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List.set_eq_take_append_cons_drop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem set_eq_take_append_cons_drop (l : List α) (n : Nat) (a : α) :
l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l | case pos
α : Type u_1
l : List α
a : α
m : Nat
h : m < l.length
h' : ¬m < m
⊢ min m l.length ≤ m | exact Nat.min_le_left m l.length | no goals | 9fb4c99baced165c |
Irreducible.dvd_iff | Mathlib/Algebra/GroupWithZero/Associated.lean | theorem Irreducible.dvd_iff [Monoid M] {x y : M} (hx : Irreducible x) :
y ∣ x ↔ IsUnit y ∨ Associated x y | case mp.intro.inr
M : Type u_1
inst✝ : Monoid M
x y : M
hx : Irreducible x
z : M
hz : x = y * z
h : IsUnit z
⊢ IsUnit y ∨ y * z ~ᵤ y | exact Or.inr (associated_mul_unit_left _ _ h) | no goals | 6a7560cd158385cb |
PerfectClosure.mul_aux_left | Mathlib/FieldTheory/PerfectClosure.lean | theorem mul_aux_left (x1 x2 y : ℕ × K) (H : R K p x1 x2) :
mk K p (x1.1 + y.1, (frobenius K p)^[y.1] x1.2 * (frobenius K p)^[x1.1] y.2) =
mk K p (x2.1 + y.1, (frobenius K p)^[y.1] x2.2 * (frobenius K p)^[x2.1] y.2) :=
match x1, x2, H with
| _, _, R.intro n x =>
Quot.sound <| by
rw [← iterate_succ_apply, iterate_succ_apply', iterate_succ_apply', ← frobenius_mul,
Nat.succ_add]
apply R.intro
| K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x1 x2 y : ℕ × K
H : R K p x1 x2
n : ℕ
x : K
⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)
((n + 1, (frobenius K p) x).1 + y.1,
(⇑(frobenius K p))^[y.1] (n + 1, (frobenius K p) x).2 * (⇑(frobenius K p))^[(n + 1, (frobenius K p) x).1] y.2) | rw [← iterate_succ_apply, iterate_succ_apply', iterate_succ_apply', ← frobenius_mul,
Nat.succ_add] | K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x1 x2 y : ℕ × K
H : R K p x1 x2
n : ℕ
x : K
⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)
((n + y.1).succ, (frobenius K p) ((⇑(frobenius K p))^[y.1] x * (⇑(frobenius K p))^[n] y.2)) | 94c5cb7db1618291 |
ENNReal.inv_strictAnti | Mathlib/Data/ENNReal/Inv.lean | theorem inv_strictAnti : StrictAnti (Inv.inv : ℝ≥0∞ → ℝ≥0∞) | ⊢ StrictAnti Inv.inv | intro a b h | a b : ℝ≥0∞
h : a < b
⊢ b⁻¹ < a⁻¹ | f6f8b260e5951a17 |
finprod_mem_finset_product | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_mem_finset_product (s : Finset (α × β)) (f : α × β → M) :
(∏ᶠ (ab) (_ : ab ∈ s), f ab) = ∏ᶠ (a) (b) (_ : (a, b) ∈ s), f (a, b) | α : Type u_1
β : Type u_2
M : Type u_5
inst✝ : CommMonoid M
s : Finset (α × β)
f : α × β → M
⊢ ∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab = ∏ᶠ (a : α) (b : β) (_ : (a, b) ∈ s), f (a, b) | classical
rw [finprod_mem_finset_product']
simp | no goals | 504da8f09a85ffae |
IsTorsion.of_surjective | Mathlib/GroupTheory/Torsion.lean | theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG : IsTorsion G) :
IsTorsion H := fun h => by
obtain ⟨g, hg⟩ := hf h
rw [← hg]
exact f.isOfFinOrder (tG g)
| G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hf : Function.Surjective ⇑f
tG : IsTorsion G
h : H
⊢ IsOfFinOrder h | obtain ⟨g, hg⟩ := hf h | case intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hf : Function.Surjective ⇑f
tG : IsTorsion G
h : H
g : G
hg : f g = h
⊢ IsOfFinOrder h | a18faf93d772be26 |
MeasureTheory.Measure.rnDeriv_smul_right' | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem rnDeriv_smul_right' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
{r : ℝ≥0} (hr : r ≠ 0) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ | α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
this :
ν.singularPart (r • μ) + (r • μ).withDensity (ν.rnDeriv (r • μ)) =
ν.singularPart (r • μ) + r⁻¹ • (r • μ).withDensity (ν.rnDeriv μ)
⊢ (↑r • μ).withDensity (ν.rnDeriv (↑r • μ)) = ↑r⁻¹ • (↑r • μ).withDensity (ν.rnDeriv μ) | rwa [add_right_inj] at this | no goals | c2338dd04dba20c8 |
CategoryTheory.frobeniusMorphism_mate | Mathlib/CategoryTheory/Closed/Functor.lean | theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)
(frobeniusMorphism F h A).natTrans = (expComparison F A).natTrans | case e_w.h.h.inv_hom_id
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
((mateEquiv h h)
(prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
(conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
(prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
F.map (prodComparison L (F.obj A) (F.obj B)) ≫
F.map (h.counit.app A ▷ L.obj (F.obj B)) ≫ F.map (A ◁ h.counit.app B) ≫ prodComparison F A B =
𝟙 (F.obj A ⊗ F.obj B) | rw [prodComparison_natural_whiskerLeft, prodComparison_natural_whiskerRight_assoc] | case e_w.h.h.inv_hom_id
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
((mateEquiv h h)
(prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
(conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
(prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
F.map (prodComparison L (F.obj A) (F.obj B)) ≫
prodComparison F (L.obj (F.obj A)) (L.obj (F.obj B)) ≫
F.map (h.counit.app A) ▷ F.obj (L.obj (F.obj B)) ≫ F.obj A ◁ F.map (h.counit.app B) =
𝟙 (F.obj A ⊗ F.obj B) | 19328243aea6bbf8 |
isPathConnected_pathComponent | Mathlib/Topology/Connected/PathConnected.lean | theorem isPathConnected_pathComponent : IsPathConnected (pathComponent x) | X : Type u_1
inst✝ : TopologicalSpace X
x : X
⊢ IsPathConnected (pathComponent x) | rw [← pathComponentIn_univ] | X : Type u_1
inst✝ : TopologicalSpace X
x : X
⊢ IsPathConnected (pathComponentIn x univ) | 7722fd5252f34002 |
hasFDerivAt_integral_of_dominated_loc_of_lip' | Mathlib/Analysis/Calculus/ParametricIntegral.lean | theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ | α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
∀ᶠ (x : H) in 𝓝 x₀,
‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
⊢ (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - (F' a) (x - x₀)‖) = fun x =>
‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))‖ | ext x | case h
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
∀ᶠ (x : H) in 𝓝 x₀,
‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
x : H
⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - (F' a) (x - x₀)‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))‖ | a8f4bcb6e23b7fde |
FermatLastTheoremForThreeGen.Solution.lambda_sq_not_dvd_a_add_eta_sq_mul_b | Mathlib/NumberTheory/FLT/Three.lean | /-- Given `(S : Solution)`, we have that `λ ^ 2` does not divide `S.a + η ^ 2 * S.b`. -/
lemma lambda_sq_not_dvd_a_add_eta_sq_mul_b : ¬ λ ^ 2 ∣ (S.a + η ^ 2 * S.b) | case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
k : 𝓞 K
hk : S.a + ↑η ^ 2 * S.b = λ ^ 2 * k
k' : 𝓞 K
hk' : S.a + S.b = λ ^ 2 * k'
⊢ False | refine S.hb ⟨(k - k') * (-η), ?_⟩ | case intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
k : 𝓞 K
hk : S.a + ↑η ^ 2 * S.b = λ ^ 2 * k
k' : 𝓞 K
hk' : S.a + S.b = λ ^ 2 * k'
⊢ S.b = λ * ((k - k') * -↑η) | 75f6387d00024605 |
DvdNotUnit.not_unit | Mathlib/Algebra/Prime/Lemmas.lean | theorem DvdNotUnit.not_unit [CommMonoidWithZero M] {p q : M} (hp : DvdNotUnit p q) : ¬IsUnit q | M : Type u_1
inst✝ : CommMonoidWithZero M
p q : M
hp : DvdNotUnit p q
⊢ ¬IsUnit q | obtain ⟨-, x, hx, rfl⟩ := hp | case intro.intro.intro
M : Type u_1
inst✝ : CommMonoidWithZero M
p x : M
hx : ¬IsUnit x
⊢ ¬IsUnit (p * x) | 370ad7c277381bb2 |
Real.ediam_eq | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem ediam_eq {s : Set ℝ} (h : Bornology.IsBounded s) :
EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) | case inr.refine_2.h
s : Set ℝ
h : Bornology.IsBounded s
hne : s.Nonempty
⊢ sSup s - sInf s ≤ (EMetric.diam s).toReal | rw [← Metric.diam, ← Metric.diam_closure] | case inr.refine_2.h
s : Set ℝ
h : Bornology.IsBounded s
hne : s.Nonempty
⊢ sSup s - sInf s ≤ Metric.diam (closure s) | d6382336badce924 |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastUdiv | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastUdiv (aig : AIG α) (lhs rhs : BitVec w) (assign : α → Bool)
(input : BinaryRefVec aig w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign⟧ = rhs.getLsbD idx) :
∀ (idx : Nat) (hidx : idx < w),
⟦(blastUdiv aig input).aig, (blastUdiv aig input).vec.get idx hidx, assign⟧
=
(lhs / rhs).getLsbD idx | case hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.rhs.get idx hidx }⟧ = rhs.getLsbD idx
idx✝ : Nat
hidx : idx✝ < w
hdiscr✝ :
¬⟦assign,
{
aig :=
(blastUdiv.go
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).aig
w { gate := ((blastConst aig 0#w).aig.mkConstCached false).ref.gate, hgate := ⋯ }
{ gate := (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).ref.gate, hgate := ⋯ }
((((input.cast ⋯).cast ⋯).cast ⋯).lhs.cast ⋯) ((((input.cast ⋯).cast ⋯).cast ⋯).rhs.cast ⋯) w 0
((((blastConst aig 0#w).vec.cast ⋯).cast ⋯).cast ⋯)
((((blastConst aig 0#w).vec.cast ⋯).cast ⋯).cast ⋯)).aig,
ref :=
{
gate :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).ref.gate,
hgate := ⋯ } }⟧ =
true
hdiscr :
¬⟦assign,
{
aig :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs, rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).aig,
ref :=
{
gate :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).ref.gate,
hgate := ⋯ } }⟧ =
true
idx : Nat
hdix : idx < w
⊢ ⟦assign,
{ aig := (blastConst aig 0#w).aig,
ref := { gate := ((blastConst aig 0#w).vec.get idx hdix).gate, hgate := ⋯ } }⟧ =
false | rw [denote_blastConst] | case hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.rhs.get idx hidx }⟧ = rhs.getLsbD idx
idx✝ : Nat
hidx : idx✝ < w
hdiscr✝ :
¬⟦assign,
{
aig :=
(blastUdiv.go
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).aig
w { gate := ((blastConst aig 0#w).aig.mkConstCached false).ref.gate, hgate := ⋯ }
