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 |
|---|---|---|---|---|---|---|
PrimeSpectrum.zeroLocus_empty_of_one_mem | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ | R : Type u
inst✝ : CommSemiring R
s : Set R
h : 1 ∈ s
x : PrimeSpectrum R
hx : s ⊆ ↑x.asIdeal
x_prime : x.asIdeal.IsPrime
⊢ 1 ∈ x.asIdeal | exact hx h | no goals | 5067e5311eeaeb23 |
Set.image_seq | Mathlib/Data/Set/Lattice.lean | theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t | α : Type u_1
β : Type u_2
γ : Type u_3
f : β → γ
s : Set (α → β)
t : Set α
⊢ f '' s.seq t = ((fun x => f ∘ x) '' s).seq t | simp only [seq, image_image2, image2_image_left, comp_apply] | no goals | f3a68a90ab8ae5d5 |
contMDiffWithinAt_iff_target | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | theorem contMDiffWithinAt_iff_target :
ContMDiffWithinAt I I' n f s x ↔
ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' (f x) ∘ f) s x | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | have cont :
ContinuousWithinAt f s x ∧ ContinuousWithinAt (extChartAt I' (f x) ∘ f) s x ↔
ContinuousWithinAt f s x :=
and_iff_left_of_imp <| (continuousAt_extChartAt _).comp_continuousWithinAt | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | 5708dc3777c22e45 |
Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing | Mathlib/RingTheory/Jacobson/Ring.lean | theorem quotient_mk_comp_C_isIntegral_of_isJacobsonRing :
((Ideal.Quotient.mk P).comp C : R →+* R[X] ⧸ P).IsIntegral | R : Type u_1
inst✝¹ : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝ : IsJacobsonRing R
P' : Ideal R := comap C P
this : P'.IsPrime
f : R[X] →+* (R ⧸ P')[X] := mapRingHom (Ideal.Quotient.mk P')
hf : Function.Surjective ⇑f
p : R[X]
hp : p ∈ comap f ⊥
n : ℕ
⊢ (Ideal.Quotient.mk (comap C P)) (p.coeff n) = 0 | simpa only [f, coeff_map, coe_mapRingHom] using (Polynomial.ext_iff.mp hp) n | no goals | a0c73d64ce92331e |
ENNReal.lt_iff_exists_rat_btwn | Mathlib/Data/ENNReal/Basic.lean | theorem lt_iff_exists_rat_btwn :
a < b ↔ ∃ q : ℚ, 0 ≤ q ∧ a < Real.toNNReal q ∧ (Real.toNNReal q : ℝ≥0∞) < b :=
⟨fun h => by
rcases lt_iff_exists_coe.1 h with ⟨p, rfl, _⟩
rcases exists_between h with ⟨c, pc, cb⟩
rcases lt_iff_exists_coe.1 cb with ⟨r, rfl, _⟩
rcases (NNReal.lt_iff_exists_rat_btwn _... | case intro.intro.intro.intro.intro.intro.intro.intro.intro
b : ℝ≥0∞
p : ℝ≥0
right✝¹ h : ↑p < b
r : ℝ≥0
right✝ : ↑r < b
pc : ↑p < ↑r
cb : ↑r < b
q : ℚ
hq0 : 0 ≤ q
pq : p < (↑q).toNNReal
qr : (↑q).toNNReal < r
⊢ ∃ q, 0 ≤ q ∧ ↑p < ↑(↑q).toNNReal ∧ ↑(↑q).toNNReal < b | exact ⟨q, hq0, coe_lt_coe.2 pq, lt_trans (coe_lt_coe.2 qr) cb⟩ | no goals | aae9f40da5b98e1c |
ProbabilityTheory.Kernel.comp_add_right | Mathlib/Probability/Kernel/Composition/Comp.lean | lemma comp_add_right (μ κ : Kernel α β) (η : Kernel β γ) :
η ∘ₖ (μ + κ) = η ∘ₖ μ + η ∘ₖ κ | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ κ : Kernel α β
η : Kernel β γ
a✝ : α
s✝ : Set γ
hs : MeasurableSet s✝
⊢ ((η ∘ₖ (μ + κ)) a✝) s✝ = ((η ∘ₖ μ + η ∘ₖ κ) a✝) s✝ | simp [comp_apply' _ _ _ hs] | no goals | 79efe5ce49ef166d |
BooleanRing.neg_eq | Mathlib/Algebra/Ring/BooleanRing.lean | theorem neg_eq : -a = a :=
calc
-a = -a + 0 | α : Type u_1
inst✝ : BooleanRing α
a : α
⊢ -a + 0 = -a + -a + a | rw [← neg_add_cancel, add_assoc] | no goals | b77c191f3c8acf07 |
MeasureTheory.integral_eq_lintegral_of_nonneg_ae | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_eq_lintegral_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f)
(hfm : AEStronglyMeasurable f μ) :
∫ a, f a ∂μ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal (f a) ∂μ) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᶠ[ae μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEMeasurable (fun a => ENNReal.ofReal (-f a)) μ | exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg | no goals | d2ae86a5dd2a5d0c |
BitVec.shiftLeft_ushiftRight | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem shiftLeft_ushiftRight {x : BitVec w} {n : Nat}:
x >>> n <<< n = x &&& BitVec.allOnes w <<< n | case neg
w n : Nat
ih : ∀ {x : BitVec w}, x >>> n <<< n = x &&& allOnes w <<< n
x : BitVec w
i : Nat
h : i < w
hw : ¬w = 0
⊢ (x >>> 1 <<< 1 &&& allOnes w <<< n <<< 1).getLsbD i = (x &&& allOnes w <<< n <<< 1).getLsbD i | by_cases hi₂ : i = 0 | case pos
w n : Nat
ih : ∀ {x : BitVec w}, x >>> n <<< n = x &&& allOnes w <<< n
x : BitVec w
i : Nat
h : i < w
hw : ¬w = 0
hi₂ : i = 0
⊢ (x >>> 1 <<< 1 &&& allOnes w <<< n <<< 1).getLsbD i = (x &&& allOnes w <<< n <<< 1).getLsbD i
case neg
w n : Nat
ih : ∀ {x : BitVec w}, x >>> n <<< n = x &&& allOnes w <<< n
x : BitV... | d74277fca2e60b6a |
norm_mahler_eq | Mathlib/NumberTheory/Padics/MahlerBasis.lean | /--
The uniform norm of the `k`-th Mahler basis function is 1, for every `k`.
