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 |
|---|---|---|---|---|---|---|
MulAction.IsBlock.of_subset | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem of_subset (a : X) (hfB : B.Finite) :
IsBlock G (⋂ (k : G) (_ : a ∈ k • B), k • B) | case inr.intro.intro.intro
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ≤ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ≤ g • B'
hag'... | rw [← smul_eq_iff_eq_inv_smul] at hkB' hgkB' | case inr.intro.intro.intro
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ≤ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ≤ g • B'
hag'... | 9d1f232c7990ecc0 |
IsLocalizedModule.lift_unique | Mathlib/Algebra/Module/LocalizedModule/Basic.lean | theorem lift_unique (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x))
(l : M' →ₗ[R] M'') (hl : l.comp f = g) : lift S f g h = l | case e_g.h
R : Type u_1
inst✝⁷ : CommSemiring R
S : Submonoid R
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : AddCommMonoid M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
g : M →ₗ[R] M''
h : ∀ (x : ↥... | simp | no goals | 6fd59fb1559945de |
IsSimpleGroup.prime_card | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | theorem prime_card [Finite α] : (Nat.card α).Prime | case intro.refine_1
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : IsSimpleGroup α
inst✝ : Finite α
h0 : 0 < Nat.card α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
n : ℕ
hn : n ∣ Nat.card α
h : Subgroup.zpowers (g ^ n) = ⊤
⊢ n = 1 | have hgo := orderOf_pow (n := n) g | case intro.refine_1
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : IsSimpleGroup α
inst✝ : Finite α
h0 : 0 < Nat.card α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
n : ℕ
hn : n ∣ Nat.card α
h : Subgroup.zpowers (g ^ n) = ⊤
hgo : orderOf (g ^ n) = orderOf g / (orderOf g).gcd n
⊢ n = 1 | 9a62ce0ba103643e |
gauge_ball | Mathlib/Analysis/Convex/Gauge.lean | theorem gauge_ball (hr : 0 ≤ r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r | case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ gauge (ball 0 0) x = ‖x‖ / 0 | simp | no goals | 500cf27f39d5582e |
MeasureTheory.support_convolution_subset_swap | Mathlib/Analysis/Convolution.lean | theorem support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst... | simp_rw [Set.mem_add, ← exists_and_left, not_exists, not_and_or, nmem_support] at hx | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst... | 6b3eeaf5338bd22b |
Ordinal.nfpFamily_lt_ord_lift | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem nfpFamily_lt_ord_lift {ι} {f : ι → Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : Cardinal.lift.{v, u} #ι < cof c) (hf : ∀ (i), ∀ b < c, f i b < c) {a} (ha : a < c) :
nfpFamily f a < c | ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Ordinal.{max u v}
hc : ℵ₀ < c.cof
hc' : Cardinal.lift.{v, u} #ι < c.cof
hf : ∀ (i : ι), ∀ b < c, f i b < c
a : Ordinal.{max u v}
ha : a < c
⊢ Cardinal.lift.{v, u} ℵ₀ < c.cof | rwa [Cardinal.lift_aleph0] | no goals | 1ab519ba3d97b754 |
ContinuousLinearMap.homothety_norm | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
(hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a | 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : Nontrivial E
f : E →SL[σ... | obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0 | case intro
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : Nontrivial E
... | 6750aa0122d27f2a |
Antitone.alternating_series_le_tendsto | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem Antitone.alternating_series_le_tendsto
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfa : Antitone f) (k : ℕ) : ∑ i ∈ range (2 * k), (-1) ^ i * f i ≤ l | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k n : ℕ
⊢ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i ≤
∑ x ∈ Finset.range (2 * n), (-1) ^ x * f x + (-1) ^ (2 * n)... | simp_rw [_root_.pow_succ', show (-1 : E) ^ (2 * n) = 1 by simp, neg_one_mul, one_mul,
← sub_eq_add_neg, le_sub_iff_add_le] | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k n : ℕ
⊢ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i + f (2 * n + 1) ≤ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i + f ... | 74802fbd069e02ac |
SimplexCategoryGenRel.factor_P_δ_σ | Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/EpiMono.lean | /-- An auxiliary lemma that shows there exists a factorization as a P_δ followed by a P_σ for
morphisms of the form `P_δ ≫ σ`. -/
private lemma factor_P_δ_σ {n : ℕ} (i : Fin (n + 1)) {x : SimplexCategoryGenRel}
(f : x ⟶ mk (n + 1)) (hf : P_δ f) : ∃ (z : SimplexCategoryGenRel) (e : x ⟶ z) (m : z ⟶ mk n)
(_ : P... | i : Fin (0 + 1)
⊢ 𝟙 (mk (0 + 1)) ≫ σ i = σ i ≫ 𝟙 (mk 0) | simp | no goals | 16ecac3bcb0756eb |
BitVec.toInt_intMin_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toInt_intMin_le (x : BitVec w) :
(intMin w).toInt ≤ x.toInt | w : Nat
x : BitVec (w + 1)
this : 0 < 2 ^ w
⊢ ↑(2 ^ w) < ↑(2 ^ (w + 1)) | omega | no goals | 659e6899233ba9d0 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftLeftConst.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftLeft.lean | theorem go_denote_eq (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(assign : α → Bool) (curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
⟦
(go aig input distance curr hcurr s).aig,
(go aig input dis... | case isTrue.inl.isFalse.isFalse.hidx
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
h✝² : curr < w
heq : curr = idx
h✝¹ : ¬curr... | rw [heq] | no goals | 625f9e09be737d85 |
Polynomial.content_mul | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content | case neg.inr.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), (p * q).degree < ↑n → (p * q).content = p.content * q.content
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : p.primPart.degree + q.primPart.degree = ↑n
⊢ q.primPart.degree ≠ ⊥ | rw [Ne, degree_eq_bot] | case neg.inr.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), (p * q).degree < ↑n → (p * q).content = p.content * q.content
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : p.primPart.degree + q.primPart.degree = ↑n
⊢ ¬q.primPart = 0 | 0a2419ef4663ed2a |
CategoryTheory.FreeBicategory.normalize_naturality | Mathlib/CategoryTheory/Bicategory/Coherence.lean | theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext (normalizeAux_congr p η))) | case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(preinclusion B).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫ (preinclusion B).map₂ (eqToHom ⋯)
... | rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih, comp_whiskerRight] | case mk.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h : b✝ ⟶ c✝
η' : Hom₂ f✝ g✝
ih :
∀ (p : Path a a✝),
(preinclusion B).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =
(normalizeIso p f✝).hom ≫ (preinclusion B).map₂ (eqToHom ⋯)
... | 362c365b74018f01 |
Set.encard_le_encard_of_injOn | Mathlib/Data/Set/Card.lean | theorem encard_le_encard_of_injOn (hf : MapsTo f s t) (f_inj : InjOn f s) :
s.encard ≤ t.encard | case h
α : Type u_1
β : Type u_2
s : Set α
t : Set β
f : α → β
hf : MapsTo f s t
f_inj : InjOn f s
⊢ f '' s ⊆ t | rintro _ ⟨x, hx, rfl⟩ | case h.intro.intro
α : Type u_1
β : Type u_2
s : Set α
t : Set β
f : α → β
hf : MapsTo f s t
f_inj : InjOn f s
x : α
hx : x ∈ s