{ gate := (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).ref.gate, hgate := ⋯ }
((((input.cast ⋯).cast ⋯).cast ⋯).lhs.cast ⋯) ((((input.cast ⋯).cast ⋯).cast ⋯).rhs.cast ⋯) w 0
((((blastConst aig 0#w).vec.cast ⋯).cast ⋯).cast ⋯)
((((blastConst aig 0#w).vec.cast ⋯).cast ⋯).cast ⋯)).aig,
ref :=
{
gate :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).ref.gate,
hgate := ⋯ } }⟧ =
true
hdiscr :
¬⟦assign,
{
aig :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs, rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).aig,
ref :=
{
gate :=
(BVPred.mkEq (((blastConst aig 0#w).aig.mkConstCached false).aig.mkConstCached true).aig
{ lhs := (((input.cast ⋯).cast ⋯).cast ⋯).rhs,
rhs := ((blastConst aig 0#w).vec.cast ⋯).cast ⋯ }).ref.gate,
hgate := ⋯ } }⟧ =
true
idx : Nat
hdix : idx < w
⊢ (0#w).getLsbD idx = false | 1e06923b53c4a62f |
QuadraticForm.comp_tensorLId_eq | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | theorem comp_tensorLId_eq (Q₂ : QuadraticForm R M₂) :
Q₂.comp (TensorProduct.lid R M₂) = QuadraticForm.tmul (sq (R := R)) Q₂ | R : Type uR
M₂ : Type uM₂
inst✝³ : CommRing R
inst✝² : AddCommGroup M₂
inst✝¹ : Module R M₂
inst✝ : Invertible 2
Q₂ : QuadraticForm R M₂
⊢ comp Q₂ ↑(TensorProduct.lid R M₂) = QuadraticForm.tmul QuadraticMap.sq Q₂ | refine (QuadraticMap.associated_rightInverse R).injective ?_ | R : Type uR
M₂ : Type uM₂
inst✝³ : CommRing R
inst✝² : AddCommGroup M₂
inst✝¹ : Module R M₂
inst✝ : Invertible 2
Q₂ : QuadraticForm R M₂
⊢ (associatedHom R) (comp Q₂ ↑(TensorProduct.lid R M₂)) = (associatedHom R) (QuadraticForm.tmul QuadraticMap.sq Q₂) | eb603c56582839d7 |
LinearIsometry.toAffineIsometry_linearIsometry | Mathlib/Analysis/Normed/Affine/Isometry.lean | theorem toAffineIsometry_linearIsometry : f.toAffineIsometry.linearIsometry = f | case h
𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
f : V →ₗᵢ[𝕜] V₂
x✝ : V
⊢ f.toAffineIsometry.linearIsometry x✝ = f x✝ | rfl | no goals | f0503555f6166a82 |
Ideal.isPrime_map_of_isLocalizationAtPrime | Mathlib/RingTheory/Localization/AtPrime.lean | lemma Ideal.isPrime_map_of_isLocalizationAtPrime {p : Ideal R} [p.IsPrime] (hpq : p ≤ q) :
(p.map (algebraMap R S)).IsPrime | R : Type u_4
inst✝⁵ : CommRing R
q : Ideal R
inst✝⁴ : q.IsPrime
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization.AtPrime S q
p : Ideal R
inst✝ : p.IsPrime
hpq : p ≤ q
⊢ Disjoint ↑q.primeCompl ↑p | simp [Ideal.primeCompl, ← le_compl_iff_disjoint_left, hpq] | no goals | a57ec231823a0625 |
mem_ball_one_iff | Mathlib/Analysis/Normed/Group/Basic.lean | theorem mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r | E : Type u_5
inst✝ : SeminormedGroup E
a : E
r : ℝ
⊢ a ∈ ball 1 r ↔ ‖a‖ < r | rw [mem_ball, dist_one_right] | no goals | e499b9b96775cf93 |
Basis.mk_coord_apply | Mathlib/LinearAlgebra/Basis/Basic.lean | theorem mk_coord_apply [DecidableEq ι] {i j : ι} :
(Basis.mk hli hsp).coord i (v j) = if j = i then 1 else 0 | case inl
ι : Type u_1
R : Type u_3
M : Type u_5
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
v : ι → M
hli : LinearIndependent R v
hsp : ⊤ ≤ span R (range v)
inst✝ : DecidableEq ι
i j : ι
h : j = i
⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0 | simp only [h, if_true, eq_self_iff_true, mk_coord_apply_eq i] | no goals | ca6b42d63a5d696c |
CategoryTheory.Subobject.map_comp | Mathlib/CategoryTheory/Subobject/Basic.lean | theorem map_comp (f : X ⟶ Y) (g : Y ⟶ Z) [Mono f] [Mono g] (x : Subobject X) :
(map (f ≫ g)).obj x = (map g).obj ((map f).obj x) | case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : Mono f
inst✝ : Mono g
t : MonoOver X
⊢ (map (f ≫ g)).obj (Quotient.mk'' t) = (map g).obj ((map f).obj (Quotient.mk'' t)) | exact Quotient.sound ⟨(MonoOver.mapComp _ _).app t⟩ | no goals | 2912982149610594 |
Polynomial.aeval_apply_smul_mem_of_le_comap' | Mathlib/Algebra/Polynomial/AlgebraMap.lean | lemma aeval_apply_smul_mem_of_le_comap'
[Semiring A] [Algebra R A] [Module A M] [IsScalarTower R A M] (hm : m ∈ q) (p : R[X]) (a : A)
(hq : q ≤ q.comap (Algebra.lsmul R R M a)) :
aeval a p • m ∈ q | case refine_2
R : Type u
A : Type z
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
q : Submodule R M
m : M
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
hm : m ∈ q
p : R[X]
a : A
hq : q ≤ Submodule.comap ((Algebra.lsmul R R M) a) q
n : ℕ
t : R
hmq : (aeval a) (C t * X ^ n) • m ∈ q
⊢ (aeval a) (C t * X ^ (n + 1)) • m ∈ q | rw [pow_succ', mul_left_comm, map_mul, aeval_X, mul_smul] | case refine_2
R : Type u
A : Type z
M : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
q : Submodule R M
m : M
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
hm : m ∈ q
p : R[X]
a : A
hq : q ≤ Submodule.comap ((Algebra.lsmul R R M) a) q
n : ℕ
t : R
hmq : (aeval a) (C t * X ^ n) • m ∈ q
⊢ a • (aeval a) (C t * X ^ n) • m ∈ q | ea53fdc1c4b58698 |
LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete | Mathlib/GroupTheory/ArchimedeanDensely.lean | lemma LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete
{G : Type*} [LinearOrderedAddCommGroup G] [Nontrivial G] {g : G} :
Set.WellFoundedOn {x : G | g ≤ x} (· < ·) ↔ Nonempty (G ≃+o ℤ) | case pos.intro.inr.refine_2
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
hd : DenselyOrdered G
y : G
hy : y ≠ 0
this : ∀ (y : G), y ≠ 0 → 0 < y → False
hy' : y ≤ 0
⊢ 0 < -y | simp [lt_of_le_of_ne hy' hy] | no goals | 59c60e31ffce1071 |
IsLocalRing.quotient_span_eq_top_iff_span_eq_top | Mathlib/RingTheory/LocalRing/Quotient.lean | theorem quotient_span_eq_top_iff_span_eq_top (s : Set S) :
span (R ⧸ p) ((Ideal.Quotient.mk (I := pS)) '' s) = ⊤ ↔ span R s = ⊤ | case mp
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) = map (IsScalarTower.toAlgHom R S (S ⧸ pS)) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
⊢ span R s = ⊤ | rw [← top_le_iff] | case mp
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) = map (IsScalarTower.toAlgHom R S (S ⧸ pS)) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
⊢ ⊤ ≤ span R s | c2f006104663815d |
pow_add_pow_le | Mathlib/Algebra/Order/Ring/Basic.lean | theorem pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y ^ n ≤ (x + y) ^ n | R : Type u_3
inst✝ : OrderedSemiring R
x y : R
hx : 0 ≤ x
hy : 0 ≤ y
k : ℕ
ih : k + 1 ≠ 0 → x ^ (k + 1) + y ^ (k + 1) ≤ (x + y) ^ (k + 1)
hn : k + 1 + 1 ≠ 0
n : ℕ := k.succ
h1 : 0 ≤ x * y ^ n + y * x ^ n
h2 : 0 ≤ x + y
⊢ (x + y) * (x ^ n + y ^ n) ≤ (x + y) ^ (n + 1) | rw [pow_succ' _ n] | R : Type u_3
inst✝ : OrderedSemiring R
x y : R
hx : 0 ≤ x
hy : 0 ≤ y
k : ℕ
ih : k + 1 ≠ 0 → x ^ (k + 1) + y ^ (k + 1) ≤ (x + y) ^ (k + 1)
hn : k + 1 + 1 ≠ 0
n : ℕ := k.succ
h1 : 0 ≤ x * y ^ n + y * x ^ n
h2 : 0 ≤ x + y
⊢ (x + y) * (x ^ n + y ^ n) ≤ (x + y) * (x + y) ^ n | 1b4313a1aa17f256 |
QPF.Wequiv.symm | Mathlib/Data/QPF/Univariate/Basic.lean | theorem Wequiv.symm (x y : q.P.W) : Wequiv x y → Wequiv y x | case trans
F : Type u → Type u
q : QPF F
x✝ y✝ x y z : (P F).W
a✝¹ : Wequiv x y
a✝ : Wequiv y z
ih₁ : Wequiv y x
ih₂ : Wequiv z y
⊢ Wequiv z x | exact QPF.Wequiv.trans _ _ _ ih₂ ih₁ | no goals | 45cc86f6cee0bf26 |
PiToModule.fromEnd_injective | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | theorem PiToModule.fromEnd_injective (hb : Submodule.span R (Set.range b) = ⊤) :
Function.Injective (PiToModule.fromEnd R b) | case h
ι : Type u_1
inst✝³ : Fintype ι
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
x y : Module.End R M
e : (fromEnd R b) x = (fromEnd R b) y
m : M
⊢ x m = y m | obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.linearCombination R R b) := by
rw [(Fintype.range_linearCombination R b).trans hb]
exact Submodule.mem_top | case h.intro
ι : Type u_1
inst✝³ : Fintype ι
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
x y : Module.End R M
e : (fromEnd R b) x = (fromEnd R b) y
m : ι → R
⊢ x (((Fintype.linearCombination R R) b) m) = y (((Fintype.linearCombination R R) b) m) | 34d5df3896496ac3 |
blimsup_cthickening_ae_le_of_eventually_mul_le_aux | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
(hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
{M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
(blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α) | α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
⊢ False | obtain ⟨d, hd, hd'⟩ : ∃ d, d ∈ W ∧ ∀ {ι : Type _} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
Tendsto δ l (𝓝[>] 0) → (∀ᶠ j in l, d ∈ closedBall (w j) (2 * δ j)) →
Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
Measure.exists_mem_of_measure_ne_zero_of_ae contra
(IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ W 2) | case intro.intro
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd : d ∈ W
hd' :
∀ {ι : Type ?u.5905} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
Tendsto δ l (𝓝[>] 0) →
(∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
⊢ False | f93df810108e5d67 |
measurableSet_pi_of_nonempty | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_pi_of_nonempty {s : Set δ} {t : ∀ i, Set (π i)} (hs : s.Countable)
(h : (pi s t).Nonempty) : MeasurableSet (pi s t) ↔ ∀ i ∈ s, MeasurableSet (t i) | case intro
δ : Type u_4
π : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (π a)
s : Set δ
t : (i : δ) → Set (π i)
hs : s.Countable
f : (i : δ) → π i
hf : f ∈ s.pi t
hst : MeasurableSet (s.pi t)
i : δ
hi : i ∈ s
⊢ MeasurableSet (t i) | convert measurable_update f (a := i) hst | case h.e'_3
δ : Type u_4
π : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (π a)
s : Set δ
t : (i : δ) → Set (π i)
hs : s.Countable
f : (i : δ) → π i
hf : f ∈ s.pi t
hst : MeasurableSet (s.pi t)
i : δ