-/
@[simp] lemma norm_mahler_eq (k : ℕ) : ‖(mahler k : C(ℤ_[p], ℚ_[p]))‖ = 1 | p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
⊢ ‖mahler k‖ = 1 | apply le_antisymm | case a
p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
⊢ ‖mahler k‖ ≤ 1
case a
p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
⊢ 1 ≤ ‖mahler k‖ | bf8ddaaf139299df |
ComplexShape.Embedding.r_eq_some | Mathlib/Algebra/Homology/Embedding/Basic.lean | lemma r_eq_some {i : ι} {i' : ι'} (hi : e.f i = i') :
e.r i' = some i | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.f i = i'
⊢ e.r i' = some i | have h : ∃ (i : ι), e.f i = i' := ⟨i, hi⟩ | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.f i = i'
h : ∃ i, e.f i = i'
⊢ e.r i' = some i | a68fee8f05e51fd2 |
Filter.exists_seq_monotone_tendsto_atTop_atTop | Mathlib/Order/Filter/AtTopBot/CountablyGenerated.lean | theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [Preorder α] [Nonempty α]
[IsDirected α (· ≤ ·)] [(atTop : Filter α).IsCountablyGenerated] :
∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop | case intro.refine_2
α : Type u_3
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : IsDirected α fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
ys : ℕ → α
h : Tendsto ys atTop atTop
c : α → α → α
hleft : ∀ (a b : α), a ≤ c a b
hright : ∀ (a b : α), b ≤ c a b
xs : ℕ → α := fun n => List.foldl (fun x n => c x (ys ... | rw [hsucc] | case intro.refine_2
α : Type u_3
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : IsDirected α fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
ys : ℕ → α
h : Tendsto ys atTop atTop
c : α → α → α
hleft : ∀ (a b : α), a ≤ c a b
hright : ∀ (a b : α), b ≤ c a b
xs : ℕ → α := fun n => List.foldl (fun x n => c x (ys ... | cfd85fba236c098c |
ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | theorem Ico_map_valMinAbs_natAbs_eq_Ico_map_id (p : ℕ) [hp : Fact p.Prime] (a : ZMod p)
(hap : a ≠ 0) : ((Ico 1 (p / 2).succ).1.map fun (x : ℕ) => (a * x).valMinAbs.natAbs) =
(Ico 1 (p / 2).succ).1.map fun a => a | p : ℕ
hp : Fact (Nat.Prime p)
a : ZMod p
hap : a ≠ 0
he : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x ≠ 0 ∧ x ≤ p / 2
hep : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x < p
hpe : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → ¬p ∣ x
⊢ ∀ x ∈ Ico 1 (p / 2).succ, (a * ↑x).valMinAbs.natAbs ∈ Ico 1 (p / 2).succ | intro x hx | p : ℕ
hp : Fact (Nat.Prime p)
a : ZMod p
hap : a ≠ 0
he : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x ≠ 0 ∧ x ≤ p / 2
hep : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x < p
hpe : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → ¬p ∣ x
x : ℕ
hx : x ∈ Ico 1 (p / 2).succ
⊢ (a * ↑x).valMinAbs.natAbs ∈ Ico 1 (p / 2).succ | e252720ce4f19196 |
Monoid.Coprod.snd_toProd | Mathlib/GroupTheory/Coprod/Basic.lean | theorem snd_toProd (x : M ∗ N) : (toProd x).2 = snd x | M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
x : M ∗ N
⊢ (toProd x).2 = snd x | rw [← snd_comp_toProd] | M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
x : M ∗ N
⊢ (toProd x).2 = ((MonoidHom.snd M N).comp toProd) x | 2fb06f44b53bed66 |
ContinuousLinearMap.coprod_add | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | @[simp] lemma coprod_add (f₁ g₁ : M₁ →L[R] M) (f₂ g₂ : M₂ →L[R] M) :
(f₁ + g₁).coprod (f₂ + g₂) = f₁.coprod f₂ + g₁.coprod g₂ | R : Type u_1
M : Type u_3
M₁ : Type u_5
M₂ : Type u_6
inst✝¹⁰ : Semiring R
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousAdd M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R... | ext <;> simp | no goals | f7b4e0746d10c1b9 |
ProbabilityTheory.gaussianReal_map_const_mul | Mathlib/Probability/Distributions/Gaussian.lean | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ Measure.dirac 0 = gaussianReal 0 (⟨0 ^ 2, ⋯⟩ * v) | convert (gaussianReal_zero_var 0).symm | case h.e'_3.h.e'_2
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ ⟨0 ^ 2, ⋯⟩ * v = 0 | 1ceed6d297abcf77 |
Stream'.Seq.terminatedAt_zero_iff | Mathlib/Data/Seq/Seq.lean | theorem terminatedAt_zero_iff {s : Seq α} : s.TerminatedAt 0 ↔ s = nil | case refine_2
α : Type u
⊢ nil.TerminatedAt 0 | simp [TerminatedAt] | no goals | 10aa63b0fda6161a |
IsGLB.mul_left | Mathlib/Algebra/Order/Field/Basic.lean | theorem IsGLB.mul_left {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :
IsGLB ((fun b => a * b) '' s) (a * b) | α : Type u_2
inst✝ : LinearOrderedSemifield α
a b : α
s : Set α
ha : 0 ≤ a
hs : IsGLB s b
⊢ IsGLB ((fun b => a * b) '' s) (a * b) | rcases lt_or_eq_of_le ha with (ha | rfl) | case inl
α : Type u_2
inst✝ : LinearOrderedSemifield α
a b : α
s : Set α
ha✝ : 0 ≤ a
hs : IsGLB s b
ha : 0 < a
⊢ IsGLB ((fun b => a * b) '' s) (a * b)
case inr
α : Type u_2
inst✝ : LinearOrderedSemifield α
b : α
s : Set α
hs : IsGLB s b
ha : 0 ≤ 0
⊢ IsGLB ((fun b => 0 * b) '' s) (0 * b) | 88bccf1fd81bb4e6 |
roth_3ap_theorem_nat | Mathlib/Combinatorics/Additive/Corner/Roth.lean | theorem roth_3ap_theorem_nat (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound (ε / 3) ≤ n)
(A : Finset ℕ) (hAn : A ⊆ range n) (hAε : ε * n ≤ #A) : ¬ ThreeAPFree (A : Set ℕ) | n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' (Nat.cast '' ↑A))
this✝¹ : ↑A = Fin.val '' (Nat.cast '' ↑A)
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x => ↑x) A)
⊢ ε / 3 * ↑(F... | calc
_ = ε / 3 * (2 * n + 1) := by simp
_ ≤ ε / 3 * (2 * n + n) := by gcongr; simp; unfold cornersTheoremBound at hG; omega
_ = ε * n := by ring
_ ≤ #A := hAε
_ = _ := by
rw [card_image_of_injOn]
exact (CharP.natCast_injOn_Iio (Fin (2 * n).succ) (2 * n).succ).mono <| hAn.trans <| by
simp; omega | no goals | c707b5b13a459d8d |
Sum.lex_inl_inl | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Sum/Basic.lean | theorem lex_inl_inl : Lex r s (inl a₁) (inl a₂) ↔ r a₁ a₂ :=
⟨fun h => by cases h; assumption, Lex.inl⟩
| α✝ : Type u_1
r : α✝ → α✝ → Prop
β✝ : Type u_2
s : β✝ → β✝ → Prop
a₁ a₂ : α✝
h : Lex r s (inl a₁) (inl a₂)
⊢ r a₁ a₂ | cases h | case inl
α✝ : Type u_1
r : α✝ → α✝ → Prop
β✝ : Type u_2
s : β✝ → β✝ → Prop
a₁ a₂ : α✝
h✝ : r a₁ a₂
⊢ r a₁ a₂ | d3aae9eca3e3cd38 |
Besicovitch.exists_goodδ | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | theorem exists_goodδ :
∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
(∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E | case intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
∀ (δ : ℝ),
0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : ... | exact fmem i | no goals | 7c59972754532a8d |
Module.Flat.exists_factorization_of_apply_eq_zero_of_free | Mathlib/RingTheory/Flat/EquationalCriterion.lean | theorem exists_factorization_of_apply_eq_zero_of_free [Flat R M] {N : Type*} [AddCommGroup N]
[Module R N] [Free R N] [Module.Finite R N] {f : N} {x : N →ₗ[R] M} (h : x f = 0) :
∃ (k : ℕ) (a : N →ₗ[R] (Fin k →₀ R)) (y : (Fin k →₀ R) →ₗ[R] M), x = y ∘ₗ a ∧ a f = 0 :=
have e := ((Module.Free.chooseBasis R N).re... | R : Type u_1
M : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Flat R M
N : Type u_3
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : N
x : N →ₗ[R] M
h : x f = 0
e : (Fin (Fintype.card (Free.ChooseBasisIndex R N)) →₀ R) ≃ₗ[R] N
k : ℕ
a : (Fi... | rwa [← comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm] | no goals | c0d139cfd91c1db8 |
LeftOrdContinuous.iterate | Mathlib/Order/OrdContinuous.lean | theorem iterate {f : α → α} (hf : LeftOrdContinuous f) (n : ℕ) :
LeftOrdContinuous f^[n] | case succ
α : Type u
inst✝ : Preorder α
f : α → α
hf : LeftOrdContinuous f
n : ℕ