⊢ f x ∈ t | e376c104d305f9f3 |
AlgebraicGeometry.HasRingHomProperty.of_stalkMap | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | /-- If `Q` is a property of ring maps that can be checked on prime ideals, the
associated property of scheme morphisms can be checked on stalks. -/
lemma of_stalkMap (hQ : OfLocalizationPrime Q) (H : ∀ x, Q (f.stalkMap x).hom) : P f | case intro.intro.intro.intro.a
P✝ : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P✝ Q
X Y : Scheme
f : X ⟶ Y
hQ : OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] => Q
hQi : RespectsIso fun {R S} [CommRing R] [CommRing S] =... | specialize H ⟨P, hP⟩ | case intro.intro.intro.intro.a
P✝ : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P✝ Q
X Y : Scheme
f : X ⟶ Y
hQ : OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] => Q
hQi : RespectsIso fun {R S} [CommRing R] [CommRing S] =... | e49c0b99d64c7542 |
Ordinal.IsFundamentalSequence.succ | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem succ : IsFundamentalSequence (succ o) 1 fun _ _ => o | o : Ordinal.{u}
⊢ (succ o).IsFundamentalSequence 1 fun x x => o | refine ⟨?_, @fun i j hi hj h => ?_, blsub_const Ordinal.one_ne_zero o⟩ | case refine_1
o : Ordinal.{u}
⊢ 1 ≤ (succ o).cof.ord
case refine_2
o i j : Ordinal.{u}
hi : i < 1
hj : j < 1
h : i < j
⊢ (fun x x => o) i hi < (fun x x => o) j hj | 718bf91bff0dd2c2 |
Real.BohrMollerup.tendsto_log_gamma | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem tendsto_log_gamma {x : ℝ} (hx : 0 < x) :
Tendsto (logGammaSeq x) atTop (𝓝 <| log (Gamma x)) | x : ℝ
hx : 0 < x
⊢ log (Gamma x) = (log ∘ Gamma) x - (log ∘ Gamma) 1 | simp_rw [Function.comp_apply, Gamma_one, log_one, sub_zero] | no goals | 0f9725f83adbdf9e |
hasFPowerSeriesOnBall_const | Mathlib/Analysis/Analytic/Constructions.lean | theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ | 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
c : F
e : E
⊢ ⊤ ≤ (constFormalMultilinearSeries 𝕜 E c).radius | simp | no goals | f625d1227a719609 |
EuclideanGeometry.dist_mul_of_eq_angle_of_dist_mul | Mathlib/Geometry/Euclidean/Triangle.lean | theorem dist_mul_of_eq_angle_of_dist_mul (a b c a' b' c' : P) (r : ℝ) (h : ∠ a' b' c' = ∠ a b c)
(hab : dist a' b' = r * dist a b) (hcb : dist c' b' = r * dist c b) :
dist a' c' = r * dist a c | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
⊢ r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * R... | simp [pow_two, ← law_cos a b c, mul_pow] | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c a' b' c' : P
r : ℝ
h : ∠ a' b' c' = ∠ a b c
hab : dist a' b' = r * dist a b
hcb : dist c' b' = r * dist c b
⊢ r * r * (dist a c * dist a c) = r * dist a c * (r * dist a c) | 21e37fd5b54d5f31 |
hasMFDerivWithinAt_congr_set' | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem hasMFDerivWithinAt_congr_set' (y : M) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
HasMFDerivWithinAt I I' f s x f' ↔ HasMFDerivWithinAt I I' f t x f' | case refine_2
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✝... | apply hasFDerivWithinAt_congr_set' (extChartAt I x x) | case refine_2
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✝... | f3481fb27de7c16f |
AdjoinRoot.minpoly_root | Mathlib/RingTheory/AdjoinRoot.lean | theorem minpoly_root (hf : f ≠ 0) : minpoly K (root f) = f * C f.leadingCoeff⁻¹ | case h₁.a
K : Type w
inst✝ : Field K
f : K[X]
hf : f ≠ 0
f'_monic : (f * C f.leadingCoeff⁻¹).Monic
q : K[X]
q_monic : q.Monic
q_aeval : (aeval (root f)) q = 0
x✝ : K
⊢ (((lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval).comp (mk q)).comp C) x✝ = ((mk f).comp C) x✝ | simp only [RingHom.comp_apply, mk_C, lift_of] | case h₁.a
K : Type w
inst✝ : Field K
f : K[X]
hf : f ≠ 0
f'_monic : (f * C f.leadingCoeff⁻¹).Monic
q : K[X]
q_monic : q.Monic
q_aeval : (aeval (root f)) q = 0
x✝ : K
⊢ (algebraMap K (AdjoinRoot f)) x✝ = (of f) x✝ | 465254e3a8d6fd47 |
DFinsupp.inner_sum | Mathlib/Analysis/InnerProductSpace/Basic.lean | theorem DFinsupp.inner_sum {ι : Type*} [DecidableEq ι] {α : ι → Type*}
[∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E)
(l : Π₀ i, α i) (x : E) : ⟪x, l.sum f⟫ = l.sum fun i a => ⟪x, f i a⟫ | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝² : DecidableEq ι
α : ι → Type u_5
inst✝¹ : (i : ι) → AddZeroClass (α i)
inst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)
f : (i : ι) → α i → E
l : Π₀ (i : ι), α i
x : E
⊢ ⟪x, l.sum f⟫_𝕜 = l... | simp +contextual only [DFinsupp.sum, inner_sum, smul_eq_mul] | no goals | e687f2314a47d5d8 |
Matrix.det_eq_of_forall_row_eq_smul_add_pred_aux | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : Fin (n + 1)) :
∀ (c : Fin n → R) (_hc : ∀ i : Fin n, k < i.succ → c i = 0)
{M N : Matrix (Fin n.succ) (Fin n.succ) R} (_h0 : ∀ j, M 0 j = N 0 j)
(_hsucc : ∀ (i : Fin n) (j), M i.succ j = N i.succ j + c i * M (Fin.castSucc i) j),
det M ... | case refine_1.e_M.a
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < i.succ → c i = 0
M N : Matrix (Fin n.succ) (Fin n.succ) R
h0 : ∀ (j : Fin n.succ), M 0 j = N 0 j
hsucc : ∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c i * M i.castSucc j
i j : Fin n.succ
⊢ M i j =... | refine Fin.cases (h0 j) (fun i => ?_) i | case refine_1.e_M.a
R : Type v
inst✝ : CommRing R
n : ℕ
k : Fin (n + 1)
c : Fin n → R
hc : ∀ (i : Fin n), 0 < i.succ → c i = 0
M N : Matrix (Fin n.succ) (Fin n.succ) R
h0 : ∀ (j : Fin n.succ), M 0 j = N 0 j
hsucc : ∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c i * M i.castSucc j
i✝ j : Fin n.succ
i : Fin ... | 8b96005e0c0eb652 |
NumberField.Embeddings.pow_eq_one_of_norm_eq_one | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ φ : K →+* A, ‖φ x‖ = 1) :
∃ (n : ℕ) (_ : 0 < n), x ^ n = 1 | K✝ : Type u_1
inst✝⁶ : Field K✝
A✝ : Type u_2
inst✝⁵ : NormedField A✝
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ = 1
a b : ℕ
habne : a ≠ b
h : x ^ (a - b) = 1 ∨ x ^ b = 0... | refine h.resolve_right fun hp => ?_ | K✝ : Type u_1
inst✝⁶ : Field K✝
A✝ : Type u_2
inst✝⁵ : NormedField A✝
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ = 1
a b : ℕ
habne : a ≠ b
h : x ^ (a - b) = 1 ∨ x ^ b = 0... | 2db53376910a5986 |
Int.max_comm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem max_comm (a b : Int) : max a b = max b a | a b : Int
⊢ (if a ≤ b then b else a) = if b ≤ a then a else b | by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂] | case pos
a b : Int
h₁ : a ≤ b
h₂ : b ≤ a
⊢ b = a
case neg
a b : Int
h₁ : ¬a ≤ b
h₂ : ¬b ≤ a
⊢ a = b | ce6e69f458db91f8 |
ContinuousMap.Homotopy.evalAt_eq | Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean | theorem evalAt_eq (x : X) : ⟦H.evalAt x⟧ = hcast (H.apply_zero x).symm ≫
(πₘ (TopCat.ofHom H.uliftMap)).map (prodToProdTopI uhpath01 (𝟙 (fromTop x))) ≫
hcast (H.apply_one x).symm.symm | case hp
X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x : ↑X
t✝ : ↑I
⊢ (H.evalAt x) t✝ = ((fun q => q.map ⋯) (unitInterval.upath01.prod (Path.refl (fromTop x).as))) t✝ | rfl | no goals | eb0c975fc82651fe |
CategoryTheory.eComp_eHomWhiskerRight | Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean | /-- Whiskering commutes with the enriched composition. -/
@[reassoc]
lemma eComp_eHomWhiskerRight {X X' : C} (f : X ⟶ X') (Y Z : C) :
eComp V X' Y Z ≫ eHomWhiskerRight V f Z =