hi : i ∈ s
⊢ t i = update f i ⁻¹' s.pi t | 63724aeb23e83fb6 |
ZMod.completedLFunction_def_even | Mathlib/NumberTheory/LSeries/ZMod.lean | lemma completedLFunction_def_even (hΦ : Φ.Even) (s : ℂ) :
completedLFunction Φ s = N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s | N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
this : ∑ j : ZMod N, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0
⊢ completedLFunction Φ s = ↑N ^ (-s) * ∑ j : ZMod N, Φ j * completedHurwitzZetaEven (toAddCircle j) s | rw [completedLFunction, this, mul_zero, add_zero] | no goals | ddd683e2a7dbb180 |
ProbabilityTheory.measure_ge_le_exp_mul_mgf | Mathlib/Probability/Moments/Basic.lean | theorem measure_ge_le_exp_mul_mgf [IsFiniteMeasure μ] (ε : ℝ) (ht : 0 ≤ t)
(h_int : Integrable (fun ω => exp (t * X ω)) μ) :
(μ {ω | ε ≤ X ω}).toReal ≤ exp (-t * ε) * mgf X μ t | case inl.hb
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
ε : ℝ
ht : 0 ≤ t
h_int : Integrable (fun ω => rexp (t * X ω)) μ
ht_zero_eq : 0 = t
⊢ μ Set.univ ≠ ⊤
case inl.h.h
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
ε : ℝ
ht : 0 ≤ t
h_int : Integrable (fun ω => rexp (t * X ω)) μ
ht_zero_eq : 0 = t
⊢ {ω | ε ≤ X ω} ⊆ Set.univ | exacts [measure_ne_top _ _, Set.subset_univ _] | no goals | d0b4656bb63daebb |
measurable_from_prod_countable' | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurable_from_prod_countable' [Countable β]
{_ : MeasurableSpace γ} {f : α × β → γ} (hf : ∀ y, Measurable fun x => f (x, y))
(h'f : ∀ y y' x, y' ∈ measurableAtom y → f (x, y') = f (x, y)) :
Measurable f := fun s hs => by
have : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ (measurableAtom y : Set β) | α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable β
x✝ : MeasurableSpace γ
f : α × β → γ
hf : ∀ (y : β), Measurable fun x => f (x, y)
h'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s) | have : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ (measurableAtom y : Set β) := by
ext1 ⟨x, y⟩
simp only [mem_preimage, mem_iUnion, mem_prod]
refine ⟨fun h ↦ ⟨y, h, mem_measurableAtom_self y⟩, ?_⟩
rintro ⟨y', hy's, hy'⟩
rwa [h'f y' y x hy'] | α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable β
x✝ : MeasurableSpace γ
f : α × β → γ
hf : ∀ (y : β), Measurable fun x => f (x, y)
h'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)
s : Set γ
hs : MeasurableSet s
this : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ measurableAtom y
⊢ MeasurableSet (f ⁻¹' s) | 8c2048365dc7fee7 |
Polynomial.cyclotomic_expand_eq_cyclotomic_mul | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
(R : Type*) [CommRing R] :
expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R | case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
⊢ (aeval (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)))
((expand ℚ p) (minpoly ℚ (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)))) =
0 | rw [aeval_def, ← eval_map, map_expand, expand_eval, ← IsRoot.def, ←
cyclotomic_eq_minpoly_rat hprim hnpos, map_cyclotomic, @isRoot_cyclotomic_iff] | case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
⊢ IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n) ^ p) n | 48914059214e6ed1 |
MeasureTheory.IsFundamentalDomain.mk_of_measure_univ_le | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
(h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
(h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)
(h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=
have aedisjoint : Pairwise (AEDisjoint μ on fun g : G => g • s) :=
pairwise_aedisjoint_of_aedisjoint_forall_ne_one h_ae_disjoint h_qmp
{ nullMeasurableSet := h_meas
aedisjoint
ae_covers | G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_meas : NullMeasurableSet s μ
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s)
⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s | replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by
rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹) | G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x => g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g => g • s)
h_meas : ∀ (g : G), NullMeasurableSet (g • s) μ
⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s | 32e90416e0da9c0b |
Polynomial.natDegree_taylor | Mathlib/Algebra/Polynomial/Taylor.lean | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p | R : Type u_1
inst✝ : Semiring R
p : R[X]
r : R
⊢ ((taylor r) p).natDegree = p.natDegree | refine map_natDegree_eq_natDegree _ ?_ | R : Type u_1
inst✝ : Semiring R
p : R[X]
r : R
⊢ ∀ (n : ℕ) (c : R), c ≠ 0 → ((taylor r) ((monomial n) c)).natDegree = n | 0c43348d43be43ae |
List.findIdx?_eq_some_iff_getElem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findIdx?_eq_some_iff_getElem {xs : List α} {p : α → Bool} {i : Nat} :
xs.findIdx? p = some i ↔
∃ h : i < xs.length, p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji h)) | case cons.isTrue.succ
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : ∀ {i : Nat}, findIdx? p xs = some i ↔ ∃ h, p xs[i] = true ∧ ∀ (j : Nat) (hji : j < i), ¬p xs[j] = true
h✝ : p x = true
i : Nat
x✝ : i + 1 < xs.length + 1
a✝ : p xs[i] = true
⊢ ∃ x_1 h, p (x :: xs)[x_1] = true | refine ⟨0, zero_lt_succ i, ‹_›⟩ | no goals | 6fc8c2dbd0d1ea38 |
ProbabilityTheory.IndepFun.integral_mul | Mathlib/Probability/Integration.lean | theorem IndepFun.integral_mul (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y | case intro
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'X : ¬X =ᶠ[ae μ] 0
h'Y : ¬Y =ᶠ[ae μ] 0
h : ¬Integrable (X * Y) μ
HX : Integrable X μ
HY : Integrable Y μ
⊢ False | exact h (hXY.integrable_mul HX HY) | no goals | d5b07299b613cdd4 |
List.map_permutationsAux2' | Mathlib/Data/List/Permutation.lean | theorem map_permutationsAux2' {α' β'} (g : α → α') (g' : β → β') (t : α) (ts ys : List α)
(r : List β) (f : List α → β) (f' : List α' → β') (H : ∀ a, g' (f a) = f' (map g a)) :
map g' (permutationsAux2 t ts r ys f).2 =
(permutationsAux2 (g t) (map g ts) (map g' r) (map g ys) f').2 | case cons
α : Type u_1
β : Type u_2
α' : Type u_3
β' : Type u_4
g : α → α'
g' : β → β'
t : α
ts : List α
r : List β
ys_hd : α
tail✝ : List α
ys_ih :
∀ (f : List α → β) (f' : List α' → β'),
(∀ (a : List α), g' (f a) = f' (map g a)) →
map g' (permutationsAux2 t ts r tail✝ f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').2
f : List α → β
f' : List α' → β'
H : ∀ (a : List α), g' (f a) = f' (map g a)
⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧
map g' (permutationsAux2 t ts r tail✝ fun x => f (ys_hd :: x)).2 =
(permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).2 | rw [ys_ih] | case cons
α : Type u_1
β : Type u_2
α' : Type u_3
β' : Type u_4
g : α → α'
g' : β → β'
t : α
ts : List α
r : List β
ys_hd : α
tail✝ : List α
ys_ih :
∀ (f : List α → β) (f' : List α' → β'),
(∀ (a : List α), g' (f a) = f' (map g a)) →
map g' (permutationsAux2 t ts r tail✝ f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').2
f : List α → β
f' : List α' → β'
H : ∀ (a : List α), g' (f a) = f' (map g a)
⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧
(permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) ?cons.f').2 =
(permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).2
case cons.f'
α : Type u_1
β : Type u_2
α' : Type u_3
β' : Type u_4
g : α → α'
g' : β → β'
t : α
ts : List α
r : List β
ys_hd : α
tail✝ : List α
ys_ih :
∀ (f : List α → β) (f' : List α' → β'),
(∀ (a : List α), g' (f a) = f' (map g a)) →
map g' (permutationsAux2 t ts r tail✝ f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').2
f : List α → β
f' : List α' → β'
H : ∀ (a : List α), g' (f a) = f' (map g a)
⊢ List α' → β'
case cons.H
α : Type u_1
β : Type u_2
α' : Type u_3
β' : Type u_4
g : α → α'
g' : β → β'
t : α
ts : List α
r : List β
ys_hd : α
tail✝ : List α
ys_ih :
∀ (f : List α → β) (f' : List α' → β'),
(∀ (a : List α), g' (f a) = f' (map g a)) →
map g' (permutationsAux2 t ts r tail✝ f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').2
f : List α → β
f' : List α' → β'
H : ∀ (a : List α), g' (f a) = f' (map g a)
⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = ?cons.f' (map g a) | 0a78be6fc0d59c01 |
Nat.Linear.PolyCnstr.denote_combine | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem PolyCnstr.denote_combine {ctx : Context} {c₁ c₂ : PolyCnstr} (h₁ : c₁.denote ctx) (h₂ : c₂.denote ctx) : (c₁.combine c₂).denote ctx | case neg
ctx : Context
eq₁ : Bool
lhs₁ rhs₁ : Poly
eq₂ : Bool
lhs₂ rhs₂ : Poly
he₁ : eq₁ = true
he₂ : ¬eq₂ = true
h₁ : Poly.denote_eq ctx (lhs₁, rhs₁)
h₂ : Poly.denote_le ctx (lhs₂, rhs₂)
⊢ Poly.denote_le ctx (lhs₁.combine lhs₂, rhs₁.combine rhs₂) | simp [Poly.denote_eq, Poly.denote_le] at h₁ h₂ |- | case neg
ctx : Context
eq₁ : Bool
lhs₁ rhs₁ : Poly
eq₂ : Bool
lhs₂ rhs₂ : Poly
he₁ : eq₁ = true
he₂ : ¬eq₂ = true
h₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁
h₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂
⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂ | 7f92a3920c333c32 |
logDeriv_mul_const | Mathlib/Analysis/Calculus/LogDeriv.lean | theorem logDeriv_mul_const {f : 𝕜 → 𝕜'} (x : 𝕜) (a : 𝕜') (ha : a ≠ 0):
logDeriv (fun z => f z * a) x = logDeriv f x | 𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
a : 𝕜'
ha : a ≠ 0
⊢ logDeriv (fun z => f z * a) x = logDeriv f x | simp only [logDeriv_apply, deriv_mul_const_field, mul_div_mul_right _ _ ha] | no goals | 9693af55c83fb31e |
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_prod_of_not_mem | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f κ μ)
(hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s) :
IndepFun (∏ j ∈ s, f j) (f i) κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : Type u_8
m : MeasurableSpace β
inst✝¹ : CommMonoid β
inst✝ : MeasurableMul₂ β
f : ι → Ω → β
hf_Indep : iIndepFun (fun x => m) f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
s : Finset ι
i : ι
hi : i ∉ s
h_right : f i = (fun p => p ⟨i, ⋯⟩) ∘ fun a j => f (↑j) a
h_meas_right : Measurable fun p => p ⟨i, ⋯⟩
⊢ ∏ j ∈ s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a | ext1 a | case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : Type u_8
m : MeasurableSpace β
inst✝¹ : CommMonoid β
inst✝ : MeasurableMul₂ β
f : ι → Ω → β
hf_Indep : iIndepFun (fun x => m) f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
s : Finset ι
i : ι
hi : i ∉ s
h_right : f i = (fun p => p ⟨i, ⋯⟩) ∘ fun a j => f (↑j) a
h_meas_right : Measurable fun p => p ⟨i, ⋯⟩
a : Ω
⊢ (∏ j ∈ s, f j) a = ((fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a) a | ee69eed743c3c705 |
isSeparable_range_derivWithin | Mathlib/Analysis/Calculus/Deriv/Slope.lean | theorem isSeparable_range_derivWithin [SeparableSpace 𝕜] (f : 𝕜 → F) (s : Set 𝕜) :
IsSeparable (range (derivWithin f s)) | 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure t
this : s ⊆ closure (s ∩ t)
⊢ IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t))) | exact (Countable.image t_count f).isSeparable.span.closure | no goals | ccec9f827683d379 |
iSupIndep_iff_supIndep_of_injOn | Mathlib/Order/CompactlyGenerated/Basic.lean | lemma iSupIndep_iff_supIndep_of_injOn {ι : Type*} {f : ι → α}
(hf : InjOn f {i | f i ≠ ⊥}) :
iSupIndep f ↔ ∀ (s : Finset ι), s.SupIndep f | α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
hf : InjOn f {i | f i ≠ ⊥}
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
⊢ ∀ (i_1 : Finset α), ↑i_1 ⊆ f '' {j | j ≠ i} → Disjoint (f i) (i_1.sup id) | intro s hs | α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
hf : InjOn f {i | f i ≠ ⊥}
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
s : Finset α
hs : ↑s ⊆ f '' {j | j ≠ i}
⊢ Disjoint (f i) (s.sup id) | ab45b25ee8be9f47 |
CategoryTheory.Limits.biprod.ext_to_iff | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | lemma biprod.ext_to_iff {f g : Z ⟶ X ⊞ Y} :
f = g ↔ f ≫ biprod.fst = g ≫ biprod.fst ∧ f ≫ biprod.snd = g ≫ biprod.snd | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
X Y : C
inst✝ : HasBinaryBiproduct X Y
Z : C
f g : Z ⟶ X ⊞ Y
⊢ f = g ↔ f ≫ fst = g ≫ fst ∧ f ≫ snd = g ≫ snd | aesop | no goals | d554989469926e92 |
MeasureTheory.addContent_union_le | Mathlib/MeasureTheory/Measure/AddContent.lean | lemma addContent_union_le (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) :
m (s ∪ t) ≤ m s + m t | α : Type u_1
C : Set (Set α)
s t : Set α
m : AddContent C
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ m s + m (t \ s) ≤ m s + m t | exact add_le_add le_rfl
(addContent_mono hC.isSetSemiring (hC.diff_mem ht hs) ht diff_subset) | no goals | 0dc54ff19c7efe9a |
RingCon.ringConGen_le | Mathlib/RingTheory/Congruence/Basic.lean | theorem ringConGen_le {r : R → R → Prop} {c : RingCon R}
(h : ∀ x y, r x y → c x y) : ringConGen r ≤ c | R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
r : R → R → Prop
c : RingCon R
h : ∀ (x y : R), r x y → c x y
⊢ ringConGen r ≤ c | rw [ringConGen_eq] | R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
r : R → R → Prop
c : RingCon R
h : ∀ (x y : R), r x y → c x y
⊢ sInf {s | ∀ (x y : R), r x y → s x y} ≤ c | 7d48517f785fc022 |
Std.Tactic.BVDecide.ofBoolExprCached.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BoolExpr/Circuit.lean | theorem go_decl_eq (idx) (aig : AIG β) (h : idx < aig.decls.size) (hbounds) :
(ofBoolExprCached.go aig expr atomHandler).val.aig.decls[idx]'hbounds = aig.decls[idx] | case gate.or
α β : Type
inst✝² : Hashable β
inst✝¹ : DecidableEq β
atomHandler : AIG β → α → Entrypoint β
inst✝ : LawfulOperator β (fun x => α) atomHandler
idx : Nat
lhs rhs : BoolExpr α
aig : AIG β
h : idx < aig.decls.size
this✝ : aig.decls.size ≤ (go aig lhs atomHandler).val.aig.decls.size
this :
(go aig lhs atomHandler).val.aig.decls.size ≤ (go (go aig lhs atomHandler).val.aig rhs atomHandler).val.aig.decls.size
hbounds : idx < (go aig (BoolExpr.gate Gate.or lhs rhs) atomHandler).val.aig.decls.size
lih : (go aig lhs atomHandler).val.aig.decls[idx] = aig.decls[idx]
rih :
(go (go aig lhs atomHandler).val.aig rhs atomHandler).val.aig.decls[idx] = (go aig lhs atomHandler).val.aig.decls[idx]
⊢ (go aig (BoolExpr.gate Gate.or lhs rhs) atomHandler).val.aig.decls[idx] = aig.decls[idx] | simp only [go] | case gate.or
α β : Type
inst✝² : Hashable β
inst✝¹ : DecidableEq β
atomHandler : AIG β → α → Entrypoint β
inst✝ : LawfulOperator β (fun x => α) atomHandler
idx : Nat
lhs rhs : BoolExpr α
aig : AIG β
h : idx < aig.decls.size
this✝ : aig.decls.size ≤ (go aig lhs atomHandler).val.aig.decls.size
this :
(go aig lhs atomHandler).val.aig.decls.size ≤ (go (go aig lhs atomHandler).val.aig rhs atomHandler).val.aig.decls.size
hbounds : idx < (go aig (BoolExpr.gate Gate.or lhs rhs) atomHandler).val.aig.decls.size
lih : (go aig lhs atomHandler).val.aig.decls[idx] = aig.decls[idx]
rih :
(go (go aig lhs atomHandler).val.aig rhs atomHandler).val.aig.decls[idx] = (go aig lhs atomHandler).val.aig.decls[idx]
⊢ ((go (go aig lhs atomHandler).1.aig rhs atomHandler).1.aig.mkOrCached
{ lhs := (go aig lhs atomHandler).1.ref.cast ⋯,
rhs := (go (go aig lhs atomHandler).1.aig rhs atomHandler).1.ref }).aig.decls[idx] =
aig.decls[idx] | 5157b7e15af3800b |
NumberField.house.asiegel_remark | Mathlib/NumberTheory/NumberField/House.lean | theorem asiegel_remark : ‖asiegel K a‖ ≤ c₂ K * A | case calc_3
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
kr : α × (K →+* ℂ)
lu : β × (K →+* ℂ)
⊢ NumberField.house.c K * house ((algebraMap (𝓞 K) K) (a kr.1 lu.1 * (NumberField.house.newBasis K) lu.2)) ≤
NumberField.house.c K * house ((algebraMap (𝓞 K) K) (a kr.1 lu.1)) *
house ((algebraMap (𝓞 K) K) ((NumberField.house.newBasis K) lu.2)) | simp only [house, _root_.map_mul, mul_assoc] | case calc_3
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
kr : α × (K →+* ℂ)
lu : β × (K →+* ℂ)
⊢ NumberField.house.c K *
‖(canonicalEmbedding K) ((algebraMap (𝓞 K) K) (a kr.1 lu.1)) *
(canonicalEmbedding K) ((algebraMap (𝓞 K) K) ((NumberField.house.newBasis K) lu.2))‖ ≤
NumberField.house.c K *
(‖(canonicalEmbedding K) ((algebraMap (𝓞 K) K) (a kr.1 lu.1))‖ *
‖(canonicalEmbedding K) ((algebraMap (𝓞 K) K) ((NumberField.house.newBasis K) lu.2))‖) | dc11ee49ed4757e5 |
FractionalIdeal.count_prod | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem count_prod {ι} (s : Finset ι) (I : ι → FractionalIdeal R⁰ K) (hS : ∀ i ∈ s, I i ≠ 0) :
count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i) | case insert
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
ι : Type u_3
I : ι → FractionalIdeal R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : (∀ i ∈ s, I i ≠ 0) → count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ insert i s, I i_1 ≠ 0
hS' : ∀ i ∈ s, I i ≠ 0
hS0 : ∏ i ∈ s, I i ≠ 0
⊢ count K v (∏ i ∈ insert i s, I i) = ∑ i ∈ insert i s, count K v (I i) | have hi0 : I i ≠ 0 := hS i (Finset.mem_insert_self i s) | case insert
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
ι : Type u_3
I : ι → FractionalIdeal R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : (∀ i ∈ s, I i ≠ 0) → count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ insert i s, I i_1 ≠ 0
hS' : ∀ i ∈ s, I i ≠ 0
hS0 : ∏ i ∈ s, I i ≠ 0
hi0 : I i ≠ 0
⊢ count K v (∏ i ∈ insert i s, I i) = ∑ i ∈ insert i s, count K v (I i) | 1035c991cce4fe30 |
HallMarriageTheorem.hall_cond_of_compl | Mathlib/Combinatorics/Hall/Finite.lean | theorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(hus : #s = #(s.biUnion t)) (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(s' : Finset (sᶜ : Set ι)) : #s' ≤ #(s'.biUnion fun x' => t x' \ s.biUnion t) | α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑(↑s)ᶜ
⊢ #s' ≤ #(s'.biUnion fun x' => t ↑x' \ s.biUnion t) | haveI := Classical.decEq ι | α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑(↑s)ᶜ
this : DecidableEq ι
⊢ #s' ≤ #(s'.biUnion fun x' => t ↑x' \ s.biUnion t) | 633ed149d2eeb7d3 |
MeasureTheory.rnDeriv_tilted_right | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma rnDeriv_tilted_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
(hf : Integrable (fun x ↦ exp (f x)) ν) :
μ.rnDeriv (ν.tilted f)
=ᵐ[ν] fun x ↦ ENNReal.ofReal (exp (- f x) * ∫ x, exp (f x) ∂ν) * μ.rnDeriv ν x | case inr.refine_4.h
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
h0 : NeZero ν
x : α
⊢ (ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂ν))⁻¹ * μ.rnDeriv ν x =
ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x | congr | case inr.refine_4.h.e_a
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
h0 : NeZero ν
x : α
⊢ (ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂ν))⁻¹ = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) | 4634d04eea4db0da |
ite_some_none_eq_none | Mathlib/.lake/packages/lean4/src/lean/Init/ByCases.lean | theorem ite_some_none_eq_none [Decidable P] :
(if P then some x else none) = none ↔ ¬ P | P : Prop
α✝ : Type u_1
x : α✝
inst✝ : Decidable P
⊢ P → False ↔ ¬P | rfl | no goals | 0b7185669c0c3453 |
Cardinal.mk_subtype_le_of_countable_eventually_mem_aux | Mathlib/SetTheory/Cardinal/CountableCover.lean | /-- If a set `t` is eventually covered by a countable family of sets, all with cardinality at
most `a`, then the cardinality of `t` is also bounded by `a`.