ihn : LeftOrdContinuous f^[n]
⊢ LeftOrdContinuous f^[n + 1] | exact ihn.comp hf | no goals | 3ce387ee4ba6ee92 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastAdd.go_get | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Add.lean | theorem go_get (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (cin : Ref aig)
(s : AIG.RefVec aig curr) (lhs rhs : AIG.RefVec aig w) :
∀ (idx : Nat) (hidx : idx < curr),
(go aig lhs rhs curr hcurr cin s).vec.get idx (by omega)
=
(s.get idx hidx).cast (by apply go_le_size) | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefVec curr
lhs rhs : aig.RefVec w
idx✝ : Nat
hidx✝ : idx✝ < curr
⊢ (go aig lhs rhs curr hcurr cin s).vec.get idx✝ ⋯ = (s.get idx✝ hidx✝).cast ⋯ | apply go_get_aux | no goals | a0b97a99a40e7a7a |
CoxeterSystem.getElem_succ_leftInvSeq_alternatingWord | Mathlib/GroupTheory/Coxeter/Inversion.lean | lemma getElem_succ_leftInvSeq_alternatingWord
(i j : B) (p k : ℕ) (h : k + 1 < 2 * p) :
(lis (alternatingWord i j (2 * p)))[k + 1]'(by simp; exact h) =
MulAut.conj (s i) ((lis (alternatingWord j i (2 * p)))[k]'(by simp; omega)) | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i j : B
p k : ℕ
h : k + 1 < 2 * p
⊢ k < (alternatingWord j i (2 * p)).length | simp[h] | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i j : B
p k : ℕ
h : k + 1 < 2 * p
⊢ k < 2 * p | 92714ccf6a2cfd6b |
RingQuot.smul_quot | Mathlib/Algebra/RingQuot.lean | theorem smul_quot [Algebra S R] {n : S} {a : R} :
(n • ⟨Quot.mk _ a⟩ : RingQuot r) = ⟨Quot.mk _ (n • a)⟩ | R : Type uR
inst✝² : Semiring R
S : Type uS
inst✝¹ : CommSemiring S
r : R → R → Prop
inst✝ : Algebra S R
n : S
a : R
⊢ n • { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) } | show smul r _ _ = _ | R : Type uR
inst✝² : Semiring R
S : Type uS
inst✝¹ : CommSemiring S
r : R → R → Prop
inst✝ : Algebra S R
n : S
a : R
⊢ RingQuot.smul r n { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) } | 1998ee7c35e5105c |
LieAlgebra.derivedSeries_baseChange | Mathlib/Algebra/Lie/Solvable.lean | theorem derivedSeries_baseChange {A : Type*} [CommRing A] [Algebra R A] (k : ℕ) :
derivedSeries A (A ⊗[R] L) k = (derivedSeries R L k).baseChange A | R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra R A
k : ℕ
⊢ derivedSeries A (A ⊗[R] L) k = LieSubmodule.baseChange A (derivedSeries R L k) | rw [derivedSeries_def, derivedSeries_def, ← derivedSeriesOfIdeal_baseChange,
LieSubmodule.baseChange_top] | no goals | 2cd3c7adb791e646 |
LieDerivation.iterate_apply_lie' | Mathlib/Algebra/Lie/Derivation/Basic.lean | theorem iterate_apply_lie' (D : LieDerivation R L L) (n : ℕ) (a b : L) :
D^[n] ⁅a, b⁆ = ∑ i ∈ range (n + 1), n.choose i • ⁅D^[i] a, D^[n - i] b⁆ | R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
D : LieDerivation R L L
n : ℕ
a b : L
⊢ (⇑D)^[n] ⁅a, b⁆ = ∑ i ∈ range (n + 1), n.choose i • ⁅(⇑D)^[i] a, (⇑D)^[n - i] b⁆ | rw [iterate_apply_lie D n a b] | R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
D : LieDerivation R L L
n : ℕ
a b : L
⊢ ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅(⇑D)^[ij.1] a, (⇑D)^[ij.2] b⁆ =
∑ i ∈ range (n + 1), n.choose i • ⁅(⇑D)^[i] a, (⇑D)^[n - i] b⁆ | 47286e6ec979c5c1 |
Doset.union_quotToDoset | Mathlib/GroupTheory/DoubleCoset.lean | theorem union_quotToDoset (H K : Subgroup G) : ⋃ q, quotToDoset H K q = Set.univ | case h.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
x h k : G
h3 : h ∈ H
h4 : k ∈ K
h5 : Quotient.out (mk H K x) = h * x * k
⊢ x = h⁻¹ * Quotient.out (mk H K x) * k⁻¹ | simp only [h5, Subgroup.coe_mk, ← mul_assoc, one_mul, inv_mul_cancel, mul_inv_cancel_right] | no goals | 8ff825a8e63bc4d6 |
MeasureTheory.continuousOn_convolution_right_with_param | Mathlib/Analysis/Convolution.lean | theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) | 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 : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | push_neg at H | 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 : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | a579c7593509cfe8 |
Fin.dfoldrM_loop | Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Fold.lean | theorem dfoldrM_loop [Monad m] [LawfulMonad m] (f : (i : Fin (n+1)) → α i.succ → m (α i.castSucc))
(x) : dfoldrM.loop (n+1) α f (i+1) h x =
dfoldrM.loop n (α ∘ succ) (f ·.succ) i (by omega) x >>= f 0 | case zero
m : Type u_1 → Type u_2
n : Nat
α : Fin (n + 1 + 1) → Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : (i : Fin (n + 1)) → α i.succ → m (α i.castSucc)
h : 0 + 1 < n + 1 + 1
x : α ⟨0 + 1, h⟩
⊢ dfoldrM.loop (n + 1) α f (0 + 1) h x = dfoldrM.loop n (α ∘ succ) (fun x => f x.succ) 0 ⋯ x >>= f 0 | rw [dfoldrM_loop_zero, dfoldrM_loop_succ, pure_bind] | case zero
m : Type u_1 → Type u_2
n : Nat
α : Fin (n + 1 + 1) → Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : (i : Fin (n + 1)) → α i.succ → m (α i.castSucc)
h : 0 + 1 < n + 1 + 1
x : α ⟨0 + 1, h⟩
⊢ f ⟨0, ⋯⟩ x >>= dfoldrM.loop (n + 1) α f 0 ⋯ = f 0 x | b74a59ce684ddcb3 |
Basis.smulTower'_apply | Mathlib/RingTheory/AlgebraTower.lean | theorem Basis.smulTower'_apply (ij) : b.smulTower' c ij = b ij.2 • c ij.1 | R : Type u_1
S : Type u_2
A : Type u_3
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
ι : Type u_5
ι' : Type u_6
b : Basis ι R S
c : Basis ι' S A
ij : ι' × ι
⊢ b ((Equiv.prodComm ι ι').symm ij).1 • c ((Equiv.prodCo... | rfl | no goals | 31e03fdbd5cbb2f1 |
Std.DHashMap.Internal.List.getValueCast!_insertList_of_contains_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast!_insertList_of_contains_eq_false [BEq α] [LawfulBEq α]
{l toInsert : List ((a : α) × β a)} {k : α} [Inhabited (β k)]
(not_contains : (toInsert.map Sigma.fst).contains k = false) :
getValueCast! k (insertList l toInsert) = getValueCast! k l | α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : LawfulBEq α
l toInsert : List ((a : α) × β a)
k : α
inst✝ : Inhabited (β k)
not_contains : (List.map Sigma.fst toInsert).contains k = false
⊢ getValueCast! k (insertList l toInsert) = getValueCast! k l | rw [getValueCast!_eq_getValueCast?, getValueCast!_eq_getValueCast?,
getValueCast?_insertList_of_contains_eq_false not_contains] | no goals | 5142a0bf7137e2ba |
Matrix.add_mul_mul_invOf_mul_eq_one | Mathlib/Data/Matrix/Invertible.lean | lemma add_mul_mul_invOf_mul_eq_one :
(A + U*C*V)*(⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A) = 1 | m : Type u_1
n : Type u_2
α : Type u_3
inst✝⁷ : Fintype n
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
inst✝³ : Ring α
A : Matrix n n α
U : Matrix n m α
C : Matrix m m α
V : Matrix m n α
inst✝² : Invertible A
inst✝¹ : Invertible C
inst✝ : Invertible (⅟C + V * ⅟A * U)
⊢ 1 - U * ⅟(⅟C + V * ⅟A * U) * V... | abel | no goals | d620666c2f32684f |
ModularFormClass.differentiableAt_cuspFunction | Mathlib/NumberTheory/ModularForms/QExpansion.lean | theorem differentiableAt_cuspFunction [NeZero n] [ModularFormClass F Γ(n) k]
{q : ℂ} (hq : ‖q‖ < 1) :
DifferentiableAt ℂ (cuspFunction n f) q | k : ℤ
F : Type u_1
inst✝² : FunLike F ℍ ℂ
n : ℕ
f : F
inst✝¹ : NeZero n
inst✝ : ModularFormClass F Γ(n) k
q : ℂ
hq : ‖q‖ < 1
npos : 0 < ↑n
⊢ DifferentiableAt ℂ (cuspFunction n f) q | rcases eq_or_ne q 0 with rfl | hq' | case inl
k : ℤ
F : Type u_1
inst✝² : FunLike F ℍ ℂ
n : ℕ
f : F
inst✝¹ : NeZero n
inst✝ : ModularFormClass F Γ(n) k
npos : 0 < ↑n
hq : ‖0‖ < 1
⊢ DifferentiableAt ℂ (cuspFunction n f) 0
case inr
k : ℤ
F : Type u_1
inst✝² : FunLike F ℍ ℂ
n : ℕ
f : F
inst✝¹ : NeZero n
inst✝ : ModularFormClass F Γ(n) k
q : ℂ
hq : ‖q‖ < 1
n... | f6e94b618ea55836 |