eHomWhiskerRight V f Y ▷ _ ≫ eComp V X Y Z | V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X X' : C
f : X ⟶ X'
Y Z : C
⊢ (λ_ (EnrichedCategory.Hom X' Y ⊗ EnrichedCategory.Hom Y Z)).inv ≫
(eHomEquiv V) f ▷ (EnrichedCategory.Hom X' Y ⊗ EnrichedCategory.Hom Y Z) ≫... | simp [e_assoc'] | no goals | 33a19e16cabe7cfa |
MeasureTheory.inducedOuterMeasure_eq_iInf | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | theorem inducedOuterMeasure_eq_iInf (s : Set α) :
inducedOuterMeasure m P0 m0 s = ⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht | α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s ... | apply le_antisymm | case a
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂... | 07b0977866260179 |
Matroid.map_freeOn | Mathlib/Data/Matroid/Map.lean | @[simp] lemma map_freeOn (f : α → β) (hf) : (freeOn E).map f hf = freeOn (f '' E) | α : Type u_1
β : Type u_2
E : Set α
f : α → β
hf : InjOn f (freeOn E).E
⊢ (freeOn E).map f hf = freeOn (f '' E) | rw [← dual_inj] | α : Type u_1
β : Type u_2
E : Set α
f : α → β
hf : InjOn f (freeOn E).E
⊢ ((freeOn E).map f hf)✶ = (freeOn (f '' E))✶ | ad16611fe8897733 |
Int.natAbs_abs | Mathlib/Algebra/Order/Group/Unbundled/Int.lean | theorem natAbs_abs (a : ℤ) : natAbs |a| = natAbs a | a : ℤ
⊢ (↑a.natAbs).natAbs = a.natAbs | rfl | no goals | a42c5edc195169e1 |
Set.ssubset_iff_sdiff_singleton | Mathlib/Data/Set/Insert.lean | lemma ssubset_iff_sdiff_singleton : s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a} | α : Type u
s t : Set α
⊢ (∃ a ∉ s, a ∈ t ∧ s ⊆ t) ↔ ∃ a ∈ t, s ⊆ t ∧ a ∉ s | aesop | no goals | 94108b8e50522049 |
MeasureTheory.Measure.dmaSMul_apply | Mathlib/MeasureTheory/Group/MeasurableEquiv.lean | lemma dmaSMul_apply (μ : Measure A) (g : Gᵈᵐᵃ) (s : Set A) :
(g • μ) s = μ (DomMulAct.mk.symm g • s) | case h.e_6.h
G : Type u_1
A : Type u_2
inst✝⁵ : Group G
inst✝⁴ : AddCommGroup A
inst✝³ : DistribMulAction G A
inst✝² : MeasurableSpace A
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableSMul G A
μ : Measure A
g : Gᵈᵐᵃ
s : Set A
⊢ ⇑(MeasurableEquiv.smul (DomMulAct.mk.symm g)⁻¹) ⁻¹' s = DomMulAct.mk.symm g • s | exact Set.preimage_smul_inv (DomMulAct.mk.symm g) s | no goals | 36ac8d9734e64a06 |
matrixEquivTensor_apply_stdBasisMatrix | Mathlib/RingTheory/MatrixAlgebra.lean | theorem matrixEquivTensor_apply_stdBasisMatrix (i j : n) (x : A) :
matrixEquivTensor n R A (stdBasisMatrix i j x) = x ⊗ₜ stdBasisMatrix i j 1 | n : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Fintype n
inst✝ : DecidableEq n
i j : n
x : A
⊢ ∀ (p : n × n), i = p.1 ∧ j = p.2 ↔ p = (i, j) | aesop | no goals | 98216ed5145dcb0c |
cauchy_map_of_uniformCauchySeqOn_fderiv | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem cauchy_map_of_uniformCauchySeqOn_fderiv {s : Set E} (hs : IsOpen s) (h's : IsPreconnected s)
(hf' : UniformCauchySeqOn f' l s) (hf : ∀ n : ι, ∀ y : E, y ∈ s → HasFDerivAt (f n) (f' n y) y)
{x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s) (hfg : Cauchy (map (fun n => f n x₀) l)) :
Cauchy (map (fun n => f n x) ... | ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen s
h's : IsPreco... | suffices H : s ⊆ t from (H hx).2 | ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen s
h's : IsPreco... | 774a9e47a2906047 |
IsUpperSet.null_frontier | Mathlib/MeasureTheory/Order/UpperLower.lean | theorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 | case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
⊢ ¬Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1) | refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_ | case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
x✝ : ℝ
⊢ x ∈ frontier sᶜ | e0b276bd026a50d3 |
IsClosed.mk_lt_continuum | Mathlib/Topology/Separation/NotNormal.lean | theorem IsClosed.mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : #s < 𝔠 | case intro.intro
X : Type u
inst✝³ : TopologicalSpace X
inst✝² : SeparableSpace X
inst✝¹ : NormalSpace X
s : Set X
hs : IsClosed s
inst✝ : DiscreteTopology ↑s
h : 𝔠 ≤ #↑s
t : Set X
htc : t.Countable
htd : Dense t
this : Countable ↑t
⊢ False | refine (Cardinal.cantor 𝔠).not_le ?_ | case intro.intro
X : Type u
inst✝³ : TopologicalSpace X
inst✝² : SeparableSpace X
inst✝¹ : NormalSpace X
s : Set X
hs : IsClosed s
inst✝ : DiscreteTopology ↑s
h : 𝔠 ≤ #↑s
t : Set X
htc : t.Countable
htd : Dense t
this : Countable ↑t
⊢ 2 ^ 𝔠 ≤ 𝔠 | 2bddcff8feea2d0c |
circleMap_not_mem_ball | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | theorem circleMap_not_mem_ball (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∉ ball c R | c : ℂ
R θ : ℝ
⊢ circleMap c R θ ∉ ball c R | simp [dist_eq, le_abs_self] | no goals | b8f19e406eb60a36 |
AffineIsometry.image_intrinsicInterior | Mathlib/Analysis/Convex/Intrinsic.lean | theorem image_intrinsicInterior (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
intrinsicInterior 𝕜 (φ '' s) = φ '' intrinsicInterior 𝕜 s | case inr
𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : SeminormedAddCommGroup W
inst✝⁵ : NormedSpace 𝕜 V
inst✝⁴ : NormedSpace 𝕜 W
inst✝³ : MetricSpace P
inst✝² : PseudoMetricSpace Q
inst✝¹ : NormedAddTorsor V P
inst✝ : NormedAddTor... | have : φ.toAffineMap ∘ (↑) ∘ f.symm = (↑) := funext isometryEquivMap.apply_symm_apply | case inr
𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : SeminormedAddCommGroup W
inst✝⁵ : NormedSpace 𝕜 V
inst✝⁴ : NormedSpace 𝕜 W
inst✝³ : MetricSpace P
inst✝² : PseudoMetricSpace Q
inst✝¹ : NormedAddTorsor V P
inst✝ : NormedAddTor... | 53334be6dead5321 |
Profinite.NobelingProof.factors_prod_eq_basis | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem factors_prod_eq_basis (x : π C (· ∈ s)) :
(factors C s x).prod = spanFinBasis C s x | I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
x : ↑(π C fun x => x ∈ s)
⊢ (factors C s x).prod = spanFinBasis C s x | ext y | case h
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
x y : ↑(π C fun x => x ∈ s)
⊢ (factors C s x).prod y = (spanFinBasis C s x) y | 65f40261e4411106 |
LipschitzWith.ae_lineDeriv_sum_eq | Mathlib/Analysis/Calculus/Rademacher.lean | theorem ae_lineDeriv_sum_eq
(hf : LipschitzWith C f) {ι : Type*} (s : Finset ι) (a : ι → ℝ) (v : ι → E) :
∀ᵐ x ∂μ, lineDeriv ℝ f x (∑ i ∈ s, a i • v i) = ∑ i ∈ s, a i • lineDeriv ℝ f x (v i) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_com... | suffices S2 : ∫ x, (∑ i ∈ s, a i * fderiv ℝ g x (v i)) * f x ∂μ =
∑ i ∈ s, a i * ∫ x, fderiv ℝ g x (v i) * f x ∂μ by
obtain ⟨D, g_lip⟩ : ∃ D, LipschitzWith D g :=
ContDiff.lipschitzWith_of_hasCompactSupport g_comp g_smooth (mod_cast le_top)
simp_rw [integral_lineDeriv_mul_eq hf g_lip g_comp]
s... | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_com... | 158292ae7c8fb6bf |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ... | positivity | no goals | e1f8d437480849f9 |
LinearIsometryEquiv.reflections_generate_dim_aux | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
(φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod | case neg.intro
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
∀ (φ : F ≃ₗᵢ[ℝ] F),
finrank ℝ
↥(LinearMap.ker
(ContinuousLinearMap.id ℝ F -
↑{ toLinearEquiv := φ.toLinearEquiv, continuous_toFun := ⋯, cont... | have H₂V : W ≤ V := by
intro w hw
apply hV
rw [hW w hw]
refine reflection_mem_subspace_eq_self ?_