Superseded by `mk_le_of_countable_eventually_mem` which does not assume
that the indexing set lives in the same universe. -/
lemma mk_subtype_le_of_countable_eventually_mem_aux {α ι : Type u} {a : Cardinal}
[Countable ι] {f : ι → Set α} {l : Filter ι} [NeBot l]
{t : Set α} (ht : ∀ x ∈ t, ∀ᶠ i in l, x ∈ f i)
(h'f : ∀ i, #(f i) ≤ a) : #t ≤ a | α ι : Type u
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
t : Set α
ht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i
n : ℕ
h'f : ∀ (i : ι), #↑(f i) ≤ ↑n
ha : ↑n < ℵ₀
s : Finset α
hs : ↑s ⊆ t
A : ∀ x ∈ s, ∀ᶠ (i : ι) in l, x ∈ f i
B : ∀ᶠ (i : ι) in l, ∀ x ∈ s, x ∈ f i
i : ι
hi : ∀ x ∈ s, x ∈ f i
this : (i : ι) → Fintype ↑(f i)
u : Finset α := (f i).toFinset
I1 : s.card ≤ u.card
⊢ ↑u.card ≤ ↑n | convert h'f i | case h.e'_3
α ι : Type u
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
t : Set α
ht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i
n : ℕ
h'f : ∀ (i : ι), #↑(f i) ≤ ↑n
ha : ↑n < ℵ₀
s : Finset α
hs : ↑s ⊆ t
A : ∀ x ∈ s, ∀ᶠ (i : ι) in l, x ∈ f i
B : ∀ᶠ (i : ι) in l, ∀ x ∈ s, x ∈ f i
i : ι
hi : ∀ x ∈ s, x ∈ f i
this : (i : ι) → Fintype ↑(f i)
u : Finset α := (f i).toFinset
I1 : s.card ≤ u.card
⊢ ↑u.card = #↑(f i) | 984a2747d98b97ba |
MeasureTheory.integrableOn_iUnion_of_summable_integral_norm | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem integrableOn_iUnion_of_summable_integral_norm {f : X → E} {s : ι → Set X}
(hs : ∀ i : ι, MeasurableSet (s i)) (hi : ∀ i : ι, IntegrableOn f (s i) μ)
(h : Summable fun i : ι => ∫ x : X in s i, ‖f x‖ ∂μ) : IntegrableOn f (iUnion s) μ | X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝¹ : Countable ι
μ : Measure X
inst✝ : NormedAddCommGroup E
f : X → E
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
hi : ∀ (i : ι), IntegrableOn f (s i) μ
h : Summable fun i => ∫ (x : X) in s i, ‖f x‖ ∂μ
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
⊢ ∑' (b : ι), ENNReal.ofReal (∫ (a : X) in s b, ↑‖f a‖₊ ∂μ) < ⊤ | have S' :
Summable fun i : ι =>
(⟨∫ x : X in s i, ‖f x‖₊ ∂μ, setIntegral_nonneg (hs i) fun x _ => NNReal.coe_nonneg _⟩ :
NNReal) := by
rw [← NNReal.summable_coe]; exact h | X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝¹ : Countable ι
μ : Measure X
inst✝ : NormedAddCommGroup E
f : X → E
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
hi : ∀ (i : ι), IntegrableOn f (s i) μ
h : Summable fun i => ∫ (x : X) in s i, ‖f x‖ ∂μ
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
S' : Summable fun i => ⟨∫ (x : X) in s i, ↑‖f x‖₊ ∂μ, ⋯⟩
⊢ ∑' (b : ι), ENNReal.ofReal (∫ (a : X) in s b, ↑‖f a‖₊ ∂μ) < ⊤ | 1b9eef9dd9f111f9 |
AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | lemma awayToΓ_ΓToStalk (f) (x) :
awayToΓ 𝒜 f ≫ (Proj| pbo f).presheaf.Γgerm x =
CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 (Submonoid.powers_le.mpr x.2)) ≫
(Proj.stalkIso' 𝒜 x.1).toCommRingCatIso.inv ≫
((Proj.toLocallyRingedSpace 𝒜).restrictStalkIso (Opens.isOpenEmbedding _) x).inv | 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
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
(structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫
Proj.presheaf.germ (⋯.functor.obj ⊤) ((ConcreteCategory.hom (pbo f).inclusion') x) ⋯ =
CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯) ≫ («Proj».stalkIso' 𝒜 ↑x).toCommRingCatIso.inv | simp only [Proj.toLocallyRingedSpace, Proj.toSheafedSpace] | 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
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
(structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫
Presheaf.germ (structureSheaf 𝒜).val (⋯.functor.obj ⊤) ((ConcreteCategory.hom (pbo f).inclusion') x) ⋯ =
CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯) ≫ («Proj».stalkIso' 𝒜 ↑x).toCommRingCatIso.inv | 20e6bafa8e9595b7 |
Finsupp.apply_single' | Mathlib/Data/Finsupp/Single.lean | lemma apply_single' [Zero N] [Zero P] (e : N → P) (he : e 0 = 0) (a : α) (n : N) (b : α) :
e ((single a n) b) = single a (e n) b | α : Type u_1
N : Type u_7
P : Type u_8
inst✝¹ : Zero N
inst✝ : Zero P
e : N → P
he : e 0 = 0
a : α
n : N
b : α
⊢ e ((single a n) b) = (single a (e n)) b | simp only [single_apply] | α : Type u_1
N : Type u_7
P : Type u_8
inst✝¹ : Zero N
inst✝ : Zero P
e : N → P
he : e 0 = 0
a : α
n : N
b : α
⊢ e (if a = b then n else 0) = if a = b then e n else 0 | 9c4fe3566f120cc2 |
Std.DHashMap.Internal.Raw.WF.out | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem WF.out [BEq α] [Hashable α] [i₁ : EquivBEq α] [i₂ : LawfulHashable α] {m : Raw α β}
(h : m.WF) : Raw.WFImp m | case empty₀
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
m : Raw α β
β✝ : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
c✝ : Nat
i₁ : EquivBEq α
i₂ : LawfulHashable α
⊢ WFImp (Raw₀.empty c✝).val | exact Raw₀.wfImp_empty | no goals | a5a1f4748871aa23 |
sign_finRotate | Mathlib/GroupTheory/Perm/Fin.lean | theorem sign_finRotate (n : ℕ) : Perm.sign (finRotate (n + 1)) = (-1) ^ n | case succ
n : ℕ
ih : sign (finRotate (n + 1)) = (-1) ^ n
⊢ sign (decomposeFin.symm (1, finRotate (n + 1))) = (-1) ^ (n + 1) | simp [ih, pow_succ] | no goals | f9f2177ec4cba59d |
ProbabilityTheory.sum_prob_mem_Ioc_le | Mathlib/Probability/StrongLaw.lean | theorem sum_prob_mem_Ioc_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) {K : ℕ} {N : ℕ}
(hKN : K ≤ N) :
∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc (j : ℝ) N} ≤ ENNReal.ofReal (𝔼[X] + 1) | Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K N : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
i : ℕ
x✝ : i ∈ range N
I : ↑i ≤ ↑(i + 1)
⊢ (↑i + 1) * ∫ (x : ℝ) in ↑i..↑(i + 1), 1 ∂ρ ≤ ∫ (x : ℝ) in ↑i..↑(i + 1), x + 1 ∂ρ | simp_rw [intervalIntegral.integral_of_le I, ← integral_mul_left] | Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K N : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
i : ℕ
x✝ : i ∈ range N
I : ↑i ≤ ↑(i + 1)
⊢ ∫ (a : ℝ) in Set.Ioc ↑i ↑(i + 1), (↑i + 1) * 1 ∂ρ ≤ ∫ (x : ℝ) in Set.Ioc ↑i ↑(i + 1), x + 1 ∂ρ | eee258637f6277e6 |
nhdsLE_sup_nhdsGE | Mathlib/Topology/Order/LeftRight.lean | theorem nhdsLE_sup_nhdsGE (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a | rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | no goals | 8101ce80549b9cb7 |
unique_topology_of_t2 | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | theorem unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @IsTopologicalAddGroup 𝕜 t _)
(h₂ : @ContinuousSMul 𝕜 𝕜 _ hnorm.toUniformSpace.toTopologicalSpace t) (h₃ : @T2Space 𝕜 t) :
t = hnorm.toUniformSpace.toTopologicalSpace | case refine_1
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
⊢ Metric.closedBall 0 ε ∈ 𝓝 0 | rcases NormedField.exists_norm_lt 𝕜 hε with ⟨ξ₀, hξ₀, hξ₀ε⟩ | case refine_1.intro.intro
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
⊢ Metric.closedBall 0 ε ∈ 𝓝 0 | fe6a4018e5de36d9 |
AlgebraicGeometry.ι_sigmaIsoGlued_inv | Mathlib/AlgebraicGeometry/Limits.lean | @[reassoc (attr := simp)]
lemma ι_sigmaIsoGlued_inv (i) : (disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv = Sigma.ι f i | ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ (disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv = Sigma.ι f i | apply Scheme.forgetToLocallyRingedSpace.map_injective | case a