dualTensorHomEquivOfBasis_symm_cancel_right | Mathlib/LinearAlgebra/Contraction.lean | theorem dualTensorHomEquivOfBasis_symm_cancel_right (x : M →ₗ[R] N) :
dualTensorHom R M N ((dualTensorHomEquivOfBasis b).symm x) = x | ι : Type w
R : Type u
M : Type v₁
N : Type v₂
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
x : M →ₗ[R] N
⊢ (dualTensorHom R M N) ((dualTensorHomEquivOfBasis b).symm x) = x | rw [← dualTensorHomEquivOfBasis_apply b, LinearEquiv.apply_symm_apply] | no goals | 98ad7dc13e73a05f |
GenContFract.dens_recurrence | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | theorem dens_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_den_eq : g.dens n = ppredB)
(succ_nth_den_eq : g.dens (n + 1) = predB) :
g.dens (n + 2) = gp.b * predB + gp.a * ppredB | K : Type u_1
g : GenContFract K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
ppredB predB : K
succ_nth_s_eq : g.s.get? (n + 1) = some gp
nth_den_eq : g.dens n = ppredB
succ_nth_den_eq : g.dens (n + 1) = predB
⊢ g.dens (n + 2) = gp.b * predB + gp.a * ppredB | obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.conts n = conts ∧ conts.b = ppredB :=
exists_conts_b_of_den nth_den_eq | case intro.intro.refl
K : Type u_1
g : GenContFract K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
predB : K
succ_nth_s_eq : g.s.get? (n + 1) = some gp
succ_nth_den_eq : g.dens (n + 1) = predB
ppredConts : Pair K
nth_conts_eq : g.conts n = ppredConts
nth_den_eq : g.dens n = ppredConts.b
⊢ g.dens (n + 2) = gp.b * predB + gp... | a1e5e5f64bf27d45 |
totallyBounded_iff_filter | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem totallyBounded_iff_filter {s : Set α} :
TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c | case mpr.intro.intro.intro.intro.intro.intro
α : Type u
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : f.HasAntitoneBasis fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝¹ ... | exact hyz (hmd ⟨hzm, hym⟩) | no goals | 8d4389700a78bd05 |
Batteries.UnionFind.rootD_eq_self | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem rootD_eq_self {self : UnionFind} {x : Nat} : self.rootD x = x ↔ self.parent x = x | self : UnionFind
x : Nat
h : self.parent x = x
⊢ (if h : x < self.size then ↑(self.root ⟨x, h⟩) else x) = x | split <;> [rw [root, dif_pos (by rwa [parent, parentD_eq ‹_›] at h)]; rfl] | no goals | 0c883697e4e0ccdf |
Sum.bnot_isLeft | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Sum/Lemmas.lean | theorem bnot_isLeft (x : α ⊕ β) : !x.isLeft = x.isRight | α : Type u_1
β : Type u_2
x : α ⊕ β
⊢ (!decide (x.isLeft = x.isRight)) = true | cases x <;> rfl | no goals | e4b00d71627b8aa9 |
Turing.TM1to0.tr_supports | Mathlib/Computability/PostTuringMachine.lean | theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
TM0.Supports (tr M) ↑(trStmts M S) | case right.mk.some
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
q' : Λ' M
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
h₁ : (q', s) ∈ tr M (some q, v) a
h₂ : (some q, v) ∈ ↑(trStmts M S)
⊢ q' ∈ ↑(trStmts M S... | obtain ⟨q', v'⟩ := q' | case right.mk.some.mk
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
h₂ : (some q, v) ∈ ↑(trStmts M S)
q' : Option (TM1.Stmt Γ Λ σ)
v' : σ
h₁ : ((q', v'), s) ∈ tr M... | c5e09604fd205550 |
MvPolynomial.eval₂_mem | Mathlib/Algebra/MvPolynomial/Eval.lean | theorem eval₂_mem {f : R →+* S} {p : MvPolynomial σ R} {s : subS}
(hs : ∀ i ∈ p.support, f (p.coeff i) ∈ s) {v : σ → S} (hv : ∀ i, v i ∈ s) :
MvPolynomial.eval₂ f v p ∈ s | R : Type u
σ : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
subS : Type u_3
inst✝² : CommSemiring S
inst✝¹ : SetLike subS S
inst✝ : SubsemiringClass subS S
f : R →+* S
p : MvPolynomial σ R
s : subS
v : σ → S
hv : ∀ (i : σ), v i ∈ s
hs : ∀ (i : σ →₀ ℕ), f (coeff i p) ∈ s
⊢ eval₂ f v p ∈ s | induction' p using MvPolynomial.induction_on''' with a a b f ha _ ih | case h_C
R : Type u
σ : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
subS : Type u_3
inst✝² : CommSemiring S
inst✝¹ : SetLike subS S
inst✝ : SubsemiringClass subS S
f : R →+* S
s : subS
v : σ → S
hv : ∀ (i : σ), v i ∈ s
a : R
hs : ∀ (i : σ →₀ ℕ), f (coeff i (C a)) ∈ s
⊢ eval₂ f v (C a) ∈ s
case h_add_weak
R : Type u
... | 84ed2e64912182c9 |
Ordnode.Sized.rotateL | Mathlib/Data/Ordmap/Ordset.lean | theorem Sized.rotateL {l x r} (hl : @Sized α l) (hr : Sized r) : Sized (rotateL l x r) | α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Sized
hr : r.Sized
⊢ (l.rotateL x r).Sized | cases r | case nil
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
hr : nil.Sized
⊢ (l.rotateL x nil).Sized
case node
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
hr : (node size✝ l✝ x✝ r✝).Sized
⊢ (l.rotateL x (node size✝ l✝ x✝ r✝)).Sized | 485e683ec0a48852 |
CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply | Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean | lemma preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply
[HasDerivedCategory.{w'} C] {X : C} {n₀ : ℕ} (x : Ext X S.X₃ n₀)
{n₁ : ℕ} (h : n₀ + 1 = n₁) :
(preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ
hS.singleTriangle n₀ n₁ (by omega) x.hom =
(x.comp hS.extC... | 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
X : C
n₀ : ℕ
x : Ext X S.X₃ n₀
n₁ : ℕ
h : n₀ + 1 = n₁
⊢ x.hom ≫
(shiftFunctor (DerivedCategory C) ↑n₀).map hS.singleTriangle.mor₃ ≫
(shiftFunctorAdd' (DerivedCategor... | rfl | no goals | e57479311dc43218 |
MeasureTheory.eLpNorm_smul_measure_of_ne_zero_of_ne_top | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | theorem eLpNorm_smul_measure_of_ne_zero_of_ne_top {p : ℝ≥0∞} (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) {f : α → ε} (c : ℝ≥0∞) :
eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ | α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α → ε
c : ℝ≥0∞
⊢ c ^ (1 / p.toReal) * eLpNorm' f p.toReal μ = c ^ (1 / p).toReal • eLpNorm' f p.toReal μ | congr | case e_a.e_a
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α → ε
c : ℝ≥0∞
⊢ 1 / p.toReal = (1 / p).toReal | 29a17d4553f41dd2 |
Finset.Colex.shadow_initSeg | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | /-- This is important for iterating Kruskal-Katona: the shadow of an initial segment is also an
initial segment. -/
lemma shadow_initSeg [Fintype α] (hs : s.Nonempty) :
∂ (initSeg s) = initSeg (erase s <| min' s hs) | case h.mpr.inr.intro.intro.intro.inr.inr.inr.h
α : Type u_1
inst✝¹ : LinearOrder α
s : Finset α
inst✝ : Fintype α
hs : s.Nonempty
t : Finset α
cards' : #(s.erase (s.min' hs)) = #t
k : α
hks : k ∈ s.erase (s.min' hs)
hkt : k ∉ t
z : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))
j : α := tᶜ.min' ⋯
hjk : j ≤ k
this... | apply (min'_le tᶜ _ _).not_lt gt | α : Type u_1
inst✝¹ : LinearOrder α
s : Finset α
inst✝ : Fintype α
hs : s.Nonempty
t : Finset α
cards' : #(s.erase (s.min' hs)) = #t
k : α
hks : k ∈ s.erase (s.min' hs)
hkt : k ∉ t
z : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))
j : α := tᶜ.min' ⋯
hjk : j ≤ k
this : j ∉ t
hcard : #s = #(insert j t)
r₁ : j = k
... | a9c55e64e04a6093 |
Batteries.RBNode.Balanced.append | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n | α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝³ x✝² a✝¹ : RBNode α
x✝¹ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝¹ x✝¹ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝¹.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
l✝ a✝ : RBNode α
x✝ : α
b✝ : ... | exact .redred nofun (.red ha hb') (.red hc' hd) | no goals | b3ecc8a07b6c9ec6 |