rw [Submodule.mem_orthogonal_singleton_iff_inner_left]
exact Submodule.sub_mem _ v.prop hφv _ hw | case neg.intro
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
∀ (φ : F ≃ₗᵢ[ℝ] F),
finrank ℝ
↥(LinearMap.ker
(ContinuousLinearMap.id ℝ F -
↑{ toLinearEquiv := φ.toLinearEquiv, continuous_toFun := ⋯, cont... | 95046b47cb73a72b |
WeierstrassCurve.b₂_of_isCharTwoJNeZeroNF_of_char_two | Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean | theorem b₂_of_isCharTwoJNeZeroNF_of_char_two : W.b₂ = 1 | R : Type u_1
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP R 2
⊢ 1 + 4 * W.a₂ = 1 | linear_combination 2 * W.a₂ * CharP.cast_eq_zero R 2 | no goals | e42eeaf01567ff6d |
MonoidHom.independent_range_of_coprime_order | Mathlib/GroupTheory/NoncommPiCoprod.lean | theorem independent_range_of_coprime_order
(hcomm : Pairwise fun i j : ι => ∀ (x : H i) (y : H j), Commute (ϕ i x) (ϕ j y))
[Finite ι] [∀ i, Fintype (H i)]
(hcoprime : Pairwise fun i j => Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :
iSupIndep fun i => (ϕ i).range | case intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
ι : Type u_2
H : ι → Type u_3
inst✝² : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype (H i)
hcoprime : Pairwise fun i j => (Fintype.card (H ... | have hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H j) := by
rw [← hgf, ← Fintype.card_pi]
exact (orderOf_map_dvd _ _).trans orderOf_dvd_card | case intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
ι : Type u_2
H : ι → Type u_3
inst✝² : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝¹ : Finite ι
inst✝ : (i : ι) → Fintype (H i)
hcoprime : Pairwise fun i j => (Fintype.card (H ... | c5d2b3b4f1f4da7d |
UniqueFactorizationMonoid.normalize_normalized_factor | Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.lean | theorem normalize_normalized_factor {a : α} :
∀ x : α, x ∈ normalizedFactors a → normalize x = x | case neg.intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : ¬a = 0
y : α
left✝ : y ∈ Classical.choose ⋯
hx : normalize y ∈ Multiset.map (fun x => normalize x) (Classical.choose ⋯)
⊢ normalize (normalize y) = normalize y | apply normalize_idem | no goals | 6feac4ff4b873185 |
MeasureTheory.addHaar_image_le_mul_of_det_lt | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
(hm : ENNReal.ofReal |A.det| < m) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' clos... | simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,
measure_preimage_add] | no goals | 35c8118ffea35652 |
Odd.ne_two_of_dvd_nat | Mathlib/Algebra/Order/Ring/Abs.lean | /-- `2` is not a factor of an odd natural number. -/
lemma Odd.ne_two_of_dvd_nat {m n : ℕ} (hn : Odd n) (hm : m ∣ n) : m ≠ 2 | m n : ℕ
hn : Odd n
hm : m ∣ n
⊢ m ≠ 2 | rintro rfl | n : ℕ
hn : Odd n
hm : 2 ∣ n
⊢ False | b1d231cdfc740293 |
CategoryTheory.Limits.PullbackCone.mono_fst_of_is_pullback_of_mono | Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean | theorem mono_fst_of_is_pullback_of_mono {t : PullbackCone f g} (ht : IsLimit t) [Mono g] :
Mono t.fst | C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
t : PullbackCone f g
ht : IsLimit t
inst✝ : Mono g
W : C
h k : W ⟶ t.pt
i : h ≫ t.fst = k ≫ t.fst
⊢ h ≫ t.snd = k ≫ t.snd | rw [← cancel_mono g, Category.assoc, Category.assoc, ← condition] | C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
t : PullbackCone f g
ht : IsLimit t
inst✝ : Mono g
W : C
h k : W ⟶ t.pt
i : h ≫ t.fst = k ≫ t.fst
⊢ h ≫ t.fst ≫ f = k ≫ t.fst ≫ f | 4c46a86933756ce9 |
BitVec.ushiftRightRec_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem ushiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
ushiftRightRec x y n = x >>> (y.setWidth (n + 1)).setWidth w₂ | case pos
w₁ w₂ n : Nat
ih : ∀ (x : BitVec w₁) (y : BitVec w₂), x.ushiftRightRec y n = x >>> setWidth w₂ (setWidth (n + 1) y)
x : BitVec w₁
y : BitVec w₂
h : y.getLsbD (n + 1) = true
⊢ setWidth w₂ (setWidth (n + 1) y) &&& twoPow w₂ (n + 1) = 0#w₂ | simp [and_twoPow] | no goals | c716927de5030eb6 |
Turing.PartrecToTM2.tr_eval | Mathlib/Computability/TMToPartrec.lean | theorem tr_eval (c v) : eval (TM2.step tr) (init c v) = halt <$> Code.eval c v | c : Code
v : List ℕ
⊢ eval (TM2.step tr) (init c v) = halt <$> c.eval v | obtain ⟨i, h₁, h₂⟩ := tr_init c v | case intro.intro
c : Code
v : List ℕ
i : Cfg'
h₁ : TrCfg (stepNormal c Cont.halt v) i
h₂ : Reaches₁ (TM2.step tr) (init c v) i
⊢ eval (TM2.step tr) (init c v) = halt <$> c.eval v | 58e74fcd24b87e49 |
Nat.range_mul_add | Mathlib/Data/ZMod/Basic.lean | /-- The range of `(m * · + k)` on natural numbers is the set of elements `≥ k` in the
residue class of `k` mod `m`. -/
lemma Nat.range_mul_add (m k : ℕ) :
Set.range (fun n : ℕ ↦ m * n + k) = {n : ℕ | (n : ZMod m) = k ∧ k ≤ n} | case h.refine_2
m k n : ℕ
x✝ : ↑n = ↑k ∧ k ≤ n
H₁ : ↑n = ↑k
H₂ : k ≤ n
⊢ ∃ y, n = k + m * y | obtain ⟨a, ha⟩ := le_iff_exists_add.mp H₂ | case h.refine_2.intro
m k n : ℕ
x✝ : ↑n = ↑k ∧ k ≤ n
H₁ : ↑n = ↑k
H₂ : k ≤ n
a : ℕ
ha : n = k + a
⊢ ∃ y, n = k + m * y | 1f09dbe2d6b4db61 |
VitaliFamily.ae_eventually_measure_zero_of_singular | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) | case h
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 ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a <... | exact ha.trans_le (mul_le_mul_right' ((ENNReal.coe_le_coe.2 hn.le).trans w_lt.le) _) | no goals | 27b6e7c43d8d9d57 |
ContMDiffSection.coe_zsmul | Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean | theorem coe_zsmul (s : Cₛ^n⟮I; F, V⟯) (z : ℤ) : ⇑(z • s : Cₛ^n⟮I; F, V⟯) = z • ⇑s | case ofNat
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ ... | refine (coe_nsmul s n).trans ?_ | case ofNat
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ ... | 10c8a741dc556a8c |
Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm | Mathlib/Topology/VectorBundle/Basic.lean | theorem symm_apply_eq_mk_continuousLinearEquivAt_symm (e : Trivialization F (π F E)) [e.IsLinear R]
(b : B) (hb : b ∈ e.baseSet) (z : F) :
e.toPartialHomeomorph.symm ⟨b, z⟩ = ⟨b, (e.continuousLinearEquivAt R b hb).symm z⟩ | case a
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → Topo... | simpa only [e.source_eq, mem_preimage] | no goals | f151ccdee46aa84d |
ZMod.intCast_eq_iff | Mathlib/Data/ZMod/Basic.lean | theorem intCast_eq_iff (p : ℕ) (n : ℤ) (z : ZMod p) [NeZero p] :
↑n = z ↔ ∃ k, n = z.val + p * k | case mpr
p : ℕ
n : ℤ
z : ZMod p
inst✝ : NeZero p
⊢ (∃ k, n = ↑z.val + ↑p * k) → ↑n = z | rintro ⟨k, rfl⟩ | case mpr.intro
p : ℕ
z : ZMod p
inst✝ : NeZero p
k : ℤ
⊢ ↑(↑z.val + ↑p * k) = z | 970ec71abe2b10bc |
Array.zipWith_map_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Zip.lean | theorem zipWith_map_right (l₁ : Array α) (l₂ : Array β) (f : β → β') (g : α → β' → γ) :
zipWith g l₁ (l₂.map f) = zipWith (fun a b => g a (f b)) l₁ l₂ | case mk
α : Type u_1
β : Type u_2
β' : Type u_3
γ : Type u_4
l₂ : Array β
f : β → β'
g : α → β' → γ
toList✝ : List α
⊢ zipWith g { toList := toList✝ } (map f l₂) = zipWith (fun a b => g a (f b)) { toList := toList✝ } l₂ | cases l₂ | case mk.mk
α : Type u_1
β : Type u_2
β' : Type u_3
γ : Type u_4
f : β → β'
g : α → β' → γ
toList✝¹ : List α
toList✝ : List β
⊢ zipWith g { toList := toList✝¹ } (map f { toList := toList✝ }) =
zipWith (fun a b => g a (f b)) { toList := toList✝¹ } { toList := toList✝ } | 576d88a4fcc73f54 |