ι : Type u
f : ι → Scheme
i : (disjointGlueData f).J
⊢ Scheme.forgetToLocallyRingedSpace.map ((disjointGlueData f).ι i ≫ (sigmaIsoGlued f).inv) =
Scheme.forgetToLocallyRingedSpace.map (Sigma.ι f i) | e703f9b449707fcf |
Finsupp.lex_lt_iff_of_unique | Mathlib/Data/Finsupp/Lex.lean | theorem lex_lt_iff_of_unique [Preorder α] [LT N] [Unique α] {a b : Lex (α →₀ N)} :
a < b ↔ ofLex a default < ofLex b default | α : Type u_1
N : Type u_2
inst✝³ : Zero N
inst✝² : Preorder α
inst✝¹ : LT N
inst✝ : Unique α
a b : Lex (α →₀ N)
⊢ a < b ↔ (ofLex a) default < (ofLex b) default | simp only [lex_lt_iff, Unique.exists_iff, and_iff_right_iff_imp] | α : Type u_1
N : Type u_2
inst✝³ : Zero N
inst✝² : Preorder α
inst✝¹ : LT N
inst✝ : Unique α
a b : Lex (α →₀ N)
⊢ (ofLex a) default < (ofLex b) default → ∀ j < default, (ofLex a) j = (ofLex b) j | eaddb5247c90979c |
neg_one_pow_expChar | Mathlib/Algebra/CharP/Lemmas.lean | lemma neg_one_pow_expChar : (-1 : R) ^ p = -1 | R : Type u_1
inst✝ : Ring R
p : ℕ
hR : ExpChar R p
⊢ (-1) ^ p + 1 ^ p = 0 | rw [← add_pow_expChar_of_commute _ (Commute.one_right _), neg_add_cancel,
zero_pow (expChar_ne_zero R p)] | no goals | 2fe32e6ee22521e8 |
pow_three | Mathlib/Algebra/Group/Defs.lean | @[to_additive three_nsmul]
lemma pow_three (a : M) : a ^ 3 = a * (a * a) | M : Type u_2
inst✝ : Monoid M
a : M
⊢ a ^ 3 = a * (a * a) | rw [pow_succ', pow_two] | no goals | e34341b09e060e4f |
TypeVec.const_append1 | Mathlib/Data/TypeVec.lean | theorem const_append1 {β γ} (x : γ) {n} (α : TypeVec n) :
TypeVec.const x (α ::: β) = appendFun (TypeVec.const x α) fun _ => x | β : Type u_1
γ : Type u_2
x : γ
n : ℕ
α : TypeVec.{u_1} n
⊢ TypeVec.const x (α ::: β) = (TypeVec.const x α ::: fun x_1 => x) | ext i : 1 | case a
β : Type u_1
γ : Type u_2
x : γ
n : ℕ
α : TypeVec.{u_1} n
i : Fin2 (n + 1)
⊢ TypeVec.const x (α ::: β) i = (TypeVec.const x α ::: fun x_1 => x) i | 9edd16b3ce182866 |
Std.DHashMap.Internal.List.getValue?_eq_some_getValue! | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValue?_eq_some_getValue! [BEq α] [Inhabited β] {l : List ((_ : α) × β)} {a : α}
(h : containsKey a l = true) : getValue? a l = some (getValue! a l) | α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : Inhabited β
l : List ((_ : α) × β)
a : α
h : containsKey a l = true
⊢ getValue? a l = some (getValue! a l) | rw [getValue?_eq_some_getValue h, getValue_eq_getValue!] | no goals | 1fa8844bd88f5a74 |
CategoryTheory.Localization.Construction.morphismProperty_is_top | Mathlib/CategoryTheory/Localization/Construction.lean | theorem morphismProperty_is_top (P : MorphismProperty W.Localization)
[P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
(hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)) :
P = ⊤ | case h.h.h.a.mpr.nil
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂ : Paths (LocQuiver W)
⊢ P (G.map Quiver.Path.nil) | simpa only [Functor.map_id] using hP₁ (𝟙 X₁.obj) | no goals | bfd1f03091fbe383 |
Turing.TM1to1.tr_supports | Mathlib/Computability/PostTuringMachine.lean | theorem tr_supports [Inhabited Λ] {S : Finset Λ} (ss : Supports M S) :
Supports (tr enc dec M) (trSupp M S) :=
⟨Finset.mem_biUnion.2 ⟨_, ss.1, Finset.mem_insert_self _ _⟩, fun q h ↦ by
suffices ∀ q, SupportsStmt S q → (∀ q' ∈ writes q, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q) ∧
∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q') by
rcases Finset.mem_biUnion.1 h with ⟨l, hl, h⟩
have :=
this _ (ss.2 _ hl) fun q' hq ↦ Finset.mem_biUnion.2 ⟨_, hl, Finset.mem_insert_of_mem hq⟩
rcases Finset.mem_insert.1 h with (rfl | h)
exacts [this.1, this.2 _ h]
intro q hs hw
induction q with
| move d q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw; refine ⟨?_, IH.2⟩
cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]
| write f q IH =>
unfold writes at hw ⊢
simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and]
at hw ⊢
replace IH := IH hs fun q hq ↦ hw q (Or.inr hq)
refine ⟨supportsStmt_read _ fun a _ s ↦ hw _ (Or.inl ⟨_, rfl⟩), fun q' hq ↦ ?_⟩
rcases hq with (⟨a, q₂, rfl⟩ | hq)
· simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]
· exact IH.2 _ hq
| load a q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw
exact ⟨supportsStmt_read _ fun _ ↦ IH.1, IH.2⟩
| branch p q₁ q₂ IH₁ IH₂ =>
unfold writes at hw ⊢
simp only [Finset.mem_union] at hw ⊢
replace IH₁ := IH₁ hs.1 fun q hq ↦ hw q (Or.inl hq)
replace IH₂ := IH₂ hs.2 fun q hq ↦ hw q (Or.inr hq)
exact ⟨supportsStmt_read _ fun _ ↦ ⟨IH₁.1, IH₂.1⟩, fun q ↦ Or.rec (IH₁.2 _) (IH₂.2 _)⟩
| goto l =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
refine supportsStmt_read _ fun a _ s ↦ ?_
exact Finset.mem_biUnion.2 ⟨_, hs _ _, Finset.mem_insert_self _ _⟩
| halt =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
simp only [SupportsStmt, supportsStmt_move, trNormal]⟩
| case write
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q✝ : Λ' Γ Λ σ
h : q✝ ∈ trSupp M S
f : Γ → σ → Γ
q : Stmt Γ Λ σ
hs : SupportsStmt S (Stmt.write f q)
hw : ∀ (q' : Λ' Γ Λ σ), (∃ a, Λ'.write a q = q') ∨ q' ∈ writes q → q' ∈ trSupp M S
IH : SupportsStmt (trSupp M S) (trNormal dec q) ∧ ∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q')
q' : Λ' Γ Λ σ
hq : (∃ a, Λ'.write a q = q') ∨ q' ∈ writes q
⊢ SupportsStmt (trSupp M S) (tr enc dec M q') | rcases hq with (⟨a, q₂, rfl⟩ | hq) | case write.inl.intro.refl
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q✝ : Λ' Γ Λ σ
h : q✝ ∈ trSupp M S
f : Γ → σ → Γ
q : Stmt Γ Λ σ
hs : SupportsStmt S (Stmt.write f q)
hw : ∀ (q' : Λ' Γ Λ σ), (∃ a, Λ'.write a q = q') ∨ q' ∈ writes q → q' ∈ trSupp M S
IH : SupportsStmt (trSupp M S) (trNormal dec q) ∧ ∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q')
a : Γ
⊢ SupportsStmt (trSupp M S) (tr enc dec M (Λ'.write a q))
case write.inr
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q✝ : Λ' Γ Λ σ
h : q✝ ∈ trSupp M S
f : Γ → σ → Γ
q : Stmt Γ Λ σ
hs : SupportsStmt S (Stmt.write f q)
hw : ∀ (q' : Λ' Γ Λ σ), (∃ a, Λ'.write a q = q') ∨ q' ∈ writes q → q' ∈ trSupp M S
IH : SupportsStmt (trSupp M S) (trNormal dec q) ∧ ∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q')
q' : Λ' Γ Λ σ
hq : q' ∈ writes q
⊢ SupportsStmt (trSupp M S) (tr enc dec M q') | bea148d15df2302d |
Polynomial.roots_C_mul_X_sub_C | Mathlib/Algebra/Polynomial/FieldDivision.lean | theorem roots_C_mul_X_sub_C (b : R) (ha : a ≠ 0) : (C a * X - C b).roots = {a⁻¹ * b} | R : Type u
a : R
inst✝ : Field R
b : R
ha : a ≠ 0
⊢ (C a * X - C b).roots = {a⁻¹ * b} | simp [roots_C_mul_X_sub_C_of_IsUnit b ⟨a, a⁻¹, mul_inv_cancel₀ ha, inv_mul_cancel₀ ha⟩] | no goals | c47245df65c343d5 |
hofer | Mathlib/Analysis/Hofer.lean | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' | X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
H : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
this : Nonempty X
⊢ False | choose! F hF using H | X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
⊢ False | 7989b2596caedc2b |
Lean.Order.Array.monotone_findSomeRevM? | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_findSomeRevM?