CategoryTheory.Functor.CommShift.ofIso_compatibility | Mathlib/CategoryTheory/Shift/CommShift.lean | lemma ofIso_compatibility :
letI := ofIso e A
NatTrans.CommShift e.hom A | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_5, u_1} C
inst✝⁴ : Category.{u_6, u_2} D
F G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
⊢ NatTrans.CommShift e.hom A | letI := ofIso e A | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_5, u_1} C
inst✝⁴ : Category.{u_6, u_2} D
F G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := ofIso e A
⊢ NatTrans.CommShift e.hom A | c2c10f3a9e743026 |
HasFPowerSeriesWithinOnBall.tendstoUniformlyOn' | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.tendstoUniformlyOn' {r' : ℝ≥0}
(hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(insert x s ∩ Metric.ball (x : E) r') | case h.e'_8.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x... | simp [dist_eq_norm] | no goals | 3f74d4280602b239 |
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
⊢ (proj 𝒜 i) a ^ m ∈ 𝒜 ... | mem_tac | no goals | e11b14794a0cd9b6 |
Nat.and_lt_two_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem and_lt_two_pow (x : Nat) {y n : Nat} (right : y < 2^n) : (x &&& y) < 2^n | case lt
x y n : Nat
right : y < 2 ^ n
i : Nat
i_ge_n : i ≥ n
⊢ 2 ^ n ≤ 2 ^ i | exact pow_le_pow_of_le_right Nat.zero_lt_two i_ge_n | no goals | c45ccfe28fec8d46 |
Set.zero_mem_smul_iff | Mathlib/Data/Set/Pointwise/SMul.lean | theorem zero_mem_smul_iff :
(0 : β) ∈ s • t ↔ (0 : α) ∈ s ∧ t.Nonempty ∨ (0 : β) ∈ t ∧ s.Nonempty | case mpr
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMulWithZero α β
s : Set α
t : Set β
inst✝ : NoZeroSMulDivisors α β
⊢ 0 ∈ s ∧ t.Nonempty ∨ 0 ∈ t ∧ s.Nonempty → 0 ∈ s • t | rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩) | case mpr.inl.intro.intro
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMulWithZero α β
s : Set α
t : Set β
inst✝ : NoZeroSMulDivisors α β
hs : 0 ∈ s
b : β
hb : b ∈ t
⊢ 0 ∈ s • t
case mpr.inr.intro.intro
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMulWithZero α β
s : Set α
... | 2eb057cfa2edc7c8 |
Monotone.seq_lt_seq_of_lt_of_le | Mathlib/Order/Iterate.lean | theorem seq_lt_seq_of_lt_of_le (hf : Monotone f) (n : ℕ) (h₀ : x 0 < y 0)
(hx : ∀ k < n, x (k + 1) < f (x k)) (hy : ∀ k < n, f (y k) ≤ y (k + 1)) : x n < y n | α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
n : ℕ
h₀ : x 0 < y 0
hx : ∀ (k : ℕ), k < n → x (k + 1) < f (x k)
hy : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)
⊢ x n < y n | cases n | case zero
α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
h₀ : x 0 < y 0
hx : ∀ (k : ℕ), k < 0 → x (k + 1) < f (x k)
hy : ∀ (k : ℕ), k < 0 → f (y k) ≤ y (k + 1)
⊢ x 0 < y 0
case succ
α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
h₀ : x 0 < y 0
n✝ : ℕ
hx : ∀ (k : ℕ), k < n✝... | 1752fa5e2b1ecc88 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.readyForRupAdd_ofArray | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem readyForRupAdd_ofArray {n : Nat} (arr : Array (Option (DefaultClause n))) :
ReadyForRupAdd (ofArray arr) | case right.intro
n : Nat
arr : Array (Option (DefaultClause n))
hsize : (ofArray arr).assignments.size = n
ModifiedAssignmentsInvariant : Array Assignment → Prop :=
fun assignments =>
∃ hsize, ∀ (i : PosFin n) (b : Bool), hasAssignment b assignments[i.val] = true → unit (i, b) ∈ (ofArray arr).toList
hb : Modified... | simp only [ofArray, ← Array.foldl_toList] at h | case right.intro
n : Nat
arr : Array (Option (DefaultClause n))
hsize : (ofArray arr).assignments.size = n
ModifiedAssignmentsInvariant : Array Assignment → Prop :=
fun assignments =>
∃ hsize, ∀ (i : PosFin n) (b : Bool), hasAssignment b assignments[i.val] = true → unit (i, b) ∈ (ofArray arr).toList
hb : Modified... | 0a237b5f7a9a7c15 |
CategoryTheory.Limits.reflectsLimit_of_reflectsIsomorphisms | Mathlib/CategoryTheory/Limits/Preserves/Basic.lean | /-- If the limit of `F` exists and `G` preserves it, then if `G` reflects isomorphisms then it
reflects the limit of `F`.
-/ -- Porting note: previous behavior of apply pushed instance holes into hypotheses, this errors
lemma reflectsLimit_of_reflectsIsomorphisms (F : J ⥤ C) (G : C ⥤ D) [G.ReflectsIsomorphisms]
[Ha... | C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
J : Type w
inst✝³ : Category.{w', w} J
F : J ⥤ C
G : C ⥤ D
inst✝² : G.ReflectsIsomorphisms
inst✝¹ : HasLimit F
inst✝ : PreservesLimit F G
c : Cone F
t : IsLimit (G.mapCone c)
this : IsIso ((limit.isLimit F).lift c)
⊢ IsLimit c | apply IsLimit.ofPointIso (limit.isLimit F) | no goals | ff0c8193ee19bd81 |
WithTop.untopD_zero_mul | Mathlib/Algebra/Order/Ring/WithTop.lean | @[simp]
lemma untopD_zero_mul (a b : WithTop α) : (a * b).untopD 0 = a.untopD 0 * b.untopD 0 | case neg.top
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithTop α
hb : ¬b = 0
ha : ¬⊤ = 0
⊢ untopD 0 (⊤ * b) = untopD 0 ⊤ * untopD 0 b | rw [top_mul hb, untopD_top, zero_mul] | no goals | ebaa5001124c917e |
Basis.mk_eq_rank'' | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | theorem Basis.mk_eq_rank'' {ι : Type v} (v : Basis ι R M) : #ι = Module.rank R M | R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
ι : Type v
v : Basis ι R M
this : Nontrivial R
⊢ LinearIndepOn R id (range ⇑v) | rw [LinearIndepOn] | R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
ι : Type v
v : Basis ι R M
this : Nontrivial R
⊢ LinearIndependent R fun (x : ↑(range ⇑v)) => id ↑x | 60944ed5994f83df |
Preorder.toLE_injective | Mathlib/Order/Basic.lean | theorem Preorder.toLE_injective : Function.Injective (@Preorder.toLE α) :=
fun A B h ↦ match A, B with
| { lt := A_lt, lt_iff_le_not_le := A_iff, .. },
{ lt := B_lt, lt_iff_le_not_le := B_iff, .. } => by
cases h
have : A_lt = B_lt | case h.h
α : Type u_2
A B : Preorder α
le✝ A_lt : α → α → Prop
le_refl✝¹ : ∀ (a : α), a ≤ a
le_trans✝¹ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
A_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
B_lt : α → α → Prop
le_refl✝ : ∀ (a : α), a ≤ a
le_trans✝ : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
B_iff : ∀ (a b : α), a < b ↔ a ≤ b ∧ ... | rw [A_iff, B_iff] | no goals | 4bb5c6c429e2e7ed |
AlgebraicGeometry.IsLocalAtTarget.of_range_subset_iSup | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | lemma of_range_subset_iSup [P.RespectsRight @IsOpenImmersion] {ι : Type*} (U : ι → Y.Opens)
(H : Set.range f.base ⊆ (⨆ i, U i : Y.Opens)) (hf : ∀ i, P (f ∣_ U i)) : P f | P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
inst✝ : P.RespectsRight @IsOpenImmersion
ι : Type u_1
U : ι → Y.Opens
H : Set.range ⇑(ConcreteCategory.hom f.base) ⊆ ↑(⨆ i, U i)
hf : ∀ (i : ι), P (f ∣_ U i)
⊢ P f | let g : X ⟶ (⨆ i, U i : Y.Opens) := IsOpenImmersion.lift (Scheme.Opens.ι _) f (by simpa using H) | P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
inst✝ : P.RespectsRight @IsOpenImmersion
ι : Type u_1
U : ι → Y.Opens
H : Set.range ⇑(ConcreteCategory.hom f.base) ⊆ ↑(⨆ i, U i)
hf : ∀ (i : ι), P (f ∣_ U i)
g : X ⟶ ↑(⨆ i, U i) := IsOpenImmersion.lift (⨆ i, U i).ι f ⋯
⊢ P f | ee63d9b5b19cceb5 |
Array.pmap_push | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (l : Array α) (h : ∀ b ∈ l.push a, P b) :
pmap f (l.push a) h =
(pmap f l (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) | α : Type u_1
β : Type u_2
P : α → Prop
f : (a : α) → P a → β
a : α
l : Array α
h : ∀ (b : α), b ∈ l.push a → P b
⊢ pmap f (l.push a) h = (pmap f l ⋯).push (f a ⋯) | simp [pmap] | no goals | ac6f1dbcf2d15765 |