Stream'.WSeq.destruct_map | Mathlib/Data/Seq/WSeq.lean | theorem destruct_map (f : α → β) (s : WSeq α) :
destruct (map f s) = Computation.map (Option.map (Prod.map f (map f))) (destruct s) | case bisim.intro
α : Type u
β : Type v
f : α → β
s✝ : WSeq α
c1 c2 : Computation (Option (β × WSeq β))
s : WSeq α
h : c1 = (map f s).destruct ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) s.destruct
⊢ Computation.BisimO
(fun c1 c2 => ∃ s, c1 = (map f s).destruct ∧ c2 = Computation.map (Option.map (Prod.m... | induction' s using WSeq.recOn with a s s <;> simp | case bisim.intro.h3
α : Type u
β : Type v
f : α → β
s✝ : WSeq α
c1 c2 : Computation (Option (β × WSeq β))
s : WSeq α
h : c1 = (map f s.think).destruct ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) s.think.destruct
⊢ ∃ s_1,
(map f s).destruct = (map f s_1).destruct ∧
Computation.map (Option.map (Pro... | 9ec5335546f5acf5 |
UV.compress_injOn | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compress_injOn : Set.InjOn (compress u v) ↑{a ∈ s | compress u v a ∉ s} | case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v : α
inst✝ : DecidableEq α
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : ¬(Disjoint u a ∧ v ≤ a)
ha : a ∈ s ∧ a ∉ s
hab : a = compress u v b
⊢ a = b | exact (ha.2 ha.1).elim | no goals | 4c8b28ba0d63c5ee |
Array.foldr_subtype' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem foldr_subtype' {p : α → Prop} {l : Array { x // p x }}
{f : { x // p x } → β → β} {g : α → β → β} {x : β}
(hf : ∀ x h b, f ⟨x, h⟩ b = g x b) (h : start = l.size) :
l.foldr f x start 0 = l.unattach.foldr g x | α : Type u_1
β : Type u_2
p : α → Prop
l : Array { x // p x }
f : { x // p x } → β → β
g : α → β → β
x : β
hf : ∀ (x : α) (h : p x) (b : β), f ⟨x, h⟩ b = g x b
⊢ foldr f x l = foldr g x l.unattach | rwa [foldr_subtype] | no goals | aa7be1d73b8fb383 |
Turing.ToPartrec.cont_eval_fix | Mathlib/Computability/TMConfig.lean | theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
Turing.eval step (stepNormal f (Cont.fix f k) v) =
f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v) | case mp
f : Code
k : Cont
v : List ℕ
fok : f.Ok
x : Cfg
⊢ ∀ (c : Cfg),
x ∈ eval step c →
∀ (v : List ℕ) (c' : Cfg),
c = c'.then (Cont.fix f k) →
Reaches step (stepNormal f Cont.halt v) c' →
∃ v₁ ∈ f.eval v,
∃ v₂ ∈ if v₁.headI = 0 then pure v₁.tail else f.fix.eval v₁... | refine fun c he => evalInduction he fun y h IH => ?_ | case mp
f : Code
k : Cont
v : List ℕ
fok : f.Ok
x c : Cfg
he : x ∈ eval step c
y : Cfg
h : x ∈ eval step y
IH :
∀ (a' : Cfg),
step y = some a' →
∀ (v : List ℕ) (c' : Cfg),
a' = c'.then (Cont.fix f k) →
Reaches step (stepNormal f Cont.halt v) c' →
∃ v₁ ∈ f.eval v,
... | 4a2318a05b9f1866 |
csSup_eq_univ_of_not_bddAbove | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | theorem csSup_eq_univ_of_not_bddAbove {s : Set α} (hs : ¬BddAbove s) : sSup s = sSup univ | α : Type u_1
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : BddAbove univ
⊢ BddAbove s | exact hs.mono (subset_univ _) | no goals | dad8d3901470e334 |
ZFSet.IsTransitive.inter | Mathlib/SetTheory/ZFC/Ordinal.lean | theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :
(x ∩ y).IsTransitive := fun z hz w hw => by
rw [mem_inter] at hz ⊢
exact ⟨hx.mem_trans hw hz.1, hy.mem_trans hw hz.2⟩
| x y : ZFSet.{u}
hx : x.IsTransitive
hy : y.IsTransitive
z : ZFSet.{u}
hz : z ∈ x ∧ z ∈ y
w : ZFSet.{u}
hw : w ∈ z
⊢ w ∈ x ∧ w ∈ y | exact ⟨hx.mem_trans hw hz.1, hy.mem_trans hw hz.2⟩ | no goals | 6dc2b6c22012f800 |
FractionalIdeal.den_mul_self_eq_num' | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem den_mul_self_eq_num' (I : FractionalIdeal S P) :
spanSingleton S (algebraMap R P I.den) * I = I.num | case h.e'_2
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
⊢ (algebraMap R P) ↑I.den • ↑I = I.den • ↑I | ext | case h.e'_2.h
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
x✝ : P
⊢ x✝ ∈ (algebraMap R P) ↑I.den • ↑I ↔ x✝ ∈ I.den • ↑I | 54d84fad9be88303 |
IsPrimitiveRoot.card_nthRootsFinset | Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean | theorem card_nthRootsFinset {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
#(nthRootsFinset n R) = n | R : Type u_4
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
⊢ #(nthRootsFinset n R) = n | rw [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), card_mk, h.card_nthRoots_one] | no goals | f034bf7a8eefb003 |
Set.Countable.isPathConnected_compl_of_one_lt_rank | Mathlib/Analysis/NormedSpace/Connected.lean | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ | E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
⊢ 2... | module | no goals | a2405a5e761a2c4d |
BitVec.append_assoc | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem append_assoc {x₁ : BitVec w₁} {x₂ : BitVec w₂} {x₃ : BitVec w₃} :
(x₁ ++ x₂) ++ x₃ = (x₁ ++ (x₂ ++ x₃)).cast (by omega) | case pred
w₂ w₃ n : Nat
x₁ : BitVec (n + 1)
x₂ : BitVec w₂
x₃ : BitVec w₃
ih : ∀ {x₂ : BitVec w₂} {x₃ : BitVec w₃}, setWidth n x₁ ++ x₂ ++ x₃ = BitVec.cast ⋯ (setWidth n x₁ ++ (x₂ ++ x₃))
j : Nat
h : j < n + 1 + w₂ + w₃
⊢ (BitVec.cast ⋯ (cons x₁.msb (BitVec.cast ⋯ (setWidth n x₁ ++ (x₂ ++ x₃))))).getLsbD j =
(BitVe... | simp [getLsbD_cons, show n + w₂ + w₃ = n + (w₂ + w₃) by omega] | no goals | ac5627f045e44cbc |
Function.Surjective.isEngelian | Mathlib/Algebra/Lie/Engel.lean | theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
(h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ | R : Type u₁
L : Type u₂
L₂ : Type u₃
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
hf : Surjective ⇑f
h : LieAlgebra.IsEngelian R L
M : Type u₄
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L₂ M
_i4 : LieModule R L₂ M
h' : ∀ (x : L₂... | aesop | no goals | 98e11bddd6196a5e |
Set.image_mul_right_Icc' | Mathlib/Algebra/Order/Group/Pointwise/Interval.lean | theorem image_mul_right_Icc' (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) '' Icc a b = Icc (a * c) (b * c) :=
((Units.mk0 c h.ne').mulRight.image_eq_preimage _).trans (by simp [h, division_def])
| α : Type u_1
inst✝ : LinearOrderedField α
a b c : α
h : 0 < c
⊢ ⇑(Equiv.symm (Units.mk0 c ⋯).mulRight) ⁻¹' Icc a b = Icc (a * c) (b * c) | simp [h, division_def] | no goals | 5a74376c2bc4ccc0 |
spectrum.nonempty_of_isAlgClosed_of_finiteDimensional | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | theorem nonempty_of_isAlgClosed_of_finiteDimensional [IsAlgClosed 𝕜] [Nontrivial A]
[I : FiniteDimensional 𝕜 A] (a : A) : (σ a).Nonempty | case intro.intro.intro.intro
𝕜 : Type u
A : Type v
inst✝⁴ : Field 𝕜
inst✝³ : Ring A
inst✝² : Algebra 𝕜 A
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Nontrivial A
I : FiniteDimensional 𝕜 A
a : A
p : 𝕜[X]
h_mon : p.Monic
h_eval_p : eval₂ ↑ₐ a p = 0
nu : ¬IsUnit ((aeval a) (Multiset.map (fun a => X - C a) p.roots).prod)
k : 𝕜
h... | exact ⟨k, hk⟩ | no goals | de343a8cea63706c |
BitVec.shiftLeft_eq' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem shiftLeft_eq' {x : BitVec w₁} {y : BitVec w₂} : x <<< y = x <<< y.toNat | w₁ w₂ : Nat
x : BitVec w₁
y : BitVec w₂
⊢ x <<< y = x <<< y.toNat | rfl | no goals | 230943b868af627d |
recursion' | Mathlib/Data/Real/Pi/Irrational.lean | /--
Auxiliary for the proof that `π` is irrational.
