3b8
|
Matrix.charpoly_sub_diagonal_degree_lt
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
theorem charpoly_sub_diagonal_degree_lt :
(M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1)
|
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1)
|
rw [charpoly, det_apply', ← insert_erase (mem_univ (Equiv.refl n)),
sum_insert (not_mem_erase (Equiv.refl n) univ), add_comm]
|
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i : n, M.charmatrix (x i) i +
↑↑(Equiv.Perm.sign (Equiv.refl n)) * ∏ i : n, M.charmatrix ((Equiv.refl n) i) i -
∏ i : n, (X - C (M i i))).degree <
↑(Fintype.card n - 1)
|
f188622eede59de8
|
List.attach_filterMap
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Attach.lean
|
theorem attach_filterMap {l : List α} {f : α → Option β} :
(l.filterMap f).attach = l.attach.filterMap
fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩)
|
case cons.h_2.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ map (fun x_1 => ⟨x_1.val, ⋯⟩) (attach ?m.63176) =
⟨a, ⋯⟩ :: filterMap ((fun x_1 => (f x_1.val).pbind fun a h => some ⟨a, ⋯⟩) ∘ fun x_1 => ⟨x_1.val, ⋯⟩) xs.attach
case cons.h_2.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ (match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs) =
?m.63176
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ List β
|
rotate_left
|
case cons.h_2.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ (match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs) =
?m.63176
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ List β
case cons.h_2.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
x : α
xs : List α
ih :
(filterMap f xs).attach =
filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
xs.attach
x✝ :
Option
{ x_1 //
x_1 ∈
match f x with
| none => filterMap f xs
| some b => b :: filterMap f xs }
a : β
h✝ : a ∈ f x
h : f x = some a
⊢ map (fun x_1 => ⟨x_1.val, ⋯⟩) (attach ?m.63176) =
⟨a, ⋯⟩ :: filterMap ((fun x_1 => (f x_1.val).pbind fun a h => some ⟨a, ⋯⟩) ∘ fun x_1 => ⟨x_1.val, ⋯⟩) xs.attach
|
689b23e1dd2280bf
|
NonUnitalSubring.closure_induction₂
|
Mathlib/RingTheory/NonUnitalSubring/Basic.lean
|
theorem closure_induction₂ {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop}
(mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy))
(zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
(neg_left : ∀ x y hx hy, p x y hx hy → p (-x) y (neg_mem hx) hy)
(neg_right : ∀ x y hx hy, p x y hx hy → p x (-y) hx (neg_mem hy))
(add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
(add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
(mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
(mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz))
{x y : R} (hx : x ∈ closure s) (hy : y ∈ closure s) :
p x y hx hy
|
case mem.mul
R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
neg_left : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p (-x) y ⋯ hy
neg_right : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p x (-y) hx ⋯
add_left :
∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x✝ y✝ : R
hx✝ : x✝ ∈ closure s
hy✝ : y✝ ∈ closure s
h₁ : p x✝ z hx✝ ⋯
h₂ : p y✝ z hy✝ ⋯
⊢ p (x✝ * y✝) z ⋯ ⋯
|
exact mul_left _ _ _ _ _ _ h₁ h₂
|
no goals
|
412c69976f3f02c1
|
PMF.restrict_toMeasure_support
|
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
theorem restrict_toMeasure_support [MeasurableSingletonClass α] (p : PMF α) :
Measure.restrict (toMeasure p) (support p) = toMeasure p
|
case h
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
p : PMF α
s : Set α
hs : MeasurableSet s
⊢ p.toMeasure (s ∩ p.support) = p.toMeasure s
|
apply toMeasure_apply_inter_support p s hs p.support_countable.measurableSet
|
no goals
|
59c875c4e07c5477
|
PowerSeries.coeff_inv_aux
|
Mathlib/RingTheory/PowerSeries/Inverse.lean
|
theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff R n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0
|
case neg
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
⊢ (-a *
∑ x ∈ antidiagonal (single () n),
if x.2 < single () n then
(MvPowerSeries.coeff R x.1) φ * (MvPowerSeries.coeff R x.2) (MvPowerSeries.inv.aux a φ)
else 0) =
-a * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff R x.1) φ * (coeff R x.2) (MvPowerSeries.inv.aux a φ) else 0
|
congr 1
|
case neg.e_a
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
⊢ (∑ x ∈ antidiagonal (single () n),
if x.2 < single () n then (MvPowerSeries.coeff R x.1) φ * (MvPowerSeries.coeff R x.2) (MvPowerSeries.inv.aux a φ)
else 0) =
∑ x ∈ antidiagonal n, if x.2 < n then (coeff R x.1) φ * (coeff R x.2) (MvPowerSeries.inv.aux a φ) else 0
|
0d1be33aa33f7301
|
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