PadicInt.isCauSeq_nthHom | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n | R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
⊢ IsCauSeq (padicNorm p) fun n => ↑(nthHom f r n) | intro ε hε | R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
ε : ℚ
hε : ε > 0
⊢ ∃ i, ∀ j ≥ i, padicNorm p ((fun n => ↑(nthHom f r n)) j - (fun n => ↑(nthHom f r n)) i) < ε | 3ca072d517dfdd09 |
MeasureTheory.integral_mul_norm_le_Lp_mul_Lq | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
(hpq : p.IsConjExponent q) (hf : MemLp f (ENNReal.ofReal p) μ)
(hg : MemLp g (ENNReal.ofReal q) μ) :
∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.IsConjExponent q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x => ‖f x‖ₑ) * fun x => ‖g x‖ₑ) a ∂μ
h_ri... | refine ENNReal.toReal_mono ?_ ?_ | case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.IsConjExponent q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x => ‖f x‖ₑ) * fun x => ‖g ... | 539c08192e88a140 |
ENNReal.eq_zero_of_le_mul_pow | Mathlib/Analysis/SpecificLimits/Basic.lean | lemma ENNReal.eq_zero_of_le_mul_pow {x r : ℝ≥0∞} {ε : ℝ≥0} (hr : r < 1)
(h : ∀ n : ℕ, x ≤ ε * r ^ n) : x = 0 | x r : ℝ≥0∞
ε : ℝ≥0
hr : r < 1
h : ∀ (n : ℕ), x ≤ ↑ε * r ^ n
⊢ x = 0 | rw [← nonpos_iff_eq_zero] | x r : ℝ≥0∞
ε : ℝ≥0
hr : r < 1
h : ∀ (n : ℕ), x ≤ ↑ε * r ^ n
⊢ x ≤ 0 | f0c4433f60379c13 |
Multiset.countP_map | Mathlib/Data/Multiset/Filter.lean | theorem countP_map (f : α → β) (s : Multiset α) (p : β → Prop) [DecidablePred p] :
countP p (map f s) = card (s.filter fun a => p (f a)) | α : Type u_1
β : Type v
f : α → β
s : Multiset α
p : β → Prop
inst✝ : DecidablePred p
⊢ countP p (map f s) = (filter (fun a => p (f a)) s).card | refine Multiset.induction_on s ?_ fun a t IH => ?_ | case refine_1
α : Type u_1
β : Type v
f : α → β
s : Multiset α
p : β → Prop
inst✝ : DecidablePred p
⊢ countP p (map f 0) = (filter (fun a => p (f a)) 0).card
case refine_2
α : Type u_1
β : Type v
f : α → β
s : Multiset α
p : β → Prop
inst✝ : DecidablePred p
a : α
t : Multiset α
IH : countP p (map f t) = (filter (fun a... | d40fb82936487e92 |
mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd [Nontrivial k] (p : ι → P) (j : ι) (q : P) :
q ∈ affineSpan k (Set.range p) ↔
∃ (s : Finset ι) (w : ι → k), q = s.weightedVSubOfPoint p (p j) w +ᵥ p j | ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p : ι → P
j : ι
s : Finset ι
w : ι → k
w' : ι → k := Function.update w j (1 - (s \ {j}).sum w)
⊢ (insert j s).sum w' = 1 | by_cases hj : j ∈ s | case pos
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p : ι → P
j : ι
s : Finset ι
w : ι → k
w' : ι → k := Function.update w j (1 - (s \ {j}).sum w)
hj : j ∈ s
⊢ (insert j s).sum w' = 1
case neg
ι : Type u_... | 40a3291069dee08b |
CategoryTheory.Idempotents.isIdempotentComplete_of_isIdempotentComplete_opposite | Mathlib/CategoryTheory/Idempotents/Basic.lean | theorem isIdempotentComplete_of_isIdempotentComplete_opposite (h : IsIdempotentComplete Cᵒᵖ) :
IsIdempotentComplete C | case h.right
C : Type u_1
inst✝ : Category.{u_2, u_1} C
h : IsIdempotentComplete Cᵒᵖ
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : Cᵒᵖ
i : Y ⟶ op X
e : op X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p.op
⊢ i.unop ≫ e.unop = p | simp only [← unop_comp, h₂] | case h.right
C : Type u_1
inst✝ : Category.{u_2, u_1} C
h : IsIdempotentComplete Cᵒᵖ
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : Cᵒᵖ
i : Y ⟶ op X
e : op X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p.op
⊢ p.op.unop = p | 4c640b5c10f1f344 |
CliffordAlgebraComplex.reverse_apply | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | theorem reverse_apply (x : CliffordAlgebra Q) : reverse (R := ℝ) x = x | case mul
x₁ x₂ : CliffordAlgebra Q
hx₁ : reverse x₁ = x₁
hx₂ : reverse x₂ = x₂
⊢ reverse (x₁ * x₂) = x₁ * x₂ | rw [reverse.map_mul, mul_comm, hx₁, hx₂] | no goals | 57b6b0031cb9b658 |
Vitali.exists_disjoint_covering_ae | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈... | case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB... | exact le_trans (ut' (vu hb)).2 (hR1 (c b)) | no goals | 4466d2fc3e80cb3f |
Submodule.span_insert_zero | Mathlib/LinearAlgebra/Span/Defs.lean | theorem span_insert_zero : span R (insert (0 : M) s) = span R s | R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ span R (insert 0 s) ≤ span R s | rw [span_le, Set.insert_subset_iff] | R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ 0 ∈ ↑(span R s) ∧ s ⊆ ↑(span R s) | 0379c410c2f6e1de |
Ideal.Filtration.submodule_fg_iff_stable | Mathlib/RingTheory/Filtration.lean | theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable | case mpr.intro.h.mk.intro
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
hF' : ∀ (i : ℕ), (F.N i).FG
n : ℕ
hn : F.submodule = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(F.N i))
i : ℕ
hi : i ∈ Finset.range n.succ
s : ... | exact ⟨_, rfl⟩ | no goals | 7d72d199901d100d |
MeasureTheory.eLpNorm_map_measure | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | theorem eLpNorm_map_measure (hg : AEStronglyMeasurable g (Measure.map f μ))
(hf : AEMeasurable f μ) : eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ | case neg
α : Type u_1
E : Type u_3
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
β : Type u_6
mβ : MeasurableSpace β
f : α → β
g : β → E
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
hp_zero : ¬p = 0
hp_top : ¬p = ⊤
⊢ (∫⁻ (x : β), ‖g x‖ₑ ^ p.toReal ∂Measure.map f μ) ^ ... | rw [lintegral_map' (hg.enorm.pow_const p.toReal) hf] | case neg
α : Type u_1
E : Type u_3
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
β : Type u_6
mβ : MeasurableSpace β
f : α → β
g : β → E
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
hp_zero : ¬p = 0
hp_top : ¬p = ⊤
⊢ (∫⁻ (a : α), ‖g (f a)‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toR... | 4caee505b835d28d |
CategoryTheory.Functor.distTriang_iff | Mathlib/CategoryTheory/Localization/Triangulated.lean | lemma distTriang_iff (T : Triangle D) :
(T ∈ distTriang D) ↔ T ∈ L.essImageDistTriang | C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{u_4, u_1} C
inst✝¹³ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
inst✝⁷ : HasShift D ℤ
inst✝⁶ : L.CommShift ℤ
inst✝⁵ : HasZeroObject D
in... | constructor | case mp
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{u_4, u_1} C
inst✝¹³ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
inst✝⁷ : HasShift D ℤ
inst✝⁶ : L.CommShift ℤ
inst✝⁵ : HasZeroObj... | 274f43afcadc904a |
EuclideanGeometry.cospherical_iff_exists_mem_of_complete | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem cospherical_iff_exists_mem_of_complete {s : AffineSubspace ℝ P} {ps : Set P} (h : ps ⊆ s)
[Nonempty s] [HasOrthogonalProjection s.direction] :
Cospherical ps ↔ ∃ center ∈ s, ∃ radius : ℝ, ∀ p ∈ ps, dist p center = radius | case mp
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
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
inst✝ : HasOrthogonalProjection s.direction
⊢ Cospherical ps → ∃ center ∈ s, ∃ radius, ∀ p ∈ ps, dist p ce... | rintro ⟨c, hcr⟩ | case mp.intro
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
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
inst✝ : HasOrthogonalProjection s.direction
c : P
hcr : ∃ radius, ∀ p ∈ ps, dist p c = radius
⊢ ∃ ce... | 53e93c5b7b48c976 |
DFinsupp.mapRange.addMonoidHom_comp | Mathlib/Data/DFinsupp/Defs.lean | theorem mapRange.addMonoidHom_comp (f : ∀ i, β₁ i →+ β₂ i) (f₂ : ∀ i, β i →+ β₁ i) :
(mapRange.addMonoidHom fun i => (f i).comp (f₂ i)) =
(mapRange.addMonoidHom f).comp (mapRange.addMonoidHom f₂) | ι : Type u