While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction... | θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)
v₂ : ℝ → ℝ... | have hv₁ (x) : HasDerivAt v₁ (v₁' x) x := (hasDerivAt_mul_const θ).sin | θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)
v₂ : ℝ → ℝ... | 834ac76d1ec60c71 |
UniqueMDiffOn.uniqueDiffOn_inter_preimage | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | theorem UniqueMDiffOn.uniqueDiffOn_inter_preimage (hs : UniqueMDiffOn I s) (x : M) (y : M'')
{f : M → M''} (hf : ContinuousOn f s) :
UniqueDiffOn 𝕜
((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) :=
haveI : UniqueMDiffOn I (s ∩ f ⁻¹' (extChartAt I' y).source) | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | apply (hs z hz.1).inter' | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | fd29766784d64fdd |
Matrix.reindexLinearEquiv_trans | Mathlib/LinearAlgebra/Matrix/Reindex.lean | theorem reindexLinearEquiv_trans (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') :
(reindexLinearEquiv R A e₁ e₂).trans (reindexLinearEquiv R A e₁' e₂') =
(reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂') : _ ≃ₗ[R] _) | case h.a
m : Type u_2
n : Type u_3
m' : Type u_6
n' : Type u_7
m'' : Type u_9
n'' : Type u_10
R : Type u_11
A : Type u_12
inst✝² : Semiring R
inst✝¹ : AddCommMonoid A
inst✝ : Module R A
e₁ : m ≃ m'
e₂ : n ≃ n'
e₁' : m' ≃ m''
e₂' : n' ≃ n''
x✝ : Matrix m n A
i✝ : m''
j✝ : n''
⊢ (reindexLinearEquiv R A e₁ e₂ ≪≫ₗ reindexL... | rfl | no goals | aeaacd5e876282fc |
TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure_absorbing | Mathlib/RingTheory/TwoSidedIdeal/Operations.lean | /-- If `s : Set R` is absorbing under multiplication, then its `TwoSidedIdeal.span` coincides with
its `AddSubgroup.closure`, as sets. -/
lemma mem_span_iff_mem_addSubgroup_closure_absorbing {s : Set R}
(h_left : ∀ x y, y ∈ s → x * y ∈ s) (h_right : ∀ y x, y ∈ s → y * x ∈ s) {z : R} :
z ∈ span s ↔ z ∈ closure s... | R : Type u_1
inst✝ : NonUnitalRing R
s : Set R
h_left : ∀ (x y : R), y ∈ s → x * y ∈ s
h_right : ∀ (y x : R), y ∈ s → y * x ∈ s
z : R
h_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s
h_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s
I : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ⋯ ⋯ ⋯
⊢ (∀ (I : TwoSidedIde... | refine ⟨fun h ↦ h I fun x hx ↦ ?mem_closure_of_forall, fun hz J hJ ↦ ?mem_ideal_of_subset⟩ | case mem_closure_of_forall
R : Type u_1
inst✝ : NonUnitalRing R
s : Set R
h_left : ∀ (x y : R), y ∈ s → x * y ∈ s
h_right : ∀ (y x : R), y ∈ s → y * x ∈ s
z : R
h_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s
h_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s
I : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ... | 82f017f42927ebef |
Submodule.basis_of_pid_aux | Mathlib/LinearAlgebra/FreeModule/PID.lean | theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z... | case neg.intro.intro.intro.refine_2.refine_2.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R... | simp only [Fin.cons_zero, Fin.castLE_zero] | case neg.intro.intro.intro.refine_2.refine_2.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R... | 48ee91e21f4317c3 |
StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | theorem liftPropWithinAt_indep_chart_target_aux {g : X → M'} {e : PartialHomeomorph X H} {x : X}
{s : Set X} (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M') (xf : g x ∈ f.source)
(hf' : f' ∈ G'.maximalAtlas M') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x) :
P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e... | H : Type u_1
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
f f' : PartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
hG : G.Local... | exact hgs.comp (e.symm.continuousAt <| e.mapsTo xe).continuousWithinAt Subset.rfl | no goals | c49041a003369f98 |
IsCyclotomicExtension.Rat.Three.Units.mem | Mathlib/NumberTheory/Cyclotomic/Three.lean | theorem Units.mem [NumberField K] [IsCyclotomicExtension {3} ℚ K] :
u ∈ [1, -1, η, -η, η ^ 2, -η ^ 2] | case h
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = ↑(x, e).1 * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i
⊢ ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i =... | rw [← Finset.prod_empty] | case h
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hrank : rank K = 0
x : ↥(torsion K)
e : Fin (rank K) → ℤ
hxu : u = ↑(x, e).1 * ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i
⊢ ∏ i : Fin (rank K), fundSystem K i ^ (x, e).2 i =... | 4597273329d964a3 |
Ideal.natAbs_det_equiv | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem natAbs_det_equiv (I : Ideal S) {E : Type*} [EquivLike E S I] [AddEquivClass E S I] (e : E) :
Int.natAbs
(LinearMap.det
((Submodule.subtype I).restrictScalars ℤ ∘ₗ AddMonoidHom.toIntLinearMap (e : S →+ I))) =
Ideal.absNorm I | case neg.inr
S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.Ch... | letI := Ideal.fintypeQuotientOfFreeOfNeBot I hI | case neg.inr
S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.Ch... | 87e25ff10e007725 |
FractionalIdeal.finprod_heightOneSpectrum_factorization_principal | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem finprod_heightOneSpectrum_factorization_principal {I : FractionalIdeal R⁰ K} (hI : I ≠ 0)
(k : K) (hk : I = spanSingleton R⁰ k) :
∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {choose
(mk'_surjective R⁰ k)} : ... | 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
k : K
hk : I = spanSingleton R⁰ k
⊢ ∏ᶠ (v : HeightOneSpectrum R),
↑v.asIdeal ^
(↑((Associates.mk v.asIdeal).count (Associates.mk (I... | set n : R := choose (mk'_surjective R⁰ k) | 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
k : K
hk : I = spanSingleton R⁰ k
n : R := choose ⋯
⊢ ∏ᶠ (v : HeightOneSpectrum R),
↑v.asIdeal ^
(↑((Associates.mk v.asIdeal).count... | 39810a3c3ef6322b |
AddCircle.coe_eq_zero_of_pos_iff | Mathlib/Topology/Instances/AddCircle.lean | theorem coe_eq_zero_of_pos_iff (hp : 0 < p) {x : 𝕜} (hx : 0 < x) :
(x : AddCircle p) = 0 ↔ ∃ n : ℕ, n • p = x | 𝕜 : Type u_1
inst✝ : LinearOrderedAddCommGroup 𝕜
p : 𝕜
hp : 0 < p
n : ℤ
hx : 0 < n