(f : γ → α → m (Option β)) (xs : Array α) (hmono : monotone f) :
monotone (fun x => xs.findSomeRevM? (f x)) | case case1
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → m (Option β)
xs : Array α
hmono : monotone f
i : Nat
x✝¹ x✝ : 0 ≤ xs.size
⊢ monotone fun x => pure none | apply monotone_const | no goals | e7f4b15caf8d55dc |
Valuation.Integers.wellFounded_gt_on_v_iff_discrete_mrange | Mathlib/RingTheory/Valuation/Archimedean.lean | lemma wellFounded_gt_on_v_iff_discrete_mrange [Nontrivial (MonoidHom.mrange v)ˣ]
(hv : Integers v O) :
WellFounded ((· > ·) on (v ∘ algebraMap O F)) ↔
Nonempty (MonoidHom.mrange v ≃*o ℤₘ₀) | case refine_3
F : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : CommRing O
inst✝¹ : Algebra O F
v : Valuation F Γ₀
inst✝ : Nontrivial (↥(MonoidHom.mrange v))ˣ
hv : v.Integers O
h : {x | x ≤ 1}.WellFoundedOn fun x1 x2 => x1 > x2
x✝ : Γ₀
⊢ x✝ ∈ Set.range (⇑v ∘ ⇑(algebraMap O F)) →
(fun x => if hx : x ∈ MonoidHom.mrange v then ⟨x, hx⟩ else 1) x✝ ∈ {x | x ≤ 1} | simp only [Set.mem_range, Function.comp_apply, MonoidHom.mem_mrange, Set.mem_setOf_eq,
forall_exists_index] | case refine_3
F : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : CommRing O
inst✝¹ : Algebra O F
v : Valuation F Γ₀
inst✝ : Nontrivial (↥(MonoidHom.mrange v))ˣ
hv : v.Integers O
h : {x | x ≤ 1}.WellFoundedOn fun x1 x2 => x1 > x2
x✝ : Γ₀
⊢ ∀ (x : O), v ((algebraMap O F) x) = x✝ → (if h : ∃ x, v x = x✝ then ⟨x✝, ⋯⟩ else 1) ≤ 1 | 21ebccc05fad3ee9 |
tendsto_inf_principal_nhds_iff_of_forall_eq | Mathlib/Topology/Basic.lean | theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α}
(h : ∀ a ∉ s, f a = x) : Tendsto f (l ⊓ 𝓟 s) (𝓝 x) ↔ Tendsto f l (𝓝 x) | X : Type u
α : Type u_1
x : X
inst✝ : TopologicalSpace X
f : α → X
l : Filter α
s : Set α
h : ∀ a ∉ s, f a = x
⊢ l ⊓ 𝓟 s ≤ comap f (𝓝 x) ↔ l ≤ comap f (𝓝 x) | replace h : 𝓟 sᶜ ≤ comap f (𝓝 x) := by
rintro U ⟨t, ht, htU⟩ x hx
have : f x ∈ t := (h x hx).symm ▸ mem_of_mem_nhds ht
exact htU this | X : Type u
α : Type u_1
x : X
inst✝ : TopologicalSpace X
f : α → X
l : Filter α
s : Set α
h : 𝓟 sᶜ ≤ comap f (𝓝 x)
⊢ l ⊓ 𝓟 s ≤ comap f (𝓝 x) ↔ l ≤ comap f (𝓝 x) | 9c6173df4d0ca04a |
AlgebraicGeometry.Scheme.IdealSheafData.support_ofIdealTop | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma support_ofIdealTop (I : Ideal Γ(X, ⊤)) : (ofIdealTop I).support = X.zeroLocus (U := ⊤) I | case h
X : Scheme
I : Ideal ↑Γ(X, ⊤)
this : ∀ (U : ↑X.affineOpens), (ofIdealTop I).support ∩ ↑↑U = X.zeroLocus ↑I ∩ ↑↑U
x : ↑↑X.toPresheafedSpace
⊢ x ∈ (ofIdealTop I).support ↔ x ∈ X.zeroLocus ↑I | obtain ⟨_, ⟨U, hU, rfl⟩, hxU, -⟩ :=
(isBasis_affine_open X).exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ | case h.intro.intro.intro.intro.intro
X : Scheme
I : Ideal ↑Γ(X, ⊤)
this : ∀ (U : ↑X.affineOpens), (ofIdealTop I).support ∩ ↑↑U = X.zeroLocus ↑I ∩ ↑↑U
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
⊢ x ∈ (ofIdealTop I).support ↔ x ∈ X.zeroLocus ↑I | 544a16b750e3fe38 |
TensorPower.mul_one | Mathlib/LinearAlgebra/TensorPower/Basic.lean | theorem mul_one {n} (a : ⨂[R]^n M) : cast R M (add_zero _) (a ₜ* ₜ1) = a | case add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
x y : ⨂[R] (i : Fin n), M
hx : (cast R M ⋯) (mulEquiv (x ⊗ₜ[R] (tprod R) Fin.elim0)) = x
hy : (cast R M ⋯) (mulEquiv (y ⊗ₜ[R] (tprod R) Fin.elim0)) = y
⊢ (cast R M ⋯) (mulEquiv ((x + y) ⊗ₜ[R] (tprod R) Fin.elim0)) = x + y | rw [TensorProduct.add_tmul, map_add, map_add, hx, hy] | no goals | e803f9f54b1f406f |
LinearMap.range_prod_eq | Mathlib/LinearAlgebra/Prod.lean | theorem range_prod_eq {f : M →ₗ[R] M₂} {g : M →ₗ[R] M₃} (h : ker f ⊔ ker g = ⊤) :
range (prod f g) = (range f).prod (range g) | case intro.intro.intro.intro.refine_2
R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₂
g : M →ₗ[R] M₃
h : ker f ⊔ ker g = ⊤
x y : M
this : y - x ∈ ker f ⊔ ker g
x' : M
hx' : x' ∈ ker f
y' : M
hy' : y' ∈ ker g
H : x' + y' = y - x
⊢ g (x' + x) = g y | simp [← eq_sub_iff_add_eq.1 H, map_add, add_left_inj, self_eq_add_right, mem_ker.mp hy'] | no goals | 8c8d54532dff4081 |
ExistsContDiffBumpBase.y_le_one | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem y_le_one {D : ℝ} (x : E) (Dpos : 0 < D) : y D x ≤ 1 | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
A : (w D ⋆[lsmul ℝ ℝ, μ] φ) x ≤ (w D ⋆[lsmul ℝ ℝ, μ] 1) x
B : (w D ⋆[lsmul ℝ ℝ, μ] fun x => 1) x = 1
⊢ y D x ≤ 1 | exact A.trans (le_of_eq B) | no goals | c45a6bd215d89453 |
Algebra.FinitePresentation.iff_quotient_mvPolynomial' | Mathlib/RingTheory/FinitePresentation.lean | theorem iff_quotient_mvPolynomial' :
FinitePresentation R A ↔
∃ (ι : Type*) (_ : Fintype ι) (f : MvPolynomial ι R →ₐ[R] A),
Surjective f ∧ f.toRingHom.ker.FG | case mpr.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
ι : Type u_1
hfintype : Fintype ι
f : MvPolynomial ι R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
⊢ FinitePresentation R A | have equiv := MvPolynomial.renameEquiv R (Fintype.equivFin ι) | case mpr.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
ι : Type u_1
hfintype : Fintype ι
f : MvPolynomial ι R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
equiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R
⊢ FinitePresentation R A | ac8c7b9e721d66d4 |
EuclideanGeometry.Sphere.two_zsmul_oangle_eq | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem two_zsmul_oangle_eq {s : Sphere P} {p₁ p₂ p₃ p₄ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
(hp₃ : p₃ ∈ s) (hp₄ : p₄ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄) (hp₃p₁ : p₃ ≠ p₁)
(hp₃p₄ : p₃ ≠ p₄) : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄ | 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 p₃ p₄ : P
hp₁ : ‖p₁ -ᵥ s.center‖ = s.radius
hp₂ : ‖p₂ -ᵥ s.center‖ = s.radius
hp₃ : ‖p₃ -ᵥ s.center‖ = s.radius
hp₄ : ‖p₄ -ᵥ s.center‖ = s.radius
hp₂p₁ : p₂ ≠ p₁
hp₂p₄ : p₂ ≠ p₄
hp₃p₁ : p₃ ≠ p₁
hp₃p₄ : p₃ ≠ p₄
⊢ 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄ | rw [oangle, oangle, ← vsub_sub_vsub_cancel_right p₁ p₂ s.center, ←
vsub_sub_vsub_cancel_right p₄ p₂ s.center,
o.two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq _ _ _ _ hp₂ hp₃ hp₁ hp₄] <;>
simp [hp₂p₁, hp₂p₄, hp₃p₁, hp₃p₄] | no goals | cfd068165848c2fd |
CategoryTheory.FreeBicategory.liftHom₂_congr | Mathlib/CategoryTheory/Bicategory/Free.lean | theorem liftHom₂_congr {a b : FreeBicategory B} {f g : a ⟶ b} {η θ : Hom₂ f g} (H : Rel η θ) :
liftHom₂ F η = liftHom₂ F θ | B : Type u₁
inst✝¹ : Quiver B
C : Type u₂
inst✝ : Bicategory C
F : B ⥤q C
a b : FreeBicategory B
f g : a ⟶ b
η θ : Hom₂ f g
H : Rel η θ
⊢ liftHom₂ F η = liftHom₂ F θ | induction H <;> (dsimp [liftHom₂]; aesop_cat) | no goals | 61a856aeb7cd21e3 |
Std.Sat.AIG.denote_congr | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Lemmas.lean | theorem denote_congr (assign1 assign2 : α → Bool) (aig : AIG α) (idx : Nat)
(hidx : idx < aig.decls.size) (h : ∀ a, a ∈ aig → assign1 a = assign2 a) :
⟦aig, ⟨idx, hidx⟩, assign1⟧ = ⟦aig, ⟨idx, hidx⟩, assign2⟧ | case hconst
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign1 assign2 : α → Bool
aig : AIG α
idx : Nat
hidx : idx < aig.decls.size
h : ∀ (a : α), a ∈ aig → assign1 a = assign2 a
b : Bool
heq : aig.decls[idx] = Decl.const b
⊢ ⟦assign1, { aig := aig, ref := { gate := idx, hgate := hidx } }⟧ =
⟦assign2, { aig := aig, ref := { gate := idx, hgate := hidx } }⟧ | simp [denote_idx_const heq] | no goals | 4309a0099ebca30e |
summable_mul_of_bigO_atTop' | Mathlib/NumberTheory/AbelSummation.lean | theorem summable_mul_of_bigO_atTop'
(hf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t)
(hf_int : LocallyIntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici 1))
(h_bdd : (fun n : ℕ ↦ ‖f n‖ * ∑ k ∈ Icc 1 n, ‖c k‖) =O[atTop] fun _ ↦ (1 : ℝ))
{g : ℝ → ℝ} (hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 1 ⌊t⌋₊, ‖c k‖) =O[atTop] g)
(hg₂ : IntegrableAtFilter g atTop) :
Summable (fun n : ℕ ↦ f n * c n) | 𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x => ‖f x‖) t
hf_int : LocallyIntegrableOn (deriv fun t => ‖f t‖) (Set.Ici 1) volume
g : ℝ → ℝ
hg₂ : IntegrableAtFilter g atTop volume
h : ∀ (n : ℕ), ∑ k ∈ Icc 1 n, ‖c k‖ = ∑ k ∈ Icc 0 n, ‖(fun n => if n = 0 then 0 else c n) k‖
h_bdd : (fun n => ‖f ↑n‖ * ∑ x ∈ Icc 0 n, ‖if x = 0 then 0 else c x‖) =O[atTop] fun x => 1
hg₁ : (fun t => deriv (fun t => ‖f t‖) t * ∑ x ∈ Icc 0 ⌊t⌋₊, ‖if x = 0 then 0 else c x‖) =O[atTop] g
n : ℕ
⊢ ‖(fun n => if n = 0 then 0 else c n) 0‖ = 0 | simp only [reduceIte, norm_zero] | no goals | c2201db1ff4cce62 |
Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s t a b : ℝ} (hs : 0 < s) (ht : 0 < t)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
Gamma (a * s + b * t) ≤ Gamma s ^ a * Gamma t ^ b | s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).IsConjExponent (1 / b)
hab' : b = 1 - a
⊢ 0 < a * s + b * t | positivity | no goals | 3f18cd4d00bf972d |
CategoryTheory.Equalizer.Presieve.Arrows.w | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | theorem w : forkMap P X π ≫ firstMap P X π = forkMap P X π ≫ secondMap P X π | C : Type u
inst✝¹ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (Presieve.ofArrows X π).hasPullbacks
⊢ forkMap P X π ≫ firstMap P X π = forkMap P X π ≫ secondMap P X π | ext x ij | case h.h
C : Type u
inst✝¹ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (Presieve.ofArrows X π).hasPullbacks
x : P.obj (op B)
ij : I × I
⊢ Pi.π (fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) ij ((forkMap P X π ≫ firstMap P X π) x) =
Pi.π (fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) ij ((forkMap P X π ≫ secondMap P X π) x) | 7a0faa4126f0b0be |
Real.tendsto_sum_pi_div_four | Mathlib/Data/Real/Pi/Leibniz.lean | theorem tendsto_sum_pi_div_four :
Tendsto (fun k => ∑ i ∈ range k, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 (π / 4)) | case hf0
⊢ Tendsto (fun i => (2 * ↑i + 1)⁻¹) atTop (𝓝 0) | apply Tendsto.inv_tendsto_atTop | case hf0.h
⊢ Tendsto (fun i => 2 * ↑i + 1) atTop atTop | a59a5c77321208c0 |