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : (i : ι) → AddZeroClass (β i)
inst✝¹ : (i : ι) → AddZeroClass (β₁ i)
inst✝ : (i : ι) → AddZeroClass (β₂ i)
f : (i : ι) → β₁ i →+ β₂ i
f₂ : (i : ι) → β i →+ β₁ i
⊢ (addMonoidHom fun i => (f i).comp (f₂ i)) = (addMonoidHom f).comp (addMonoidHom f₂) | refine AddMonoidHom.ext <| mapRange_comp (fun i x => f i x) (fun i x => f₂ i x) ?_ ?_ ?_ | case refine_1
ι : Type u
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : (i : ι) → AddZeroClass (β i)
inst✝¹ : (i : ι) → AddZeroClass (β₁ i)
inst✝ : (i : ι) → AddZeroClass (β₂ i)
f : (i : ι) → β₁ i →+ β₂ i
f₂ : (i : ι) → β i →+ β₁ i
⊢ ∀ (i : ι), (fun i x => (f i) x) i 0 = 0
case refine_2
ι : Type u
β : ι → T... | f5b74fc335cfe878 |
VitaliFamily.measure_limRatioMeas_zero | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem measure_limRatioMeas_zero : ρ {x | v.limRatioMeas hρ x = 0} = 0 | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = 0}
o : Set α
xo : x ∈ o
o_open ... | simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos] | no goals | 1ff6cbb082b8eb59 |
SetTheory.PGame.Numeric.add | Mathlib/SetTheory/Surreal/Basic.lean | theorem add : ∀ {x y : PGame} (_ : Numeric x) (_ : Numeric y), Numeric (x + y)
| ⟨xl, xr, xL, xR⟩, ⟨yl, yr, yL, yR⟩, ox, oy =>
⟨by
rintro (ix | iy) (jx | jy)
· exact add_lt_add_right (ox.1 ix jx) _
· exact (add_lf_add_of_lf_of_le (lf_mk _ _ ix) (oy.le_moveRight jy)).lt
((ox.moveLeft ix... | case left
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
ox : (PGame.mk xl xr xL xR).Numeric
oy : (PGame.mk yl yr yL yR).Numeric
⊢ ∀ (i : xl ⊕ yl),
((fun t =>
Sum.rec
(fun i =>
(fun a =>
rec (motive := fun x => ... | rintro (ix | iy) | case left.inl
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
ox : (PGame.mk xl xr xL xR).Numeric
oy : (PGame.mk yl yr yL yR).Numeric
ix : xl
⊢ ((fun t =>
Sum.rec
(fun i =>
(fun a =>
rec (motive := fun x => PGame → PGame)
... | fb960894a76bd8f8 |
MonomialOrder.Monic.prod | Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean | theorem Monic.prod {ι : Type*} {P : ι → MvPolynomial σ R} {s : Finset ι}
(H : ∀ i ∈ s, m.Monic (P i)) :
m.Monic (∏ i ∈ s, P i) | σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, m.Monic (P i)
⊢ ∀ i ∈ s, IsRegular (m.leadingCoeff (P i)) | intro i hi | σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, m.Monic (P i)
i : ι
hi : i ∈ s
⊢ IsRegular (m.leadingCoeff (P i)) | b6dfccd4b783abc5 |
CategoryTheory.IsCofilteredOrEmpty.of_left_adjoint | Mathlib/CategoryTheory/Filtered/Basic.lean | theorem of_left_adjoint {L : C ⥤ D} {R : D ⥤ C} (h : L ⊣ R) : IsCofilteredOrEmpty D :=
{ cone_objs := fun X Y =>
⟨L.obj (min (R.obj X) (R.obj Y)), (h.homEquiv _ X).symm (minToLeft _ _),
(h.homEquiv _ Y).symm (minToRight _ _), ⟨⟩⟩
cone_maps := fun X Y f g =>
⟨L.obj (eq (R.map f) (R.map g)), (h.... | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : IsCofilteredOrEmpty C
D : Type u₁
inst✝ : Category.{v₁, u₁} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
X Y : D
f g : X ⟶ Y
⊢ (h.homEquiv (eq (R.map f) (R.map g)) X).symm (eqHom (R.map f) (R.map g)) ≫ f =
(h.homEquiv (eq (R.map f) (R.map g)) X).symm (eqHom (R.map f) (R.map g)) ≫ g | rw [← h.homEquiv_naturality_right_symm, ← h.homEquiv_naturality_right_symm, eq_condition] | no goals | 15cbf31b2fe278bb |
IsDenseInducing.extend_Z_bilin_key | Mathlib/Topology/Algebra/UniformGroup/Defs.lean | theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
in... | exact hφ.comp continuous_swap | no goals | 7e13666b5fc1d57e |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | have k_val_ne_zero : k.1 ≠ 0 := by
intro k_eq_zero
simp only [List.length_cons, ← k_eq_zero, ne_eq, not_true] at k_ne_zero | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | 02ca1706b397b254 |
isAlgebraic_of_isFractionRing | Mathlib/RingTheory/Localization/Integral.lean | lemma isAlgebraic_of_isFractionRing {R S} (K L) [CommRing R] [CommRing S] [Field K] [CommRing L]
[Algebra R S] [Algebra R K] [Algebra R L] [Algebra S L] [Algebra K L] [IsScalarTower R S L]
[IsScalarTower R K L] [IsFractionRing S L]
[Algebra.IsIntegral R S] : Algebra.IsAlgebraic K L | case isAlgebraic.intro.intro.a.hy
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Field K
inst✝⁹ : CommRing L
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R K
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra S L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTow... | show IsIntegral _ _ | case isAlgebraic.intro.intro.a.hy
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Field K
inst✝⁹ : CommRing L
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R K
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra S L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTow... | ae4f70dc8f48707d |
Submodule.spanRank_toENat_eq_iInf_finset_card | Mathlib/Algebra/Module/SpanRank.lean | lemma spanRank_toENat_eq_iInf_finset_card (p : Submodule R M) :
p.spanRank.toENat =
⨅ (s : {s : Set M // s.Finite ∧ span R s = p}), (s.2.1.toFinset.card : ℕ∞) | case inl
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h1 : ⨅ s, ⨅ (_ : span R s = p), s.encard = ⊤
⊢ ⨅ s, ↑⋯.toFinset.card = ⊤ | simp_rw [iInf_eq_top] at h1 ⊢ | case inl
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h1 : ∀ (i : Set M), span R i = p → i.encard = ⊤
⊢ ∀ (i : { s // s.Finite ∧ span R s = p }), ↑⋯.toFinset.card = ⊤ | f74636938a98e2b3 |
Finsupp.filter_pos_add_filter_neg | Mathlib/Data/Finsupp/Basic.lean | theorem filter_pos_add_filter_neg [AddZeroClass M] (f : α →₀ M) (p : α → Prop) [DecidablePred p] :
(f.filter p + f.filter fun a => ¬p a) = f :=
DFunLike.coe_injective <| by
simp only [coe_add, filter_eq_indicator]
exact Set.indicator_self_add_compl { x | p x } f
| α : Type u_1
M : Type u_5
inst✝¹ : AddZeroClass M
f : α →₀ M
p : α → Prop
inst✝ : DecidablePred p
⊢ (fun f => ⇑f) (filter p f + filter (fun a => ¬p a) f) = (fun f => ⇑f) f | simp only [coe_add, filter_eq_indicator] | α : Type u_1
M : Type u_5
inst✝¹ : AddZeroClass M
f : α →₀ M
p : α → Prop
inst✝ : DecidablePred p
⊢ {x | p x}.indicator ⇑f + {a | ¬p a}.indicator ⇑f = ⇑f | 08c2ba5abccc381f |
Nimber.add_eq_zero | Mathlib/SetTheory/Nimber/Basic.lean | theorem add_eq_zero {a b : Nimber} : a + b = 0 ↔ a = b | case mp.inr.inl
a : Nimber
hab : a + a = 0
⊢ a = a | rfl | no goals | 0d57b0f83f981add |
AlgebraicGeometry.pointsPi_surjective | Mathlib/AlgebraicGeometry/PointsPi.lean | lemma pointsPi_surjective [CompactSpace X] [∀ i, IsLocalRing (R i)] :
Function.Surjective (pointsPi R X) | ι : Type u
R : ι → CommRingCat
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : ∀ (i : ι), IsLocalRing ↑(R i)
f : (i : ι) → Spec (R i) ⟶ X
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
j : ι → 𝒰.J
hj : ∀ (i : ι), Set.range ⇑(ConcreteCategory.hom (f i).base) ⊆ ↑... | ext i : 1 | case h
ι : Type u
R : ι → CommRingCat
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : ∀ (i : ι), IsLocalRing ↑(R i)
f : (i : ι) → Spec (R i) ⟶ X
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
j : ι → 𝒰.J
hj : ∀ (i : ι), Set.range ⇑(ConcreteCategory.hom (f i).ba... | 35d67c613e419cbd |
SimpleGraph.ConnectedComponent.mem_coe_supp_of_adj | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma mem_coe_supp_of_adj {v w : V} {H : Subgraph G} {c : ConnectedComponent H.coe}
(hv : v ∈ (↑) '' (c : Set H.verts)) (hw : w ∈ H.verts)
(hadj : H.Adj v w) : w ∈ (↑) '' (c : Set H.verts) | case h
V : Type u
G : SimpleGraph V
v w : V
H : G.Subgraph
c : H.coe.ConnectedComponent