⊢ n.toNat • p = n • p | rw [← natCast_zsmul, Int.toNat_of_nonneg hx.le] | no goals | 4c3ef473bd7cb704 |
Polynomial.eq_zero_of_mul_eq_zero_of_smul | Mathlib/Algebra/Polynomial/AlgebraMap.lean | theorem eq_zero_of_mul_eq_zero_of_smul (P : R[X]) (h : ∀ r : R, r • P = 0 → r = 0) (Q : R[X])
(hQ : P * Q = 0) : Q = 0 | case step.inr
R : Type u
inst✝ : CommSemiring R
P : R[X]
h : ∀ (r : R), r • P = 0 → r = 0
Q : R[X]
hQ : P * Q = 0
l : ℕ
IH : ∀ m > l, P.coeff m • Q = 0
hl : (P.coeff l • Q).natDegree = Q.natDegree
m : ℕ := Q.natDegree
⊢ P.coeff l * Q.leadingCoeff = 0 | suffices (P * Q).coeff (l + m) = P.coeff l * Q.leadingCoeff by rw [← this, hQ, coeff_zero] | case step.inr
R : Type u
inst✝ : CommSemiring R
P : R[X]
h : ∀ (r : R), r • P = 0 → r = 0
Q : R[X]
hQ : P * Q = 0
l : ℕ
IH : ∀ m > l, P.coeff m • Q = 0
hl : (P.coeff l • Q).natDegree = Q.natDegree
m : ℕ := Q.natDegree
⊢ (P * Q).coeff (l + m) = P.coeff l * Q.leadingCoeff | 00213fbe9af3ea6e |
Submodule.sup_eq_sup_smul_of_le_smul_of_le_jacobson | Mathlib/RingTheory/Nakayama.lean | theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
(hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N' ≤ N ⊔ I • N') : N ⊔ N' = N ⊔ J • N' | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I J : Ideal R
N N' : Submodule R M
hN' : N'.FG
hIJ : I ≤ J.jacobson
hNN : N' ≤ N ⊔ I • N'
⊢ N ⊔ N' = N ⊔ J • N' | have hNN' : N ⊔ N' = N ⊔ I • N' :=
le_antisymm (sup_le le_sup_left hNN)
(sup_le_sup_left (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _) | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I J : Ideal R
N N' : Submodule R M
hN' : N'.FG
hIJ : I ≤ J.jacobson
hNN : N' ≤ N ⊔ I • N'
hNN' : N ⊔ N' = N ⊔ I • N'
⊢ N ⊔ N' = N ⊔ J • N' | 7ad4ec5458cce455 |
Option.or_of_isSome | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean | theorem or_of_isSome {o o' : Option α} (h : o.isSome) : o.or o' = o | α : Type u_1
o o' : Option α
h : o.isSome = true
val✝ : α
x✝ : (some val✝).isSome = true
⊢ (some val✝).or o' = some val✝ | simp | no goals | 0b3b0483a2b0019c |
Std.Sat.AIG.toCNF.projectLeftAssign_property | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CNF.lean | theorem projectLeftAssign_property : (projectLeftAssign assign) x = (assign <| .inl x) | aig✝ : AIG Nat
assign : aig✝.CNFVar → Bool
x : Nat
⊢ projectLeftAssign assign x = assign (Sum.inl x) | simp [projectLeftAssign] | no goals | c55c10bee6cd36f6 |
Int.le_bmod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem le_bmod {x : Int} {m : Nat} (h : 0 < m) : - (m/2) ≤ Int.bmod x m | case isFalse.inl
x : Int
m : Nat
h✝ : 0 < m
w : (↑m + 1) / 2 ≤ x % ↑m
q : Int
h : ↑m % 2 = 0
⊢ -((2 * q + 0) / 2) ≤ (2 * q + 0 + 1) / 2 - (2 * q + 0) | rw [Int.add_zero, Int.mul_ediv_cancel_left, Int.add_ediv_of_dvd_left,
Int.mul_ediv_cancel_left, show (1 / 2 : Int) = 0 by decide, Int.add_zero,
Int.neg_eq_neg_one_mul] | case isFalse.inl
x : Int
m : Nat
h✝ : 0 < m
w : (↑m + 1) / 2 ≤ x % ↑m
q : Int
h : ↑m % 2 = 0
⊢ -1 * q ≤ q - 2 * q
case isFalse.inl.H
x : Int
m : Nat
h✝ : 0 < m
w : (↑m + 1) / 2 ≤ x % ↑m
q : Int
h : ↑m % 2 = 0
⊢ 2 ≠ 0
case isFalse.inl
x : Int
m : Nat
h✝ : 0 < m
w : (↑m + 1) / 2 ≤ x % ↑m
q : Int
h : ↑m % 2 = 0
⊢ 2 ∣ 2 ... | c8cfe301519f3ee0 |
List.rotate_nil | Mathlib/Data/List/Rotate.lean | theorem rotate_nil (n : ℕ) : ([] : List α).rotate n = [] | α : Type u
n : ℕ
⊢ [].rotate n = [] | simp [rotate] | no goals | f8563d13aad770ba |
CategoryTheory.InjectiveResolution.desc_commutes | Mathlib/CategoryTheory/Abelian/Injective/Resolution.lean | theorem desc_commutes {Y Z : C} (f : Z ⟶ Y) (I : InjectiveResolution Y)
(J : InjectiveResolution Z) : J.ι ≫ desc f I J = (CochainComplex.single₀ C).map f ≫ I.ι | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
Y Z : C
f : Z ⟶ Y
I : InjectiveResolution Y
J : InjectiveResolution Z
⊢ J.ι ≫ desc f I J = (CochainComplex.single₀ C).map f ≫ I.ι | ext | case hfg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
Y Z : C
f : Z ⟶ Y
I : InjectiveResolution Y
J : InjectiveResolution Z
⊢ (J.ι ≫ desc f I J).f 0 = ((CochainComplex.single₀ C).map f ≫ I.ι).f 0 | 92fc0f33cd96df9f |
MvPolynomial.prime_C_iff | Mathlib/RingTheory/Polynomial/Basic.lean | theorem prime_C_iff : Prime (C r : MvPolynomial σ R) ↔ Prime r :=
⟨comap_prime C constantCoeff (constantCoeff_C _), fun hr =>
⟨fun h => hr.1 <| by
rw [← C_inj, h]
simp,
fun h =>
hr.2.1 <| by
rw [← constantCoeff_C _ r]
exact h.map _,
fun a b hd => by
obtain ⟨... | case intro.intro.intro.intro
R : Type u
σ : Type v
inst✝ : CommRing R
r : R
hr : Prime r
s : Finset σ
a' b' : MvPolynomial { x // x ∈ s } R
hd : (algebraMap R (MvPolynomial σ R)) r ∣ (rename Subtype.val) a' * (rename Subtype.val) b'
this : (algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'
f : MvPolynomial { x ... | exact (((prime_C_iff_of_fintype s).2 hr).2.2 a' b' this).imp f.map_dvd f.map_dvd | no goals | 595a4850ef2d6a25 |
IsRetrocompact_iff_isSpectralMap_subtypeVal | Mathlib/Topology/Constructible.lean | lemma IsRetrocompact_iff_isSpectralMap_subtypeVal :
IsRetrocompact s ↔ IsSpectralMap (Subtype.val : s → X) | case refine_1
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsRetrocompact s
t : Set X
htopen : IsOpen t
htcomp : IsCompact t
⊢ IsCompact (Subtype.val ⁻¹' t) | rw [IsEmbedding.subtypeVal.isCompact_iff, image_preimage_eq_inter_range,
Subtype.range_coe_subtype, setOf_mem_eq, inter_comm] | case refine_1
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsRetrocompact s
t : Set X
htopen : IsOpen t
htcomp : IsCompact t
⊢ IsCompact (s ∩ t) | 0527d902b413e441 |
Algebra.FormallyEtale.iff_exists_algEquiv_prod | Mathlib/RingTheory/Etale/Field.lean | theorem iff_exists_algEquiv_prod [EssFiniteType K A] :
FormallyEtale K A ↔
∃ (I : Type u) (_ : Finite I) (Ai : I → Type u) (_ : ∀ i, Field (Ai i))
(_ : ∀ i, Algebra K (Ai i)) (_ : A ≃ₐ[K] Π i, Ai i),
∀ i, Algebra.IsSeparable K (Ai i) | case mp
K A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
H : FormallyEtale K A
this✝² : Module.Finite K A
this✝¹ : IsReduced A
this✝ : IsArtinianRing A
this : Fintype (MaximalSpectrum A) := ⋯.some
⊢ ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (... | let v (i : MaximalSpectrum A) : A := (IsArtinianRing.equivPi A).symm (Pi.single i 1) | case mp