Complex.tan_arctan | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | theorem tan_arctan {z : ℂ} (h₁ : z ≠ I) (h₂ : z ≠ -I) : tan (arctan z) = z | z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
⊢ tan z.arctan = z | unfold tan sin cos | z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
⊢ (cexp (-z.arctan * I) - cexp (z.arctan * I)) * I / 2 / ((cexp (z.arctan * I) + cexp (-z.arctan * I)) / 2) = z | b2b39565dc385b44 |
limsup_finset_sup' | Mathlib/Order/LiminfLimsup.lean | theorem limsup_finset_sup' [ConditionallyCompleteLinearOrder β] {f : Filter α}
{F : ι → α → β} {s : Finset ι} (hs : s.Nonempty)
(h₁ : ∀ i ∈ s, f.IsCoboundedUnder (· ≤ ·) (F i) | case a
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
F : ι → α → β
s : Finset ι
hs : s.Nonempty
h₁ : autoParam (∀ i ∈ s, IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)) _auto✝
h₂ : autoParam (∀ i ∈ s, IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)) _auto✝
bddsup : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f fun a => s.sup' hs fun i => F i a
h₃ : ∃ i ∈ s, IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)
cobddsup : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun a => s.sup' hs fun i => F i a
b : β
hb : b > s.sup' hs fun i => limsup (F i) f
⊢ ∀ᶠ (a : α) in f, (s.sup' hs fun i => F i a) < b | rw [eventually_iff_exists_mem] | case a
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
F : ι → α → β
s : Finset ι
hs : s.Nonempty
h₁ : autoParam (∀ i ∈ s, IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)) _auto✝
h₂ : autoParam (∀ i ∈ s, IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)) _auto✝
bddsup : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f fun a => s.sup' hs fun i => F i a
h₃ : ∃ i ∈ s, IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f (F i)
cobddsup : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun a => s.sup' hs fun i => F i a
b : β
hb : b > s.sup' hs fun i => limsup (F i) f
⊢ ∃ v ∈ f, ∀ y ∈ v, (s.sup' hs fun i => F i y) < b | 63b957bf23cf7390 |
exists_squarefree_dvd_pow_of_ne_zero | Mathlib/Algebra/Squarefree/Basic.lean | lemma _root_.exists_squarefree_dvd_pow_of_ne_zero {x : R} (hx : x ≠ 0) :
∃ (y : R) (n : ℕ), Squarefree y ∧ y ∣ x ∧ x ∣ y ^ n | case hi.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
z p : R
hz : z ≠ 0
hp : Irreducible p
ih : z ≠ 0 → ∃ y n, Squarefree y ∧ y ∣ z ∧ z ∣ y ^ n
hx : p * z ≠ 0
y : R
n : ℕ
hy : Squarefree y
hyx : y ∣ z
hy' : z ∣ y ^ n
⊢ ∃ y n, Squarefree y ∧ y ∣ p * z ∧ p * z ∣ y ^ n | rcases n.eq_zero_or_pos with rfl | hn | case hi.intro.intro.intro.intro.inl
R : Type u_1
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
z p : R
hz : z ≠ 0
hp : Irreducible p
ih : z ≠ 0 → ∃ y n, Squarefree y ∧ y ∣ z ∧ z ∣ y ^ n
hx : p * z ≠ 0
y : R
hy : Squarefree y
hyx : y ∣ z
hy' : z ∣ y ^ 0
⊢ ∃ y n, Squarefree y ∧ y ∣ p * z ∧ p * z ∣ y ^ n
case hi.intro.intro.intro.intro.inr
R : Type u_1
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
z p : R
hz : z ≠ 0
hp : Irreducible p
ih : z ≠ 0 → ∃ y n, Squarefree y ∧ y ∣ z ∧ z ∣ y ^ n
hx : p * z ≠ 0
y : R
n : ℕ
hy : Squarefree y
hyx : y ∣ z
hy' : z ∣ y ^ n
hn : n > 0
⊢ ∃ y n, Squarefree y ∧ y ∣ p * z ∧ p * z ∣ y ^ n | 2f11f1533d70f050 |
Nat.primeFactorsList_count_eq | Mathlib/Data/Nat/Factorization/Defs.lean | theorem primeFactorsList_count_eq {n p : ℕ} : n.primeFactorsList.count p = n.factorization p | case inr
n p : ℕ
hn0 : n > 0
pp : Prime p
⊢ count p n.primeFactorsList = n.factorization p | simp only [factorization_def _ pp] | case inr
n p : ℕ
hn0 : n > 0
pp : Prime p
⊢ count p n.primeFactorsList = padicValNat p n | 742c414288dd44f1 |
AddMonoidAlgebra.supDegree_sub_lt_of_leadingCoeff_eq | Mathlib/Algebra/MonoidAlgebra/Degree.lean | lemma supDegree_sub_lt_of_leadingCoeff_eq (hD : D.Injective) {R} [CommRing R] {p q : R[A]}
(hd : p.supDegree D = q.supDegree D) (hc : p.leadingCoeff D = q.leadingCoeff D) :
(p - q).supDegree D < p.supDegree D ∨ p = q | A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : CommRing R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
⊢ ¬p = q → supDegree D (p - q) < supDegree D p | refine fun he => (supDegree_sub_le.trans ?_).lt_of_ne ?_ | case refine_1
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : CommRing R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬p = q
⊢ supDegree D p ⊔ supDegree D q ≤ supDegree D p
case refine_2
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : CommRing R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬p = q
⊢ supDegree D (p - q) ≠ supDegree D p | 8a45d765e6091c8f |
ContextFreeRule.rewrites_iff | Mathlib/Computability/ContextFreeGrammar.lean | theorem rewrites_iff :
r.Rewrites u v ↔ ∃ p q : List (Symbol T N),
u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q :=
⟨Rewrites.exists_parts, by rintro ⟨p, q, rfl, rfl⟩; apply rewrites_of_exists_parts⟩
| case intro.intro.intro
T : Type u_1
N : Type u_2
r : ContextFreeRule T N
p q : List (Symbol T N)
⊢ r.Rewrites (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q) | apply rewrites_of_exists_parts | no goals | 018f0091ad90e8b6 |
Real.logb_nonneg_iff | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem logb_nonneg_iff (hx : 0 < x) : 0 ≤ logb b x ↔ 1 ≤ x | b x : ℝ
hb : 1 < b
hx : 0 < x
⊢ 0 ≤ logb b x ↔ 1 ≤ x | rw [← not_lt, logb_neg_iff hb hx, not_lt] | no goals | 9bebd044aa988f2d |
MeasureTheory.integral_comp_rpow_Ioi | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
(∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y | case h.mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
p : ℝ
hp : p ≠ 0
S : Set ℝ := Ioi 0
a1 : ∀ x ∈ S, HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x
a2 : InjOn (fun x => x ^ p) S
x : ℝ
hx : x ∈ S
⊢ ∃ x_1 ∈ S, x_1 ^ p = x | refine ⟨x ^ (1 / p), rpow_pos_of_pos hx _, ?_⟩ | case h.mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
p : ℝ
hp : p ≠ 0
S : Set ℝ := Ioi 0
a1 : ∀ x ∈ S, HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x
a2 : InjOn (fun x => x ^ p) S
x : ℝ
hx : x ∈ S
⊢ (x ^ (1 / p)) ^ p = x | 2e1724a0aef0efd0 |
Algebra.Generators.CotangentSpace.fst_compEquiv | Mathlib/RingTheory/Kaehler/JacobiZariski.lean | lemma CotangentSpace.fst_compEquiv :
LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ
(compEquiv Q P).toLinearMap = Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom | R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type uT
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
⊢ LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P) =
Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i) := i <;>
simp only [comp_vars, Basis.repr_symm_apply, Finsupp.linearCombination_single, Basis.prod_apply,
LinearMap.coe_inl, LinearMap.coe_inr, Sum.elim_inl, Function.comp_apply, one_smul,
Basis.repr_self, Finsupp.single_apply, pderiv_X, Pi.single_apply, RingHom.map_ite_one_zero,
Sum.elim_inr, Function.comp_apply, Basis.baseChange_apply, one_smul,
map_zero, Finsupp.coe_zero, Pi.zero_apply, derivation_C] | no goals | f446de9ba069a3ea |
Finset.sup'_const | Mathlib/Data/Finset/Lattice/Fold.lean | theorem sup'_const (a : α) : s.sup' H (fun _ => a) = a | α : Type u_2
β : Type u_3
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
a : α
⊢ (s.sup' H fun x => a) = a | apply le_antisymm | case a
α : Type u_2
β : Type u_3
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
a : α
⊢ (s.sup' H fun x => a) ≤ a
case a
α : Type u_2
β : Type u_3
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
a : α
⊢ a ≤ s.sup' H fun x => a | 19a233cc520bfbb7 |
IsTopologicallyNilpotent.map | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | theorem map {F : Type*} [FunLike F R S] [MonoidWithZeroHomClass F R S]
{φ : F} (hφ : Continuous φ) {a : R} (ha : IsTopologicallyNilpotent a) :
IsTopologicallyNilpotent (φ a) | R : Type u_1
S : Type u_2
inst✝⁵ : TopologicalSpace R
inst✝⁴ : MonoidWithZero R
inst✝³ : MonoidWithZero S
inst✝² : TopologicalSpace S
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : MonoidWithZeroHomClass F R S
φ : F
hφ : Continuous ⇑φ
a : R
ha : Tendsto (fun x => a ^ x) atTop (𝓝 0)
⊢ Tendsto (fun x => φ (a ^ x)) atTop (𝓝 0) | exact (map_zero φ ▸ hφ.tendsto 0).comp ha | no goals | 3b2b48869271a747 |
Array.getElem_reverse | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_reverse (as : Array α) (i : Nat) (hi : i < as.reverse.size) :
(as.reverse)[i] = as[as.size - 1 - i]'(by simp at hi; omega) | α : Type ?u.328456
as : Array α
i : Nat
hi : i < as.reverse.size
⊢ as.size - 1 - i < as.size | simp at hi | α : Type ?u.328456
as : Array α
i : Nat
hi : i < as.size
⊢ as.size - 1 - i < as.size | af4a1d0749ff35ea |
AddCircle.closedBall_eq_univ_of_half_period_le | Mathlib/Analysis/Normed/Group/AddCircle.lean | theorem closedBall_eq_univ_of_half_period_le (hp : p ≠ 0) (x : AddCircle p) {ε : ℝ}
(hε : |p| / 2 ≤ ε) : closedBall x ε = univ :=
eq_univ_iff_forall.mpr fun x => by
simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε
| p : ℝ
hp : p ≠ 0
x✝ : AddCircle p
ε : ℝ
hε : |p| / 2 ≤ ε
x : AddCircle p
⊢ x ∈ closedBall x✝ ε | simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε | no goals | e005dcb1cc4af2ea |
List.Nodup.union | Mathlib/Data/List/Nodup.lean | theorem Nodup.union [DecidableEq α] (l₁ : List α) (h : Nodup l₂) : (l₁ ∪ l₂).Nodup | case cons
α : Type u
inst✝ : DecidableEq α
a : α
l₁ : List α
ih : ∀ {l₂ : List α}, l₂.Nodup → (l₁ ∪ l₂).Nodup
l₂ : List α
h : l₂.Nodup
⊢ (a :: l₁ ∪ l₂).Nodup | exact (ih h).insert | no goals | 703b8be42fe97c42 |
MeasureTheory.integral_prod | Mathlib/MeasureTheory/Integral/Prod.lean | theorem integral_prod (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
∫ z, f z ∂μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ | case pos.h_ind
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
c : E
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s < ⊤
⊢ ((μ.prod ν) s).toReal • c = (∫⁻ (a : α), ν (Prod.mk a ⁻¹' s) ∂μ).toReal • c | rw [prod_apply hs] | no goals | fa8a81f609cb5cf4 |
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