hw : w ∈ H.verts
hadj : H.Adj v w
w✝ : ↑H.verts
h : w✝ ∈ ↑c ∧ ↑w✝ = v
⊢ ⟨w, hw⟩ ∈ ↑(H.coe.connectedComponentMk w✝) ∧ ↑⟨w, hw⟩ = w | exact ⟨connectedComponentMk_eq_of_adj <| Subgraph.Adj.coe <| h.2 ▸ hadj.symm, rfl⟩ | no goals | d0853eeb7056ceb2 |
InnerProductSpaceable.I_prop | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | theorem I_prop : innerProp' E (I : 𝕜) | case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ 4⁻¹ *
(𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ -
... | have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add] | case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ 4⁻¹ *
(𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖-... | a4c1a41ce4d42a42 |
Fin.coe_orderIso_apply | Mathlib/Order/Fin/Basic.lean | /-- If `e` is an `orderIso` between `Fin n` and `Fin m`, then `n = m` and `e` is the identity
map. In this lemma we state that for each `i : Fin n` we have `(e i : ℕ) = (i : ℕ)`. -/
@[simp] lemma coe_orderIso_apply (e : Fin n ≃o Fin m) (i : Fin n) : (e i : ℕ) = i | case mk
m n : ℕ
e : Fin n ≃o Fin m
i : ℕ
hi : i < n
⊢ ↑(e ⟨i, hi⟩) = i | induction' i using Nat.strong_induction_on with i h | case mk.h
m n : ℕ
e : Fin n ≃o Fin m
i : ℕ
h : ∀ m_1 < i, ∀ (hi : m_1 < n), ↑(e ⟨m_1, hi⟩) = m_1
hi : i < n
⊢ ↑(e ⟨i, hi⟩) = i | f22dc0d8cb79ada4 |
Polynomial.Sequence.degree_strictMono | Mathlib/Algebra/Polynomial/Sequence.lean | /-- `S i` has strictly monotone degree. -/
lemma degree_strictMono : StrictMono <| degree ∘ S := fun _ _ ↦ by simp
| R : Type u_1
inst✝ : Semiring R
S : Sequence R
x✝¹ x✝ : ℕ
⊢ x✝¹ < x✝ → (degree ∘ ↑S) x✝¹ < (degree ∘ ↑S) x✝ | simp | no goals | c82e90676df63101 |
CategoryTheory.NonPreadditiveAbelian.sub_sub_sub | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | theorem sub_sub_sub {X Y : C} (a b c d : X ⟶ Y) : a - c - (b - d) = a - b - (c - d) | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c d : X ⟶ Y
⊢ a - c - (b - d) = a - b - (c - d) | rw [sub_def, ← lift_sub_lift, sub_def, Category.assoc, σ_comp, prod.lift_map_assoc] | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c d : X ⟶ Y
⊢ prod.lift (prod.lift a b ≫ σ) (prod.lift c d ≫ σ) ≫ σ = a - b - (c - d) | f10b3bcf911da574 |
integral_Ioi_cpow_of_lt | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
(∫ t : ℝ in Ioi c, (t : ℂ) ^ a) = -(c : ℂ) ^ (a + 1) / (a + 1) | a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
⊢ 0 < -(a.re + 1) | linarith | no goals | 548595cb49c80036 |
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) / Int.fract a + b = (b * a + c) / Int.fract a | field_simp [fract_a_ne_zero] | 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 | 4bc31c2d1c2c0d60 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.derivedLitsInvariant_performRupCheck | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem derivedLitsInvariant_performRupCheck {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(rupHints : Array Nat)
(f'_assignments_size : (performRupCheck f rupHints).1.assignments.size = n) :
let rupCheckRes := performRupCheck f rupHints
DerivedLitsInvariant f f_assignme... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f'_assignments_size : (f.performRupCheck rupHints).fst.assignments.size = n
motive : Nat → Array Assignment × CNF.Clause (PosFin n) × Bool × Bool → Prop :=
fun x acc => ∃ hsize, f.DerivedLitsInvariant f_assignments_size acc.... | apply Or.inl | case h
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f'_assignments_size : (f.performRupCheck rupHints).fst.assignments.size = n
motive : Nat → Array Assignment × CNF.Clause (PosFin n) × Bool × Bool → Prop :=
fun x acc => ∃ hsize, f.DerivedLitsInvariant f_assignments_si... | de25fdd0385200ff |
toIocDiv_wcovBy_toIcoDiv | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIocDiv_wcovBy_toIcoDiv (a b : α) : toIocDiv hp a b ⩿ toIcoDiv hp a b | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ toIocDiv hp a b ⩿ toIcoDiv hp a b | suffices toIocDiv hp a b = toIcoDiv hp a b ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b by
rwa [wcovBy_iff_eq_or_covBy, ← Order.succ_eq_iff_covBy] | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ toIocDiv hp a b = toIcoDiv hp a b ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b | e9d5b7f8dec9f69e |
MeasureTheory.Measure.restrict_eq_self | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem restrict_eq_self (h : s ⊆ t) : μ.restrict t s = μ s :=
(le_iff'.1 restrict_le_self s).antisymm <|
calc
μ s ≤ μ (toMeasurable (μ.restrict t) s ∩ t) :=
measure_mono (subset_inter (subset_toMeasurable _ _) h)
_ = μ.restrict t s | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
⊢ μ (toMeasurable (μ.restrict t) s ∩ t) = (μ.restrict t) s | rw [← restrict_apply (measurableSet_toMeasurable _ _), measure_toMeasurable] | no goals | fdc48b9df9dc2d43 |
FormalMultilinearSeries.applyComposition_single | Mathlib/Analysis/Analytic/Composition.lean | theorem applyComposition_single (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n)
(v : Fin n → E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v | case h.e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
i... | convert Composition.single_embedding hn ⟨i, hi2⟩ using 1 | case h.e'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
... | 3c651f8937b91732 |
ENNReal.log_surjective | Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean | theorem log_surjective : Function.Surjective log | case h
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
exp_y_pos : ¬Real.exp y.toReal ≤ 0
⊢ (ENNReal.ofReal (Real.exp y.toReal)).log = y | simp only [log, ofReal_eq_zero, exp_y_pos, ↓reduceIte, ofReal_ne_top,
ENNReal.toReal_ofReal (Real.exp_pos y.toReal).le, Real.log_exp y.toReal] | case h
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
exp_y_pos : ¬Real.exp y.toReal ≤ 0
⊢ ↑y.toReal = y | 1d58ac00fdba3020 |
UniformContinuous.pow_const | Mathlib/Topology/Algebra/UniformGroup/Defs.lean | theorem UniformContinuous.pow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
∀ n : ℕ, UniformContinuous fun x => f x ^ n
| 0 => by
simp_rw [pow_zero]
exact uniformContinuous_const
| n + 1 => by
simp_rw [pow_succ']
exact hf.mul (hf.pow_const n)
| α : Type u_1
β : Type u_2
inst✝³ : UniformSpace α
inst✝² : Group α
inst✝¹ : UniformGroup α
inst✝ : UniformSpace β
f : β → α
hf : UniformContinuous f
n : ℕ
⊢ UniformContinuous fun x => f x ^ (n + 1) | simp_rw [pow_succ'] | α : Type u_1
β : Type u_2
inst✝³ : UniformSpace α
inst✝² : Group α
inst✝¹ : UniformGroup α
inst✝ : UniformSpace β
f : β → α
hf : UniformContinuous f
n : ℕ
⊢ UniformContinuous fun x => f x * f x ^ n | 5db82d8e2dacf3d4 |
Equiv.Perm.support_mul_le | Mathlib/GroupTheory/Perm/Support.lean | theorem support_mul_le (f g : Perm α) : (f * g).support ≤ f.support ⊔ g.support := fun x => by
simp only [sup_eq_union]
rw [mem_union, mem_support, mem_support, mem_support, mul_apply, ← not_and_or, not_imp_not]
rintro ⟨hf, hg⟩
rw [hg, hf]
| α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
x : α
⊢ x ∈ (f * g).support → x ∈ f.support ∪ g.support | rw [mem_union, mem_support, mem_support, mem_support, mul_apply, ← not_and_or, not_imp_not] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
x : α
⊢ f x = x ∧ g x = x → f (g x) = x | 6d98e5a02bd70b7c |
not_disjoint_segment_convexHull_triple | Mathlib/Analysis/Convex/StoneSeparation.lean | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < ... | simp [w, Fin.sum_univ_succ, Fin.sum_univ_zero] | no goals | b7449eac49163f66 |
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