K A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
H : FormallyEtale K A
this✝² : Module.Finite K A
this✝¹ : IsReduced A
this✝ : IsArtinianRing A
this : Fintype (MaximalSpectrum A) := ⋯.some
v : MaximalSpectrum A → A := fun i => (IsArtinianRing.equivPi A).symm (Pi.s... | bed71e21534c428f |
Multiset.prod_map_prod_map | Mathlib/Algebra/BigOperators/Group/Multiset/Basic.lean | theorem prod_map_prod_map (m : Multiset β') (n : Multiset γ) {f : β' → γ → α} :
prod (m.map fun a => prod <| n.map fun b => f a b) =
prod (n.map fun b => prod <| m.map fun a => f a b) :=
Multiset.induction_on m (by simp) fun a m ih => by simp [ih]
| α : Type u_3
β' : Type u_5
γ : Type u_6
inst✝ : CommMonoid α
m✝ : Multiset β'
n : Multiset γ
f : β' → γ → α
a : β'
m : Multiset β'
ih : (map (fun a => (map (fun b => f a b) n).prod) m).prod = (map (fun b => (map (fun a => f a b) m).prod) n).prod
⊢ (map (fun a => (map (fun b => f a b) n).prod) (a ::ₘ m)).prod =
(map... | simp [ih] | no goals | 11ed2bd77ceea997 |
TopologicalSpace.IsOpenCover.isOpenMap_iff_restrictPreimage | Mathlib/Topology/LocalAtTarget.lean | theorem isOpenMap_iff_restrictPreimage :
IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
H : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
s : Set α
hs : IsOpen s
⊢ IsOpen (f '' s) | rw [hU.isOpen_iff_coe_preimage] | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
H : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
s : Set α
hs : IsOpen s
⊢ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' (f '' s)) | 344d9e2126901f20 |
Vector.countP_mkVector | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Count.lean | theorem countP_mkVector (p : α → Bool) (a : α) (n : Nat) :
countP p (mkVector n a) = if p a then n else 0 | α : Type u_1
p : α → Bool
a : α
n : Nat
⊢ countP p (mkVector n a) = if p a = true then n else 0 | simp only [mkVector_eq_mk_mkArray, countP_cast, countP_mk] | α : Type u_1
p : α → Bool
a : α
n : Nat
⊢ Array.countP p (mkArray n a) = if p a = true then n else 0 | e00d22b4b3ff15b0 |
Finset.coe_cons | Mathlib/Data/Finset/Insert.lean | theorem coe_cons {a s h} : (@cons α a s h : Set α) = insert a (s : Set α) | α : Type u_1
a : α
s : Finset α
h : a ∉ s
⊢ ↑(cons a s h) = insert a ↑s | ext | case h
α : Type u_1
a : α
s : Finset α
h : a ∉ s
x✝ : α
⊢ x✝ ∈ ↑(cons a s h) ↔ x✝ ∈ insert a ↑s | c6d47bb34ea0463d |
exists_continuous_one_zero_of_isCompact_of_isGδ | Mathlib/Topology/UrysohnsLemma.lean | theorem exists_continuous_one_zero_of_isCompact_of_isGδ [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), s = f ⁻¹' {1} ∧ EqOn f 0 t ∧ HasCompactSupport f
∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsComp... | have hgmc : EqOn g 0 mᶜ := by
intro x hx
have B n : f n x = 0 := by
have : mᶜ ⊆ (U n ∩ interior m)ᶜ := by
simpa using inter_subset_right.trans interior_subset
exact fm n (this hx)
simp [g, B] | case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsComp... | 06fdcc97382f69bb |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.derivedLitsInvariant_confirmRupHint | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem derivedLitsInvariant_confirmRupHint {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(rupHints : Array Nat) (i : Fin rupHints.size)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
(ih : ∃ hsize : acc.1.size = n, DerivedLitsInvariant f f_assignments_size a... | case intro.isFalse.inr.inr.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (co... | rcases reduce_c_options with hencounteredBoth | hreducedToEmpty | ⟨l, hreducedToUnit⟩ | hreducedToNonunit | case intro.isFalse.inr.inr.intro.inl
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' :... | 0f20fdb04006be12 |
List.set_replicate_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem set_replicate_self : (replicate n a).set i a = replicate n a | n : Nat
α✝ : Type u_1
a : α✝
i : Nat
⊢ (replicate n a).set i a = replicate n a | apply ext_getElem | case hl
n : Nat
α✝ : Type u_1
a : α✝
i : Nat
⊢ ((replicate n a).set i a).length = (replicate n a).length
case h
n : Nat
α✝ : Type u_1
a : α✝
i : Nat
⊢ ∀ (i_1 : Nat) (h₁ : i_1 < ((replicate n a).set i a).length) (h₂ : i_1 < (replicate n a).length),
((replicate n a).set i a)[i_1] = (replicate n a)[i_1] | 806280df4d55bf71 |
IntermediateField.exists_algHom_adjoin_of_splits'' | Mathlib/FieldTheory/Extension.lean | theorem exists_algHom_adjoin_of_splits'' {L : IntermediateField F E}
(f : L →ₐ[F] K) (hK : ∀ s ∈ S, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom) :
∃ φ : adjoin L S →ₐ[F] K, φ.restrictDomain L = f | case intro.intro.refine_1
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
S : Set E
L : IntermediateField F E
f : ↥L →ₐ[F] K
hK : ∀ s ∈ S, IsIntegral (↥L) s ∧ Splits f.toRingHom (minpoly (↥L) s)
φ : Lifts F E K
hfφ : { carrier := L, emb ... | convert (hK s h).2 | case h.e'_5
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
S : Set E
L : IntermediateField F E
f : ↥L →ₐ[F] K
hK : ∀ s ∈ S, IsIntegral (↥L) s ∧ Splits f.toRingHom (minpoly (↥L) s)
φ : Lifts F E K
hfφ : { carrier := L, emb := f } ≤ φ
hφ ... | bdc96de1fec7cebe |
nhds_basis_opens' | Mathlib/Topology/Basic.lean | theorem nhds_basis_opens' (x : X) :
(𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsOpen s) fun x => x | X : Type u
inst✝ : TopologicalSpace X
x : X
⊢ (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsOpen s) fun x => x | convert nhds_basis_opens x using 2 | case h.e'_4.h.a
X : Type u
inst✝ : TopologicalSpace X
x : X
x✝ : Set X
⊢ x✝ ∈ 𝓝 x ∧ IsOpen x✝ ↔ x ∈ x✝ ∧ IsOpen x✝ | b7d6fea9cd7f4a74 |
ProbabilityTheory.Kernel.withDensity_add_right | Mathlib/Probability/Kernel/WithDensity.lean | lemma withDensity_add_right [IsSFiniteKernel κ] {f g : α → β → ℝ≥0∞}
(hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) :
withDensity κ (f + g) = withDensity κ f + withDensity κ g | case h.h.hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ Measurable (Function.uncurry (f + g)) | exact hf.add hg | no goals | 30ddbaf52c285553 |
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